Q Chem 5.1 User's Manual

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Q-Chem 5.1 User’s Manual

Version 5.1
May, 2018

Q-C HEM User’s Manual

Version 5.1 was edited by:
Dr. Andrew Gilbert
Prof. John Herbert
Version 5.0 was edited by:
Dr. Andrew Gilbert
Version 4 editors:
Prof. John Herbert
Prof. Anna Krylov
Dr. Narbe Mardirossian
Prof. Martin Head-Gordon
Dr. Emil Proynov
Dr. Andrew Gilbert
Dr. Jing Kong
The contributions of individual developers to each version are highlighted in “New Features”,
Section 1.3

Published by:
Q-Chem, Inc.
6601 Owens Dr.
Suite 105
Pleasanton, CA 94588

Customer Support:
Telephone: (412) 687-0695
Facsimile: (412) 687-0698
email:
support@q-chem.com
website:
www.q-chem.com

Q-C HEM is a trademark of Q-Chem, Inc. All rights reserved.
The information in this document applies to version 5.1 of Q-C HEM.
This document version generated on October 20, 2018.

© Copyright 2000–2018 Q-Chem, Inc. This document is protected under the U.S. Copyright Act of 1976 and state
trade secret laws. Unauthorized disclosure, reproduction, distribution, or use is prohibited and may violate federal and
state laws.

3

CONTENTS

Contents
1

2

3

Introduction
1.1
About This Manual . . . . . . . . . . . . . . . . . .
1.1.1 Overview . . . . . . . . . . . . . . . . . . . .
1.1.2 Chapter Summaries . . . . . . . . . . . . . .
1.2
Q-C HEM, Inc. . . . . . . . . . . . . . . . . . . . . .
1.2.1 Contact Information and Customer Support . .
1.2.2 About the Company . . . . . . . . . . . . . .
1.2.3 Company Mission . . . . . . . . . . . . . . .
1.3
Q-C HEM Features . . . . . . . . . . . . . . . . . . .
1.3.1 New Features in Q-C HEM 5.1 . . . . . . . . .
1.3.2 New Features in Q-C HEM 5.0 . . . . . . . . .
1.3.3 New Features in Q-C HEM 4.4 . . . . . . . . .
1.3.4 New Features in Q-C HEM 4.3 . . . . . . . . .
1.3.5 New Features in Q-C HEM 4.2 . . . . . . . . .
1.3.6 New Features in Q-C HEM 4.1 . . . . . . . . .
1.3.7 New Features in Q-C HEM 4.0.1 . . . . . . . .
1.3.8 New Features in Q-C HEM 4.0 . . . . . . . . .
1.3.9 Summary of Features in Q-C HEM versions 3. x
1.3.10 Summary of Features Prior to Q-C HEM 3.0 . .
1.4
Citing Q-C HEM . . . . . . . . . . . . . . . . . . . .
References and Further Reading . . . . . . . . . . . . . . .

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15
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23
23
26
28
29
31

Installation, Customization, and Execution
2.1
Installation Requirements . . . . . . . . . . . . . . . .
2.1.1 Execution Environment . . . . . . . . . . . . .
2.1.2 Hardware Platforms and Operating Systems . .
2.1.3 Memory and Disk Requirements . . . . . . . .
2.2
Installing Q-C HEM . . . . . . . . . . . . . . . . . . .
2.3
Q-C HEM Auxiliary files ($QCAUX) . . . . . . . . . .
2.4
Q-C HEM Run-time Environment Variables . . . . . . .
2.5
User Account Adjustments . . . . . . . . . . . . . . .
2.6
Further Customization: .qchemrc and preferences Files
2.7
Running Q-C HEM . . . . . . . . . . . . . . . . . . . .
2.8
Parallel Q-C HEM Jobs . . . . . . . . . . . . . . . . .
2.9
IQ MOL Installation Requirements . . . . . . . . . . .
2.10 Testing and Exploring Q-C HEM . . . . . . . . . . . .

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32
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35
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37
38
39

Q-C HEM Inputs
3.1
IQ MOL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2
General Form . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3
Molecular Coordinate Input ($molecule) . . . . . . . . . . . . . .
3.3.1 Specifying the Molecular Coordinates Manually . . . . . .
3.3.2 Reading Molecular Coordinates from a Previous Job or File

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40
40
40
42
42
46

4

CONTENTS
3.4
3.5

3.6
3.7
4

Job Specification: The $rem Input Section . . . . . . . . . . . . . . . . . . . . . . . . .
Additional Input Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5.1 Comments ($comment) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5.2 User-Defined Basis Sets ($basis and $aux_basis) . . . . . . . . . . . . . . . . . .
3.5.3 User-Defined Effective Core Potential ($ecp) . . . . . . . . . . . . . . . . . . . .
3.5.4 User-Defined Exchange-Correlation Density Functionals ($xc_functional) . . . .
3.5.5 User-defined Parameters for DFT Dispersion Correction ($empirical_dispersion) .
3.5.6 Addition of External Point Charges ($external_charges) . . . . . . . . . . . . . .
3.5.7 Applying a Multipole Field ($multipole_field) . . . . . . . . . . . . . . . . . . .
3.5.8 User-Defined Occupied Guess Orbitals ($occupied and $swap_occupied_virtual) .
3.5.9 Polarizable Continuum Solvation Models ($pcm) . . . . . . . . . . . . . . . . . .
3.5.10 SS(V)PE Solvation Modeling ($svp and $svpirf ) . . . . . . . . . . . . . . . . . .
3.5.11 User-Defined van der Waals Radii ($van_der_waals) . . . . . . . . . . . . . . . .
3.5.12 Effective Fragment Potential Calculations ($efp_fragments and $efp_params) . .
3.5.13 Natural Bond Orbital Package ($nbo) . . . . . . . . . . . . . . . . . . . . . . . .
3.5.14 Orbitals, Densities and Electrostatic Potentials on a Mesh ($plots) . . . . . . . . .
3.5.15 Intracules ($intracule) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5.16 Geometry Optimization with Constraints ($opt) . . . . . . . . . . . . . . . . . .
3.5.17 Isotopic Substitutions ($isotopes) . . . . . . . . . . . . . . . . . . . . . . . . . .
Multiple Jobs in a Single File: Q-C HEM Batch Jobs . . . . . . . . . . . . . . . . . . . .
Q-C HEM Output File . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Self-Consistent Field Ground-State Methods
4.1
Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2
Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 SCF and LCAO Approximations . . . . . . . . . . . . . . . . . . . . .
4.2.2 Hartree-Fock Theory . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3
Basic SCF Job Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.2 Additional Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.4 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4
SCF Initial Guess . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.2 Simple Initial Guesses . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.3 Reading MOs from Disk . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.4 Modifying the Occupied Molecular Orbitals . . . . . . . . . . . . . . .
4.4.5 Basis Set Projection . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5
Converging SCF Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5.2 Basic Convergence Control Options . . . . . . . . . . . . . . . . . . . .
4.5.3 Direct Inversion in the Iterative Subspace (DIIS) . . . . . . . . . . . . .
4.5.4 Geometric Direct Minimization (GDM) . . . . . . . . . . . . . . . . . .
4.5.5 Direct Minimization (DM) . . . . . . . . . . . . . . . . . . . . . . . .
4.5.6 Maximum Overlap Method (MOM) . . . . . . . . . . . . . . . . . . . .
4.5.7 Relaxed Constraint Algorithm (RCA) . . . . . . . . . . . . . . . . . . .
4.5.8 User-Customized Hybrid SCF Algorithm . . . . . . . . . . . . . . . . .
4.5.9 Internal Stability Analysis and Automated Correction for Energy Minima
4.5.10 Small-Gap Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5.11 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6
Large Molecules and Linear Scaling Methods . . . . . . . . . . . . . . . . . .
4.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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46
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54
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65
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94

5

CONTENTS
4.6.2 Continuous Fast Multipole Method (CFMM) . . . . . .
4.6.3 Linear Scaling Exchange (LinK) Matrix Evaluation . .
4.6.4 Incremental and Variable Thresh Fock Matrix Building
4.6.5 Fourier Transform Coulomb Method . . . . . . . . . .
4.6.6 Resolution of the Identity Fock Matrix Methods . . . .
4.6.7 PARI-K Fast Exchange Algorithm . . . . . . . . . . .
4.6.8 CASE Approximation . . . . . . . . . . . . . . . . . .
4.6.9 occ-RI-K Exchange Algorithm . . . . . . . . . . . . .
4.6.10 Examples . . . . . . . . . . . . . . . . . . . . . . . . .
4.7
Dual-Basis Self-Consistent Field Calculations . . . . . . . . .
4.7.1 Dual-Basis MP2 . . . . . . . . . . . . . . . . . . . . .
4.7.2 Dual-Basis Dynamics . . . . . . . . . . . . . . . . . .
4.7.3 Basis-Set Pairings . . . . . . . . . . . . . . . . . . . .
4.7.4 Job Control . . . . . . . . . . . . . . . . . . . . . . . .
4.7.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . .
4.8
Hartree-Fock and Density-Functional Perturbative Corrections
4.8.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . .
4.8.2 Job Control . . . . . . . . . . . . . . . . . . . . . . . .
4.8.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . .
4.9
Unconventional SCF Calculations . . . . . . . . . . . . . . .
4.9.1 Polarized Atomic Orbital (PAO) Calculations . . . . . .
4.9.2 SCF Meta-Dynamics . . . . . . . . . . . . . . . . . .
4.10 Ground State Method Summary . . . . . . . . . . . . . . . .
References and Further Reading . . . . . . . . . . . . . . . . . . . .
5

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Density Functional Theory
5.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2
Kohn-Sham Density Functional Theory . . . . . . . . . . . . . . . . . . . . . . .
5.3
Overview of Available Functionals . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.1 Suggested Density Functionals . . . . . . . . . . . . . . . . . . . . . . . .
5.3.2 Exchange Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.3 Correlation Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.4 Exchange-Correlation Functionals . . . . . . . . . . . . . . . . . . . . . .
5.3.5 Specialized Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.6 User-Defined Density Functionals . . . . . . . . . . . . . . . . . . . . . . .
5.4
Basic DFT Job Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5
DFT Numerical Quadrature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5.1 Angular Grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5.2 Standard Quadrature Grids . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5.3 Consistency Check and Cutoffs . . . . . . . . . . . . . . . . . . . . . . . .
5.5.4 Multi-resolution Exchange-Correlation (MRXC) Method . . . . . . . . . .
5.5.5 Incremental DFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.6
Range-Separated Hybrid Density Functionals . . . . . . . . . . . . . . . . . . . .
5.6.1 Semi-Empirical RSH Functionals . . . . . . . . . . . . . . . . . . . . . . .
5.6.2 User-Defined RSH Functionals . . . . . . . . . . . . . . . . . . . . . . . .
5.6.3 Tuned RSH Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.6.4 Tuned RSH Functionals Based on the Global Density-Dependent Condition
5.7
DFT Methods for van der Waals Interactions . . . . . . . . . . . . . . . . . . . . .
5.7.1 Non-Local Correlation (NLC) Functionals . . . . . . . . . . . . . . . . . .
5.7.2 Empirical Dispersion Corrections: DFT-D . . . . . . . . . . . . . . . . . .
5.7.3 Exchange-Dipole Model (XDM) . . . . . . . . . . . . . . . . . . . . . . .
5.7.4 Tkatchenko-Scheffler van der Waals Model (TS-vdW) . . . . . . . . . . . .
5.7.5 Many-Body Dispersion (MBD) Method . . . . . . . . . . . . . . . . . . .

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95
97
97
98
100
102
102
103
106
107
107
107
107
108
110
112
112
112
114
114
114
115
120
121

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124
124
124
126
128
129
130
131
136
137
140
142
142
143
145
145
147
148
148
149
153
154
155
155
158
164
168
170

6

CONTENTS
5.8
5.9
5.10

Empirical Corrections for Basis Set Superposition Error . . . . . . . . .
Double-Hybrid Density Functional Theory . . . . . . . . . . . . . . . .
Asymptotically Corrected Exchange-Correlation Potentials . . . . . . .
5.10.1 LB94 Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.10.2 Localized Fermi-Amaldi (LFA) Schemes . . . . . . . . . . . . .
5.11 Derivative Discontinuity Restoration . . . . . . . . . . . . . . . . . . .
5.12 Thermally-Assisted-Occupation Density Functional Theory (TAO-DFT)
5.13 Methods Based on “Constrained” DFT . . . . . . . . . . . . . . . . . .
5.13.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.13.2 CDFT Job Control and Examples . . . . . . . . . . . . . . . . .
5.13.3 Configuration Interaction with Constrained DFT (CDFT-CI) . . .
5.13.4 CDFT-CI Job Control and Examples . . . . . . . . . . . . . . .
References and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . .
6

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172
173
178
178
179
180
182
185
185
187
190
192
196

Wave Function-Based Correlation Methods
6.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2
Treatment and the Definition of Core Electrons . . . . . . . . . . . . . . . . .
6.3
Møller-Plesset Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . .
6.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3.2 Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4
Exact MP2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4.1 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4.2 Algorithm Control and Customization . . . . . . . . . . . . . . . . . .
6.4.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.5
Local MP2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.5.1 Local Triatomics in Molecules (TRIM) Model . . . . . . . . . . . . . .
6.5.2 EPAO Evaluation Options . . . . . . . . . . . . . . . . . . . . . . . . .
6.5.3 Algorithm Control and Customization . . . . . . . . . . . . . . . . . .
6.5.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.6
Auxiliary Basis (Resolution of the Identity) MP2 Methods . . . . . . . . . . .
6.6.1 RI-MP2 Energies and Gradients. . . . . . . . . . . . . . . . . . . . . .
6.6.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.6.3 OpenMP Implementation of RI-MP2 . . . . . . . . . . . . . . . . . . .
6.6.4 GPU Implementation of RI-MP2 . . . . . . . . . . . . . . . . . . . . .
6.6.5 Spin-Biased MP2 Methods (SCS-MP2, SOS-MP2, MOS-MP2, and O2)
6.6.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.6.7 RI-TRIM MP2 Energies . . . . . . . . . . . . . . . . . . . . . . . . . .
6.6.8 Dual-Basis MP2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.7
Attenuated MP2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.8
Coupled-Cluster Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.8.1 Coupled Cluster Singles and Doubles (CCSD) . . . . . . . . . . . . . .
6.8.2 Quadratic Configuration Interaction (QCISD) . . . . . . . . . . . . . .
6.8.3 Optimized Orbital Coupled Cluster Doubles (OD) . . . . . . . . . . . .
6.8.4 Quadratic Coupled Cluster Doubles (QCCD) . . . . . . . . . . . . . . .
6.8.5 Resolution of the Identity with CC (RI-CC) . . . . . . . . . . . . . . . .
6.8.6 Cholesky decomposition with CC (CD-CC) . . . . . . . . . . . . . . .
6.8.7 Job Control Options . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.8.8 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.9
Non-Iterative Corrections to Coupled Cluster Energies . . . . . . . . . . . . .
6.9.1 (T) Triples Corrections . . . . . . . . . . . . . . . . . . . . . . . . . .
6.9.2 (2) Triples and Quadruples Corrections . . . . . . . . . . . . . . . . . .
6.9.3
(dT) and (fT) corrections . . . . . . . . . . . . . . . . . . . . . . . . .
6.9.4 Job Control Options . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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205
205
207
208
208
208
209
209
211
212
212
212
214
215
217
217
218
219
219
220
222
225
227
228
228
229
230
231
232
233
233
234
234
237
238
238
239
239
239

7

CONTENTS
6.9.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Coupled Cluster Active Space Methods . . . . . . . . . . . . . . . . . . . . .
6.10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.10.2 VOD and VOD(2) Methods . . . . . . . . . . . . . . . . . . . . . . . .
6.10.3 VQCCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.10.4 CCVB-SD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.10.5 Local Pair Models for Valence Correlations Beyond Doubles . . . . . .
6.10.6 Convergence Strategies and More Advanced Options . . . . . . . . . . .
6.10.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.11 Frozen Natural Orbitals in CCD, CCSD, OD, QCCD, and QCISD Calculations
6.11.1 Job Control Options . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.11.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.12 Non-Hartree-Fock Orbitals in Correlated Calculations . . . . . . . . . . . . . .
6.13 Analytic Gradients and Properties for Coupled-Cluster Methods . . . . . . . .
6.13.1 Job Control Options . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.13.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.14 Memory Options and Parallelization of Coupled-Cluster Calculations . . . . .
6.14.1 Serial and Shared Memory Parallel Jobs . . . . . . . . . . . . . . . . .
6.14.2 Distributed Memory Parallel Jobs . . . . . . . . . . . . . . . . . . . . .
6.14.3 Summary of Keywords . . . . . . . . . . . . . . . . . . . . . . . . . .
6.15 Simplified Coupled-Cluster Methods Based on a Perfect-Pairing Active Space .
6.15.1 Perfect pairing (PP) . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.15.2 Coupled Cluster Valence Bond (CCVB) . . . . . . . . . . . . . . . . .
6.15.3 Second-order Correction to Perfect Pairing: PP(2) . . . . . . . . . . . .
6.15.4 Other GVBMAN Methods and Options . . . . . . . . . . . . . . . . . .
6.16 Geminal Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.16.1 Reference Wave Function . . . . . . . . . . . . . . . . . . . . . . . . .
6.16.2 Perturbative Corrections . . . . . . . . . . . . . . . . . . . . . . . . . .
6.17 Variational Two-Electron Reduced-Density-Matrix Methods . . . . . . . . . .
6.17.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.17.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.17.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.17.4 v2RDM Job Control . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Open-Shell and Excited-State Methods
7.1
General Excited-State Features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2
Uncorrelated Wave Function Methods . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2.1 Single Excitation Configuration Interaction (CIS) . . . . . . . . . . . . . . . . .
7.2.2 Random Phase Approximation (RPA) . . . . . . . . . . . . . . . . . . . . . . . .
7.2.3 Extended CIS (XCIS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2.4 Spin-Flip Extended CIS (SF-XCIS) . . . . . . . . . . . . . . . . . . . . . . . . .
7.2.5 Spin-Adapted Spin-Flip CIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2.6 CIS Analytical Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2.7 Non-Orthogonal Configuration Interaction . . . . . . . . . . . . . . . . . . . . .
7.2.8 Basic CIS Job Control Options . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2.9 CIS Job Customization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2.10 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3
Time-Dependent Density Functional Theory (TDDFT) . . . . . . . . . . . . . . . . . .
7.3.1 Brief Introduction to TDDFT . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3.2 TDDFT within a Reduced Single-Excitation Space . . . . . . . . . . . . . . . . .
7.3.3 Job Control for TDDFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3.4 TDDFT Coupled with C-PCM for Excitation Energies and Properties Calculations

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310

6.10

7

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8

CONTENTS

7.4
7.5
7.6

7.7

7.8

7.9

7.3.5 Analytical Excited-State Hessian in TDDFT . . . . . . . . . . . . . . . . . . . . . . . . .
7.3.6 Calculations of Spin-Orbit Couplings Between TDDFT States . . . . . . . . . . . . . . . .
7.3.7 Various TDDFT-Based Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Maximum Overlap Method (MOM) for SCF Excited States . . . . . . . . . . . . . . . . . . . . .
Restricted Open-Shell Kohn-Sham Method for ∆-SCF Calculations of Excited States . . . . . . .
Correlated Excited State Methods: The CIS(D) Family . . . . . . . . . . . . . . . . . . . . . . .
7.6.1 CIS(D) Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.6.2 Resolution of the Identity CIS(D) Methods . . . . . . . . . . . . . . . . . . . . . . . . . .
7.6.3 SOS-CIS(D) Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.6.4 SOS-CIS(D0 ) Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.6.5 CIS(D) Job Control and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.6.6 RI-CIS(D), SOS-CIS(D), and SOS-CIS(D0 ): Job Control . . . . . . . . . . . . . . . . . .
7.6.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Coupled-Cluster Excited-State and Open-Shell Methods . . . . . . . . . . . . . . . . . . . . . . .
7.7.1 Excited States via EOM-EE-CCSD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.7.2 EOM-XX-CCSD and CI Suite of Methods . . . . . . . . . . . . . . . . . . . . . . . . . .
7.7.3 Spin-Flip Methods for Di- and Triradicals . . . . . . . . . . . . . . . . . . . . . . . . . .
7.7.4 EOM-DIP-CCSD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.7.5 EOM-CC Calculations of Core-Level States: Core-Valence Separation within EOM-CCSD
7.7.6 EOM-CC Calculations of Metastable States: Super-Excited Electronic States, Temporary
Anions, and More . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.7.7 Charge Stabilization for EOM-DIP and Other Methods . . . . . . . . . . . . . . . . . . .
7.7.8 Frozen Natural Orbitals in CC and IP-CC Calculations . . . . . . . . . . . . . . . . . . . .
7.7.9 Approximate EOM-CC Methods: EOM-MP2 and EOM-MP2T . . . . . . . . . . . . . . .
7.7.10 Approximate EOM-CC Methods: EOM-CCSD-S(D) and EOM-MP2-S(D) . . . . . . . . .
7.7.11 Implicit solvent models in EOM-CC/MP2 calculations. . . . . . . . . . . . . . . . . . . .
7.7.12 EOM-CC Jobs: Controlling Guess Formation and Iterative Diagonalizers . . . . . . . . . .
7.7.13 Equation-of-Motion Coupled-Cluster Job Control . . . . . . . . . . . . . . . . . . . . . .
7.7.14 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.7.15 Non-Hartree-Fock Orbitals in EOM Calculations . . . . . . . . . . . . . . . . . . . . . . .
7.7.16 Analytic Gradients and Properties for the CCSD and EOM-XX-CCSD Methods . . . . . .
7.7.17 EOM-CC Optimization and Properties Job Control . . . . . . . . . . . . . . . . . . . . . .
7.7.18 EOM(2,3) Methods for Higher-Accuracy and Problematic Situations (CCMAN only) . . .
7.7.19 Active-Space EOM-CC(2,3): Tricks of the Trade (CCMAN only) . . . . . . . . . . . . . .
7.7.20 Job Control for EOM-CC(2,3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.7.21 Non-Iterative Triples Corrections to EOM-CCSD and CCSD . . . . . . . . . . . . . . . .
7.7.22 Potential Energy Surface Crossing Minimization . . . . . . . . . . . . . . . . . . . . . . .
7.7.23 Dyson Orbitals for Ionized or Attached States within the EOM-CCSD Formalism . . . . .
7.7.24 Interpretation of EOM/CI Wave Functions and Orbital Numbering . . . . . . . . . . . . .
Correlated Excited State Methods: The ADC(n) Family . . . . . . . . . . . . . . . . . . . . . . .
7.8.1 The Algebraic Diagrammatic Construction (ADC) Scheme . . . . . . . . . . . . . . . . .
7.8.2 Resolution of the Identity ADC Methods . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.8.3 Spin Opposite Scaling ADC(2) Models . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.8.4 Core-Excitation ADC Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.8.5 Spin-Flip ADC Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.8.6 Properties and Visualization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.8.7 Excited States in Solution with ADC/SS-PCM . . . . . . . . . . . . . . . . . . . . . . . .
7.8.8 Frozen-Density Embedding: FDE-ADC methods . . . . . . . . . . . . . . . . . . . . . . .
7.8.9 ADC Job Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.8.10 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Restricted Active Space Spin-Flip (RAS-SF) and Configuration Interaction (RAS-CI) . . . . . . .
7.9.1 The Restricted Active Space (RAS) Scheme . . . . . . . . . . . . . . . . . . . . . . . . .
7.9.2 Second-Order Perturbative Corrections to RAS-CI . . . . . . . . . . . . . . . . . . . . . .

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9

CONTENTS

8

9

7.9.3 Short-Range Density Functional Correlation within RAS-CI . . . . . . . .
7.9.4 Excitonic Analysis of the RAS-CI Wave Function . . . . . . . . . . . . .
7.9.5 Job Control for the RASCI1 Implementation . . . . . . . . . . . . . . . .
7.9.6 Job Control Options for RASCI2 . . . . . . . . . . . . . . . . . . . . . .
7.9.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.10 Core Ionization Energies and Core-Excited States . . . . . . . . . . . . . . . . .
7.10.1 Calculations of States Involving Core Excitation/Ionization with (TD)DFT
7.11 Real-Time SCF Methods (RT-TDDFT, RT-HF, OSCF2) . . . . . . . . . . . . . .
7.12 Visualization of Excited States . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.12.1 Attachment/Detachment Density Analysis . . . . . . . . . . . . . . . . .
7.12.2 Natural Transition Orbitals . . . . . . . . . . . . . . . . . . . . . . . . .
References and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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455

Basis Sets
8.1
Introduction . . . . . . . . . . . . . . . . . . .
8.2
Built-In Basis Sets . . . . . . . . . . . . . . .
8.3
Basis Set Symbolic Representation . . . . . . .
8.3.1 Customization . . . . . . . . . . . . . .
8.4
User-Defined Basis Sets ($basis) . . . . . . . .
8.4.1 Introduction . . . . . . . . . . . . . . .
8.4.2 Job Control . . . . . . . . . . . . . . . .
8.4.3 Format for User-Defined Basis Sets . . .
8.4.4 Example . . . . . . . . . . . . . . . . .
8.5
Mixed Basis Sets . . . . . . . . . . . . . . . .
8.6
Dual Basis Sets . . . . . . . . . . . . . . . . .
8.7
Auxiliary Basis Sets for RI (Density Fitting) . .
8.8
Ghost Atoms and Basis Set Superposition Error
References and Further Reading . . . . . . . . . . . .

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461
461
461
462
462
466
466
466
467
468
468
470
471
473
475

Effective Core Potentials
9.1
Introduction . . . . . . . . . . . . . . . . . .
9.2
ECP Fitting . . . . . . . . . . . . . . . . . .
9.3
Built-In ECPs . . . . . . . . . . . . . . . . .
9.3.1 Overview . . . . . . . . . . . . . . . .
9.3.2 Combining ECPs . . . . . . . . . . .
9.3.3 Examples . . . . . . . . . . . . . . . .
9.4
User-Defined ECPs . . . . . . . . . . . . . .
9.4.1 Job Control for User-Defined ECPs . .
9.4.2 Example . . . . . . . . . . . . . . . .
9.5
ECPs and Electron Correlation . . . . . . . .
9.6
Forces and Vibrational Frequencies with ECPs
9.7
A Brief Guide to Q-C HEM’s Built-In ECPs .
9.7.1 The fit-HWMB ECP at a Glance . . .
9.7.2 The fit-LANL2DZ ECP at a Glance . .
9.7.3 The fit-SBKJC ECP at a Glance . . . .
9.7.4 The fit-CRENBS ECP at a Glance . . .
9.7.5 The fit-CRENBL ECP at a Glance . .
9.7.6 The SRLC ECP at a Glance . . . . . .
9.7.7 The SRSC ECP at a Glance . . . . . .
9.7.8 The Karlsruhe “def2” ECP at a Glance
References and Further Reading . . . . . . . . . . .

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476
476
477
477
477
478
478
480
480
482
483
483
484
485
486
487
488
489
490
491
492
493

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10 Exploring Potential Energy Surfaces: Critical Points and Molecular Dynamics

495

10

CONTENTS
10.1

Equilibrium Geometries and Transition-State Structures . . . . . . . . . . . . .
10.1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.1.2 Job Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.1.3 Hessian-Free Characterization of Stationary Points . . . . . . . . . . . .
10.2 Improved Algorithms for Transition-Structure Optimization . . . . . . . . . . .
10.2.1 Freezing String Method . . . . . . . . . . . . . . . . . . . . . . . . . .
10.2.2 Hessian-Free Transition-State Search . . . . . . . . . . . . . . . . . . .
10.2.3 Improved Dimer Method . . . . . . . . . . . . . . . . . . . . . . . . .
10.3 Constrained Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.3.1 Geometry Optimization with General Constraints . . . . . . . . . . . .
10.3.2 Frozen Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.3.3 Dummy Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.3.4 Dummy Atom Placement in Dihedral Constraints . . . . . . . . . . . .
10.3.5 Additional Atom Connectivity . . . . . . . . . . . . . . . . . . . . . . .
10.3.6 Application of External Forces . . . . . . . . . . . . . . . . . . . . . .
10.4 Potential Energy Scans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.5 Intrinsic Reaction Coordinate . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.6 Nonadiabatic Couplings and Optimization of Minimum-Energy Crossing Points
10.6.1 Nonadiabatic Couplings . . . . . . . . . . . . . . . . . . . . . . . . . .
10.6.2 Job Control and Examples . . . . . . . . . . . . . . . . . . . . . . . . .
10.6.3 Minimum-Energy Crossing Points . . . . . . . . . . . . . . . . . . . .
10.6.4 Job Control and Examples . . . . . . . . . . . . . . . . . . . . . . . . .
10.6.5 State-Tracking Algorithm . . . . . . . . . . . . . . . . . . . . . . . . .
10.7 Ab Initio Molecular Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . .
10.7.1 Overview and Basic Job Control . . . . . . . . . . . . . . . . . . . . .
10.7.2 Additional Job Control and Examples . . . . . . . . . . . . . . . . . . .
10.7.3 Thermostats: Sampling the NVT Ensemble . . . . . . . . . . . . . . . .
10.7.4 Vibrational Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.7.5 Quasi-Classical Molecular Dynamics . . . . . . . . . . . . . . . . . . .
10.7.6 Fewest-Switches Surface Hopping . . . . . . . . . . . . . . . . . . . .
10.8 Ab initio Path Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.8.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.8.2 Job Control and Examples . . . . . . . . . . . . . . . . . . . . . . . . .
References and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11 Molecular Properties and Analysis
11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.2 Wave Function Analysis . . . . . . . . . . . . . . . . . . . . . . . . .
11.2.1 Population Analysis . . . . . . . . . . . . . . . . . . . . . . . .
11.2.2 Multipole Moments . . . . . . . . . . . . . . . . . . . . . . . .
11.2.3 Symmetry Decomposition . . . . . . . . . . . . . . . . . . . . .
11.2.4 Localized Orbital Bonding Analysis . . . . . . . . . . . . . . .
11.2.5 Basic Excited-State Analysis of CIS and TDDFT Wave Functions
11.2.6 General Excited-State Analysis . . . . . . . . . . . . . . . . . .
11.3 Interface to the NBO Package . . . . . . . . . . . . . . . . . . . . . . .
11.4 Orbital Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.5 Visualizing and Plotting Orbitals, Densities, and Other Volumetric Data
11.5.1 Visualizing Orbitals Using M OL D EN and M AC M OL P LT . . . .
11.5.2 Visualization of Natural Transition Orbitals . . . . . . . . . . . .
11.5.3 Generation of Volumetric Data Using $plots . . . . . . . . . . .
11.5.4 Direct Generation of “Cube” Files . . . . . . . . . . . . . . . .
11.5.5 NCI Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.5.6 Electrostatic Potentials . . . . . . . . . . . . . . . . . . . . . . .

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495
495
496
502
505
505
507
508
509
510
510
511
511
512
513
514
518
521
521
522
525
526
531
532
532
537
540
543
545
548
553
553
555
558

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561
561
561
562
567
568
569
570
572
574
575
576
576
578
579
584
587
587

11

CONTENTS
11.6
11.7
11.8
11.9

Spin and Charge Densities at the Nuclei . . . . . . . . .
Atoms in Molecules . . . . . . . . . . . . . . . . . . . .
Distributed Multipole Analysis . . . . . . . . . . . . . .
Intracules . . . . . . . . . . . . . . . . . . . . . . . . .
11.9.1 Position Intracules . . . . . . . . . . . . . . . . .
11.9.2 Momentum Intracules . . . . . . . . . . . . . . .
11.9.3 Wigner Intracules . . . . . . . . . . . . . . . . .
11.9.4 Intracule Job Control . . . . . . . . . . . . . . .
11.9.5 Format for the $intracule Section . . . . . . . . .
11.10 Harmonic Vibrational Analysis . . . . . . . . . . . . . .
11.10.1 Job Control . . . . . . . . . . . . . . . . . . . . .
11.10.2 Isotopic Substitutions . . . . . . . . . . . . . . .
11.10.3 Partial Hessian Vibrational Analysis . . . . . . .
11.10.4 Localized Mode Vibrational Analysis . . . . . . .
11.11 Anharmonic Vibrational Frequencies . . . . . . . . . . .
11.11.1 Vibration Configuration Interaction Theory . . . .
11.11.2 Vibrational Perturbation Theory . . . . . . . . . .
11.11.3 Transition-Optimized Shifted Hermite Theory . .
11.11.4 Job Control . . . . . . . . . . . . . . . . . . . . .
11.12 Linear-Scaling Computation of Electric Properties . . . .
11.12.1 $fdpfreq Input Section . . . . . . . . . . . . . . .
11.12.2 Job Control for the MOProp Module . . . . . . .
11.12.3 Examples . . . . . . . . . . . . . . . . . . . . . .
11.13 NMR and Other Magnetic Properties . . . . . . . . . . .
11.13.1 NMR Chemical Shifts and J-Couplings . . . . . .
11.13.2 Linear-Scaling NMR Chemical Shift Calculations
11.13.3 Additional Magnetic Field-Related Properties . .
11.14 Finite-Field Calculation of (Hyper)Polarizabilities . . . .
11.14.1 Numerical Calculation of Static Polarizabilities . .
11.14.2 Romberg Finite-Field Procedure . . . . . . . . .
11.15 General Response Theory . . . . . . . . . . . . . . . . .
11.15.1 Job Control . . . . . . . . . . . . . . . . . . . . .
11.15.2 $response Section and Operator Specification . .
11.15.3 Examples Including $response Section . . . . . .
11.16 Electronic Couplings for Electron- and Energy Transfer .
11.16.1 Eigenstate-Based Methods . . . . . . . . . . . . .
11.16.2 Diabatic-State-Based Methods . . . . . . . . . .
11.17 Population of Effectively Unpaired Electrons . . . . . .
11.18 Molecular Junctions . . . . . . . . . . . . . . . . . . . .
References and Further Reading . . . . . . . . . . . . . . . . .

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589
590
590
590
591
592
593
594
596
597
598
599
601
604
606
607
608
608
609
612
613
614
619
619
619
625
627
629
629
630
633
634
639
641
642
642
649
657
660
675

12 Molecules in Complex Environments: Solvent Models, QM/MM and QM/EFP Features, Density
Embedding
12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.2 Chemical Solvent Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.2.1 Kirkwood-Onsager Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.2.2 Polarizable Continuum Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.2.3 PCM Job Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.2.4 Linear-Scaling QM/MM/PCM Calculations . . . . . . . . . . . . . . . . . . . . . . . .
12.2.5 Isodensity Implementation of SS(V)PE . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.2.6 Composite Method for Implicit Representation of Solvent (CMIRS) . . . . . . . . . . . .
12.2.7 COSMO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.2.8 SM8, SM12, and SMD Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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681
681
681
684
686
691
710
713
721
724
724

12

CONTENTS
12.2.9 Langevin Dipoles Model . . . . . . . . . . . . . . . . .
12.2.10 Poisson Boundary Conditions . . . . . . . . . . . . . . .
12.3 Stand-Alone QM/MM Calculations . . . . . . . . . . . . . . . .
12.3.1 Available QM/MM Methods and Features . . . . . . . .
12.3.2 Using the Stand-Alone QM/MM Features . . . . . . . .
12.3.3 Additional Job Control Variables . . . . . . . . . . . . .
12.3.4 QM/MM Examples . . . . . . . . . . . . . . . . . . . .
12.4 Q-CHEM/CHARMM Interface . . . . . . . . . . . . . . . . . .
12.5 Effective Fragment Potential Method . . . . . . . . . . . . . . .
12.5.1 Theoretical Background . . . . . . . . . . . . . . . . . .
12.5.2 Excited-State Calculations with EFP . . . . . . . . . . .
12.5.3 Extension to Macromolecules: Fragmented EFP Scheme .
12.5.4 Running EFP Jobs . . . . . . . . . . . . . . . . . . . . .
12.5.5 Library of Fragments . . . . . . . . . . . . . . . . . . .
12.5.6 Calculation of User-Defined EFP Potentials . . . . . . .
12.5.7 fEFP Input Structure . . . . . . . . . . . . . . . . . . . .
12.5.8 Input keywords . . . . . . . . . . . . . . . . . . . . . .
12.5.9 Examples . . . . . . . . . . . . . . . . . . . . . . . . . .
12.6 Projector-Based Density Embedding . . . . . . . . . . . . . . .
12.6.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.6.2 Job Control for Density Embedding Calculations . . . . .
12.7 Frozen-Density Embedding Theory based methods . . . . . . .
12.7.1 FDE-ADC . . . . . . . . . . . . . . . . . . . . . . . . .
References and Further Reading . . . . . . . . . . . . . . . . . . . . .

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13 Fragment-Based Methods
13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.2 Specifying Fragments in the $molecule Section . . . . . . . . . . . . . . . . . . . . . . .
13.3 FRAGMO Initial Guess for SCF Methods . . . . . . . . . . . . . . . . . . . . . . . . . .
13.4 Locally-Projected SCF Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.4.1 Locally-Projected SCF Methods with Single Roothaan-Step Correction . . . . . . .
13.4.2 Roothaan-Step Corrections to the FRAGMO Initial Guess . . . . . . . . . . . . . .
13.4.3 Automated Evaluation of the Basis-Set Superposition Error . . . . . . . . . . . . .
13.5 The First-Generation ALMO-EDA and Charge-Transfer Analysis (CTA) . . . . . . . . . .
13.5.1 Energy Decomposition Analysis Based on Absolutely Localized Molecular Orbitals
13.5.2 Analysis of Charge-Transfer Based on Complementary Occupied/Virtual Pairs . . .
13.6 Job Control for Locally-Projected SCF Methods . . . . . . . . . . . . . . . . . . . . . . .
13.7 The Second-Generation ALMO-EDA Method . . . . . . . . . . . . . . . . . . . . . . . .
13.7.1 Generalized SCFMI Calculations and Additional Features . . . . . . . . . . . . . .
13.7.2 Polarization Energy with a Well-defined Basis Set Limit . . . . . . . . . . . . . . .
13.7.3 Further Decomposition of the Frozen Interaction Energy . . . . . . . . . . . . . . .
13.7.4 Job Control for EDA2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.8 The MP2 ALMO-EDA Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.9 The Adiabatic ALMO-EDA Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.10 ALMO-EDA Involving Excited-State Molecules . . . . . . . . . . . . . . . . . . . . . . .
13.10.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.10.2 Job Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.11 The Explicit Polarization (XPol) Method . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.11.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.11.2 Supplementing XPol with Empirical Potentials . . . . . . . . . . . . . . . . . . . .
13.11.3 Job Control Variables for XPol . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.11.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.12 Symmetry-Adapted Perturbation Theory (SAPT) . . . . . . . . . . . . . . . . . . . . . .

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730
733
746
746
747
755
757
760
764
764
767
768
769
770
772
773
775
780
782
783
783
786
786
792

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798
798
799
800
804
805
806
806
807
807
810
812
815
815
817
819
822
826
827
831
831
833
835
835
836
837
838
840

13

CONTENTS
13.12.1 Theory . . . . . . . . . . . . . . . . . . . . . .
13.12.2 Job Control for SAPT Calculations . . . . . . .
13.13 The XPol+SAPT (XSAPT) Method . . . . . . . . . .
13.13.1 Theory . . . . . . . . . . . . . . . . . . . . . .
13.13.2 AO-XSAPT(KS)+aiD . . . . . . . . . . . . . .
13.14 Energy Decomposition Analysis based on SAPT/cDFT
13.15 The Many-Body Expansion Method . . . . . . . . . .
13.15.1 Theory and Implementation Details . . . . . . .
13.15.2 Job Control and Examples . . . . . . . . . . . .
13.16 Ab Initio Frenkel Davydov Exciton Model (AIFDEM) .
13.16.1 Theory . . . . . . . . . . . . . . . . . . . . . .
13.16.2 Job Control . . . . . . . . . . . . . . . . . . . .
13.16.3 Derivative Couplings . . . . . . . . . . . . . .
13.16.4 Job Control for AIFDEM Derivative Couplings .
13.17 TDDFT for Molecular Interactions . . . . . . . . . . .
13.17.1 Theory . . . . . . . . . . . . . . . . . . . . . .
13.17.2 Job Control . . . . . . . . . . . . . . . . . . . .
13.18 The ALMO-CIS and ALMO-CIS+CT Methods . . . .
13.18.1 Theory . . . . . . . . . . . . . . . . . . . . . .
13.18.2 Job Control . . . . . . . . . . . . . . . . . . . .
References and Further Reading . . . . . . . . . . . . . . . .
A Geometry Optimization with Q-C HEM
A.1
Introduction . . . . . . . . . . . . . .
A.2
Theoretical Background . . . . . . . .
A.3
Eigenvector-Following (EF) Algorithm
A.4
Delocalized Internal Coordinates . . .
A.5
Constrained Optimization . . . . . . .
A.6
Delocalized Internal Coordinates . . .
A.7
GDIIS . . . . . . . . . . . . . . . . .
References and Further Reading . . . . . . .

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840
843
846
846
848
851
854
854
855
859
859
861
862
863
864
864
864
865
865
866
868

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871
871
872
874
875
878
880
881
882

B AOI NTS
B.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . .
B.2
Historical Perspective . . . . . . . . . . . . . . . . . . . . . .
B.3
AOI NTS: Calculating ERIs with Q-C HEM . . . . . . . . . . .
B.4
Shell-Pair Data . . . . . . . . . . . . . . . . . . . . . . . . .
B.5
Shell-Quartets and Integral Classes . . . . . . . . . . . . . . .
B.6
Fundamental ERI . . . . . . . . . . . . . . . . . . . . . . . .
B.7
Angular Momentum Problem . . . . . . . . . . . . . . . . . .
B.8
Contraction Problem . . . . . . . . . . . . . . . . . . . . . .
B.9
Quadratic Scaling . . . . . . . . . . . . . . . . . . . . . . . .
B.10 Algorithm Selection . . . . . . . . . . . . . . . . . . . . . . .
B.11 More Efficient Hartree–Fock Gradient and Hessian Evaluations
B.12 User-Controllable Variables . . . . . . . . . . . . . . . . . . .
References and Further Reading . . . . . . . . . . . . . . . . . . . .

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884
884
884
885
886
886
886
887
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887
888
888
888
888

C Q-C HEM Quick Reference
C.1
Q-C HEM Text Input Summary
C.1.1 Keyword: $molecule . .
C.1.2 Keyword: $rem . . . .
C.1.3 Keyword: $basis . . . .
C.1.4 Keyword: $comment . .

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14

Chapter 0: CONTENTS
C.1.5 Keyword: $ecp . . . . . . . . . . . . . . . . . . . . . . . . . . . .
C.1.6 Keyword: $empirical_dispersion . . . . . . . . . . . . . . . . . .
C.1.7 Keyword: $external_charges . . . . . . . . . . . . . . . . . . . .
C.1.8 Keyword: $intracule . . . . . . . . . . . . . . . . . . . . . . . . .
C.1.9 Keyword: $isotopes . . . . . . . . . . . . . . . . . . . . . . . . .
C.1.10 Keyword: $multipole_field . . . . . . . . . . . . . . . . . . . . .
C.1.11 Keyword: $nbo . . . . . . . . . . . . . . . . . . . . . . . . . . .
C.1.12 Keyword: $occupied . . . . . . . . . . . . . . . . . . . . . . . . .
C.1.13 Keyword: $opt . . . . . . . . . . . . . . . . . . . . . . . . . . . .
C.1.14 Keyword: $svp . . . . . . . . . . . . . . . . . . . . . . . . . . . .
C.1.15 Keyword: $svpirf . . . . . . . . . . . . . . . . . . . . . . . . . .
C.1.16 Keyword: $plots . . . . . . . . . . . . . . . . . . . . . . . . . . .
C.1.17 Keyword: $localized_diabatization . . . . . . . . . . . . . . . . .
C.1.18 Keyword: $van_der_waals . . . . . . . . . . . . . . . . . . . . .
C.1.19 Keyword: $xc_functional . . . . . . . . . . . . . . . . . . . . . .
C.2
Geometry Optimization with General Constraints . . . . . . . . . . . . .
C.3
$rem Variable List . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
C.3.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
C.3.2 SCF Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
C.3.3 DFT Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
C.3.4 Large Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . .
C.3.5 Correlated Methods . . . . . . . . . . . . . . . . . . . . . . . . .
C.3.6 Correlated Methods Handled by CCMAN and CCMAN2 . . . . .
C.3.7 Perfect pairing, Coupled cluster valence bond, and related methods
C.3.8 Excited States: CIS, TDDFT, SF-XCIS and SOS-CIS(D) . . . . .
C.3.9 Excited States: EOM-CC and CI Methods . . . . . . . . . . . . .
C.3.10 Geometry Optimizations . . . . . . . . . . . . . . . . . . . . . . .
C.3.11 Vibrational Analysis . . . . . . . . . . . . . . . . . . . . . . . . .
C.3.12 Reaction Coordinate Following . . . . . . . . . . . . . . . . . . .
C.3.13 NMR Calculations . . . . . . . . . . . . . . . . . . . . . . . . . .
C.3.14 Wave function Analysis and Molecular Properties . . . . . . . . .
C.3.15 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
C.3.16 Printing Options . . . . . . . . . . . . . . . . . . . . . . . . . . .
C.3.17 Resource Control . . . . . . . . . . . . . . . . . . . . . . . . . .
C.4
Alphabetical Listing of $rem Variables . . . . . . . . . . . . . . . . . . .
References and Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . .

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. 893
. 893
. 893
. 894
. 894
. 894
. 894
. 895
. 895
. 895
. 896
. 896
. 896
. 896
. 897
. 897
. 898
. 898
. 898
. 899
. 899
. 899
. 899
. 900
. 900
. 900
. 901
. 901
. 901
. 901
. 901
. 902
. 902
. 902
. 902
.1090

Chapter 1

Introduction
1.1
1.1.1

About This Manual
Overview

This manual is intended as a general-purpose user’s guide for Q-C HEM, a modern electronic structure program. The
manual contains background information that describes Q-C HEM methods and user-selected parameters. It is assumed
that the user has some familiarity with the Unix/Linux environment, an ASCII file editor, and a basic understanding of
quantum chemistry.
After installing Q-C HEM and making necessary adjustments to your user account, it is recommended that particular
attention be given to Chapters 3 and 4. The latter, which describes Q-C HEM’s self-consistent field capabilities, has
been formatted so that advanced users can quickly find the information they require while supplying new users with
a moderate level of important background information. This format has been maintained throughout the manual, and
every attempt has been made to guide the user forward and backward to other relevant information so that a logical
progression through this manual is not necessary.
Documentation for IQ MOL, a graphical user interface designed for use with Q-C HEM, can be found on the www.
iqmol.org. IQ MOL functions as a molecular structure builder, as an interface for local or remote submission of
Q-C HEM jobs, and as a post-calculation visualization program for densities and molecular orbitals.

1.1.2

Chapter Summaries

Ch. 1: General overview of Q-C HEM’s features, contributors, and contact information.
Ch. 2: Procedures to install, test, and run Q-C HEM on your machine.
Ch. 3: Overview of the Q-C HEM command-line input.
Ch. 4: Running ground-state self-consistent field calculations.
Ch. 5: Details specific to running density functional theory (DFT) calculations.
Ch. 6: Running post-Hartree-Fock correlated wave function calculations for ground states.
Ch. 7: Running calculations for excited states and open-shell species.
Ch. 8: Using Q-C HEM’s built-in basis sets, or specifying a user-defined basis set.
Ch. 9: Using Q-C HEM’s effective core potential capabilities.
Ch. 10: Options available for exploring potential energy surfaces, such as determining critical points (transition states
and local minima on a single surface, or minimum-energy crossing points between surfaces) as well as ab initio
molecular dynamics.

Chapter 1: Introduction

16

Ch. 11: Molecular properties and a posteriori wave function analysis.
Ch. 12: Methods for molecules in complex environments, including implicit solvation models, QM/MM models, the
Effective Fragment Potential, and density embedding.
Ch. 13: Fragment-based approaches for efficient calculations on large systems, calculation of non-covalent interactions, and energy decomposition analysis.
App. A: Overview of the O PTIMIZE package used for determining molecular geometry critical points.
App. B: Overview of the AOI NTS library, which contains some of the fastest two-electron integral code currently
available.
App. C: Quick-reference section containing an alphabetized list of job control variables.

1.2

Q-C HEM, Inc.

1.2.1

Contact Information and Customer Support

For general information regarding Q-C HEM program, visit www.q-chem.com. Full customer support is promptly
provided via telephone or email (support@q-chem.com) for those customers who have purchased Q-C HEM’s
“QMP” maintenance contract. In addition to free customer support, this contract provides discounts on future updates
and releases of Q-C HEM. For details of the maintenance contract please see www.q-chem.com.

1.2.2

About the Company

Q-C HEM, Inc. was founded in 1993 and was based in Pittsburgh, PA until 2013, when it relocated to Pleasanton,
CA. Q-C HEM’s scientific contributors include leading quantum chemists around the world. The company is governed
by the Board of Directors which currently consists of Peter Gill (Canberra), Anna Krylov (USC), John Herbert (Ohio
State), and Hilary Pople. Fritz Schaefer (Georgia) is a Board Member Emeritus. Martin Head-Gordon is a Scientific
Advisor to the Board. The close coupling between leading university research groups and Q-C HEM Inc. ensures that
the methods and algorithms available in Q-C HEM are state-of-the-art.
In order to create this technology, the founders of Q-C HEM, Inc. built entirely new methodologies from the ground up,
using the latest algorithms and modern programming techniques. Since 1993, well over 300 person-years have been
devoted to the development of the Q-C HEM program. The author list of the program shows the full list of contributors
to the current version, and the journal citations for Q-C HEM versions 2, 3, and 4 1,3,4 illustrate the breadth of the QC HEM developer community. The current group of developers consist of more than 100 people in 9 countries. A brief
history of Q-C HEM is given in the recent article Q-Chem: An Engine for Innovation. 2

1.2.3

Company Mission

The mission of Q-C HEM, Inc. is to develop, distribute, and support innovative and sustainable quantum chemistry software for industrial, government and academic researchers in the chemical, petrochemical, biochemical, pharmaceutical
and material sciences.

1.3

Q-C HEM Features

Quantum chemistry methods have proven invaluable for studying chemical and physical properties of molecules. The
Q-C HEM system brings together a variety of advanced computational methods and tools in an integrated ab initio
software package, greatly improving the speed and accuracy of calculations being performed. In addition, Q-C HEM
will accommodate larger molecular structures than previously possible, with no loss in accuracy, thereby bringing the

Chapter 1: Introduction

17

power of quantum chemistry to critical research projects for which this tool was previously unavailable. Below is a
reverse-chronological listing of new features added to Q-C HEM.

1.3.1

New Features in Q-C HEM 5.1

• Improved OpenMP parallelization for:
– SCF vibrational frequency calculations (Z. Gan)
– RIMP2 gradient (F. Rob, Joonho Lee, X. Feng, & E. Epifanovsky)
• Complete active space self-consistent field (CASSCF) and adaptive sampling CI (D. Levine, M. Head-Gordon)
• Tkatchenko-Scheffler van der Waals method (Section 5.7.4) and many-body dispersion method (Section 5.7.5)
(D. Barton, Ka Un Lao, & R. DiStasio)
• Enhancements to the coupled-cluster package:
– Core/valence separation for EOM-CCSD core-level excited and ionized states (M. Vidal, A.I. Krylov, X.
Feng, E. Epifanovsky & S. Coriani), Section 7.7.5.
– NTO analysis of two-photon transitions (K. Nanda & A.I. Krylov), Section 7.7.16.1.
– NTO analysis of the complex-valued EOM wave functions (A.I. Krylov, W. Skomorowski), Section 7.7.16.
– Analytic gradients for Cholesky-decomposed and resolution-of-identity CCSD and EOM-CCSD (X. Feng,
A.I. Krylov).
– Improved performance, reduced disk usage by coupled-cluster methods (E. Epifanovsky, I. Kaliman, & X.
Feng).
• New features in NTO analysis: Energies of NTOs (A.I. Krylov), Section 11.2.6.
• Finite-difference evaluation of non-linear properties (M. de Wergifosse & A.I. Krylov), Section 11.14.2.
• Poisson boundary conditions for SCF calculations (M. Coons & J. Herbert), Section 12.2.10.
– Enables quantum chemistry calculations in an arbitrary (anisotropic and inhomogeneous) dielectric environment.
– Nonequilibrium solvent corrections for vertical ionization energies.
• Energy decomposition analysis (EDA):
– EDA based on symmetry-adapted perturbation theory and constrained DFT (SAPT/cDFT-EDA), Section 13.14
(Ka Un Lao, K. Fenk, & J. Herbert)
– ALMO-EDA for CIS and TDDFT/TDA excited states, Section 13.10 (Qinghui Ge, Yuezhi Mao, & M.
Head-Gordon)
– Perturbative ALMO-CTA and COVP analysis in EDA2 (Yuezhi Mao & M. Head-Gordon)
• Analytic derivative couplings for computing excitation/vibration energy couplings within the ab initio FrenkelDavydov exciton model (A. Morrison & J. Herbert), Section 13.16.3.
• Hyperfine spin-spin couplings and nuclear electric quadrupole couplings, Section 11.13.3 (E. Berquist & D.
Lambrecht)
• Variational two-electron reduced-density-matrix (v2RDM) and v2RDM-driven complete active space self-consistent
field (v2RDM-CASSCF) method (G. Gidofalvi, L. Koulias, J.W. Mullinax, & A.E. DePrince III)
• Frozen and restrained potential energy scans, Section 10.4 (Yihan Shao)
• Extended ESP charge fitting procedure to the computation of RESP charges (Yihan Shao)

Chapter 1: Introduction

1.3.2

18

New Features in Q-C HEM 5.0

• Enhancements to the coupled-cluster package:
– Analytic gradients for Cholesky-decomposed CCSD and EOM-CCSD; efficiency improvement for canonical CCSD and EOM-CCSD gradients (X. Feng, E. Epifanovsky).
– CAP-EOM-CCSD analytic gradients (Z. Benda and T.-C. Jagau) and Dyson orbitals for metastable states
(T.-C. Jagau, A.I. Krylov), Section 7.7.6).
– CAP-EOM-MP2 method (A. Kunitsa, K. Bravaya).
– Evaluation of polarizabilities using CCSD and EOM-CCSD (EE and SF) wave functions using full derivative formulation (K. Nanda and A. Krylov, Section 7.7.16.4).
– Evaluation of hS 2 i for EOM-CCSD wave functions (X. Feng).
– Evaluation of NACs for EOM-CCSD wave functions (S. Faraji, A. Krylov, E. Epifanovski, X. Feng, Section
7.7.16.3).
– Efficiency improvement and new multicore-parallel code for (T) correction (I. Kaliman).
– New coupled-cluster based methods for core states (A. Krylov).
• New capabilities for implicit solvation modeling:
– PCM capabilities for computing vertical excitation, ionization, and electron attachment energies at EOMCC and MP2 levels (Section 7.7.11).
– State-specific equilibrium and non-equilibrium solvation for all orders and variants of ADC (J. M. Mewes
and A. Dreuw; Section 7.8.7).
– Poisson equation boundary conditions allowing use of an arbitrary, anisotropic dielectric function ε(r),
with full treatment of volume polarization (M. P. Coons and J. M. Herbert; Section 12.2.10).
– Composite Model for Implicit Representation of Solvent (CMIRS), an accurate model for free energies of
solvation (Section 12.2.6)
• New density functionals (N. Mardirossian and M. Head-Gordon; Section 5.3):
– GGA functionals: BEEF-vdW, HLE16, KT1, KT2, KT3, rVV10
– Meta-GGA functionals: B97M-rV, BLOC, mBEEF, oTPSS, TM
– Hybrids: CAM-QTP(00), CAM-QTP(01), HSE-HJS, LC-ωPBE08, MN15, rCAM-B3LYP, WC04, WP04
– Double hybrids: B2GP-PLYP, DSD-PBEB95-D3, DSD-PBEP86-D3, DSD-PBEPBE-D3, LS1DH-PBE,
PBE-QIDH, PTPSS-D3, PWPB95-D3
– Grimme’s PBEh-3c “low-cost” composite method
– rVV10 non-local correlation functional
• Additional DFT developments:
– New forms of DFT-D3 (J. Witte; Section 5.7.2).
– New standard integration grids, SG-2 and SG-3 (S. Dasgupta and J. M. Herbert; Section 5.5.2).
– More efficient propagator algorithms for real-time TDDFT (Y. Zhu and J. M. Herbert; Section 7.11).
• New integral package for for computing effective core potential (ECP) integrals (S. C. McKenzie, E. Epifanovsky; Chapter 9).
– More efficient analytic algorithms for energies and first derivatives.
– Support for arbitrary projector angular momentum.
– Support up to h angular momentum in the basis set.

Chapter 1: Introduction

19

• Analytic derivative couplings for the ab initio Frenkel-Davydov exciton model (A. F. Morrison and J. M. Herbert;
Section 13.16.3).
• New ALMO-based energy decomposition analysis (EDA) methods:
– The second-generation ALMO-EDA methods for DFT (P. R. Horn, Y. Mao and M. Head-Gordon; Section 13.7)
– The extension of ALMO-EDA to RIMP2 theory (J. Thirman and M. Head-Gordon; Section 13.8)
– The “adiabatic" EDA method for decomposing changes in molecular properties (Y. Mao, P. R. Horn and
M. Head-Gordon; Section 13.9)
• Wave function correlation capabilities:
– Coupled cluster valence bond (CCVB) method for describing open-shell molecules with strong spin correlations (D. W. Small and M. Head-Gordon; Section 6.15.2).
– Implementation of coupled-cluster valence bond with singles and doubles (CCVB-SD) for closed-shell
species (J. Lee, D. W. Small and M. Head-Gordon; Section 6.10.4).
Note: Several important changes in Q-C HEM’s default settings have occurred since version 4.4.
• Core electrons are now frozen by default in most post-Hartree-Fock calculations; see Section 6.2.
• The keywords for calculation of SOCs and NACs were renamed for consistency between different methods.
• Some newer density functionals now use either the SG-2 or SG-3 quadrature grid by default, whereas
all functionals used SG-1 by default in v. 4.4. Table 5.3 lists the default grid for various classes of
functionals.

1.3.3

New Features in Q-C HEM 4.4

• occ-RI-K algorithm for the evaluation of exact exchange in energy and force calculations (S. Manzer, F. Rob and
M. Head-Gordon; Section 4.6.9)
• Combinatorially-optimized exchange-correlation functionals (N. Mardirossian and M. Head-Gordon; Section 5.3):
– ωB97M-V (range-separated hybrid, meta-GGA functional with VV10 non-local correlation)
– B97M-V (meta-GGA functional with VV10 non-local correlation)
– ωB97X-V (range-separated hybrid functional with VV10 non-local correlation)
• Implementation of new exchange-correlation functionals from the literature (N. Mardirossian and M. HeadGordon; Section 5.3). These include:
– MGGA_MS0, MGGA_MS1, MGGA_MS2, MGGA_MS2h, MGGA_MVS, MGGA_MVSh, PKZB, revTPSS,
revTPSSh, SCAN, SCAN0, PBEsol, revPBE, revPBE0
– N12, N12-SX, GAM, MN12-L, MN12-SX, MN15-L, dlDF
– VV10, LC-VV10
– B97-K, B97-D3(0), B97-3, τ -HCTH, τ -HCTHh
– SRC1-R1, SRC1-R2, SRC2-R1, SRC2-R2
– B1LYP, B1PW91, MPW1K, LRC-BOP, BHH, BB1K, PW6B95, PWB6K, B2PLYP
• Hessian-free minimum point verification (S. M. Sharada and M. Head-Gordon; Section 10.2.2)
• Exciton-based excited-state models:

Chapter 1: Introduction

20

– Ab initio Frenkel-Davydov model for coupled excitations in multi-chromophore systems (A. F. Morrison
and J. M. Herbert; Section 13.16).
– TDDFT for molecular interactions [TDDFT(MI)], a set of local excitation approximations for efficient
TDDFT calculations in multi-chromophore systems and for single chromophores in the presence of explicit
solvent molecules (J. Liu and J. M. Herbert; Section 13.17).
• Improvements to many-body and XSAPT methods (K. U. Lao and J. M. Herbert)
– MPI-parallelized many-body expansion with analytic gradient (Section 13.15).
– Efficient atomic orbital implementation of XSAPT for both closed- and open-shell systems (Section 13.13.2).
• Thermostats for ab initio molecular dynamics (R. P. Steele and J. M. Herbert).
• Analytic energy gradient for the Ewald summation in QM/MM calculations (Z. C. Holden and J. M. Herbert)
• Zeolite QM/MM methods (J. Gomes and M. Head-Gordon).
• EOM-MP2 methods for excitation, ionization and electron attachment energies (A. Kunitsa and K. Bravaya;
Section 7.7.9).
• Evaluation of polarizabilities using CCSD and EOM-CCSD wave functions (Section 7.7.16.4, K. Nanda and A. I.
Krylov)
• Distributed-memory parallel implementation of CC and EOM-CC methods and performance improvements in
disk-based algorithms (E. Epifanovsky, I. Kaliman, and A. I. Krylov)
• Improvements to the maximum overlap method (MOM) for SCF calculations (A. T. B. Gilbert; Section 7.4).
• Non-equilibrium PCM method to describe solvent effects in ADC excited-state calculations (J.-M. Mewes and
A. Dreuw; Section 7.8.7).
• Spin-flip ADC method (D. Lefrancois and A. Dreuw; Section 7.8.5).

1.3.4

New Features in Q-C HEM 4.3

• Analytic derivative couplings (i.e., non-adiabatic couplings) between electronic states computed at the CIS, spinflip CIS, TDDFT, and spin-flip TDDFT levels (S. Fatehi, Q. Ou, J. E. Subotnik, X. Zhang, and J. M. Herbert;
Section 10.6).
• A third-generation (“+D3”) dispersion potential for XSAPT (K. U. Lao and J. M. Herbert; Section 13.13).
• Non-equilibrium PCM for computing vertical excitation energies (at the TDDFT level) and ionization energies
in solution (Z.-Q. You and J. M. Herbert; Section 12.2.2.3).
• Spin-orbit couplings between electronic states for CC and EOM-CC wave functions (E. Epifanovsky, J. Gauss,
and A. I. Krylov; Section 7.7.16.2).
• PARI-K method for evaluation of exact exchange, which affords dramatic speed-ups for triple-ζ and larger basis
sets in hybrid DFT calculations (S. Manzer and M. Head-Gordon).
• Transition moments and cross sections for two-photon absorption using EOM-CC wave functions (K. Nanda and
A. I. Krylov; Section 7.7.16.1).
• New excited-state analysis for ADC and CC/EOM-CC methods (M. Wormit; Section 11.2.6).
• New Dyson orbital code for EOM-IP-CCSD and EOM-EA-CCSD (A. Gunina and A. I. Krylov; Section 7.7.23).
• Transition moments, state dipole moments, and Dyson orbitals for CAP-EOM-CCSD (T.-C. Jagau and A. I. Krylov;
Sections 7.7.6 and 7.7.23).

Chapter 1: Introduction

21

• TAO-DFT: Thermally-assisted-occupation density functional theory (J.-D. Chai; Section 5.12).
• MP2[V], a dual basis method that approximates the MP2 energy (J. Deng and A. Gilbert).
• Iterative Hirshfeld population analysis for charged systems, and CM5 semi-empirical charge scheme (K. U. Lao
and J. M. Herbert; Section 11.2.1).
• New DFT functionals: (Section 5.3):
– Long-range corrected functionals with empirical dispersion-: ωM05-D, ωB97X-D3 and ωM06-D3 (Y.-S.
Lin, K. Hui, and J.-D. Chai.
– PBE0_DH and PBE0_2 double-hybrid functionals (K. Hui and J.-D. Chai; Section 5.9).
– AK13 (K. Hui and J.-D. Chai).
– LFAs asymptotic correction scheme (P.-T. Fang and J.-D. Chai).
• LDA/GGA fundamental gap using a frozen-orbital approximation (K. Hui and J.-D. Chai; Section 5.11).

1.3.5

New Features in Q-C HEM 4.2

• Input file changes:
– New keyword METHOD simplifies input in most cases by replacing the pair of keywords EXCHANGE and
CORRELATION (see Chapter 4).
– Keywords for requesting excited-state calculations have been modified and simplified (see Chapter 7 for
details).
– Keywords for solvation models have been modified and simplified (see Section 12.2 for details).
• New features for NMR calculations including spin-spin couplings (J. Kussmann, A. Luenser, and C. Ochsenfeld;
Section 11.13.1).
• New built-in basis sets (see Chapter 8).
• New features and performance improvements in EOM-CC:
– EOM-CC methods extended to treat meta-stable electronic states (resonances) via complex scaling and
complex absorbing potentials (D. Zuev, T.-C. Jagau, Y. Shao, and A. I. Krylov; Section 7.7.6).
– New features added to EOM-CC iterative solvers, such as methods for interior eigenvalues and userspecified guesses (D. Zuev; Section 7.7.12).
– Multi-threaded parallel code for (EOM-)CC gradients and improved CCSD(T) performance.
• New features and performance improvements in ADC methods (M. Wormit, A. Dreuw):
– RI-ADC can tackle much larger systems at reduced cost (Section 7.8.2).
– SOS-ADC methods (Section 7.8.3).
– State-to-state properties for ADC (Section 7.8.6).
• SM12 implicit solvation model (A. V. Marenich, D. G. Truhlar, and Y. Shao; Section 12.2.8.1).
• Interface to NBO v. 6 (Section 11.3).
• Optimization of MECPs between electronic states at the SOS-CIS(D) and TDDFT levels (X. Zhang and J. M.
Herbert; Section 10.6.3).
• ROKS method for ∆SCF calculations of excited states (T. Kowalczyk and T. Van Voorhis; Section 7.5).
• Fragment-based initial guess for SCF methods (Section 13.3).

Chapter 1: Introduction

22

• Pseudo-fractional occupation number method for improved SCF convergence in small-gap systems (D. S. Lambrecht; Section 4.5.10).
• Density embedding scheme (B. J. Albrecht, E. Berquist, and D. S. Lambrecht; Section 12.6).
• New features and enhancements in fragment-based many-body expansion methods (K. U. Lao and J. M. Herbert):
– XSAPT(KS)+D: A dispersion corrected version of symmetry-adapted perturbation theory for fast and accurate calculation of interaction energies in non-covalent clusters (Section 13.13).
– Many-body expansion and fragment molecular orbital (FMO) methods for clusters (Section 13.15).
• Periodic boundary conditions with proper Ewald summation, for energies only (Z. C. Holden and J. M. Herbert;
Section 12.3).

1.3.6

New Features in Q-C HEM 4.1

• Fundamental algorithms:
– Improved parallel performance at all levels including new OpenMP capabilities for Hartree-Fock, DFT,
MP2, and coupled cluster theory (Z. Gan, E. Epifanovsky, M. Goldey, and Y. Shao; Section 2.8).
– Significantly enhanced ECP capabilities, including gradients and frequencies in all basis sets for which the
energy can be evaluated (Y. Shao and M. Head-Gordon; Chap. 9).
• SCF and DFT capabilities:
– TDDFT energy with the M06, M08, and M11 series of functionals.
– XYGJ-OS analytical energy gradient.
– TDDFT/C-PCM excitation energies, gradient, and Hessian (J. Liu and W. Liang; Section 7.3.4).
– Additional features in the maximum overlap method (MOM) approach for converging difficult SCF calculations (N. A. Besley; Section 4.5.6).
• Wave function correlation capabilities:
– RI and Cholesky decomposition implementation of all CC and EOM-CC methods enabling applications to
larger systems with reduced disk and memory requirements and improved performance (E. Epifanovsky,
X. Feng, D. Zuev, Y. Shao, and A. I. Krylov; Sections 6.8.5 and 6.8.6).
– Attenuated MP2 theory in the aug-cc-pVDZ and aug-cc-pVTZ basis sets, which truncates two-electron
integrals to cancel basis set superposition error, yielding results for intermolecular interactions that are much
more accurate than standard MP2 in the same basis set (M. Goldey and M. Head-Gordon; Section 6.7).
– Extended RAS-nSF methodology for ground and excited states involving strong non-dynamical correlation
(P. M. Zimmerman, D. Casanova, and M. Head-Gordon; Section 7.9).
– Coupled cluster valence bond (CCVB) method for describing molecules with strong spin correlations (D. W.
Small and M. Head-Gordon; Section 6.15.2).
• Searching and scanning potential energy surfaces:
– Potential energy surface scans (Y. Shao; Section 10.4).
– Improvements in automatic transition structure searching via the “freezing string” method, including the
ability to perform such calculations without a Hessian calculation (S. M. Sharada and M. Head-Gordon;
Section 10.2.2).
– Enhancements to partial Hessian vibrational analysis (N. A. Besley; Section 11.10.3).
• Calculating and characterizing inter- and intramolecular interactions

Chapter 1: Introduction

23

– Extension of EFP to macromolecules: fEFP approach (A. Laurent, D. Ghosh, A. I. Krylov, and L. V.
Slipchenko; Section 12.5.3).
– Symmetry-adapted perturbation theory level at the “SAPT0” level, for intermolecular interaction energy decomposition analysis into physically-meaningful components such as electrostatics, induction, dispersion,
and exchange. An RI version is also available (L. D. Jacobson, J. M. Herbert; Section 13.12).
– The “explicit polarization” (XPol) monomer-based SCF calculations to compute many-body polarization
effects in linear-scaling time via charge embedding (Section 13.11), which can be combined either with
empirical potentials (e.g., Lennard-Jones) for the non-polarization parts of the intermolecular interactions,
or better yet, with SAPT for an ab initio approach called XSAPT that extends SAPT to systems containing
more that two monomers (L. D. Jacobson and J. M. Herbert; Section 13.13).
– Extension of the absolutely-localized molecular orbital (ALMO)-based energy decomposition analysis to
unrestricted cases (P. R. Horn and M. Head-Gordon; Section 13.5).
– Calculation of the populations of “effectively unpaired electrons” in low-spin state using DFT, a new
method of evaluating localized atomic magnetic moments within Kohn-Sham without symmetry breaking, and Mayer-type bond order analysis with inclusion of static correlation effects (E. I. Proynov; Section 11.17).
• Quantum transport calculations including electron transmission functions and electron tunneling currents under
applied bias voltage (B. D. Dunietz and N. Sergueev; Section 11.18).
• Searchable online version of the Q-C HEM PDF manual (J. M. Herbert and E. Epifanovsky).

1.3.7

New Features in Q-C HEM 4.0.1

• Remote submission capability in IQ MOL (A. T. B. Gilbert).
• Scaled nuclear charge and charge-cage stabilization capabilities (T. Kús and A. I. Krylov; Section 7.7.7).
• Calculations of excited state properties including transition dipole moments between different excited states in
CIS and TDDFT as well as couplings for electron and energy transfer (Z.-Q. You and C.-P. Hsu; Section 11.16).

1.3.8

New Features in Q-C HEM 4.0

• New exchange-correlation functionals (Section 5.3):
– Density-functional dispersion using Becke and Johnson’s XDM model in an efficient, analytic form (Z. Gan,
E. I. Proynov, and J. Kong; Section 5.7.3).
– Van der Waals density functionals vdW-DF-04 and vdW-DF-10 of Langreth and coworkers (O. Vydrov;
Section 5.7.1).
– VV09 and VV10, new analytic dispersion functionals (O. Vydrov, T. Van Voorhis; Section 5.7.1)
– DFT-D3 empirical dispersion methods for non-covalent interactions (S.-P. Mao and J.-D. Chai; Section 5.7.2).
– ωB97X-2, a double-hybrid functional based on the long-range corrected B97 functional, with improved
accounting for medium- and long-range interactions (J.-D. Chai and M. Head-Gordon; Section 5.9).
– XYGJ-OS, a double-hybrid functional for predictions of non-bonded interactions and thermochemistry at
nearly chemical accuracy (X. Xu, W. A. Goddard, and Y. Jung; Section 5.9).
– Short-range corrected functional for calculation of near-edge X-ray absorption spectra (N. A. Besley; Section 7.10.1).
– LB94 asymptotically-corrected exchange-correlation functional for TDDFT (Y.-C. Su and J.-D. Chai; Section 5.10.1).

Chapter 1: Introduction

24

– Non-dynamical correlation in DFT with an efficient RI implementation of the Becke05 model in a fully
analytic formulation (E. I. Proynov, Y. Shao, F. Liu, and J. Kong; Section 5.3).
– TPSS and its hybrid version TPSSh, and rPW86 (F. Liu and O. Vydrov).
– Double-hybrid functional B2PLYP-D (J.-D. Chai).
– Hyper-GGA functional MCY2 from Mori-Sánchez, Cohen, and Yang (F. Liu).
– SOGGA, SOGGA11 and SOGGA11-X family of GGA functionals (R. Peverati, Y. Zhao, and D. G. Truhlar).
– M08-HX and M08-SO suites of high HF exchange meta-GGA functionals (Y. Zhao and D. G. Truhlar).
– M11-L and M11 suites of meta-GGA functionals (R. Peverati, Y. Zhao, D. G. Truhlar).
• Improved DFT algorithms:
– Multi-resolution exchange-correlation (mrXC) for fast calculation of grid-based XC quadrature (S. T.
Brown, C.-M. Chang, and J. Kong; Section 5.5.4).
– Efficient computation of the XC part of the dual basis DFT (Z. Gan and J. Kong; Section 4.4.5).
– Fast DFT calculation with “triple jumps” between different sizes of basis set and grid, and different levels
of functional (J. Deng, A. T. B. Gilbert, and P. M. W. Gill; Section 4.8).
– Faster DFT and HF calculation with an atomic resolution-of-identity algorithm (A. Sodt and M. HeadGordon; Section 4.6.6).
• Post-Hartree–Fock methods:
– Significantly enhanced coupled-cluster code rewritten for better performance on multi-core architectures,
including energy and gradient calculations with CCSD and energy calculations with EOM-EE/SF/IP/EACCSD, and CCSD(T) energy calculations (E. Epifanovsky, M. Wormit, T. Kús, A. Landau, D. Zuev,
K. Khistyaev, I. Kaliman, A. I. Krylov, and A. Dreuw; Chaps. 6 and 7).
– Fast and accurate coupled-cluster calculations with frozen natural orbitals (A. Landau, D. Zuev, and A. I.
Krylov; Section 6.11).
– Correlated excited states with the perturbation-theory based, size-consistent ADC scheme (M. Wormit and
A. Dreuw; Section 7.8).
– Restricted active space, spin-flip method for multi-configurational ground states and multi-electron excited
states (P. M. Zimmerman, F. Bell, D. Casanova, and M. Head-Gordon; Section 7.2.4).
• Post-Hartree–Fock methods for describing strong correlation:
– “Perfect quadruples” and “perfect hextuples” methods for strong correlation problems (J. A. Parkhill and
M. Head-Gordon; Section 6.10.5).
– Coupled-cluster valence bond (CCVB) methods for multiple-bond breaking (D. W. Small, K. V. Lawler,
and M. Head-Gordon; Section 6.15).
• TDDFT for excited states:
– Nuclear gradients for TDDFT (Z. Gan, C.-P. Hsu, A. Dreuw, M. Head-Gordon, and J. Kong; Section 7.3.1).
– Direct coupling of charged states for study of charge transfer reactions (Z.-Q. You and C.-P. Hsu; Section 11.16.2).
– Analytical excited-state Hessian for TDDFT within the Tamm-Dancoff approximation (J. Liu and W. Liang;
Section 7.3.5).
– Self-consistent excited-states with the maximum overlap method (A. T. B. Gilbert, N. A. Besley, and
P. M. W. Gill; Section 7.4).

Chapter 1: Introduction

25

– Calculation of reactions via configuration interactions of charge-constrained states computed with constrained DFT (Q. Wu, B. Kaduk and T. Van Voorhis; Section 5.13).
– Overlap analysis of the charge transfer in a TDDFT excited state (N. A. Besley; Section 7.3.2).
– Localizing diabatic states with Boys or Edmiston-Ruedenberg localization, for charge or energy transfer
(J. E Subotnik, R. P. Steele, N. Shenvi, and A. Sodt; Section 11.16.1.2).
– Non-collinear formalism for spin-flip TDDFT (Y. Shao, Y. A. Bernard, and A. I. Krylov; Section 7.3)
• Solvation and condensed-phase modeling
– Smooth free energy surface for solvated molecules via SWIG-PCMs, for QM and QM/MM calculations,
including a linear-scaling QM/MM/PCM algorithm (A. W. Lange and J. M. Herbert; Sections 12.2.2 and
12.2.4).
– Klamt’s COSMO solvation model with DFT energy and gradient (Y. Shao; Section 12.2.7).
– Polarizable explicit solvent via EFP, for ground- and excited-state calculations at the DFT/TDDFT and
CCSD/EOM-CCSD levels, as well as CIS and CIS(D). A library of effective fragments for common solvents is also available, along with energy and gradient for EFP–EFP calculations (V. Vanovschi, D. Ghosh,
I. Kaliman, D. Kosenkov, C. F. Williams, J. M. Herbert, M. S. Gordon, M. W. Schmidt, Y. Shao, L. V.
Slipchenko, and A. I. Krylov; Section 12.5).
• Optimizations, vibrations, and dynamics:
– “Freezing” and “growing” string methods for efficient automated reaction-path finding (A. Behn, P. M.
Zimmerman, A. T. Bell, and M. Head-Gordon; Section 10.2.1).
– Improved robustness of the intrinsic reaction coordinate (IRC)-following code (M. Head-Gordon).
– Quantum-mechanical treatment of nuclear motion at equilibrium via path integrals (R. P. Steele; Section 10.8).
– Calculation of local vibrational modes of interest with partial Hessian vibrational analysis (N. A. Besley;
Section 11.10.3).
– Accelerated ab initio molecular dynamics MP2 and/or dual-basis methods, based on Z-vector extrapolation
(R. P. Steele; Section 4.7.2).
– Quasi-classical ab initio molecular dynamics (D. S. Lambrecht and M. Head-Gordon; Section 10.7.5).
• Fragment-based methods:
– Symmetry-adapted perturbation theory (SAPT) for computing and analyzing dimer interaction energies
(L. D. Jacobson, M. A. Rohrdanz, and J. M. Herbert; Section 13.12).
– Many-body generalization of SAPT (“XSAPT”), with empirical dispersion corrections for high accuracy
and low cost in large clusters (L. D. Jacobson, K. U. Lao, and J. M. Herbert; Section 13.13).
– Methods based on a truncated many-body expansion, including the fragment molecular orbital (FMO)
method (K. U. Lao and J. M. Herbert; Section 13.15).
• Properties and wave function analysis:
– Analysis of metal oxidation states via localized orbital bonding analysis (A. J. W. Thom, E. J. Sundstrom,
and M. Head-Gordon; Section 11.2.4).
– Hirshfeld population analysis (S. Yeganeh; Section 11.2.1).
– Visualization of non-covalent bonding using Johnson and Yang’s NCI algorithm (Y. Shao; Section 11.5.5).
– Electrostatic potential on a grid for transition densities (Y. Shao; Section 11.5.6).
• Support for modern computing platforms

Chapter 1: Introduction

26

– Efficient multi-threaded parallel performance for CC, EOM, and ADC methods.
– Better performance for multi-core systems with shared-memory parallel DFT and Hartree-Fock (Z. Gan,
Y. Shao, and J. Kong) and RI-MP2 (M. Goldey and M. Head-Gordon; Section 6.14).
– Accelerated RI-MP2 calculation on GPUs (R. Olivares-Amaya, M. Watson, R. Edgar, L. Vogt, Y. Shao, and
A. Aspuru-Guzik; Section 6.6.4).
• Graphical user interfaces:
– Input file generation, Q-C HEM job submission, and visualization is supported by IQ MOL, a fully integrated
GUI developed by Andrew Gilbert. IQ MOL is a free software and does not require purchasing a Q-C HEM
license. See www.iqmol.org for details and installation instructions.
– Other graphical interfaces are also available, including M OL D EN, M AC M OL P LT, and AVOGADRO (Chapter 11 and elsewhere).

1.3.9

Summary of Features in Q-C HEM versions 3. x

• DFT functionals and algorithms:
– Long-ranged corrected (LRC) functionals, also known as range-separated hybrid functionals (M. A. Rohrdanz
and J. M. Herbert)
– Constrained DFT (Q. Wu and T. Van Voorhis)
– Grimme’s “DFT-D” empirical dispersion corrections (C.-D. Sherrill)
– “Incremental” DFT method that significantly accelerates exchange-correlation quadrature in later SCF cycles (S. T. Brown)
– Efficient SG-0 quadrature grid with approximately half the number of grid points relative to SG-1 (S.-H.
Chien)
• Solvation models:
– SM8 model (A. V. Marenich, R. M. Olson, C. P. Kelly, C. J. Cramer, and D. G. Truhlar)
– Onsager reaction-field model (C.-L. Cheng, T. Van Voorhis, K. Thanthiriwatte, and S. R. Gwaltney)
– Chipman’s SS(V)PE model (S. T. Brown)
• Second-order perturbation theory algorithms for ground and excited states:
– Dual-basis RIMP2 energy and analytical gradient (R. P. Steele, R. A. DiStasio Jr., and M. Head-Gordon)
– O2 energy and gradient (R. C. Lochan and M. Head-Gordon)
– SOS-CIS(D), SOS-CIS(D0 ), and RI-CIS(D) for excited states (D. Casanova, Y. M. Rhee, and M. HeadGordon)
– Efficient resolution-of-identity (RI) implementations of MP2 and SOS-MP2 (including both energies and
gradients), and of RI-TRIM and RI-CIS(D) energies (Y. Jung, R. A. DiStasio, Jr., R. C. Lochan, and Y. M.
Rhee)
• Coupled-cluster methods (P. A. Pieniazek, E. Epifanovsky, A. I. Krylov):
– IP-CISD and EOM-IP-CCSD energy and gradient
– Multi-threaded (OpenMP) parallel coupled-cluster calculations
– Potential energy surface crossing minimization with CCSD and EOM-CCSD methods (E. Epifanovsky)
– Dyson orbitals for ionization from the ground and excited states within CCSD and EOM-CCSD methods
(M. Oana)
• QM/MM methods (H. L. Woodcock, A. Ghysels, Y. Shao, J. Kong, and H. B. Brooks)

Chapter 1: Introduction

27

– Q-C HEM/C HARMM interface (H. L. Woodcock)
– Full QM/MM Hessian evaluation and approximate mobile-block-Hessian evaluation
– Two-layer ONIOM model (Y. Shao).
– Integration with the M OLARIS simulation package (E. Rosta).
• Improved two-electron integrals package
– Rewrite of the Head-Gordon–Pople algorithm for modern computer architectures (Y. Shao)
– Fourier Transform Coulomb method for linear-scaling construction of the Coulomb matrix, even for basis
sets with high angular moment and diffuse functions (L. Fusti-Molnar)
• Dual basis self-consistent field calculations, offering an order-of-magnitude reduction in the cost of large-basis
DFT calculations (J. Kong and R. P. Steele)
• Enhancements to the correlation package including:
– Most extensive range of EOM-CCSD methods available including EOM-SF-CCSD, EOM-EE-CCSD, EOMDIP-CCSD, EOM-IP/EA-CCSD (A. I. Krylov).
– Available for RHF, UHF, and ROHF references.
– Analytic gradients and properties calculations (permanent and transition dipoles etc..).
– Full use of Abelian point-group symmetry.
• Coupled-cluster perfect-paring methods applicable to systems with > 100 active electrons (M. Head-Gordon)
• Transition structure search using the “growing string” algorithm (A. Heyden and B. Peters):
• Ab initio molecular dynamics (J. M. Herbert)
• Linear scaling properties for large systems (J. Kussmann, C. Ochsenfeld):
– NMR chemical shifts
– Static and dynamic polarizabilities
– Static hyper-polarizabilities, optical rectification, and electro-optical Pockels effect
• Anharmonic frequencies (C. Y. Lin)
• Wave function analysis tools:
– Analysis of intermolecular interactions with ALMO-EDA (R. Z. Khaliullin and M. Head-Gordon)
– Electron transfer analysis (Z.-Q. You and C.-P. Hsu)
– Spin densities at the nuclei (V. A. Rassolov)
– Position, momentum, and Wigner intracules (N. A. Besley and D. P. O’Neill)
• Graphical user interface options:
– IQ MOL, a fully integrated GUI. IQ MOL includes input file generator and contextual help, molecular builder,
job submission tool, and visualization kit (molecular orbital and density viewer, frequencies, etc). For
the latest version and download/installation instructions, please see the IQ MOL homepage (www.iqmol.
org).
– Seamless integration with the S PARTAN package (see www.wavefun.com).
– Support for several other public-domain visualization programs:
* W EB MO
www.webmo.net
* AVOGADRO
https://avogadro.cc

Chapter 1: Introduction

28

* M OL D EN
http://www.cmbi.ru.nl/molden
* M AC M OL P LT (via a M OL D EN-formatted input file)
https://brettbode.github.io/wxmacmolplt
* JM OL
www.sourceforge.net/project/showfiles.php?group_id=23629&release_id=
66897

1.3.10

Summary of Features Prior to Q-C HEM 3.0

• Efficient algorithms for large-molecule density functional calculations:
– CFMM for linear scaling Coulomb interactions (energies and gradients) (C. A. White).
– Second-generation J-engine and J-force engine (Y. Shao).
– LinK for exchange energies and forces (C. Ochsenfeld and C. A. White).
– Linear scaling DFT exchange-correlation quadrature.
• Local, gradient-corrected, and hybrid DFT functionals:
– Slater, Becke, GGA91 and Gill ‘96 exchange functionals.
– VWN, PZ81, Wigner, Perdew86, LYP and GGA91 correlation functionals.
– EDF1 exchange-correlation functional (R. Adamson).
– B3LYP, B3P and user-definable hybrid functionals.
– Analytical gradients and analytical frequencies.
– SG-0 standard quadrature grid (S.-H. Chien).
– Lebedev grids up to 5294 points (S. T. Brown).
• High level wave function-based electron correlation methods
– Efficient semi-direct MP2 energies and gradients.
– MP3, MP4, QCISD, CCSD energies.
– OD and QCCD energies and analytical gradients.
– Triples corrections (QCISD(T), CCSD(T) and OD(T) energies).
– CCSD(2) and OD(2) energies.
– Active space coupled cluster methods: VOD, VQCCD, VOD(2).
– Local second order Møller-Plesset (MP2) methods (DIM and TRIM).
– Improved definitions of core electrons for post-HF correlation (V. A. Rassolov).
• Extensive excited state capabilities:
– CIS energies, analytical gradients and analytical frequencies.
– CIS(D) energies.
– Time-dependent density functional theory energies (TDDFT).
– Coupled cluster excited state energies, OD and VOD (A. I. Krylov).
– Coupled-cluster excited-state geometry optimizations.
– Coupled-cluster property calculations (dipoles, transition dipoles).
– Spin-flip calculations for CCSD and TDDFT excited states (A. I. Krylov and Y. Shao).
• High performance geometry and transition structure optimization (J. Baker):

Chapter 1: Introduction

29

– Optimizes in Cartesian, Z-matrix or delocalized internal coordinates.
– Impose bond angle, dihedral angle (torsion) or out-of-plane bend constraints.
– Freezes atoms in Cartesian coordinates.
– Constraints do not need to be satisfied in the starting structure.
– Geometry optimization in the presence of fixed point charges.
– Intrinsic reaction coordinate (IRC) following code.
• Evaluation and visualization of molecular properties
– Onsager, SS(V)PE and Langevin dipoles solvation models.
– Evaluate densities, electrostatic potentials, orbitals over cubes for plotting.
– Natural Bond Orbital (NBO) analysis.
– Attachment/detachment densities for excited states via CIS, TDDFT.
– Vibrational analysis after evaluation of the nuclear coordinate Hessian.
– Isotopic substitution for frequency calculations (R. Doerksen).
– NMR chemical shifts (J. Kussmann).
– Atoms in Molecules (AIMPAC) support (J. Ritchie).
– Stability analysis of SCF wave functions (Y. Shao).
– Calculation of position and momentum molecular intracules A. Lee, N. A. Besley, and D. P. O’Neill).
• Flexible basis set and effective core potential (ECP) functionality: (Ross Adamson and Peter Gill)
– Wide range of built-in basis sets and ECPs.
– Basis set superposition error correction.
– Support for mixed and user-defined basis sets.
– Effective core potentials for energies and gradients.
– Highly efficient PRISM-based algorithms to evaluate ECP matrix elements.
– Faster and more accurate ECP second derivatives for frequencies.

1.4

Citing Q-C HEM

Users who publish papers based on Q-C HEM calculations are asked to cite the official peer-reviewed literature citation
for the software. For versions corresponding to 4.0 and later, this is:
Y. Shao, Z. Gan, E. Epifanovsky, A. T. B. Gilbert, M. Wormit, J. Kussmann, A. W. Lange, A. Behn, J. Deng,
X. Feng, D. Ghosh, M. Goldey P. R. Horn, L. D. Jacobson, I. Kaliman, R. Z. Khaliullin, T. Kús, A. Landau, J. Liu,
E. I. Proynov, Y. M. Rhee, R. M. Richard, M. A. Rohrdanz, R. P. Steele, E. J. Sundstrom, H. L. Woodcock III,
P. M. Zimmerman, D. Zuev, B. Albrecht, E. Alguire, B. Austin, G. J. O. Beran, Y. A. Bernard, E. Berquist,
K. Brandhorst, K. B. Bravaya, S. T. Brown, D. Casanova, C.-M. Chang, Y. Chen, S. H. Chien, K. D. Closser,
D. L. Crittenden, M. Diedenhofen, R. A. DiStasio Jr., H. Dop, A. D. Dutoi, R. G. Edgar, S. Fatehi, L. FustiMolnar, A. Ghysels, A. Golubeva-Zadorozhnaya, J. Gomes, M. W. D. Hanson-Heine, P. H. P. Harbach, A. W.
Hauser, E. G. Hohenstein, Z. C. Holden, T.-C. Jagau, H. Ji, B. Kaduk, K. Khistyaev, J. Kim, J. Kim, R. A. King,
P. Klunzinger, D. Kosenkov, T. Kowalczyk, C. M. Krauter, K. U. Lao, A. Laurent, K. V. Lawler, S. V. Levchenko,
C. Y. Lin, F. Liu, E. Livshits, R. C. Lochan, A. Luenser, P. Manohar, S. F. Manzer, S.-P. Mao, N. Mardirossian,
A. V. Marenich, S. A. Maurer, N. J. Mayhall, C. M. Oana, R. Olivares-Amaya, D. P. O’Neill, J. A. Parkhill,
T. M. Perrine, R. Peverati, P. A. Pieniazek, A. Prociuk, D. R. Rehn, E. Rosta, N. J. Russ, N. Sergueev, S. M.
Sharada, S. Sharmaa, D. W. Small, A. Sodt, T. Stein, D. Stück, Y.-C. Su, A. J. W. Thom, T. Tsuchimochi, L. Vogt,
O. Vydrov, T. Wang, M. A. Watson, J. Wenzel, A. White, C. F. Williams, V. Vanovschi, S. Yeganeh, S. R. Yost,

Chapter 1: Introduction

30

Z.-Q. You, I. Y. Zhang, X. Zhang, Y. Zhou, B. R. Brooks, G. K. L. Chan, D. M. Chipman, C. J. Cramer, W. A.
Goddard III, M. S. Gordon, W. J. Hehre, A. Klamt, H. F. Schaefer III, M. W. Schmidt, C. D. Sherrill, D. G.
Truhlar, A. Warshel, X. Xua, A. Aspuru-Guzik, R. Baer, A. T. Bell, N. A. Besley, J.-D. Chai, A. Dreuw, B. D.
Dunietz, T. R. Furlani, S. R. Gwaltney, C.-P. Hsu, Y. Jung, J. Kong, D. S. Lambrecht, W. Liang, C. Ochsenfeld,
V. A. Rassolov, L. V. Slipchenko, J. E. Subotnik, T. Van Voorhis, J. M. Herbert, A. I. Krylov, P. M. W. Gill,
and M. Head-Gordon. Advances in molecular quantum chemistry contained in the Q-Chem 4 program package.
[Mol. Phys. 113, 184–215 (2015)]
Literature citations for Q-C HEM v. 2.0 1 and v. 3.0 3 are also available, and the most current list of Q-C HEM authors
can always be found on the website, www.q-chem.com. The primary literature is extensively referenced throughout
this manual, and users are urged to cite the original literature for particular theoretical methods. This is how our large
community of academic developers gets credit for its effort.

Chapter 1: Introduction

31

References and Further Reading
[1] J. Kong, C. A. White, A. I. Krylov, D. Sherrill, R. D. Adamson, T. R. Furlani, M. S. Lee, A. M. Lee, S. R. Gwaltney,
T. R. Adams, C. Ochsenfeld, A. T. B. Gilbert, G. S. Kedziora, V. A. Rassolov, D. R. Maurice, N. Nair, Y. Shao,
N. A. Besley, P. E. Maslen, J. P. Dombroski, H. Daschel, W. Zhang, P. P. Korambath, J. Baker, E. F. C. Byrd, T. Van
Voorhis, M. Oumi, S. Hirata, C.-P. Hsu, N. Ishikawa, J. Florian, A. Warshel, B. G. Johnson, P. M. W. Gill, M. HeadGordon, and J. A. Pople. J. Comput. Chem., 21:1532, 2000. DOI: 10.1002/1096-987X(200012)21:16<1532::AIDJCC10>3.0.CO;2-W.
[2] A. I. Krylov and P. M. W. Gill. Wiley Interdiscip. Rev.: Comput. Mol. Sci., 3:317, 2013. DOI: 10.1002/wcms.1122.
[3] Y. Shao, L. Fusti-Molnar, Y. Jung, J. Kussmann, C. Ochsenfeld, S. T. Brown, A. T. B. Gilbert, L. V. Slipchenko,
S. V. Levchenko, D. P. O’Neill, R. A. DiStasio Jr., R. C. Lochan, T. Wang, G. J. O. Beran, N. A. Besley, J. M.
Herbert, C. Y. Lin, T. Van Voorhis, S. H. Chien, A. Sodt, R. P. Steele, V. A. Rassolov, P. E. Maslen, P. P. Korambath,
R. D. Adamson, B. Austin, J. Baker, E. F. C. Byrd, H. Dachsel, R. J. Doerksen, A. Dreuw, B. D. Dunietz, A. D.
Dutoi, T. R. Furlani, S. R. Gwaltney, A. Heyden, S. Hirata, C.-P. Hsu, G. Kedziora, R. Z. Khalliulin, P. Klunzinger,
A. M. Lee, M. S. Lee, W. Liang, I. Lotan, N. Nair, B. Peters, E. I. Proynov, P. A. Pieniazek, Y. M. Rhee, J. Ritchie,
E. Rosta, C. D. Sherrill, A. C. Simmonett, J. E. Subotnik, H. L. Woodcock III, W. Zhang, A. T. Bell, A. K.
Chakraborty, D. M. Chipman, F. J. Keil, A. Warshel, W. J. Hehre, H. F. Schaefer III, J. Kong, A. I. Krylov, P. M. W.
Gill, and M. Head-Gordon. Phys. Chem. Chem. Phys., 8:3172, 2006. DOI: 10.1039/B517914A.
[4] Y. Shao, Z. Gan, E. Epifanovsky, A. T. B. Gilbert, M. Wormit, J. Kussmann, A. W. Lange, A. Behn, J. Deng,
X. Feng, D. Ghosh, M. Goldey, P. R. Horn, L. D. Jacobson, I. Kaliman, R. Z. Khaliullin, T. Kús, A. Landau, J. Liu,
E. I. Proynov, Y. M. Rhee, R. M. Richard, M. A. Rohrdanz, R. P. Steele, E. J. Sundstrom, H. L. Woodcock III, P. M.
Zimmerman, D. Zuev, B. Albrecht, E. Alguire, B. Austin, G. J. O. Beran, Y. A. Bernard, E. Berquist, K. Brandhorst,
K. B. Bravaya, S. T. Brown, D. Casanova, C.-M. Chang, Y. Chen, S. H. Chien, K. D. Closser, D. L. Crittenden,
M. Diedenhofen, R. A. DiStasio Jr., H. Dop, A. D. Dutoi, R. G. Edgar, S. Fatehi, L. Fusti-Molnar, A. Ghysels,
A. Golubeva-Zadorozhnaya, J. Gomes, M. W. D. Hanson-Heine, P. H. P. Harbach, A. W. Hauser, E. G. Hohenstein,
Z. C. Holden, T.-C. Jagau, H. Ji, B. Kaduk, K. Khistyaev, J. Kim, J. Kim, R. A. King, P. Klunzinger, D. Kosenkov,
T. Kowalczyk, C. M. Krauter, K. U. Lao, A. Laurent, K. V. Lawler, S. V. Levchenko, C. Y. Lin, F. Liu, E. Livshits,
R. C. Lochan, A. Luenser, P. Manohar, S. F. Manzer, S.-P. Mao, N. Mardirossian, A. V. Marenich, S. A. Maurer,
N. J. Mayhall, C. M. Oana, R. Olivares-Amaya, D. P. O’Neill, J. A. Parkhill, T. M. Perrine, R. Peverati, P. A.
Pieniazek, A. Prociuk, D. R. Rehn, E. Rosta, N. J. Russ, N. Sergueev, S. M. Sharada, S. Sharmaa, D. W. Small,
A. Sodt, T. Stein, D. Stück, Y.-C. Su, A. J. W. Thom, T. Tsuchimochi, L. Vogt, O. Vydrov, T. Wang, M. A. Watson,
J. Wenzel, A. White, C. F. Williams, V. Vanovschi, S. Yeganeh, S. R. Yost, Z.-Q. You, I. Y. Zhang, X. Zhang,
Y. Zhou, B. R. Brooks, G. K. L. Chan, D. M. Chipman, C. J. Cramer, W. A. Goddard III, M. S. Gordon, W. J.
Hehre, A. Klamt, H. F. Schaefer III, M. W. Schmidt, C. D. Sherrill, D. G. Truhlar, A. Warshel, X. Xua, A. AspuruGuzik, R. Baer, A. T. Bell, N. A. Besley, J.-D. Chai, A. Dreuw, B. D. Dunietz, T. R. Furlani, S. R. Gwaltney, C.-P.
Hsu, Y. Jung, J. Kong, D. S. Lambrecht, W. Liang, C. Ochsenfeld, V. A. Rassolov, L. V. Slipchenko, J. E. Subotnik,
T. Van Voorhis, J. M. Herbert, A. I. Krylov, P. M. W. Gill, and M. Head-Gordon. Mol. Phys., 113:184, 2015. DOI:
10.1080/00268976.2014.952696.

Chapter 2

Installation, Customization, and Execution
2.1
2.1.1

Installation Requirements
Execution Environment

Q-C HEM is shipped as a single executable along with several scripts. No compilation is required. Once the package is
installed it is ready to run. Please refer to the installation notes for your particular platform, which are distributed with
the software. The system software required to run Q-C HEM on your platform is minimal, and includes:
• A suitable operating system.
• Run-time libraries (usually provided with your operating system).
• Vendor implementation of MPI or MPICH libraries (for the MPI-based parallel version only).
Please check the Q-C HEM web site (www.q-chem.com), or contact Q-C HEM support (support@q-chem.com)
if further details are required.

2.1.2

Hardware Platforms and Operating Systems

Q-C HEM runs on a wide varieties of computer systems, ranging from Intel and AMD microprocessor-based PCs and
workstations, to high-performance server nodes used in clusters and supercomputers. Q-C HEM supports the Linux,
Mac, and Windows operating systems. To determine the availability of a specific platform or operating system, please
contact support@q-chem.com.

2.1.3

Memory and Disk Requirements

Memory
Q-C HEM, Inc. has endeavored to minimize memory requirements and maximize the efficiency of its use. Still, the
larger the structure or the higher the level of theory, the more memory is needed. Although Q-C HEM can be run
successfully in very small-memory environments, this is seldom an issue nowadays and we recommend 1 Gb as a
minimum. Q-C HEM also offers the ability for user control of important, memory-intensive aspects of the program. In
general, the more memory your system has, the larger the calculation you will be able to perform.
Q-C HEM uses two types of memory: a chunk of static memory that is used by multiple data sets and managed by the
code, and dynamic memory which is allocated using system calls. The size of the static memory is specified by the
user through the $rem variable MEM_STATIC and has a default value of 64 Mb.

Chapter 2: Installation, Customization, and Execution

33

The $rem word MEM_TOTAL specifies the limit of the total memory the user’s job can use. The default value is sufficiently large that on most machines it will allow Q-C HEM to use all the available memory. This value should be
reduced on machines where this is undesirable (for example if the machine is used by multiple users). The limit for
the dynamic memory allocation is given by (MEM_TOTAL − MEM_STATIC). The amount of MEM_STATIC needed
depends on the size of the user’s particular job. Please note that one should not specify an excessively large value for
MEM_STATIC, otherwise it will reduce the available memory for dynamic allocation. Memory settings in CC, EOM,
and ADC calculations are described in Section 6.14. The use of $rem variables will be discussed in the next Chapter.
Disk
The Q-C HEM executables, shell scripts, auxiliary files, samples and documentation require between 360–400 Mb of
disk space, depending on the platform. The default Q-C HEM output, which is printed to the designated output file,
is usually only a few kilobytes. This will be exceeded, of course, in difficult geometry optimizations, QM/MM and
QM/EFP jobs, as well as in cases where users invoke non-default print options. In order to maximize the capabilities
of your copy of Q-C HEM, additional disk space is required for scratch files created during execution, and these are
automatically deleted on normal termination of a job. The amount of disk space required for scratch files depends
critically on the type of job, the size of the molecule and the basis set chosen.
Q-C HEM uses direct methods for Hartree-Fock and density functional theory calculations, which do not require large
amount of scratch disk space. Wave function-based correlation methods, such as MP2 and coupled-cluster theory
require substantial amounts of temporary (scratch) disk storage, and the faster the access speeds, the better these jobs
will perform. With the low cost of disk drives, it is feasible to have between 100 and 1000 Gb of scratch space available
as a dedicated file system for these large temporary job files. The more you have available, the larger the jobs that will
be feasible and in the case of some jobs, like MP2, the jobs will also run faster as two-electron integrals are computed
less often.
Although the size of any one of the Q-C HEM temporary files will not exceed 2 Gb, a user’s job will not be limited by
this. Q-C HEM writes large temporary data sets to multiple files so that it is not bounded by the 2 Gb file size limitation
on some operating systems.

2.2

Installing Q-C HEM

Users are referred to the detailed installation instructions distributed with your copy of Q-C HEM.
An encrypted license file, qchem.license.dat, must be obtained from your vendor before you will be able to use QC HEM. This file should be placed in the directory $QCAUX/license and must be able to be read by all users of the
software. This file is node-locked, i.e., it will only operate correctly on the machine for which it was generated. Further
details about obtaining this file, can be found in the installation instructions.
Do not alter the license file unless directed by Q-C HEM, Inc.

2.3

Q-C HEM Auxiliary files ($QCAUX)

The $QCAUX environment variable determines the directory where Q-C HEM searches for auxiliary files and the machine license. If not set explicitly, it defaults to $QC/qcaux.
The $QCAUX directory contains files required to run Q-C HEM calculations, including basis set and ECP specifications, SAD guesses (see Chapter 4), library of standard effective fragments (see Section 12.5), and instructions for the
AOI NTS package for generating two-electron integrals efficiently.

Chapter 2: Installation, Customization, and Execution

2.4

34

Q-C HEM Run-time Environment Variables

Q-C HEM requires the following shell environment variables setup prior to running any calculations:
QC
QCAUX

QCSCRATCH

QCLOCALSCR

2.5

Defines the location of the Q-C HEM directory structure. The qchem.install shell script
determines this automatically.
Defines the location of the auxiliary information required by Q-C HEM, which includes
the license required to run Q-C HEM. If not explicitly set by the user, this defaults to
$QC/qcaux.
Defines the directory in which Q-C HEM will store temporary files. Q-C HEM will
usually remove these files on successful completion of the job, but they can be saved,
if so wished. Therefore, $QCSCRATCH should not reside in a directory that will
be automatically removed at the end of a job, if the files are to be kept for further
calculations.
Note that many of these files can be very large, and it should be ensured that the
volume that contains this directory has sufficient disk space available. The $QCSCRATCH directory should be periodically checked for scratch files remaining from
abnormally terminated jobs. $QCSCRATCH defaults to the working directory if not
explicitly set. Please see section 2.7 for details on saving temporary files and consult
your systems administrator.
On certain platforms, such as Linux clusters, it is sometimes preferable to write the
temporary files to a disk local to the node. $QCLOCALSCR specifies this directory.
The temporary files will be copied to $QCSCRATCH at the end of the job, unless the
job is terminated abnormally. In such cases Q-C HEM will attempt to remove the files
in $QCLOCALSCR, but may not be able to due to access restrictions. Please specify
this variable only if required.

User Account Adjustments

In order for individual users to run Q-C HEM, User file access permissions must be set correctly so that the user can
read, write and execute the necessary Q-C HEM files. It may be advantageous to create a qchem user group on your
machine and recursively change the group ownership of the Q-C HEM directory to qchem group.
The Q-C HEM run-time environment need to be initiated prior to running any Q-C HEM calculations, which is done
by sourcing the environment setup script qcenv.sh (for bash) or qcenv.csh (for csh and tcsh) placed in your Q-C HEM
top directory after a successful installation. It might be more convenient for user to include the Q-C HEM environment
setup in their shell startup script, e.g., .cshrc/.tcshrc for csh/tcsh or .bashrc for bash.
If using the csh or tcsh shell, add the following lines to the .cshrc file in the user’s home directory:
#
setenv
setenv
source
#

QC
qchem_root_directory_name
QCSCRATCH scratch_directory_name
$QC/qcenv.csh

If using the Bourne-again shell (bash), add the following lines to the .bashrc file in the user’s home directory:
#
export QC=qchem_root_directory_name
export QCSCRATCH=scratch_directory_name
. $QC/qcenv.sh
#

Chapter 2: Installation, Customization, and Execution

2.6

35

Further Customization: .qchemrc and preferences Files

Q-C HEM has developed a simple mechanism for users to set user-defined long-term defaults to override the built-in
program defaults. Such defaults may be most suited to machine specific features such as memory allocation, as the total
available memory will vary from machine to machine depending on specific hardware and accounting configurations.
However, users may identify other important uses for this customization feature. Q-C HEM obtains input initialization
variables from four sources:
1. User input file
2. $HOME/.qchemrc file
3. $QC/config/preferences file
4. “Factory installed” program defaults
Input mechanisms higher in this list override those that are lower. Mechanisms #2 and #3 allow the user to specify
alternative default settings for certain variables that will override the Q-C HEM “factory-installed” defaults. This can
be done by a system administrator via a preferences file added to the $QC/config directory, or by an individual user by
means of a .qchemrc file in her home directory.
Note: The .qchemrc and preferences files are not requisites for running Q-C HEM and currently only support keywords
in the $rem input section.
The format of the .qchemrc and preferences files consists of a $rem keyword section, as in the Q-C HEM input file,
terminated with the usual $end keyword. Any other $whatever section will be ignored. To aid in reproducibility, a
copy of the .qchemrc file (if present) is included near the top of the job’s output file. (The .qchemrc and preferences
files must have file permissions such that they are readable by the user invoking Q-C HEM.) The format of both of these
files is as follows:
$rem
rem_variable
rem_variable
...
$end

option
option

comment
comment

Example 2.1 An example of a .qchemrc file to override default $rem settings for all of the user’s Q-C HEM jobs.
$rem
INCORE_INTS_BUFFER
DIIS_SUBSPACE_SIZE
THRESH
MAX_SCF_CYCLES
$end

4000000
5
10
100

More integrals in memory
Modify max DIIS subspace size
10**(-10) threshold
More than the default of 50

The following $rem variables are specifically recommended as those that a user might want to customize:
• AO2MO_DISK
• INCORE_INTS_BUFFER
• MEM_STATIC
• SCF_CONVERGENCE
• THRESH
• MAX_SCF_CYCLES
• GEOM_OPT_MAX_CYCLES

Chapter 2: Installation, Customization, and Execution

2.7

36

Running Q-C HEM

Once installation is complete, and any necessary adjustments are made to the user account, the user is now able to run
Q-C HEM. There are several ways to invoke Q-C HEM:
1. IQ MOL offers a fully integrated graphical interface for the Q-C HEM package and includes a sophisticated input
generator with contextual help which is able to guide you through the many Q-C HEM options available. It also
provides a molecular builder, job submission and monitoring tools, and is able to visualize molecular orbitals,
densities and vibrational frequencies. For the latest version and download/installation instructions, please see the
IQ MOL homepage (www.iqmol.org).
2. qchem command line shell script. The simple format for command line execution is given below. The remainder
of this manual covers the creation of input files in detail.
3. Via a third-party GUI. The two most popular ones are:
• A general web-based interface for electronic structure software, W EB MO
(www.webmo.net).
• Wavefunction’s S PARTAN user interface on some platforms. Contact Wavefunction, Inc.
(www.wavefun.com) or Q-C HEM for full details of current availability.
Using the Q-C HEM command line shell script (qchem) is straightforward provided Q-C HEM has been correctly installed on your machine and the necessary environment variables have been set in your .cshrc, .profile, or equivalent
login file. If done correctly, the necessary changes will have been made to the $PATH variable automatically on login
so that Q-C HEM can be invoked from your working directory.
The qchem shell script can be used in either of the following ways:
qchem infile outfile
qchem infile outfile savename
qchem -save infile outfile savename
where infile is the name of a suitably formatted Q-C HEM input file (detailed in Chapter 3, and the remainder of this
manual), and the outfile is the name of the file to which Q-C HEM will place the job output information.
Note: If the outfile already exists in the working directory, it will be overwritten.
The use of the savename command line variable allows the saving of a few key scratch files between runs, and is
necessary when instructing Q-C HEM to read information from previous jobs. If the savename argument is not given,
Q-C HEM deletes all temporary scratch files at the end of a run. The saved files are in $QCSCRATCH/savename/, and
include files with the current molecular geometry, the current molecular orbitals and density matrix and the current
force constants (if available). The –save option in conjunction with savename means that all temporary files are saved,
rather than just the few essential files described above. Normally this is not required. When $QCLOCALSCR has
been specified, the temporary files will be stored there and copied to $QCSCRATCH/savename/ at the end of normal
termination.
The name of the input parameters infile, outfile and save can be chosen at the discretion of the user (usual UNIX file
and directory name restrictions apply). It maybe helpful to use the same job name for infile and outfile, but with varying
suffixes. For example:
localhost-1> qchem water.in water.out &
invokes Q-C HEM where the input is taken from water.in and the output is placed into water.out. The & places the job
into the background so that you may continue to work in the current shell.

37

Chapter 2: Installation, Customization, and Execution
localhost-2> qchem water.com water.log water &

invokes Q-C HEM where the input is assumed to reside in water.com, the output is placed into water.log and the key
scratch files are saved in a directory $QCSCRATCH/water/.
Note: A checkpoint file can be requested by setting GUI = 2 in the $rem section of the input. The checkpoint file name
is determined by the $GUIFILE environment variable which by default is set to ${input}.fchk

2.8

Parallel Q-C HEM Jobs

Parallel execution of Q-C HEM can be threaded across multiple processors on a single node, using the OpenMP protocol,
or using the message-passing interface (MPI) protocol to parallelize over multiple processor cores and/or multiple
compute nodes. A hybrid MPI + OpenMP scheme is also available for certain calculations, in which each MPI process
spawns several OpenMP threads. In this hybrid scheme, cross-node communication is handled by the MPI protocol
and intra-node communication is done implicitly using OpenMP threading for efficient utilization of shared-memory
parallel (SMP) systems. This parallelization strategy reflects current trends towards multi-core architectures in cluster
computing.
As of the v. 4.2 release, the OpenMP parallelization is fully supported by HF/DFT, RIMP2, CC, EOM-CC, and
ADC methods. The MPI parallel capability is available for SCF, DFT, CIS, and TDDFT methods. The hybrid
MPI+OpenMP parallelization is introduced in v. 4.2 for HF/DFT energy and gradient calculations only. Distributed
memory MPI+OpenMP parallelization of CC and EOM-CC methods was added in Q-C HEM v. 4.3. Table 2.1 summarizes the parallel capabilities of Q-C HEM v. 5.0.
Method
HF energy & gradient
DFT energy & gradient
CDFT/CDFT-CI
RI-MP2 energy
Attenuated RI-MP2 energy
Integral transformation
CCMAN & CCMAN2 methods
ADC methods
CIS energy & gradient
TDDFT energy & gradient
HF & DFT analytical Hessian

OpenMP
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes
yes

MPI
yes
yes
no
no
no
no
yes
no
yes
yes
yes

MPI+OpenMP
yes
yes
no
no
no
no
yes
no
no
no
no

Table 2.1: Parallel capabilities of Q-C HEM v. 5.0
To run Q-C HEM calculation with OpenMP threads specify the number of threads (nthreads) using qchem command
option -seq -nt. Since each thread uses one CPU core, you should not specify more threads than the total number of
available CPU cores for performance reason. When unspecified, the number of threads defaults to 1 (serial calculation).
qchem -seq -nt nthreads infile outfile
qchem -seq -nt nthreads infile outfile save
qchem -save -seq -nt nthreads infile outfile save
Similarly, to run parallel calculations with MPI use the option -np to specify the number of MPI processes to be
spawned.
qchem -np nprocs infile outfile
qchem -np nprocs infile outfile savename
qchem -save -np nprocs infile outfile savename

Chapter 2: Installation, Customization, and Execution

38

where nprocs is the number of processors to use. If the -np switch is not given, Q-C HEM will default to running locally
on a single node.
To run hybrid MPI+OpenMP HF/DFT calculations use combined options -np and -nt together, where -np followed by
the number of MPI processes to be spawned and -nt followed by the number of OpenMP threads used in each MPI
process.
qchem -np nprocs -nt nthreads infile outfile
qchem -np nprocs -nt nthreads infile outfile savename
qchem -save -np nprocs -nt nthreads infile outfile savename
When the additional argument savename is specified, the temporary files for MPI-parallel Q-C HEM are stored in
$QCSCRATCH/savename.0 At the start of a job, any existing files will be copied into this directory, and on successful
completion of the job, be copied to $QCSCRATCH/savename/ for future use. If the job terminates abnormally, the files
will not be copied.
To run parallel Q-C HEM using a batch scheduler such as PBS, users may need to set QCMPIRUN environment variable
to point to the mpirun command used in the system. For further details users should read the $QC/README.Parallel
file, and contact Q-C HEM if any problems are encountered (support@q-chem.com).

2.9

IQ MOL Installation Requirements

IQ MOL provides a fully integrated molecular builder and viewer for the Q-C HEM package. It is available for the
Windows, Linux, and Mac OS X platforms and instructions for downloading and installing the latest version can be
found at www.iqmol.org/downloads.html.
IQ MOL can be run as a stand-alone package which is able to open existing Q-C HEM input/output files, but it can also be
used as a fully functional front end which is able to submit and monitor Q-C HEM jobs, and to analyze the resulting output. By default, IQ MOL submits Q-C HEM jobs to a server that is owned by Q-C HEM, Inc., which provides prospective
users with the opportunity to run short Q-C HEM demonstration jobs for free simply by downloading IQ MOL, without
the need to install Q-C HEM.
For customers who own Q-C HEM, it is necessary to configure IQ MOL to submit jobs to an appropriate server. To do
this, first ensure Q-C HEM has been correctly installed on the target machine and can be run from the command line.
Second, open IQ MOL and carry out the following steps:
1. Select the Calculation→Edit Servers menu option. A dialog will appear with a list of configured servers (which
will initially be empty).
2. Click the Add New Server button with the ‘+’ icon. This opens a dialog which allows the new server to be
configured. The server is the machine which has your Q-C HEM installation.
3. Give the server a name (this is simply used to identify the current server configuration and does not have to match
the actual machine name) and select if the machine is local (i.e. the same machine as IQ MOL is running on) or
remote.
4. If there is PBS software running on the server, select the PBS ‘Type’ option, otherwise in most cases the Basic
option should be sufficient. Please note that the server must be Linux based and cannot be a Windows server.
5. If required, the server can be further configured using the Configure button. Details on this can be found in the
embedded IQ MOL help which can be accessed via the Help→Show Help menu option.
6. For non-PBS servers the number of concurrent Q-C HEM jobs can be limited using a simple inbuilt queuing
system. The maximum number of jobs is set by the Job Limit control. If the Job Limit is set to zero the queue is
disabled and any number of jobs can be run concurrently. Please note that this limit applies to the current IQ MOL
session and does not account for jobs submitted by other users or by other IQ MOL sessions.

Chapter 2: Installation, Customization, and Execution

39

7. The $QC environment variable should be entered in the given box.
8. For remote servers the address of the machine and your user name are also required. IQ MOL uses SSH2 to
connect to remote machines and the most convenient way to set this up is by using authorized keys () for details
on how these can be set up). IQ MOL can then connect via the SSH Agent and will not have to prompt you for
your password. If you are not able to use an SSH Agent, several other authentication methods are offered:
• Public Key This requires you to enter your SSH passphrase (if any) to unlock your private key file. The
passphrase is stored in memory, not disk, so you will need to re-enter this each time IQ MOL is run.
• Password Prompt This requires each server password to be entered each time IQ MOL is run. Once the
connection has been established the memory used to hold the password is overwritten to reduce the risk of
recovery from a core dump.
Further configuration of SSH options should not be required unless your public/private keys are stored in a
non-standard location.
It is recommended that you test the server configuration to ensure everything is working before attempting to submit a
job. Multiple servers can be configured if you have access to more than one copy of Q-C HEM or have different account
configurations. In this case the default server is the first on the list and if you want to change this you should use the
arrow buttons in the Server List dialog. The list of configured servers will be displayed when submitting Q-C HEM jobs
and you will be able to select the desired server for each job.
Please note that while Q-C HEM is file-based, as of version 2.1 IQ MOL uses a directory to keep the various files from a
calculation. More details can be found in the IQ MOL user manual.

2.10

Testing and Exploring Q-C HEM

Q-C HEM is shipped with a small number of test jobs which are located in the $QC/samples directory. If you wish to
test your version of Q-C HEM, run the test jobs in the samples directory and compare the output files with the reference
files (suffixed .out) of the same name.
These test jobs are not an exhaustive quality control test (a small subset of the test suite used at Q-C HEM, Inc.), but
they should all run correctly on your platform. If any fault is identified in these, or any output files created by your
version, do not hesitate to contact customer service immediately.
These jobs are also an excellent way to begin learning about Q-C HEM’s text-based input and output formats in detail.
In many cases you can use these inputs as starting points for building your own input files, if you wish to avoid reading
the rest of this manual!
Please check the Q-C HEM web page (www.q-chem.com) and the README files in the $QC/bin directory for
updated information.

Chapter 3

Q-C HEM Inputs
3.1

IQ MOL

The easiest way to run Q-C HEM is by using the IQ MOL interface which can be downloaded for free from www.
iqmol.org. Before submitting a Q-C HEM job from you will need to configure a Q-C HEM server and details on how
to do this are given in Section 2.9 of this manual.
IQ MOL provides a free-form molecular builder and a comprehensive interface for setting up the input for Q-C HEM
jobs. Additionally calculations can be submitted to either the local or a remote machine and monitored using the
built in job monitor. The output can also be analyzed allowing visualization of molecular orbitals and densities, and
animation of vibrational modes and reaction pathways. A more complete list of features can be found at www.iqmol.
org/features.html.
The IQ MOL program comes with a built-in help system that details how to set up and submit Q-C HEM calculations.
This help can be accessed via the Help→Show Help menu option.

3.2

General Form

IQ MOL (or another graphical interface) is the simplest way to control Q-C HEM. However, the low level command
line interface is available to enable maximum customization and allow the user to exploit all Q-C HEM’s features. The
command line interface requires a Q-C HEM input file which is simply an ASCII text file. This input file can be created
using your favorite editor (e.g., vi, emacs, jot, etc.) following the basic steps outlined in the next few chapters.
Q-C HEM’s input mechanism uses a series of keywords to signal user input sections of the input file. As required, the
Q-C HEM program searches the input file for supported keywords. When Q-C HEM finds a keyword, it then reads the
section of the input file beginning at the keyword until that keyword section is terminated the $end keyword. A short
description of all Q-C HEM keywords is provided in Table 3.1 and the following sections. The user must understand
the function and format of the $molecule (Section 3.3) and $rem (Section 3.4) keywords, as these keyword sections are
where the user places the molecular geometry information and job specification details.
The keywords $rem and $molecule are required in any Q-C HEM input file
As each keyword has a different function, the format required for specific keywords varies somewhat, to account for
these differences (format requirements are summarized in Appendix C). However, because each keyword in the input
file is sought out independently by the program, the overall format requirements of Q-C HEM input files are much less
stringent. For example, the $molecule section does not have to occur at the very beginning of the input file.

41

Chapter 3: Q-C HEM Inputs

Section Name
$molecule
$rem
$basis
$cdft
$chem_sol
$comment
$complex_ccman
$ecp
$efei
$efp_fragments
$efp_params
$empirical_dispersion
$eom_user_guess
$external_charges
$force_field_params
$intracule
$isotopes
$localized_diabatization
$magnet
$multipole_field
$nbo
$occupied
$opt
$pcm
$plots
$qct_active_modes
$qct_vib_distribution
$qct_vib_phase
$qm_atoms
$response
$solvent
$smx
$swap_occupied_virtual
$svp
$svpirf
$2pa
$van_der_waals
$xc_functional

Description
Contains the molecular coordinate input (input file requisite).
Job specification and customization parameters (input file requisite).
User-defined basis set information (Chapter 8).
Options for the constrained DFT method (Section 5.13).
Job control for the Q-C HEM/C HEM S OL interface (Langevin dipoles
model; Section 12.2.9).
User comments for inclusion into output file.
Contains parameters for complex-scaled and CAP-augmented EOM-CC
calculations (Chapter 7.7).
User-defined effective core potentials (Chapter 9).
Application of external forces in a geometry optimization (Section 10.3.6).
Specifies labels and positions of EFP fragments (Section 12.5).
Contains user-defined parameters for effective fragments (Section 12.5).
User-defined van der Waals parameters for DFT dispersion correction
(Section 5.7.2).
User-defined guess for EOM-CC calculations (Chapter 7.7).
Specifies external point charges and their positions.
Force-field parameters for QM/MM calculations (Section 12.3).
Intracule parameters (Section 11.9).
Isotopic substitutions for vibrational calculations (Section 11.10.2).
Information for mixing together multiple adiabatic states into diabatic
states (Chapter 11).
Job control for magnetic field-related response properties (Section 11.13.3).
Details of an external multipole field (Section 3.5.7).
Options for the Natural Bond Orbital package (Section 11.3).
Guess orbitals to be occupied (Section 4.4.4).
Constraint definitions for geometry optimizations (Section 10.3).
Job control for polarizable continuum models (Section 12.2.3).
Generate plotting information over a grid of points (Section 11.5).
Information for quasi-classical trajectory calculations (Section 10.7.5).
Specify the QM region for QM/MM calculations (Section 12.3).
Job control for the generalized response solver (Section 11.15).
Additional parameters and variables for implicit solvent models
(Section 12.2).
Job control for SMx implicit solvent models (Section 12.2.8).
Guess orbitals to be swapped (Section 4.4.4).
Special parameters for the iso-density SS(V)PE module (Section 12.2.5).
Initial guess for the iso-density SS(V)PE module (Section 12.2.5).
Additional parameters for two-photon absorption calculations
(Section 7.7.16.1).
User-defined atomic radii for Langevin dipoles solvation (Section 12.2.9)
and PCMs (Section 12.2.2).
User-defined DFT exchange-correlation functional (Section 5.3.6).

Table 3.1: A list of Q-C HEM input sections; the first two ($molecule and $rem) are required for all jobs, whereas the
rest are required only for certain job types, or else are optional places to specify additional job-control variables. Each
input section (“$section”) should be terminated with $end. See the $QC/samples directory that is included with your
release for specific examples of Q-C HEM input files using these keywords.

Chapter 3: Q-C HEM Inputs

42

Note: (1) Users are able to enter keyword sections in any order.
(2) Each keyword section must be terminated with the $end keyword.
(3) The $rem and $molecule sections must be included.
(4) It is not necessary to have all keywords in an input file.
(5) Each keyword section is described in Appendix C.
(6) The entire Q-C HEM input is case-insensitive.
The second general aspect of Q-C HEM input is that there are effectively four input sources:
• User input file (required)
• .qchemrc file in $HOME (optional)
• preferences file in $QC/config (optional)
• Internal program defaults and calculation results (built-in)
The order of preference is as shown, i.e., the input mechanism offers a program default override for all users, default
override for individual users and, of course, the input file provided by the user overrides all defaults. Refer to Section 2.6 for details of .qchemrc and preferences. Currently, Q-C HEM only supports the $rem keyword in .qchemrc and
preferences files.
In general, users will need to enter variables for the $molecule and $rem keyword section and are encouraged to add a
$comment for future reference. The necessity of other keyword input will become apparent throughout the manual.

3.3

Molecular Coordinate Input ($molecule)

The $molecule section communicates to the program the charge, spin multiplicity, and geometry of the molecule being
considered. The molecular coordinates input begins with two integers: the net charge and the spin multiplicity of the
molecule. The net charge must be between −50 and 50, inclusive (0 for neutral molecules, 1 for cations, −1 for anions,
etc.). The multiplicity must be between 1 and 10, inclusive (1 for a singlet, 2 for a doublet, 3 for a triplet, etc.). Each
subsequent line of the molecular coordinate input corresponds to a single atom in the molecule (or dummy atom),
regardless of whether using Z-matrix internal coordinates or Cartesian coordinates.
Note: The coordinate system used for declaring an initial molecular geometry by default does not affect that used in
a geometry optimization procedure. See Appendix A which discusses the O PTIMIZE package in further detail.
Q-C HEM begins all calculations by rotating and translating the user-defined molecular geometry into a Standard Nuclear Orientation whereby the center of nuclear charge is placed at the origin. This is a standard feature of most quantum
chemistry programs. This action can be turned off by using SYM_IGNORE TRUE.
Note: SYM_IGNORE = TRUE will also turn off determining and using of the point group symmetry.
Note: Q-C HEM ignores commas and equal signs, and requires all distances, positions and angles to be entered as
Ångstroms and degrees unless the INPUT_BOHR $rem variable is set to TRUE, in which case all lengths are
assumed to be in bohr.

3.3.1

Specifying the Molecular Coordinates Manually

3.3.1.1

Cartesian Coordinates

Q-C HEM can accept a list of N atoms and their 3N Cartesian coordinates. The atoms can be entered either as atomic
numbers or atomic symbols where each line corresponds to a single atom. The Q-C HEM format for declaring a
molecular geometry using Cartesian coordinates (in Ångstroms) is:

43

Chapter 3: Q-C HEM Inputs
atom

x-coordinate

y-coordinate

z-coordinate

Note: The geometry can by specified in bohr by setting the $rem variable INPUT_BOHR equal to TRUE.
Example 3.1 Atomic number Cartesian coordinate input for H2 O. The first line species the molecular charge and
multiplicity, respectively.
$molecule
0 1
8
0.000000
1
1.370265
1 -1.370265
$end

0.000000
0.000000
0.000000

-0.212195
0.848778
0.848778

Example 3.2 Atomic symbol Cartesian coordinate input for H2 O.
$molecule
0 1
O
0.000000
H
1.370265
H -1.370265
$end

0.000000
0.000000
0.000000

-0.212195
0.848778
0.848778

Note: (1) Atoms can be declared by either atomic number or symbol.
(2) Coordinates can be entered either as variables/parameters or real numbers.
(3) Variables/parameters can be declared in any order.
(4) A single blank line separates parameters from the atom declaration.
Once all the molecular Cartesian coordinates have been entered, terminate the molecular coordinate input with the $end
keyword.

3.3.1.2

Z-matrix Coordinates

For small molecules, Z-matrix notation is a common input format. The Z-matrix defines the positions of atoms relative
to previously defined atoms using a length, an angle and a dihedral angle. Again, note that all bond lengths and angles
must be in Ångstroms and degrees, unless INPUT_BOHR is set to TRUE, in which case bond lengths are specified in
bohr.
Note: As with the Cartesian coordinate input method, Q-C HEM begins a calculation by taking the user-defined coordinates and translating and rotating them into a Standard Nuclear Orientation.
The first three atom entries of a Z-matrix are different from the subsequent entries. The first Z-matrix line declares
a single atom. The second line of the Z-matrix input declares a second atom, refers to the first atom and gives the
distance between them. The third line declares the third atom, refers to either the first or second atom, gives the
distance between them, refers to the remaining atom and gives the angle between them. All subsequent entries begin
with an atom declaration, a reference atom and a distance, a second reference atom and an angle, a third reference atom
and a dihedral angle. This can be summarized as:
1. First atom.
2. Second atom, reference atom, distance.
3. Third atom, reference atom A, distance between A and the third atom, reference atom B, angle defined by atoms
A, B and the third atom.
4. Fourth atom, reference atom A, distance, reference atom B, angle, reference atom C, dihedral angle (A, B, C and
the fourth atom).

44

Chapter 3: Q-C HEM Inputs
5. All subsequent atoms follow the same basic form as (4)
Example 3.3 Z-matrix for hydrogen peroxide
O1
O2
H1
H2

O1
O1
O2

oo
ho
ho

O2
O1

hoo
hoo

H1

hooh

Line 1 declares an oxygen atom (O1). Line 2 declares the second oxygen atom (O2), followed by a reference to the
first atom (O1) and a distance between them denoted oo. Line 3 declares the first hydrogen atom (H1), indicates it is
separated from the first oxygen atom (O1) by a distance HO and makes an angle with the second oxygen atom (O2)
of hoo. Line 4 declares the fourth atom and the second hydrogen atom (H2), indicates it is separated from the second
oxygen atom (O2) by a distance HO and makes an angle with the first oxygen atom (O1) of hoo and makes a dihedral
angle with the first hydrogen atom (H1) of hooh.
Some further points to note are:
• Atoms can be declared by either atomic number or symbol.
– If declared by atomic number, connectivity needs to be indicated by Z-matrix line number.
– If declared by atomic symbol either number similar atoms (e.g., H1, H2, O1, O2 etc.) and refer connectivity
using this symbol, or indicate connectivity by the line number of the referred atom.
• Bond lengths and angles can be entered either as variables/parameters or real numbers.
– Variables/parameters can be declared in any order.
– A single blank line separates parameters from the Z-matrix.

45

Chapter 3: Q-C HEM Inputs
All the following examples are equivalent in the information forwarded to the Q-C HEM program.

Example 3.4 Using parameters to define bond lengths and angles, and using numbered symbols to define atoms and
indicate connectivity.
$molecule
0 1
O1
O2 O1
H1 O1
H2 O2
oo
oh
hoo
hooh
$end

oo
ho
ho

O2
O1

hoo
hoo

H1

hooh

=
1.5
=
1.0
= 120.0
= 180.0

Example 3.5 Not using parameters to define bond lengths and angles, and using numbered symbols to define atoms
and indicate connectivity.
$molecule
0 1
O1
O2 O1
H1 O1
H2 O2
$end

1.5
1.0
1.0

O2
O1

120.0
120.0

H1

180.0

Example 3.6 Using parameters to define bond lengths and angles, and referring to atom connectivities by line number.
$molecule
0 1
8
8 1 oo
1 1 ho
1 2 ho
oo
oh
hoo
hooh
$end

2
1

hoo
hoo

3

hooh

=
1.5
=
1.0
= 120.0
= 180.0

Example 3.7 Referring to atom connectivities by line number, and entering bond length and angles directly.
$molecule
0 1
8
8 1 1.5
1 1 1.0
1 2 1.0
$end

2
1

120.0
120.0

3

180.0

Obviously, a number of the formats outlined above are less appealing to the eye and more difficult for us to interpret
than the others, but each communicates exactly the same Z-matrix to the Q-C HEM program.

Chapter 3: Q-C HEM Inputs
3.3.1.3

46

Dummy Atoms

Dummy atoms are indicated by the identifier X and followed, if necessary, by an integer. (e.g., X1, X2. Dummy
atoms are often useful for molecules where symmetry axes and planes are not centered on a real atom, and have also
been useful in the past for choosing variables for structure optimization and introducing symmetry constraints.
Note: Dummy atoms play no role in the quantum mechanical calculation, and are used merely for convenience in
specifying other atomic positions or geometric variables.

3.3.2

Reading Molecular Coordinates from a Previous Job or File

Often users wish to perform several calculations in sequence, where the later calculations rely on results obtained from
the previous ones. For example, a geometry optimization at a low level of theory, followed by a vibrational analysis and
then, perhaps, single-point energy at a higher level. Rather than having the user manually transfer the coordinates from
the output of the optimization to the input file of a vibrational analysis or single point energy calculation, Q-C HEM can
transfer them directly from job to job.
To achieve this requires that:
• The READ variable is entered into the molecular coordinate input
• Scratch files from a previous calculation have been saved. These may be obtained explicitly by using the save
option across multiple job runs as described below and in Chapter 2, or implicitly when running multiple calculations in one input file, as described in Section 3.6.
Example 3.8 Reading a geometry from a prior calculation.
$molecule
READ
$end

localhost-1> qchem job1.in job1.out job1
localhost-2> qchem job2.in job2.out job1
In this example, the job1 scratch files are saved in a directory $QCSCRATCH/job1 and are then made available to the
job2 calculation.
Note: The program must be instructed to read specific scratch files by the input of job2.
The READ function can also be used to read molecular coordinates from a second input file. The format for the
coordinates in the second file follows that for standard Q-C HEM input, and must be delimited with the $molecule and
$end keywords.
Example 3.9 Reading molecular coordinates from another file. filename may be given either as the full file path, or
path relative to the working directory.
$molecule
READ filename
$end

3.4

Job Specification: The $rem Input Section

The $rem section in the input file is the means by which users specify the type of calculation that they wish to perform
(i.e., level of theory, basis set, convergence criteria, additional special features, etc.). The keyword $rem signals the

47

Chapter 3: Q-C HEM Inputs

beginning of the overall job specification. Within the $rem section the user inserts $rem variables (one per line) which
define the essential details of the calculation. The allowed format is either
REM_VARIABLE

VALUE

[ comment ]

or alternatively
REM_VARIABLE

=

VALUE

[ comment ]

The “=” sign is automatically discarded and only the first two remaining arguments are read, so that all remaining text is
ignored and can be used to place comments in the input file. Thus the $rem section that provides Q-C HEM job control
takes the form shown in the following example.
Example 3.10 General format of the $rem section of the text input file.
$rem
REM_VARIABLE
REM_VARIABLE
...
$end

value
value

[ comment ]
[ comment ]

Note: (1) Tab stops can be used to format input.
(2) A line prefixed with an exclamation mark ‘!’ is treated as a comment and will be ignored by the program.
(3) $rem variables are case-insensitive (as is the whole Q-C HEM input file).
(4) Depending on the particular $rem variable, “value” may be a keyword (string), an integer, or a logical
value (true or false).
(5) A complete list of $rem variables can be found in Appendix C.
In this manual, $rem variables will be described using the following format:
REM_VARIABLE_NAME
A short description of what the variable controls.
TYPE:
The type of variable (INTEGER, LOGICAL or STRING)
DEFAULT:
The default value, if any.
OPTIONS:
A list of the options available to the user.
RECOMMENDATION:
A brief recommendation, where appropriate.

If a default setting is indicated for a particular $rem variable, then it is not necessary to declare that variable in order for
the default setting to be used. For example, the default value for the variable JOBTYPE is SP, indicating a single-point
energy calculation, so to perform such a calculation the user does not need to set the JOBTYPE variable. To perform a
geometry optimization, however, it is necessary to override this default by setting JOBTYPE = OPT. System administrator preferences for default $rem settings can be specified in the $QC/config/preferences file, and user preferences in
a $HOME/.qchemrc file, both of which are described in Section 2.6.
Q-C HEM provides defaults for most $rem variables, but the user will always have to stipulate a few others. In a single
point energy calculation, for example, the minimum requirements will be BASIS (defining the basis set) and METHOD

48

Chapter 3: Q-C HEM Inputs

(defining the level of theory for correlation and exchange). For example, METHOD = HF invokes a Hartree-Fock
calculation, whereas METHOD = CIS specifies a CIS excited-state calculation.
Example 3.11 Example of minimal $rem requirements to run an MP2/6-31G* single-point energy calculation.
$rem
BASIS
METHOD
$end

6-31G*
mp2

Just a small basis set
MP2

The level of theory can alternatively be specified by setting values for two other $rem variables, EXCHANGE (defining
the level of theory to treat exchange) and CORRELATION (defining the level of theory to treat electron correlation, if
required). For excited states computed using equation-of-motion (EOM) methods (Chapter 7), there is a third $rem
variable, EOM_CORR, which specifies the level of correlation for the target states.
For DFT calculations, METHOD specifies an exchange-correlation functional; see Section 5.4 for a list of supported
functionals. For wave function approaches, supported values of METHOD can be found in Section 6.1 for ground-state
methods and in Section 7.1 for excited-state methods. If a wave function-based correlation treatment such as MP2 or
CC is requested using the CORRELATION keyword, then HF is taken as the default for EXCHANGE.

3.5

Additional Input Sections

The $molecule and $rem sections are required for all Q-C HEM jobs, but depending on the details of the job a number of
other input sections may be required. These are summarized briefly below, with references to more detailed descriptions
to be found later in this manual.

3.5.1

Comments ($comment)

Users are able to add comments to the input file outside keyword input sections, which will be ignored by the program.
This can be useful as reminders to the user, or perhaps, when teaching another user to set up inputs. Comments can also
be provided in a $comment block, which is actually redundant given that the entire input deck is copied to the output
file.

3.5.2

User-Defined Basis Sets ($basis and $aux_basis)

By setting the $rem keyword BASIS = GEN, the user indicates that the basis set will be user defined. In that case, the
$basis input section is used to specify the basis set. Similarly, if AUX_BASIS = GEN then the $aux_basis input section
is used to specify the auxiliary basis set. See Chapter 8 for details on how to input a user-defined basis set.

3.5.3

User-Defined Effective Core Potential ($ecp)

By setting ECP = GEN, the user indicates that the effective core potentials (pseudopotentials, which replace explicit core
electrons) to be used will be defined by the user. In that case, the $ecp section is used to specify these pseudopotentials.
See Chapter 9 for further details.

3.5.4

User-Defined Exchange-Correlation Density Functionals
($xc_functional)

If the keyword EXCHANGE = GEN then a DFT calculation will be performed using a user-specified combination of
exchange and correlation functional(s), as described in Chapter 4. Custom functionals of this sort can be constructed

49

Chapter 3: Q-C HEM Inputs

as any linear combination of exchange and/or correlation functionals that are supported by Q-C HEM; see Section 5.4
for a list of supported functionals. The format for the $xc_functional input section is the following:
$xc_functional
X exchange_symbol coefficient
X exchange_symbol coefficient
...
C correlation_symbol coefficient
C correlation_symbol coefficient
...
K coefficient
$end

Note: The coefficients must be real numbers.

3.5.5

User-defined Parameters for DFT Dispersion Correction
($empirical_dispersion)

If a user wants to change from the default values recommended by Grimme, then user-defined parameters can be
specified using the $empirical_dispersion input section. See Section 5.7.2 for details.

3.5.6

Addition of External Point Charges ($external_charges)

If the $external_charges keyword is present, Q-C HEM scans for a set of external charges to be incorporated into a
calculation. The format is shown below and consists of Cartesian coordinates and the value of the point charge, with
one charge per line. The charge is in atomic units and the coordinates are in Ångstroms, unless bohrs are selected by
setting the $rem keyword INPUT_BOHR to TRUE. The external charges are rotated with the molecule into the standard
nuclear orientation.
Example 3.12 General format for incorporating a set of external charges.
$external_charges
x-coord1 y-coord1
x-coord2 y-coord2
x-coord3 y-coord3
$end

z-coord1
z-coord2
z-coord3

charge1
charge2
charge3

In addition, the user can request to add a charged cage around the molecule (for so-called “charge stabilization” calculations) using the keyword ADD_CHARGED_CAGE. See Section 7.7.7 for details.

3.5.7

Applying a Multipole Field ($multipole_field)

A multipole field can be applied to the molecule under investigation by specifying the $multipole_field input section.
Each line in this section consists of a single component of the applied field, in the following format.
Example 3.13 General format for imposing a multipole field.
$multipole_field
field_component_1
field_component_2
$end

value_1
value_2

Each field_component is stipulated using the Cartesian representation e.g., X, Y, and/or Z, (dipole field components);

Chapter 3: Q-C HEM Inputs

50

XX, XY, and/or YY (quadrupole field components); XXX, XXY, etc.. The value (magnitude) of each field component
should be provided in atomic units.

3.5.8

User-Defined Occupied Guess Orbitals ($occupied and
$swap_occupied_virtual)

It is sometimes useful for the occupied guess orbitals to be different from the lowest Nα (or Nα + Nβ ) orbitals.
Q-C HEM allows the occupied guess orbitals to be defined using the $occupied keyword. Using the $occupied input
section, the user can choose which orbitals (by number) to occupy by specifying the α-spin orbitals on the first line of
the $occupied section and the β-spin orbitals on the second line. For large molecules where only a few occupied →
virtual promotions are desired, it is simpler to use the $swap_occupied_virtual input section. Details can be found in
Section 4.4.4.

3.5.9

Polarizable Continuum Solvation Models ($pcm)

The $pcm section provides fine-tuning of the job control for polarizable continuum models (PCMs), which are requested
by setting the $rem keyword SOLVENT_METHOD equal to PCM. Supported PCMs include C-PCM, IEF-PCM, and
SS(V)PE, which share a common set of job-control variables. Details are provided in Section 12.2.2.

3.5.10

SS(V)PE Solvation Modeling ($svp and $svpirf )

The $svp section is available to specify special parameters to the solvation module such as cavity grid parameters and
modifications to the numerical integration procedure. The $svpirf section allows the user to specify an initial guess for
the solution of the cavity charges. As discussed in section 12.2.5, the $svp and $svpirf input sections are used to specify
parameters for the iso-density implementation of SS(V)PE. An alternative implementation of the SS(V)PE mode, based
on a more empirical definition of the solute cavity, is available in the PCM (see Section 12.2.2) and controlled from
within the $pcm input section.

3.5.11

User-Defined van der Waals Radii ($van_der_waals)

The $van_der_waals section of the input enables the user to customize the van der Waals radii that are important
parameters in the Langevin dipoles solvation model; see Section 12.2.

3.5.12

Effective Fragment Potential Calculations ($efp_fragments and $efp_params)

These keywords are used to specify positions and parameters for effective fragments in EFP calculations. Details are
provided in Section 12.5.

3.5.13

Natural Bond Orbital Package ($nbo)

When NBO is set to TRUE in the $rem section, a natural bond orbital (NBO) calculation is performed, using the
Q-C HEM interface to the NBO 5.0 and NBO 6.0 packages. In such cases, the $nbo section may contain standard
parameters and keywords for the NBO program.

Chapter 3: Q-C HEM Inputs

3.5.14

51

Orbitals, Densities and Electrostatic Potentials on a Mesh ($plots)

The $plots part of the input permits the evaluation of molecular orbitals, densities, electrostatic potentials, transition
densities, electron attachment and detachment densities on a user-defined mesh of points. Q-C HEM will print out the
raw data, but can also format these data into the form of a “cube” file that is a standard input format for volumetric data
that can be read various visualization programs. See Section 11.5 for details.

3.5.15

Intracules ($intracule)

Setting the $rem keyword INTRACULE = TRUE requests a molecular intracule calculation, in which case additional
customization is possible using the $intracule input section. See Section 11.9.

3.5.16

Geometry Optimization with Constraints ($opt)

For JOBTYPE = OPT, Q-C HEM scans the input file for the $opt section. Here, the user may specify distance, angle,
dihedral and out-of-plane bend constraints to be imposed on the optimization procedure, as described in Chapter 10.

3.5.17

Isotopic Substitutions ($isotopes)

For vibrational frequency calculations (JOBTYPE = FREQ), nuclear masses are set by default to be those corresponding
to the most abundant naturally-occurring isotopes. Alternative masses for one or more nuclei can be requested by
setting ISOTOPES = TRUE in the $rem section, in which case the $isotopes section is used to specify the desired masses
as described in Section 11.10.2. Isotopic substitutions incur negligible additional cost in a frequency calculation.

3.6

Multiple Jobs in a Single File: Q-C HEM Batch Jobs

It is sometimes useful to place a sequence of jobs into a single Q-C HEM input file, where the individual inputs should
be separated from one another by a line consisting of the string @@@. The output from these jobs is then appended
sequentially to a single output file. This is useful to (a) use information obtained in a prior job (i.e., an optimized
geometry) in a subsequent job; or (b) keep related calculations together in a single output file.
Some limitations should be kept in mind:
• The first job will overwrite any existing output file of the same name in the working directory. Restarting the job
will also overwrite any existing file.
• Q-C HEM reads all the jobs from the input file immediately and stores them. Therefore no changes can be made
to the details of subsequent jobs following command-line initiation of Q-C HEM, even if these subsequent jobs
have not yet run.
• If any single job fails, Q-C HEM proceeds to the next job in the batch file, for good or ill.
• No check is made to ensure that dependencies are satisfied, or that information is consistent. For example, in a
geometry optimization followed by a frequency calculation, no attempt is made by the latter to check that the
optimization was successful. When reading MO coefficients from a previous job, it is the user’s responsibility to
ensure that the basis set is the same in both calculations, as this is assumed by the program.
• Scratch files are saved from one job to the next in a batch job, so that information from previous jobs can be shared
with subsequent ones, but are deleted upon completion of the entire batch job unless the –save command-line
argument is supplied, as discussed in Chapter 2.

52

Chapter 3: Q-C HEM Inputs

The following example requests a batch job consisting of (i) a HF/6-31G* geometry optimization; followed by (ii) a
frequency calculation at the same level of theory that uses the previously-optimized geometry (and also reads in the
final MOs from the optimization job); and finally (iii) a single-point calculation at the same geometry but at a higher
level of theory, MP2/6-311G(d,p).
Example 3.14 Example of using information from previous jobs in a single input file.
$comment
Optimize H-H at HF/6-31G*
$end
$molecule
0 1
H
H 1 r
r = 1.1
$end
$rem
JOBTYPE
METHOD
BASIS
$end

opt
Optimize the bond length
hf
6-31G*

@@@
$comment
Now calculate the frequency of H-H at the same level of theory.
$end
$molecule
read
$end
$rem
JOBTYPE
METHOD
BASIS
SCF_GUESS
$end

freq
hf
6-31G*
read

Calculate vibrational frequency

Read the MOs from disk

@@@
$comment
Now a single point calculation at at MP2/6-311G(d,p)//HF/6-31G*
$end
$molecule
read
$end
$rem
METHOD
BASIS
$end

3.7

mp2
6-311G(d,p)

Q-C HEM Output File

When Q-C HEM is invoked using

Chapter 3: Q-C HEM Inputs

53

qchem infile outfile
the output file outfile contains a variety of information, depending on the type of job(s), but in general consists of the
following.
• Q-C HEM citation
• User input (for record-keeping purposes)
• Molecular geometry in Cartesian coordinates
• Molecular point group, nuclear repulsion energy, number of α- and β-spin electrons
• Basis set information (number of functions, shells and function pairs)
• SCF details (method, guess, and convergence procedure)
• Energy and DIIS error for each SCF iteration
• Results of any post-SCF calculation that is requested
• Results of any excited-state calculation that is requested
• Molecular orbital symmetries and energies
• Wave function analysis
• Message signaling successful job completion
Note: If outfile above already exists when the job is started, then the existing file is overwritten with the results of the
new calculation.

Chapter 4

Self-Consistent Field Ground-State Methods
4.1

Overview

Theoretical “model chemistries" 28 involve two principle approximations. One must specify, first of all, the type of
atomic orbital (AO) basis set that will be used to construct molecular orbitals (MOs), via the “linear combination of
atomic orbitals” (LCAO) ansatz, available options for which are discussed in Chapters 8 and 9. Second, one must
specify the manner in which the instantaneous interactions between electrons (“electron correlation”) are to be treated.
Self-consistent field (SCF) methods, in which electron correlation is described in a mean-field way, represent the simplest, most affordable, and most widely-used electronic structure methods. The SCF category of methods includes both
Hartree-Fock (HF) theory as well as Kohn-Sham (KS) density functional theory (DFT). This Chapter summarizes QC HEM’s SCF capabilities, while Chapter 5 provides further details specific to DFT calculations. Chapter 6 describes the
more sophisticated (but also more computationally expensive!) post-HF, wave function-based methods for describing
electron correlation. If you are new to quantum chemistry, we recommend an introductory textbook such as Refs. 28,
62, or 31.
Section 4.2 provides the theoretical background behind SCF methods, including both HF and KS-DFT. In some sense,
the former may be considered as a special case of the latter, and job-control $rem variables are much the same in both
cases. Basic SCF job control is described in Section 4.3. Later sections introduce more specialized options that can be
consulted as needed. Of particular note are the following:
• Initial guesses for SCF calculations (Section 4.4). Modification of the guess is recommended in cases where the
SCF calculation fails to converge.
• Changing the SCF convergence algorithm (Section 4.5) is also a good strategy when the SCF calculation fails to
converge.
• Linear-scaling [“O(N )”] and other reduced-cost methods are available for large systems (see Section 4.6).
• Unconventional SCF calculations. Some non-standard SCF methods with novel physical and mathematical features are available. These include:
– Dual-basis SCF calculations (Section 4.7) and DFT perturbation theory (Section 4.8), which facilitate largebasis quality results but require self-consistent iterations only in a smaller basis set.
– SCF meta-dynamics (Section 4.9.2), which can be used to locate multiple solutions to the SCF equations
and to help check that the solution obtained is actually the lowest minimum.
Some of these unconventional SCF methods are available exclusively in Q-C HEM.

55

Chapter 4: Self-Consistent Field Ground-State Methods

4.2
4.2.1

Theoretical Background
SCF and LCAO Approximations

The fundamental equation of non-relativistic quantum chemistry is the time-independent Schrödinger equation,
Ĥ(R, r) Ψ(R, r) = E(R) Ψ(R, r) .

(4.1)

In quantum chemistry, this equation is solved as a function of the electronic variables (r), for fixed values of the nuclear
coordinates (R). The Hamiltonian operator in Eq. (4.1), in atomic units, is
Ĥ = −

N
M
N X
N N
M X
M
M
X
X
1X 2 1X 1
ZA ZB
ZA X X 1
∇i −
∇2A −
+
+
2 i=1
2
MA
riA i=1 j>i rij
RAB
i=1

(4.2)

∂2
∂2
∂2
+ 2+ 2 .
2
∂x
∂y
∂z

(4.3)

A=1

A=1 B>A

A=1

where
∇2 =

In Eq. (4.2), Z is the nuclear charge, MA is the ratio of the mass of nucleus A to the mass of an electron, RAB =
|RA − RB | is the distance between nuclei A and B, rij = |ri − rj | is the distance between the ith and jth electrons,
riA = |ri − RA | is the distance between the ith electron and the Ath nucleus, M is the number of nuclei and N is the
number of electrons. The total energy E is an eigenvalue of Ĥ, with a corresponding eigenfunction (wave function) Ψ.
Separating the motions of the electrons from that of the nuclei, an idea originally due to Born and Oppenheimer, 7 yields
the electronic Hamiltonian operator:
Ĥelec = −

N M
N
N N
1 X 2 X X ZA X X 1
∇i −
+
2 i=1
riA i=1 j>i rij
i=1

(4.4)

A=1

The solution of the corresponding electronic Schrödinger equation,
Ĥelec Ψelec = Eelec Ψelec ,

(4.5)

affords the total electronic energy, Eelec , and electronic wave function, Ψelec , which describes the distribution of the
electrons for fixed nuclear positions. The total energy is obtained by simply adding the nuclear–nuclear repulsion
energy [the fifth term in Eq. (4.2)] to the total electronic energy:
Etot = Eelec + Enuc .

(4.6)

Solving the eigenvalue problem in Eq. (4.5) yields a set of eigenfunctions (Ψ0 , Ψ1 , Ψ2 . . .) with corresponding eigenvalues E0 ≤ E1 ≤ E2 ≤ . . ..
Our interest lies in determining the lowest eigenvalue and associated eigenfunction which correspond to the ground
state energy and wave function of the molecule. However, solving Eq. (4.5) for other than the most trivial systems is
extremely difficult and the best we can do in practice is to find approximate solutions.
The first approximation used to solve Eq. (4.5) is the independent-electron (mean-field) approximation, in which the
wave function is approximated as an antisymmetrized product of one-electron functions, namely, the MOs. Each MO
is determined by considering the electron as moving within an average field of all the other electrons. This affords the
well-known Slater determinant wave function 52,53

1
Ψ= √
n!

χ1 (1)
χ1 (2)
..
.

χ2 (1)
χ2 (2)
..
.

···
···

χ1 (n) χ2 (n) · · ·

χn (1)
χn (2)
..
.

,

χn (n)

where χi , a spin orbital, is the product of a molecular orbital ψi and a spin function (α or β).

(4.7)

56

Chapter 4: Self-Consistent Field Ground-State Methods

One obtains the optimum set of MOs by variationally minimizing the energy in what is called a “self-consistent field”
or SCF approximation to the many-electron problem. The archetypal SCF method is the Hartree-Fock (HF) approximation, but these SCF methods also include KS-DFT (Chapter 5). All SCF methods lead to equations of the form
fˆ(i) χ(xi ) = ε χ(xi ) ,

(4.8)

where the Fock operator fˆ(i) for the ith electron is
1
fˆ(i) = − ∇2i + υeff (i) .
2

(4.9)

Here xi are spin and spatial coordinates of the ith electron, the functions χ are spin orbitals and υeff is the effective
potential “seen” by the ith electron, which depends on the spin orbitals of the other electrons. The nature of the effective
potential υeff depends on the SCF methodology, i.e., on the choice of density-functional approximation.
The second approximation usually introduced when solving Eq. (4.5) is the introduction of an AO basis {φµ } linear
combinations of which will then determine the MOs. There are many standardized, atom-centered Gaussian basis sets
and details of these are discussed in Chapter 8.
After eliminating the spin components in Eq. (4.8) and introducing a finite basis,
X
ψi =
cµi φµ ,

(4.10)

µ

Eq. (4.8) reduces to the Roothaan-Hall matrix equation
FC = εSC .

(4.11)

Here, F is the Fock matrix, C is a square matrix of molecular orbital coefficients, S is the AO overlap matrix with
elements
Z
Sµν = φµ (r)φν (r)dr
(4.12)
and ε is a diagonal matrix containing the orbital energies. Generalizing to an unrestricted formalism by introducing
separate spatial orbitals for α and β spin in Eq. (4.7) yields the Pople-Nesbet equations 42
Fα Cα = εα SCα
Fβ Cβ = εβ SCβ

(4.13)

In SCF methods, an initial guess is for the MOs is first determined, and from this, an average field seen by each electron can be calculated. A new set of MOs can be obtained by solving the Roothaan-Hall or Pople-Nesbet eigenvalue
equations, resulting in the restricted or unrestricted finite-basis SCF approximation. This procedure is repeated until
the new MOs differ negligibly from those of the previous iteration. The Hartree-Fock approximation for the effective
potential in Eq. (4.9) inherently neglects the instantaneous electron-electron correlations that are averaged out by the
SCF procedure, and while the chemistry resulting from HF calculations often offers valuable qualitative insight, quantitative energetics are often poor. In principle, the DFT methodologies are able to capture all the correlation energy,
i.e., the difference in energy between the HF energy and the true energy. In practice, the best-available density functionals perform well but not perfectly, and conventional post-HF approaches to calculating the correlation energy (see
Chapter 6) are often required.
That said, because SCF methods often yield acceptably accurate chemical predictions at low- to moderate computational cost, self-consistent field methods are the cornerstone of most quantum-chemical programs and calculations. The
formal costs of many SCF algorithms is O(N 4 ), that is, they grow with the fourth power of system size, N . This is
slower than the growth of the cheapest conventional correlated methods, which scale as O(N 5 ) or worse, algorithmic
advances available in Q-C HEM can reduce the SCF cost to O(N ) in favorable cases, an improvement that allows SCF
methods to be applied to molecules previously considered beyond the scope of ab initio quantum chemistry.
Types of ground-state energy calculations currently available in Q-C HEM are summarized in Table 4.1.

57

Chapter 4: Self-Consistent Field Ground-State Methods
Calculation
Single point energy (default)
Force (energy + gradient)
Equilibrium structure search
Transition structure search
Intrinsic reaction pathway
Potential energy scan
Vibrational frequency calculation
Polarizability and relaxed dipole
NMR chemical shift
Indirect nuclear spin-spin coupling
Ab initio molecular dynamics
Ab initio path integrals
BSSE (counterpoise) correction
Energy decomposition analysis

$rem Variable JOBTYPE
SINGLE_POINT or SP
FORCE
OPTIMIZATION or OPT
TS
RPATH
PES_SCAN
FREQUENCY or FREQ
POLARIZABILITY, DIPOLE
NMR
ISSC
AIMD
PIMD, PIMC
BSSE
EDA

(Ch. 10)
(Ch. 10)
(Sec. 10.5)
(Sec. 10.4)
(Sec. 11.10 and 11.11)
(Sec. 11.14.1)
(Sec. 11.13.1)
(Sec. 11.13.1)
(Sec. 10.7)
(Sec. 10.8)
(Sec. 13.4.3)
(Sec. 13.5)

Table 4.1: The type of calculation to be run by Q-C HEM is controlled by the $rem variable JOBTYPE.

4.2.2

Hartree-Fock Theory

As with much of the theory underlying modern quantum chemistry, the HF approximation was developed shortly after
publication of the Schrödinger equation, but remained a qualitative theory until the advent of the computer. Although
the HF approximation tends to yield qualitative chemical accuracy, rather than quantitative information, and is generally
inferior to many of the DFT approaches available, it remains as a useful tool in the quantum chemist’s toolkit. In
particular, for organic chemistry, HF predictions of molecular structure are very useful.
Consider once more the Roothaan-Hall equations, Eq. (4.11), or the Pople-Nesbet equations, Eq. (4.13), which can
be traced back to Eq. (4.8), in which the effective potential υeff depends on the SCF methodology. In a restricted HF
(RHF) formalism, the effective potential can be written as
N/2

υeff =

M
X
 X
ZA
2Jˆa (1) − K̂a (1) −
r1A
a

(4.14)

A=1

where the Coulomb and exchange operators are defined as
Z
1
Jˆa (1) = ψa∗ (2)
ψa (2) dr2
r12
and

Z
K̂a (1)ψi (1) =

ψa∗ (2)


1
ψi (2) dr2 ψa (1)
r12

(4.15)

(4.16)

respectively. By introducing an atomic orbital basis, we obtain Fock matrix elements
core
Fµν = Hµν
+ Jµν − Kµν

(4.17)

core
Hµν
= Tµν + Vµν

(4.18)

where the core Hamiltonian matrix elements

consist of kinetic energy elements
Z
Tµν =



1 2
φµ (r) − ∇ φν (r) dr
2

(4.19)

and nuclear attraction elements
Z
Vµν =

φµ (r) −

X
A

ZA
|RA − r|

!
φν (r) dr

(4.20)

58

Chapter 4: Self-Consistent Field Ground-State Methods
The Coulomb and exchange elements are given by
Jµν =

X

Pλσ (µν|λσ)

(4.21)

λσ

and
Kµν =

1X
Pλσ (µλ|νσ)
2

(4.22)

λσ

respectively, where the density matrix elements are
N/2

Pµν = 2

X

Cµa Cνa

(4.23)

a=1

and the two electron integrals are


Z Z
(µν|λσ) =

φµ (r1 )φν (r1 )

1
r12


φλ (r2 )φσ (r2 ) dr1 dr2 .

(4.24)

Note: The formation and utilization of two-electron integrals is a topic central to the overall performance of SCF
methodologies. The performance of the SCF methods in new quantum chemistry software programs can be
quickly estimated simply by considering the quality of their atomic orbital integrals packages. See Appendix B
for details of Q-C HEM’s AOI NTS package.
Substituting the matrix element in Eq. (4.17) back into the Roothaan-Hall equations, Eq. (4.11), and iterating until
self-consistency is achieved will yield the RHF energy and wave function. Alternatively, one could have adopted the
unrestricted form of the wave function by defining separate α and β density matrices:
α
Pµν
=

nα
X

α
α
Cµa
Cνa

a=1
nβ
β
Pµν

=

X

(4.25)
β
β
Cµa
Cνa

a=1

The total electron density matrix P = Pα + Pβ . The unrestricted α Fock matrix,
α
core
α
Fµν
= Hµν
+ Jµν − Kµν
,

(4.26)

differs from the restricted one only in the exchange contributions, where the α exchange matrix elements are given by
α
Kµν

=

N X
N
X
λ

4.3
4.3.1

α
Pλσ
(µλ|νσ)

(4.27)

σ

Basic SCF Job Control
Overview

As of version 5.1, Q-C HEM uses a new SCF package, GEN_SCFMAN, developed by E. J. Sundstrom, P. R. Horn
and many other coworkers. In addition to supporting the basic features of the previous SCF package (e.g. restricted,
unrestricted and restricted open-shell HF/KS-DFT calculations), many new features are now available in Q-C HEM,
including:
• Addition of several useful SCF convergence algorithms and support for user-specified hybrid algorithm (Sect.
4.5.8).
• More general and user-friendly internal stability analysis and automatic correction for the energy minimum (Sect.
4.5.9).

Chapter 4: Self-Consistent Field Ground-State Methods

59

GEN_SCFMAN also supports a wider range of orbital types, including complex orbitals. A full list of supported
orbitals is:
• Restricted (R): typically appropriate for closed shell molecules at their equilibrium geometry, where electrons
occupy orbitals in pairs.
• Unrestricted (U): - appropriate for radicals with an odd number of electrons, and also for molecules with even
numbers of electrons where not all electrons are paired, e.g., stretched bonds and diradicals.
• Restricted open-shell (RO): for open-shell molecules, where the α and β orbitals are constrained to be identical.
• Open-shell singlet ROSCF (OS_RO): see the “ROKS" method documented in Section 7.5.
• Generalized (G): i.e., each MO is associated with both α and β spin components.
• The use of complex orbitals (with Hartree-Fock only): restricted (CR), unrestricted (CU), and generalized (CG).
Aspects of an SCF calculation such as the SCF guess, the use of efficient algorithms to construct the Fock matrix like
occ-RI-K (see Section 4.6.9), are unaffected by the use of GEN_SCFMAN. Likewise, using GEN_SCFMAN does not
make any difference to the post-SCF procedures such as correlated methods, excited state calculations and evaluation
of molecular properties.
It should be noted that many special features (e.g. dual-basis SCF, CDFT, etc.) based on Q-C HEM’s old SCF code are
not yet supported in GEN_SCFMAN. They will become available in the future.

4.3.1.1

Job Control

The following two $rem variables must be specified in order to run HF calculations:
METHOD
Specifies the exchange-correlation functional.
TYPE:
STRING
DEFAULT:
No default
OPTIONS:
NAME Use METHOD = NAME, where NAME is either HF for Hartree-Fock theory or
else one of the DFT methods listed in Section 5.3.4.
RECOMMENDATION:
In general, consult the literature to guide your selection. Our recommendations for DFT are
indicated in bold in Section 5.3.4.
BASIS
Specifies the basis sets to be used.
TYPE:
STRING
DEFAULT:
No default basis set
OPTIONS:
General, Gen User defined ($basis keyword required).
Symbol
Use standard basis sets as per Chapter 8.
Mixed
Use a mixture of basis sets (see Chapter 8).
RECOMMENDATION:
Consult literature and reviews to aid your selection.

In addition, the following $rem variables can be used to customize the SCF calculation:

Chapter 4: Self-Consistent Field Ground-State Methods
GEN_SCFMAN
Use GEN_SCFMAN for the present SCF calculation.
TYPE:
BOOLEAN
DEFAULT:
TRUE
OPTIONS:
FALSE Use the previous SCF code.
TRUE
Use GEN_SCFMAN.
RECOMMENDATION:
Set to FALSE in cases where features not yet supported by GEN_SCFMAN are needed.
PRINT_ORBITALS
Prints orbital coefficients with atom labels in analysis part of output.
TYPE:
INTEGER/LOGICAL
DEFAULT:
FALSE
OPTIONS:
FALSE Do not print any orbitals.
TRUE
Prints occupied orbitals plus 5 virtual orbitals.
NVIRT Number of virtual orbitals to print.
RECOMMENDATION:
Use true unless more virtual orbitals are desired.
SCF_CONVERGENCE
SCF is considered converged when the wave function error is less that 10−SCF_CONVERGENCE .
Adjust the value of THRESH at the same time. (Starting with Q-C HEM 3.0, the DIIS error is
measured by the maximum error rather than the RMS error as in earlier versions.)
TYPE:
INTEGER
DEFAULT:
5 For single point energy calculations.
8 For geometry optimizations and vibrational analysis.
8 For SSG calculations, see Chapter 6.
OPTIONS:
User-defined
RECOMMENDATION:
Tighter criteria for geometry optimization and vibration analysis. Larger values provide more
significant figures, at greater computational cost.
UNRESTRICTED
Controls the use of restricted or unrestricted orbitals.
TYPE:
LOGICAL
DEFAULT:
FALSE Closed-shell systems.
TRUE
Open-shell systems.
OPTIONS:
FALSE Constrain the spatial part of the alpha and beta orbitals to be the same.
TRUE
Do not Constrain the spatial part of the alpha and beta orbitals.
RECOMMENDATION:
Use the default unless ROHF is desired. Note that for unrestricted calculations on systems with
an even number of electrons it is usually necessary to break α/β symmetry in the initial guess, by
using SCF_GUESS_MIX or providing $occupied information (see Section 4.4 on initial guesses).

60

Chapter 4: Self-Consistent Field Ground-State Methods

61

The calculations using other more special orbital types are controlled by the following $rem variables (they are not
effective if GEN_SCFMAN = FALSE):
OS_ROSCF
Run an open-shell singlet ROSCF calculation with GEN_SCFMAN.
TYPE:
BOOLEAN
DEFAULT:
FALSE
OPTIONS:
TRUE
OS_ROSCF calculation is performed.
FALSE Do not run OS_ROSCF (it will run a close-shell RSCF calculation instead).
RECOMMENDATION:
Set to TRUE if desired.
GHF
Run a generalized Hartree-Fock calculation with GEN_SCFMAN.
TYPE:
BOOLEAN
DEFAULT:
FALSE
OPTIONS:
TRUE
Run a GHF calculation.
FALSE Do not use GHF.
RECOMMENDATION:
Set to TRUE if desired.
COMPLEX
Run an SCF calculation with complex MOs using GEN_SCFMAN.
TYPE:
BOOLEAN
DEFAULT:
FALSE
OPTIONS:
TRUE
Use complex orbitals.
FALSE Use real orbitals.
RECOMMENDATION:
Set to TRUE if desired.
COMPLEX_MIX
Mix a certain percentage of the real part of the HOMO to the imaginary part of the LUMO.
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
0–100 The mix angle = π·COMPLEX_MIX/100.
RECOMMENDATION:
It may help find the stable complex solution (similar idea as SCF_GUESS_MIX).

Chapter 4: Self-Consistent Field Ground-State Methods

62

Example 4.1 Restricted open-shell singlet ROSCF calculation for the first excited state of formaldehyde using GEN_SCFMAN.
The first job provides the guess orbitals through a restricted SCF calculation.
$molecule
0 1
H -0.940372
H 0.940372
C 0.000000
O 0.000000
$end

0.000000 1.268098
0.000000 1.268098
0.000000 0.682557
0.000000 -0.518752

$rem
GEN_SCFMAN
METHOD
BASIS
THRESH
SCF_CONVERGENCE
SYM_IGNORE
$end

true
wb97x-d
def2-svpd
14
9
true

@@@
$molecule
read
$end
$rem
JOBTYPE
METHOD
BASIS
GEN_SCFMAN
OS_ROSCF
THRESH
SCF_CONVERGENCE
SCF_ALGORITHM
SYM_IGNORE
SCF_GUESS
$end

4.3.2

sp
wb97x-d
def2-svpd
true
true
14
9
diis
true
read

Additional Options

Listed below are a number of useful options to customize an SCF calculation. This is only a short summary of the
function of these $rem variables. A full list of all SCF-related variables is provided in Appendix C. Several important
sub-topics are discussed separately, including O(N ) methods for large molecules (Section 4.6), customizing the initial
guess (Section 4.4), and converging the SCF calculation (Section 4.5).
INTEGRALS_BUFFER
Controls the size of in-core integral storage buffer.
TYPE:
INTEGER
DEFAULT:
15 15 Megabytes.
OPTIONS:
User defined size.
RECOMMENDATION:
Use the default, or consult your systems administrator for hardware limits.

Chapter 4: Self-Consistent Field Ground-State Methods
DIRECT_SCF
Controls direct SCF.
TYPE:
LOGICAL
DEFAULT:
Determined by program.
OPTIONS:
TRUE
Forces direct SCF.
FALSE Do not use direct SCF.
RECOMMENDATION:
Use the default; direct SCF switches off in-core integrals.
METECO
Sets the threshold criteria for discarding shell-pairs.
TYPE:
INTEGER
DEFAULT:
2 Discard shell-pairs below 10−THRESH .
OPTIONS:
1 Discard shell-pairs four orders of magnitude below machine precision.
2 Discard shell-pairs below 10−THRESH .
RECOMMENDATION:
Use the default.
THRESH
Cutoff for neglect of two electron integrals. 10−THRESH (THRESH ≤ 14).
TYPE:
INTEGER
DEFAULT:
8
For single point energies.
10 For optimizations and frequency calculations.
14 For coupled-cluster calculations.
OPTIONS:
n for a threshold of 10−n .
RECOMMENDATION:
Should be at least three greater than SCF_CONVERGENCE. Increase for more significant figures,
at greater computational cost.
STABILITY_ANALYSIS
Performs stability analysis for a HF or DFT solution.
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
TRUE
Perform stability analysis.
FALSE Do not perform stability analysis.
RECOMMENDATION:
Set to TRUE when a HF or DFT solution is suspected to be unstable.

63

Chapter 4: Self-Consistent Field Ground-State Methods
SCF_PRINT
Controls level of output from SCF procedure to Q-C HEM output file.
TYPE:
INTEGER
DEFAULT:
0 Minimal, concise, useful and necessary output.
OPTIONS:
0 Minimal, concise, useful and necessary output.
1 Level 0 plus component breakdown of SCF electronic energy.
2 Level 1 plus density, Fock and MO matrices on each cycle.
3 Level 2 plus two-electron Fock matrix components (Coulomb, HF exchange
and DFT exchange-correlation matrices) on each cycle.
RECOMMENDATION:
Proceed with care; can result in extremely large output files at level 2 or higher. These levels are
primarily for program debugging.
SCF_FINAL_PRINT
Controls level of output from SCF procedure to Q-C HEM output file at the end of the SCF.
TYPE:
INTEGER
DEFAULT:
0 No extra print out.
OPTIONS:
0 No extra print out.
1 Orbital energies and break-down of SCF energy.
2 Level 1 plus MOs and density matrices.
3 Level 2 plus Fock and density matrices.
RECOMMENDATION:
The break-down of energies is often useful (level 1).

64

Chapter 4: Self-Consistent Field Ground-State Methods

4.3.3

65

Examples

Provided below are examples of Q-C HEM input files to run ground state, HF single point energy calculations.
Example 4.2 Example Q-C HEM input for a single point energy calculation on water. Note that the declaration of the
single point $rem variable is redundant because it is the same as the Q-C HEM default.
$molecule
0 1
O
H1 O oh
H2 O oh

H1

hoh

oh =
1.2
hoh = 120.0
$end
$rem
JOBTYPE
METHOD
BASIS
$end

sp
hf
sto-3g

Single Point energy
Hartree-Fock
Basis set

Example 4.3 UHF/6-311G calculation on the Li atom. Note that correlation and the job type were not indicated
because Q-C HEM defaults automatically to no correlation and single point energies. Note also that, since the number
of α and β electron differ, MOs default to an unrestricted formalism.
$molecule
0,2
Li
$end
$rem
METHOD
BASIS
$end

HF
6-311G

Hartree-Fock
Basis set

Example 4.4 ROHF/6-311G calculation on the Lithium atom.
$molecule
0,2
3
$end
$rem
METHOD
UNRESTRICTED
BASIS
$end

4.3.4

hf
false
6-311G

Hartree-Fock
Restricted MOs
Basis set

Symmetry

Symmetry is a powerful branch of mathematics and is often exploited in quantum chemistry, both to reduce the computational workload and to classify the final results obtained. 20,21,63 Q-C HEM is able to determine the point group
symmetry of the molecular nuclei and, on completion of the SCF procedure, classify the symmetry of molecular orbitals, and provide symmetry decomposition of kinetic and nuclear attraction energy (see Chapter 11).
Molecular systems possessing point group symmetry offer the possibility of large savings of computational time, by
avoiding calculations of integrals which are equivalent i.e., those integrals which can be mapped on to one another

Chapter 4: Self-Consistent Field Ground-State Methods

66

under one of the symmetry operations of the molecular point group. The Q-C HEM default is to use symmetry to reduce
computational time, when possible.
There are several keywords that are related to symmetry, which causes frequent confusion. SYM_IGNORE controls
symmetry throughout all modules. The default is FALSE. In some cases it may be desirable to turn off symmetry
altogether, for example if you do not want Q-C HEM to reorient the molecule into the standard nuclear orientation,
or if you want to turn it off for finite difference calculations. If the SYM_IGNORE keyword is set to TRUE then the
coordinates will not be altered from the input, and the point group will be set to C1 .
The SYMMETRY keyword controls symmetry in some integral routines. It is set to FALSE by default. Note that
setting it to FALSE does not turn point group symmetry off, and does not disable symmetry in the coupled-cluster suite
(CCMAN and CCMAN2), which is controlled by CC_SYMMETRY (see Chapters 6 and 7), although we noticed that
sometimes it may interfere with the determination of orbital symmetries, possibly due to numerical noise. In some
cases, SYMMETRY = TRUE can cause problems (poor convergence and wildly incorrect SCF energies) and turning it
off can avoid these problems.
Note: The user should be aware about different conventions for defining symmetry elements. The arbitrariness affects, for example, C2v point group. The specific choice affects how the irreducible representations in the affected groups are labeled. For example, b1 and b2 irreducible representations in C2v
are flipped when using different conventions. Q-C HEM uses non-Mulliken symmetry convention. See
http://iopenshell.usc.edu/howto/symmetry for detailed explanations.
SYMMETRY
Controls the efficiency through the use of point group symmetry for calculating integrals.
TYPE:
LOGICAL
DEFAULT:
TRUE Use symmetry for computing integrals.
OPTIONS:
TRUE
Use symmetry when available.
FALSE Do not use symmetry. This is always the case for RIMP2 jobs
RECOMMENDATION:
Use the default unless benchmarking. Note that symmetry usage is disabled for RIMP2, FFT,
and QM/MM jobs.
SYM_IGNORE
Controls whether or not Q-C HEM determines the point group of the molecule and reorients the
molecule to the standard orientation.
TYPE:
LOGICAL
DEFAULT:
FALSE Do determine the point group (disabled for RIMP2 jobs).
OPTIONS:
TRUE/FALSE
RECOMMENDATION:
Use the default unless you do not want the molecule to be reoriented. Note that symmetry usage
is disabled for RIMP2 jobs.

Chapter 4: Self-Consistent Field Ground-State Methods

67

SYM_TOL
Controls the tolerance for determining point group symmetry. Differences in atom locations less
than 10−SYM_TOL are treated as zero.
TYPE:
INTEGER
DEFAULT:
5 Corresponding to 10−5 .
OPTIONS:
User defined.
RECOMMENDATION:
Use the default unless the molecule has high symmetry which is not being correctly identified.
Note that relaxing this tolerance too much may introduce errors into the calculation.

4.4
4.4.1

SCF Initial Guess
Introduction

The Roothaan-Hall and Pople-Nesbet equations of SCF theory are non-linear in the molecular orbital coefficients. Like
many mathematical problems involving non-linear equations, prior to the application of a technique to search for a
numerical solution, an initial guess for the solution must be generated. If the guess is poor, the iterative procedure
applied to determine the numerical solutions may converge very slowly, requiring a large number of iterations, or at
worst, the procedure may diverge.
Thus, in an ab initio SCF procedure, the quality of the initial guess is of utmost importance for (at least) two main
reasons:
• To ensure that the SCF converges to an appropriate ground state. Often SCF calculations can converge to different
local minima in wave function space, depending upon which part of “LCAO space” in which the initial guess
lands.
• When considering jobs with many basis functions requiring the recalculation of ERIs at each iteration, using a
good initial guess that is close to the final solution can reduce the total job time significantly by decreasing the
number of SCF iterations.
For these reasons, sooner or later most users will find it helpful to have some understanding of the different options
available for customizing the initial guess. Q-C HEM currently offers six options for the initial guess:
• Superposition of Atomic Density (SAD)
• Purified SAD guess (provides molecular orbitals; SADMO)
• Core Hamiltonian (CORE)
• Generalized Wolfsberg-Helmholtz (GWH)
• Reading previously obtained MOs from disk. (READ)
• Basis set projection (BASIS2)
The first four of these guesses are built-in, and are briefly described in Section 4.4.2. The option of reading MOs from
disk is described in Section 4.4.3. The initial guess MOs can be modified, either by mixing, or altering the order of
occupation. These options are discussed in Section 4.4.4. Finally, Q-C HEM’s novel basis set projection method is
discussed in Section 4.4.5.

Chapter 4: Self-Consistent Field Ground-State Methods

4.4.2

68

Simple Initial Guesses

There are four simple initial guesses available in Q-C HEM. While they are all simple, they are by no means equal in
quality, as we discuss below.
1. Superposition of Atomic Densities (SAD): The SAD guess is almost trivially constructed by summing together
atomic densities that have been spherically averaged to yield a trial density matrix. The SAD guess is far superior
to the other two options below, particularly when large basis sets and/or large molecules are employed. There
are three issues associated with the SAD guess to be aware of:
(a) No molecular orbitals are obtained, which means that SCF algorithms requiring orbitals (the direct minimization methods discussed in Section 4.5) cannot directly use the SAD guess, and,
(b) The SAD guess is not available for general (read-in) basis sets. All internal basis sets support the SAD
guess.
(c) The SAD guess is not idempotent and thus requires at least two SCF iterations to ensure proper SCF
convergence (idempotency of the density).
2. Purified Superposition of Atomic Densities (SADMO): This guess is similar to the SAD guess, with two
critical differences, namely, the removal of issues 1a and 1c above. The functional difference to the SAD guess
is that the density matrix obtained from the superposition is diagonalized to obtain natural molecular orbitals,
after which an idempotent density matrix is created by aufbau occupation of the natural orbitals. Since the initial
density matrix is created with the SAD guess, the SADMO guess is not available either for a general (read-in)
basis set.
3. Generalized Wolfsberg-Helmholtz (GWH): The GWH guess procedure 74 uses a combination of the overlap
matrix elements in Eq. (4.12), and the diagonal elements of the Core Hamiltonian matrix in Eq. (4.18). This
initial guess is most satisfactory in small basis sets for small molecules. It is constructed according to the relation
given below, where cx is a constant typically chosen as cx = 1.75.
Hµυ = cx Sµυ (Hµµ + Hυυ )/2.

(4.28)

4. Core Hamiltonian: The core Hamiltonian guess simply obtains the guess MO coefficients by diagonalizing the
core Hamiltonian matrix in Eq. (4.18). This approach works best with small basis sets, and degrades as both the
molecule size and the basis set size are increased.
The selection of these choices (or whether to read in the orbitals) is controlled by the following $rem variables:

Chapter 4: Self-Consistent Field Ground-State Methods

69

SCF_GUESS
Specifies the initial guess procedure to use for the SCF.
TYPE:
STRING
DEFAULT:
SAD
Superposition of atomic densities (available only with standard basis sets)
GWH
For ROHF where a set of orbitals are required.
FRAGMO For a fragment MO calculation
OPTIONS:
CORE
Diagonalize core Hamiltonian
SAD
Superposition of atomic density
SADMO
Purified superposition of atomic densities (available only with standard basis sets)
GWH
Apply generalized Wolfsberg-Helmholtz approximation
READ
Read previous MOs from disk
FRAGMO Superimposing converged fragment MOs
RECOMMENDATION:
SAD or SADMO guess for standard basis sets. For general basis sets, it is best to use the BASIS2
$rem. Alternatively, try the GWH or core Hamiltonian guess. For ROHF it can be useful to
READ guesses from an SCF calculation on the corresponding cation or anion. Note that because
the density is made spherical, this may favor an undesired state for atomic systems, especially
transition metals. Use FRAGMO in a fragment MO calculation.
SCF_GUESS_ALWAYS
Switch to force the regeneration of a new initial guess for each series of SCF iterations (for use
in geometry optimization).
TYPE:
LOGICAL
DEFAULT:
False
OPTIONS:
False Do not generate a new guess for each series of SCF iterations in an
optimization; use MOs from the previous SCF calculation for the guess,
if available.
True Generate a new guess for each series of SCF iterations in a geometry
optimization.
RECOMMENDATION:
Use the default unless SCF convergence issues arise

4.4.3

Reading MOs from Disk

There are two methods by which MO coefficients can be used from a previous job by reading them from disk:
1. Running two independent jobs sequentially invoking Q-C HEM with three command line variables:.
localhost-1> qchem job1.in job1.out save
localhost-2> qchem job2.in job2.out save
Note: (1) The $rem variable SCF_GUESS must be set to READ in job2.in.
(2) Scratch files remain in $QCSCRATCH/save on exit.
2. Running a batch job where two jobs are placed into a single input file separated by the string @@@ on a single line.
Note: (1) SCF_GUESS must be set to READ in the second job of the batch file.
(2) A third qchem command line variable is not necessary.
(3) As for the SAD guess, Q-C HEM requires at least two SCF cycles to ensure proper
SCF convergence (idempotency of the density).

Chapter 4: Self-Consistent Field Ground-State Methods

70

Note: It is up to the user to make sure that the basis sets match between the two jobs. There is no internal checking
for this, although the occupied orbitals are re-orthogonalized in the current basis after being read in. If you
want to project from a smaller basis into a larger basis, consult section 4.4.5.

4.4.4

Modifying the Occupied Molecular Orbitals

It is sometimes useful for the occupied guess orbitals to be other than the lowest Nα (or Nβ ) orbitals. Reasons why
one may need to do this include:
• To converge to a state of different symmetry or orbital occupation.
• To break spatial symmetry.
• To break spin symmetry, as in unrestricted calculations on molecules with an even number of electrons.
There are two mechanisms for modifying a set of guess orbitals: either by SCF_GUESS_MIX, or by specifying the orbitals to occupy. Q-C HEM users may define the occupied guess orbitals using the $occupied or $swap_occupied_virtual
keywords. In the former, occupied guess orbitals are defined by listing the α orbitals to be occupied on the first line
and β on the second. In the former, only pair of orbitals that needs to be swapped is specified.

Chapter 4: Self-Consistent Field Ground-State Methods

71

Note: (1) To prevent Q-C HEM to change orbital occupation during SCF procedure, MOM_START option is often used
in combination with $occupied or $swap_occupied_virtual keywords.
(2) The need for orbitals renders these options incompatible with the SAD guess. Most often, they are used
with SCF_GUESS = READ.
Example 4.5 Format for modifying occupied guess orbitals.
$occupied
1 2 3
1 2 3
$end

4 ...
4 ...

NAlpha
NBeta

Example 4.6 Alternative format for modifying occupied guess orbitals.
$swap_occupied_virtual
  
  
$end

Example 4.7 Example of swapping guess orbitals.
$swap_occupied_virtual
alpha 5 6
beta
6 7
$end

This is identical to:
Example 4.8 Example of specifying occupied guess orbitals.
$occupied
1 2 3 4 6 5 7
1 2 3 4 5 7 6
$end

or
Example 4.9 Example of specifying occupied guess orbitals.
$occupied
1:4 6 5 7
1:5 7 6
$end

The other $rem variables related to altering the orbital occupancies are:

Chapter 4: Self-Consistent Field Ground-State Methods

72

SCF_GUESS_PRINT
Controls printing of guess MOs, Fock and density matrices.
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
0
Do not print guesses.
SAD
1
Atomic density matrices and molecular matrix.
2
Level 1 plus density matrices.
CORE and GWH
1
No extra output.
2
Level 1 plus Fock and density matrices and, MO coefficients and
eigenvalues.
READ
1
No extra output
2
Level 1 plus density matrices, MO coefficients and eigenvalues.
RECOMMENDATION:
None
SCF_GUESS_MIX
Controls mixing of LUMO and HOMO to break symmetry in the initial guess. For unrestricted
jobs, the mixing is performed only for the alpha orbitals.
TYPE:
INTEGER
DEFAULT:
0 (FALSE) Do not mix HOMO and LUMO in SCF guess.
OPTIONS:
0 (FALSE) Do not mix HOMO and LUMO in SCF guess.
1 (TRUE)
Add 10% of LUMO to HOMO to break symmetry.
n
Add n × 10% of LUMO to HOMO (0 < n < 10).
RECOMMENDATION:
When performing unrestricted calculations on molecules with an even number of electrons, it is
often necessary to break alpha/beta symmetry in the initial guess with this option, or by specifying input for $occupied.

4.4.5

Basis Set Projection

Q-C HEM also includes a novel basis set projection method developed by Dr Jing Kong of Q-C HEM Inc. It permits a
calculation in a large basis set to bootstrap itself up via a calculation in a small basis set that is automatically spawned
when the user requests this option. When basis set projection is requested (by providing a valid small basis for BASIS2),
the program executes the following steps:
• A simple DFT calculation is performed in the small basis, BASIS2, yielding a converged density matrix in this
basis.
• The large basis set SCF calculation (with different values of EXCHANGE and CORRELATION set by the input)
begins by constructing the DFT Fock operator in the large basis but with the density matrix obtained from the
small basis set.
• By diagonalizing this matrix, an accurate initial guess for the density matrix in the large basis is obtained, and
the target SCF calculation commences.

Chapter 4: Self-Consistent Field Ground-State Methods

73

Two different methods of projection are available and can be set using the BASISPROJTYPE $rem. The OVPROJECTION
option expands the MOs from the BASIS2 calculation in the larger basis, while the FOPPROJECTION option constructs
the Fock matrix in the larger basis using the density matrix from the initial, smaller basis set calculation. Basis set
projection is a very effective option for general basis sets, where the SAD guess is not available. In detail, this initial
guess is controlled by the following $rem variables:
BASIS2
Sets the small basis set to use in basis set projection.
TYPE:
STRING
DEFAULT:
No second basis set default.
OPTIONS:
Symbol. Use standard basis sets as per Chapter 8.
BASIS2_GEN
General BASIS2
BASIS2_MIXED
Mixed BASIS2
RECOMMENDATION:
BASIS2 should be smaller than BASIS. There is little advantage to using a basis larger than a
minimal basis when BASIS2 is used for initial guess purposes. Larger, standardized BASIS2
options are available for dual-basis calculations (see Section 4.7).
BASISPROJTYPE
Determines which method to use when projecting the density matrix of BASIS2
TYPE:
STRING
DEFAULT:
FOPPROJECTION (when DUAL_BASIS_ENERGY=false)
OVPROJECTION (when DUAL_BASIS_ENERGY=true)
OPTIONS:
FOPPROJECTION Construct the Fock matrix in the second basis
OVPROJECTION
Projects MOs from BASIS2 to BASIS.
RECOMMENDATION:
None

Note: BASIS2 sometimes affects post-Hartree-Fock calculations. It is recommended to split such jobs into two subsequent one, such that in the first job a desired Hartree-Fock solution is found using BASIS2, and in the second
job, which performs a post-HF calculation, SCF_GUESS = READ is invoked.

74

Chapter 4: Self-Consistent Field Ground-State Methods

4.4.6

Examples

Example 4.10 Input where basis set projection is used to generate a good initial guess for a calculation employing a
general basis set, for which the default initial guess is not available.
$molecule
0 1
O
H 1 r
H 1 r
r
a
$end

$basis
O
0
S
3

2

SP

1

SP

1

****
H
S

S
****
$end

a

0.9
104.0

$rem
METHOD
BASIS
BASIS2
$end

SP

2

0
2

1

mp2
general
sto-3g

1.000000
3.22037000E+02
4.84308000E+01
1.04206000E+01
1.000000
7.40294000E+00
1.57620000E+00
1.000000
3.73684000E-01
1.000000
8.45000000E-02

1.000000
5.44717800E+00
8.24547000E-01
1.000000
1.83192000E-01

5.92394000E-02
3.51500000E-01
7.07658000E-01
-4.04453000E-01
1.22156000E+00

2.44586000E-01
8.53955000E-01

1.00000000E+00

1.00000000E+00

1.00000000E+00

1.00000000E+00

1.56285000E-01
9.04691000E-01
1.00000000E+00

Chapter 4: Self-Consistent Field Ground-State Methods

75

Example 4.11 Input for an ROHF calculation on the OH radical. One SCF cycle is initially performed on the cation,
to get reasonably good initial guess orbitals, which are then read in as the guess for the radical. This avoids the use of
Q-C HEM’s default GWH guess for ROHF, which is often poor.
$comment
OH radical, part 1. Do 1 iteration of cation orbitals.
$end
$molecule
1 1
O 0.000
H 0.000
$end

0.000
0.000

$rem
BASIS
METHOD
MAX_SCF_CYCLES
THRESH
$end

0.000
1.000

=
=
=
=

6-311++G(2df)
hf
1
10

@@@
$comment
OH radical, part 2. Read cation orbitals, do the radical
$end
$molecule
0 2
O 0.000
H 0.000
$end

0.000
0.000

$rem
BASIS
METHOD
UNRESTRICTED
SCF_ALGORITHM
SCF_CONVERGENCE
SCF_GUESS
THRESH
$end

0.000
1.000

=
=
=
=
=
=
=

6-311++G(2df)
hf
false
dm
7
read
10

Example 4.12 Input for an unrestricted HF calculation on H2 in the dissociation limit, showing the use of SCF_GUESS_MIX
= 2 (corresponding to 20% of the alpha LUMO mixed with the alpha HOMO). Geometric direct minimization with DIIS
is used to converge the SCF, together with MAX_DIIS_CYCLES = 1 (using the default value for MAX_DIIS_CYCLES,
the DIIS procedure just oscillates).
$molecule
0 1
H 0.000
H 0.000
$end

0.000
0.0
0.000 -10.0

$rem
UNRESTRICTED
METHOD
BASIS
SCF_ALGORITHM
MAX_DIIS_CYCLES
SCF_GUESS
SCF_GUESS_MIX
$end

=
=
=
=
=
=
=

true
hf
6-31g**
diis_gdm
1
gwh
2

Chapter 4: Self-Consistent Field Ground-State Methods

4.5
4.5.1

76

Converging SCF Calculations
Introduction

As for any numerical optimization procedure, the rate of convergence of the SCF procedure is dependent on the initial
guess and on the algorithm used to step towards the stationary point. Q-C HEM features a number of SCF optimization
algorithms which can be selected via the $rem variable SCF_ALGORITHM, including:
Methods that are based on extrapolation/interpolation:
• The highly successful DIIS procedures. These are the default (except for restricted open-shell SCF calculations)
and are available for all orbital types (see Section 4.5.3).
• ADIIS (the augmented DIIS algorithm developed by Hu and Yang, 30 available for R and U only).
Methods that make use of orbital gradient:
• Direct Minimization (DM), which has been re-implemented as simple steepest descent with line search, and is
available for all orbital types. DM can be invoked after a few DIIS iterations.
• Geometric Direct Minimization (GDM) which is an improved and highly robust version of DM and is the recommended fall-back when DIIS fails. Like DM, It can also be invoked after a few iterations with DIIS to improve
the initial guess. GDM is the default algorithm for restricted open-shell SCF calculations and is available for all
orbital types (see Section 4.5.4).
• GDM_LS (it is essentially a preconditioned (using orbital energy differences as the preconditioner) L-BFGS
algorithm with line search, available for R, U, RO and OS_RO).
Methods that require orbital Hessian:
• NEWTON_CG/NEWTON_MINRES (solve Hd = −g for the update direction with CG/MINRES solvers).
• SF_NEWTON_CG (the “saddle-free" version of NEWTON_CG).
The analytical orbital Hessian is available for R/U/RO/G/CR unless special density functionals (e.g. those containing
VV10) are used, while the use of finite-difference Hessian is available for all orbital types by setting FD_MAT_VEC_PROD
= TRUE.
In addition to these algorithms, there is also the maximum overlap method (MOM) which ensures that DIIS always
occupies a continuous set of orbitals and does not oscillate between different occupancies. MOM can also be used to
obtain higher-energy solutions of the SCF equations (see Section 7.4). The relaxed constraint algorithm (RCA), which
guarantees that the energy goes down at every step, is also available via the old SCF code (set GEN_SCFMAN = FALSE).
Nevertheless, the performance of the ADIIS algorithm should be similar to it.
Since the code in GEN_SCFMAN is highly modular, the availability of different SCF algorithms to different SCF
(orbital) types is largely extended in general. For example, the old ROSCF implementation requires the use of the
GWH guess and the GDM algorithm exclusively. Such a limitation has been eliminated in GEN_SCFMAN based RO
calculations.

4.5.2

Basic Convergence Control Options

See also more detailed options in the following sections, and note that the SCF convergence criterion and the integral
threshold must be set in a compatible manner, (this usually means THRESH should be set to at least 3 higher than
SCF_CONVERGENCE).

Chapter 4: Self-Consistent Field Ground-State Methods

77

MAX_SCF_CYCLES
Controls the maximum number of SCF iterations permitted.
TYPE:
INTEGER
DEFAULT:
50
OPTIONS:
n n > 0 User-selected.
RECOMMENDATION:
Increase for slowly converging systems such as those containing transition metals.
SCF_ALGORITHM
Algorithm used for converging the SCF.
TYPE:
STRING
DEFAULT:
DIIS Pulay DIIS.
OPTIONS:
DIIS
Pulay DIIS.
DM
Direct minimizer.
DIIS_DM
Uses DIIS initially, switching to direct minimizer for later iterations
(See THRESH_DIIS_SWITCH, MAX_DIIS_CYCLES).
DIIS_GDM
Use DIIS and then later switch to geometric direct minimization
(See THRESH_DIIS_SWITCH, MAX_DIIS_CYCLES).
GDM
Geometric Direct Minimization.
RCA
Relaxed constraint algorithm
RCA_DIIS
Use RCA initially, switching to DIIS for later iterations (see
THRESH_RCA_SWITCH and MAX_RCA_CYCLES described
later in this chapter)
ROOTHAAN Roothaan repeated diagonalization.
RECOMMENDATION:
Use DIIS unless performing a restricted open-shell calculation, in which case GDM is recommended. If DIIS fails to find a reasonable approximate solution in the initial iterations,
RCA_DIIS is the recommended fallback option. If DIIS approaches the correct solution but
fails to finally converge, DIIS_GDM is the recommended fallback.
SCF_CONVERGENCE
SCF is considered converged when the wave function error is less that 10−SCF_CONVERGENCE .
Adjust the value of THRESH at the same time. Note as of Q-C HEM 3.0 the DIIS error is measured
by the maximum error rather than the RMS error.
TYPE:
INTEGER
DEFAULT:
5 For single point energy calculations.
7 For geometry optimizations and vibrational analysis.
8 For SSG calculations, see Chapter 6.
OPTIONS:
n Corresponding to 10−n
RECOMMENDATION:
Tighter criteria for geometry optimization and vibration analysis. Larger values provide more
significant figures, at greater computational cost.
In some cases besides the total SCF energy, one needs its separate energy components, like kinetic energy, exchange energy, correlation energy, etc. The values of these components are printed at each SCF cycle if one specifies SCF_PRINT

78

Chapter 4: Self-Consistent Field Ground-State Methods
= 1 in the input.

4.5.3

Direct Inversion in the Iterative Subspace (DIIS)

The SCF implementation of the Direct Inversion in the Iterative Subspace (DIIS) method 43,44 uses the property of an
SCF solution that requires the density matrix to commute with the Fock matrix:
SPF − FPS = 0 .

(4.29)

During the SCF cycles, prior to achieving self-consistency, it is therefore possible to define an error vector ei , which is
non-zero except at convergence:
SPi Fi − Fi Pi S = ei
(4.30)
Here Pi is obtained by diagonalizing Fi , and
Fk =

k−1
X

cj Fj

(4.31)

j=1

The DIIS coefficients ck , are obtained by a least-squares constrained minimization of the error vectors, viz
!
!
X
X
Z=
ck ek ·
ck ek
k

where the constraint

P

k ck

(4.32)

k

= 1 is imposed to yield a set of linear equations, of dimension N + 1:
e1 · e1
..


.

 eN · e1


1

···
..
.
···
···

e1 · eN
..
.
eN · eN
1


 
1
c1
..   ..  

 
. 
 .  = 


1
cN  
0
λ


0
.. 
. 
 .
0 

(4.33)

1

Convergence criteria require the largest element of the N th error vector to be below a cutoff threshold, usually 10−5
a.u. for single point energies, but often increased to 10−8 a.u. for optimizations and frequency calculations.
The rate of convergence may be improved by restricting the number of previous Fock matrices used for determining
the DIIS coefficients,
k−1
X
Fk =
c j Fj .
(4.34)
j=k−(L+1)

Here L is the size of the DIIS subspace, which is set using the $rem variable DIIS_SUBSPACE_SIZE. As the Fock matrix
nears self-consistency, the linear matrix equations in Eq. (4.33) tend to become severely ill-conditioned and it is often
necessary to reset the DIIS subspace (this is automatically carried out by the program).
Finally, on a practical note, we observe that DIIS has a tendency to converge to global minima rather than local minima
when employed for SCF calculations. This seems to be because only at convergence is the density matrix in the DIIS
iterations idempotent. On the way to convergence, one is not on the true energy surface, and this seems to permit DIIS
to “tunnel” through barriers in wave function space. This is usually a desirable property, and is the motivation for
the options that permit initial DIIS iterations before switching to direct minimization to converge to the minimum in
difficult cases.
The following $rem variables permit some customization of the DIIS iterations:

Chapter 4: Self-Consistent Field Ground-State Methods

79

DIIS_SUBSPACE_SIZE
Controls the size of the DIIS and/or RCA subspace during the SCF.
TYPE:
INTEGER
DEFAULT:
15
OPTIONS:
User-defined
RECOMMENDATION:
None
DIIS_PRINT
Controls the output from DIIS SCF optimization.
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
0 Minimal print out.
1 Chosen method and DIIS coefficients and solutions.
2 Level 1 plus changes in multipole moments.
3 Level 2 plus Multipole moments.
4 Level 3 plus extrapolated Fock matrices.
RECOMMENDATION:
Use the default

Note: In Q-C HEM 3.0 the DIIS error is determined by the maximum error rather than the RMS error. For backward
compatibility the RMS error can be forced by using the following $rem:
DIIS_ERR_RMS
Changes the DIIS convergence metric from the maximum to the RMS error.
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
TRUE, FALSE
RECOMMENDATION:
Use the default, the maximum error provides a more reliable criterion.

Chapter 4: Self-Consistent Field Ground-State Methods

80

DIIS_SEPARATE_ERRVEC
Control optimization of DIIS error vector in unrestricted calculations.
TYPE:
LOGICAL
DEFAULT:
FALSE Use a combined α and β error vector.
OPTIONS:
FALSE Use a combined α and β error vector.
TRUE
Use separate error vectors for the α and β spaces.
RECOMMENDATION:
When using DIIS in Q-C HEM a convenient optimization for unrestricted calculations is to sum
the α and β error vectors into a single vector which is used for extrapolation. This is often
extremely effective, but in some pathological systems with symmetry breaking, can lead to
false solutions being detected, where the α and β components of the error vector cancel exactly giving a zero DIIS error. While an extremely uncommon occurrence, if it is suspected, set
DIIS_SEPARATE_ERRVEC = TRUE to check.

4.5.4

Geometric Direct Minimization (GDM)

Troy Van Voorhis, working at Berkeley with Martin Head-Gordon, has developed a novel direct minimization method
that is extremely robust, and at the same time is only slightly less efficient than DIIS. This method is called geometric
direct minimization (GDM) because it takes steps in an orbital rotation space that correspond properly to the hyperspherical geometry of that space. In other words, rotations are variables that describe a space which is curved like a
many-dimensional sphere. Just like the optimum flight paths for airplanes are not straight lines but great circles, so
too are the optimum steps in orbital rotation space. GDM takes this correctly into account, which is the origin of its
efficiency and its robustness. For full details, we refer the reader to Ref. 66. GDM is a good alternative to DIIS for SCF
jobs that exhibit convergence difficulties with DIIS.
Recently, Barry Dunietz, also working at Berkeley with Martin Head-Gordon, has extended the GDM approach to
restricted open-shell SCF calculations. Their results indicate that GDM is much more efficient than the older direct
minimization method (DM).
In section 4.5.3, we discussed the fact that DIIS can efficiently head towards the global SCF minimum in the early
iterations. This can be true even if DIIS fails to converge in later iterations. For this reason, a hybrid scheme has been
implemented which uses the DIIS minimization procedure to achieve convergence to an intermediate cutoff threshold.
Thereafter, the geometric direct minimization algorithm is used. This scheme combines the strengths of the two methods quite nicely: the ability of DIIS to recover from initial guesses that may not be close to the global minimum, and
the ability of GDM to robustly converge to a local minimum, even when the local surface topology is challenging for
DIIS. This is the recommended procedure with which to invoke GDM (i.e., setting SCF_ALGORITHM = DIIS_GDM).
This hybrid procedure is also compatible with the SAD guess, while GDM itself is not, because it requires an initial
guess set of orbitals. If one wishes to disturb the initial guess as little as possible before switching on GDM, one should
additionally specify MAX_DIIS_CYCLES = 1 to obtain only a single Roothaan step (which also serves up a properly
orthogonalized set of orbitals).
$rem options relevant to GDM are SCF_ALGORITHM which should be set to either GDM or DIIS_GDM and the following:

Chapter 4: Self-Consistent Field Ground-State Methods

81

MAX_DIIS_CYCLES
The maximum number of DIIS iterations before switching to (geometric) direct minimization
when SCF_ALGORITHM is DIIS_GDM or DIIS_DM. See also THRESH_DIIS_SWITCH.
TYPE:
INTEGER
DEFAULT:
50
OPTIONS:
1 Only a single Roothaan step before switching to (G)DM
n n DIIS iterations before switching to (G)DM.
RECOMMENDATION:
None
THRESH_DIIS_SWITCH
The threshold for switching between DIIS extrapolation and direct minimization of the SCF
energy is 10−THRESH_DIIS_SWITCH when SCF_ALGORITHM is DIIS_GDM or DIIS_DM. See
also MAX_DIIS_CYCLES
TYPE:
INTEGER
DEFAULT:
2
OPTIONS:
User-defined.
RECOMMENDATION:
None

4.5.5

Direct Minimization (DM)

Direct minimization (DM) is a less sophisticated forerunner of the geometric direct minimization (GDM) method
discussed in the previous section. DM does not properly step along great circles in the hyper-spherical space of orbital
rotations, and therefore converges less rapidly and less robustly than GDM, in general. DM is retained in Q-C HEM
only for legacy purposes. In general, the input options are the same as for GDM, with the exception of the specification
of SCF_ALGORITHM, which can be either DIIS_DM (recommended) or DM.
PSEUDO_CANONICAL
When SCF_ALGORITHM = DM, this controls the way the initial step, and steps after subspace
resets are taken.
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
FALSE Use Roothaan steps when (re)initializing
TRUE
Use a steepest descent step when (re)initializing
RECOMMENDATION:
The default is usually more efficient, but choosing TRUE sometimes avoids problems with orbital
reordering.

4.5.6

Maximum Overlap Method (MOM)

In general, the DIIS procedure is remarkably successful. One difficulty that is occasionally encountered is the problem
of an SCF that occupies two different sets of orbitals on alternating iterations, and therefore oscillates and fails to

Chapter 4: Self-Consistent Field Ground-State Methods

82

converge. This can be overcome by choosing orbital occupancies that maximize the overlap of the new occupied
orbitals with the set previously occupied. Q-C HEM contains the maximum overlap method (MOM), 26 developed by
Andrew Gilbert and Peter Gill.
MOM is therefore is a useful adjunct to DIIS in convergence problems involving flipping of orbital occupancies. It
is controlled by the $rem variable MOM_START, which specifies the SCF iteration on which the MOM procedure is
first enabled. There are two strategies that are useful in setting a value for MOM_START. To help maintain an initial
configuration it should be set to start on the first cycle. On the other hand, to assist convergence it should come on later
to avoid holding on to an initial configuration that may be far from the converged one.
The MOM-related $rem variables in full are the following:.
MOM_PRINT
Switches printing on within the MOM procedure.
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
FALSE Printing is turned off
TRUE
Printing is turned on.
RECOMMENDATION:
None
MOM_START
Determines when MOM is switched on to stabilize DIIS iterations.
TYPE:
INTEGER
DEFAULT:
0 (FALSE)
OPTIONS:
0 (FALSE) MOM is not used
n
MOM begins on cycle n.
RECOMMENDATION:
Set to 1 if preservation of initial orbitals is desired. If MOM is to be used to aid convergence, an
SCF without MOM should be run to determine when the SCF starts oscillating. MOM should be
set to start just before the oscillations.
MOM_METHOD
Determines the target orbitals with which to maximize the overlap on each SCF cycle.
TYPE:
INTEGER
DEFAULT:
3
OPTIONS:
3
Maximize overlap with the orbitals from the previous SCF cycle.
13 Maximize overlap with the initial guess orbitals.
RECOMMENDATION:
If appropriate guess orbitals can be obtained, then MOM_METHOD = 13 can provide more
reliable convergence to the desired solution.

4.5.7

Relaxed Constraint Algorithm (RCA)

The relaxed constraint algorithm (RCA) is an ingenious and simple means of minimizing the SCF energy that is
particularly effective in cases where the initial guess is poor. The latter is true, for example, when employing a user-

83

Chapter 4: Self-Consistent Field Ground-State Methods

specified basis (when the “core” or GWH guess must be employed) or when near-degeneracy effects imply that the
initial guess will likely occupy the wrong orbitals relative to the desired converged solution.
Briefly, RCA begins with the SCF problem as a constrained minimization of the energy as a function of the density
matrix, E(P). 8,9 The constraint is that the density matrix be idempotent, P · P = P, which basically forces the occupation numbers to be either zero or one. The fundamental realization of RCA is that this constraint can be relaxed to
allow sub-idempotent density matrices, P · P ≤ P. This condition forces the occupation numbers to be between zero
and one. Physically, we expect that any state with fractional occupations can lower its energy by moving electrons from
higher energy orbitals to lower ones. Thus, if we solve for the minimum of E(P) subject to the relaxed sub-idempotent
constraint, we expect that the ultimate solution will nonetheless be idempotent. In fact, for Hartree-Fock this can be
rigorously proven. For density functional theory, it is possible that the minimum will have fractional occupation numbers but these occupations have a physical interpretation in terms of ensemble DFT. The reason the relaxed constraint is
easier to deal with is that it is easy to prove that a linear combination of sub-idempotent matrices is also sub-idempotent
as long as the linear coefficients are between zero and one. By exploiting this property, convergence can be accelerated
in a way that guarantees the energy will go down at every step.
The implementation of RCA in Q-C HEM closely follows the “Energy DIIS” implementation of the RCA algorithm. 32
Here, the current density matrix is written as a linear combination of the previous density matrices:
X
xi Pi
(4.35)
P(x) =
i

To a very good approximation (exact for Hartree-Fock) the energy for P(x) can be written as a quadratic function of x:
E(x) =

X
i

Ei xi +

1X
xi (Pi − Pj ) · (Fi − Fj )xj
2 i

(4.36)

At each iteration, x is chosen to minimize E(x) subject to the constraint that all of the xi are between zero and one.
The Fock matrix for P(x) is further written as a linear combination of the previous Fock matrices,
X
F(x) =
xi Fi + δFxc (x)
(4.37)
i

where δFxc (x) denotes a (usually quite small) change in the exchange-correlation part that is computed once x has been
determined. We note that this extrapolation is very similar to that used by DIIS. However, this procedure is guaranteed
to reduce the energy E(x) at every iteration, unlike DIIS.
In practice, the RCA approach is ideally suited to difficult convergence situations because it is immune to the erratic
orbital swapping that can occur in DIIS. On the other hand, RCA appears to perform relatively poorly near convergence, requiring a relatively large number of steps to improve the precision of a good approximate solution. It is thus
advantageous in many cases to run RCA for the initial steps and then switch to DIIS either after some specified number
of iterations or after some target convergence threshold has been reached. Finally, note that by its nature RCA considers
the energy as a function of the density matrix. As a result, it cannot be applied to restricted open shell calculations
which are explicitly orbital-based. Note: RCA interacts poorly with INCDFT, so INCDFT is disabled by default when
an RCA or RCA_DIIS calculation is requested. To enable INCDFT with such a calculation, set INCDFT = 2 in the
$rem section. RCA may also have poor interactions with incremental Fock builds; if RCA fails to converge, setting
INCFOCK = FALSE may improve convergence in some cases.
Job-control variables for RCA are listed below, and an example input can be found in Section 4.5.11.

Chapter 4: Self-Consistent Field Ground-State Methods

84

RCA_PRINT
Controls the output from RCA SCF optimizations.
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
0 No print out
1 RCA summary information
2 Level 1 plus RCA coefficients
3 Level 2 plus RCA iteration details
RECOMMENDATION:
None
MAX_RCA_CYCLES
The maximum number of RCA iterations before switching to DIIS when SCF_ALGORITHM is
RCA_DIIS.
TYPE:
INTEGER
DEFAULT:
50
OPTIONS:
N N RCA iterations before switching to DIIS
RECOMMENDATION:
None
THRESH_RCA_SWITCH
The threshold for switching between RCA and DIIS when SCF_ALGORITHM is RCA_DIIS.
TYPE:
INTEGER
DEFAULT:
3
OPTIONS:
N Algorithm changes from RCA to DIIS when Error is less than 10−N .
RECOMMENDATION:
None

4.5.8

User-Customized Hybrid SCF Algorithm

It is often the case that a single algorithm is not able to guarantee SCF convergence. Meanwhile, some SCF algorithms
(e.g., ADIIS) can accelerate convergence at the beginning of an SCF calculation but becomes less efficient near the
convergence. While a few hybrid algorithms (DIIS_GDM, RCA_DIIS) have been enabled in Q-C HEM’s original SCF
implementation, in GEN_SCFMAN, we seek for a more flexible setup for the use of multiple SCF algorithms so that
users can have a more precise control on the SCF procedure. With the current implementation, at most four distinct
algorithms (usually more than enough) can be employed in one single SCF calculation based on GEN_SCFMAN, and
the basic job control is as follows:

Chapter 4: Self-Consistent Field Ground-State Methods
GEN_SCFMAN_HYBRID_ALGO
Use multiple algorithms in an SCF calculation based on GEN_SCFMAN.
TYPE:
BOOLEAN
DEFAULT:
FALSE
OPTIONS:
FALSE Use a single SCF algorithm (given by SCF_ALGORITHM).
TRUE
Use multiple SCF algorithms (to be specified).
RECOMMENDATION:
Set it to TRUE when the use of more than one algorithm is desired.
GEN_SCFMAN_ALGO_1
The first algorithm to be used in a hybrid-algorithm calculation.
TYPE:
STRING
DEFAULT:
0
OPTIONS:
All the available SCF_ALGORITHM options, including the GEN_SCFMAN additions (Section 4.3.1).
RECOMMENDATION:
None
GEN_SCFMAN_ITER_1
Maximum number of iterations given to the first algorithm. If used up, switch to the next algorithm.
TYPE:
INTEGER
DEFAULT:
50
OPTIONS:
User-defined
RECOMMENDATION:
None
GEN_SCFMAN_CONV_1
The convergence criterion given to the first algorithm. If reached, switch to the next algorithm.
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
n 10−n
RECOMMENDATION:
None

85

Chapter 4: Self-Consistent Field Ground-State Methods

86

Note: $rem variables GEN_SCFMAN_ALGO_X, GEN_SCFMAN_ITER_X, GEN_SCFMAN_CONV_X (X = 2, 3, 4) are
defined and used in a similar way.
Example 4.13 B3LYP/3-21G calculation for a cadmium-imidazole complex using the ADIIS + DIIS algorithm (an
example from Ref. 30). Due to the poor quality of the CORE guess, using a single algorithm such as DIIS or GDM fails
to converge.
$molecule
2 1
Cd
0.000000
N
0.000000
N
-0.685444
C
0.676053
C
1.085240
C
-1.044752
H
1.231530
H
2.088641
H
-2.068750
H
-1.313170
$end

0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000

$rem
JOBTYPE
EXCHANGE
BASIS
UNRESTRICTED
SYMMETRY
SYM_IGNORE
THRESH
SCF_GUESS
GEN_SCFMAN
GEN_SCFMAN_HYBRID_ALGO
GEN_SCFMAN_ALGO_1
GEN_SCFMAN_CONV_1
GEN_SCFMAN_ITER_1
GEN_SCFMAN_ALGO_2
GEN_SCFMAN_CONV_2
GEN_SCFMAN_ITER_2
$end

4.5.9

0.000000
-2.260001
-4.348035
-4.385069
-3.091231
-3.060220
-5.300759
-2.711077
-2.726515
-5.174718

SP
B3LYP
3-21g
FALSE
FALSE
TRUE
14
CORE
TRUE
TRUE
ADIIS
3 !switch to DIIS when error < 1E-3
50
DIIS
8
50

Internal Stability Analysis and Automated Correction for Energy Minima

At convergence, the SCF energy will be at a stationary point with respect to changes in the MO coefficients. However,
this stationary point is not guaranteed to be an energy minimum, and in cases where it is not, the wave function is said
to be unstable. Even if the wave function is at a minimum, this minimum may be an artifact of the constraints placed
on the form of the wave function. For example, an unrestricted calculation will usually give a lower energy than the
corresponding restricted calculation, and this can give rise to an RHF → UHF instability.
Based on our experience, even for very simple data set such as the G2 atomization energies, 11 using the default algorithm (DIIS) produces unstable solutions for several species (even for single atoms with some density functionals). In
such cases, failure to check the internal stability of SCF solutions can result in flawed benchmark results. Although in
general the use of gradient-based algorithms such as GDM is more likely to locate the true minimum, it still cannot
entirely eliminate the possibility of finding an unstable solution.
To understand what instabilities can occur, it is useful to consider the most general form possible for the spin orbitals:
χi (r, ζ) = ψiα (r)α(ζ) + ψiβ (r)β(ζ) .

(4.38)

Here, ψiα and ψiβ are complex-valued functions of the Cartesian coordinates r, and α and β are spin eigenfunctions of
the spin-variable ζ. The first constraint that is almost universally applied is to assume the spin orbitals depend only on

87

Chapter 4: Self-Consistent Field Ground-State Methods
one or other of the spin-functions α or β. Thus, the spin-functions take the form
χi (r, ζ) = ψiα (r)α(ζ)

or χi (r, ζ) = ψiβ (r)β(ζ) .

(4.39)

In addition, most SCF calculations use real functions, and this places an additional constraint on the form of the wave
function. If there exists a complex solution to the SCF equations that has a lower energy, the wave function exhibits a
real → complex instability. The final constraint that is commonly placed on the spin-functions is that ψiα = ψiβ , i.e.,
that the spatial parts of the spin-up and spin-down orbitals are the same. This gives the familiar restricted formalism
and can lead to an RHF → UHF instability as mentioned above. Further details about the possible instabilities can be
found in Ref. 48.
Wave function instabilities can arise for several reasons, but frequently occur if
• There exists a singlet diradical at a lower energy then the closed-shell singlet state.
• There exists a triplet state at a lower energy than the lowest singlet state.
• There are multiple solutions to the SCF equations, and the calculation has not found the lowest energy solution.
Q-C HEM’s previous stability analysis package suffered from the following limitations:
• It is only available for restricted (close-shell) and unrestricted SCF calculations.
• It requires the analytical orbital Hessian of the wave function energy.
• The calculation terminates after the corrected MOs are generated, and a second job is needed to read in these
orbitals and run another SCF calculation.
The implementation of internal stability analysis in GEN_SCFMAN overcomes almost all these shortcomings. Its
availability has been extended to all the implemented orbital types. As in the old code, when the analytical Hessian
of the given orbital type and theory (e.g. RO/B3LYP) is available, it computes matrix-vector products analytically for
the Davidson algorithm. 14 If the analytical Hessian is not available, users can still run stability analysis by using the
finite-difference matrix-vector product technique developed by Sharada et al., 51 which requires the gradient (related to
the Fock matrix) only:
∇E(X0 + ξb1 ) − ∇E(X0 − ξb1 )
Hb1 =
(4.40)
2ξ
where H is the Hessian matrix, b1 is a trial vector, X0 stands for the current stationary point, and ξ is the finite step
size. With this method, internal stability analysis is available for all the implemented orbital types in GEN_SCFMAN.
It should be noted that since the second derivative of NLC functionals such as VV10 is not available in Q-C HEM, this
finite-difference method will be used by default for the evaluation of Hessian-vector products.
GEN_SCFMAN allows multiple SCF calculations and stability analyses to be performed in a single job so that it can
make use of the corrected MOs and locate the true minimum automatically. The MOs are displaced along the direction
of the lowest-energy eigenvector (with line search) if an SCF solution is found to be unstable. A new SCF calculation
that reads in these corrected MOs as initial guess will be launched automatically if INTERNAL_STABILITY_ITER > 0.
Such macro-loops will keep going until a stable solution is reached.
Note: The stability analysis package can be used to analyze both HF and DFT wave functions.

Chapter 4: Self-Consistent Field Ground-State Methods
4.5.9.1

Job Control
INTERNAL_STABILITY
Perform internal stability analysis in GEN_SCFMAN.
TYPE:
BOOLEAN
DEFAULT:
FALSE
OPTIONS:
FALSE Do not perform internal stability analysis after convergence.
TRUE
Perform internal stability analysis and generate the corrected MOs.
RECOMMENDATION:
Turn it on when the SCF solution is prone to unstable solutions, especially for open-shell species.
FD_MAT_VEC_PROD
Compute Hessian-vector product using the finite difference technique.
TYPE:
BOOLEAN
DEFAULT:
FALSE (TRUE when the employed functional contains NLC)
OPTIONS:
FALSE Compute Hessian-vector product analytically.
TRUE
Use finite difference to compute Hessian-vector product.
RECOMMENDATION:
Set it to TRUE when analytical Hessian is not available.
Note: For simple R and U calculations, it can always be set to FALSE, which indicates that
only the NLC part will be computed with finite difference.
INTERNAL_STABILITY_ITER
Maximum number of new SCF calculations permitted after the first stability analysis is performed.
TYPE:
INTEGER
DEFAULT:
0 (automatically set to 1 if INTERNAL_STABILITY = TRUE)
OPTIONS:
n n new SCF calculations permitted.
RECOMMENDATION:
Give a larger number if 1 is not enough (still unstable).
INTERNAL_STABILITY_DAVIDSON_ITER
Maximum number of Davidson iterations allowed in one stability analysis.
TYPE:
INTEGER
DEFAULT:
50
OPTIONS:
n Perform up to n Davidson iterations.
RECOMMENDATION:
Use the default.

88

Chapter 4: Self-Consistent Field Ground-State Methods

89

INTERNAL_STABILITY_CONV
Convergence criterion for the Davidson solver (for the lowest eigenvalues).
TYPE:
INTEGER
DEFAULT:
4 (3 when FD_MAT_ON_VECS = TRUE)
OPTIONS:
n Terminate Davidson iterations when the norm of the residual vector is below 10−n .
RECOMMENDATION:
Use the default.
INTERNAL_STABILITY_ROOTS
Number of lowest Hessian eigenvalues to solve for.
TYPE:
INTEGER
DEFAULT:
2
OPTIONS:
n Solve for n lowest eigenvalues.
RECOMMENDATION:
Use the default.

Example 4.14 Unrestricted SCF calculation of triplet B2 using B97M-V/6-31g with the GDM algorithm. A displacement is performed when the first solution is characterized as a saddle point, and the second SCF gives a stable
solution.
$molecule
0 3
b
b 1 R
R = 1.587553
$end
$rem
JOBTYPE
METHOD
BASIS
UNRESTRICTED
THRESH
SYMMETRY
SYM_IGNORE
SCF_FINAL_PRINT
SCF_ALGORITHM
SCF_CONVERGENCE
INTERNAL_STABILITY
FD_MAT_VEC_PROD
$end

4.5.10

sp
b97m-v
6-31g
true
14
false
true
1
gdm
8
true !turn on internal stability analysis
false !use finite-diff for the vv10 part only

Small-Gap Systems

SCF calculations for systems with zero or small HOMO-LUMO gap (such as metals) can exhibit very slow convergence
or may even fail to converge. This problem arises because the energetic ordering of orbitals and states can switch during
the SCF optimization leading to discontinuities in the optimization. Using fractional MO occupation numbers can

90

Chapter 4: Self-Consistent Field Ground-State Methods

improve the convergence for small-gap systems. In this approach, the occupation numbers of MOs around the Fermi
level are allowed to assume non-integer values. This “occupation smearing” allows one to include multiple electron
configurations in the same optimization, which improves the stability of the optimization.
We follow the pseudo-Fractional Occupation Number (pFON) method of Rabuck and Scuseria 45 that scales the MO
occupation used to construct the AO density:
Pµν =

N
X

np Cµp Cνp .

(4.41)

p=1

For a conventional (integer occupation number) SCF run, the occupation number np is either one (occupied) or zero
(virtual). In pFON, the occupation numbers are following a Fermi-Dirac distribution,
np = 1 + e(p −F )/kT


−1

,

(4.42)

where p is the respective orbital energy and kT the Boltzmann constant and temperature, respectively. The Fermi
energy F is set to (HOMO +LUMO )/2 in our implementation. To ensure conservation of the total number of electrons,
P
the pFON approach re-scales the occupation numbers so that p np = Nel .
There are several parameters to control the electronic temperature T throughout a pFON SCF run. The temperature
can either be held constant at finite temperature (Tinit = Tfinal ), or the system can be cooled from a higher temperature
down to the final temperature. So far, no zero-temperature extrapolation has been implemented.
OCCUPATIONS
Activates pFON calculation.
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
0 Integer occupation numbers
1 Not yet implemented
2 Pseudo-fractional occupation numbers (pFON)
RECOMMENDATION:
Use pFON to improve convergence for small-gap systems.
FON_T_START
Initial electronic temperature (in K) for FON calculation.
TYPE:
INTEGER
DEFAULT:
1000
OPTIONS:
Any desired initial temperature.
RECOMMENDATION:
Pick the temperature to either reproduce experimental conditions (e.g. room temperature) or as
low as possible to approach zero-temperature.

Chapter 4: Self-Consistent Field Ground-State Methods
FON_T_END
Final electronic temperature for FON calculation.
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
Any desired final temperature.
RECOMMENDATION:
Pick the temperature to either reproduce experimental conditions (e.g. room temperature) or as
low as possible to approach zero-temperature.
FON_NORB
Number of orbitals above and below the Fermi level that are allowed to have fractional occupancies.
TYPE:
INTEGER
DEFAULT:
4
OPTIONS:
n number of active orbitals
RECOMMENDATION:
The number of valence orbitals is a reasonable choice.
FON_T_SCALE
Determines the step size for the cooling.
TYPE:
INTEGER
DEFAULT:
90
OPTIONS:
n temperature is scaled by 0.01 · n in each cycle (cooling method 1)
n temperature is decreased by n K in each cycle (cooling method 2)
RECOMMENDATION:
The cooling rate should be neither too slow nor too fast. Too slow may lead to final energies
that are at undesirably high temperatures. Too fast may lead to convergence issues. Reasonable
choices for methods 1 and 2 are 98 and 50, respectively. When in doubt, use constant temperature.
FON_E_THRESH
DIIS error below which occupations will be kept constant.
TYPE:
INTEGER
DEFAULT:
4
OPTIONS:
n freeze occupations below DIIS error of 10−n
RECOMMENDATION:
This should be one or two numbers bigger than the desired SCF convergence threshold.

91

Chapter 4: Self-Consistent Field Ground-State Methods

92

FON_T_METHOD
Selects cooling algorithm.
TYPE:
INTEGER
DEFAULT:
1
OPTIONS:
1 temperature is scaled by a factor in each cycle
2 temperature is decreased by a constant number in each cycle
RECOMMENDATION:
We have made slightly better experience with a constant cooling rate. However, choose constant
temperature when in doubt.

4.5.11

Examples

Example 4.15 Input for a UHF calculation using geometric direct minimization (GDM) on the phenyl radical, after
initial iterations with DIIS.
$molecule
0
2
c1
x1 c1
c2 c1
x2 c2
c3 c1
c4 c1
c5 c3
c6 c4
h1 c2
h2 c3
h3 c4
h4 c5
h5 c6
rc2
rc3
tc3
rc5
ac5
rh1
rh2
ah2
rh4
ah4
$end

1.0
rc2
1.0
rc3
rc3
rc5
rc5
rh1
rh2
rh2
rh4
rh4

x1
c1
x1
x1
c1
c1
x2
c1
c1
c3
c4

90.0
90.0
90.0
90.0
ac5
ac5
90.0
ah2
ah2
ah4
ah4

x1
c2
c2
x1
x1
c1
x1
x1
c1
c1

=
2.672986
=
1.354498
= 62.851505
=
1.372904
= 116.454370
=
1.085735
=
1.085342
= 122.157328
=
1.087216
= 119.523496

$rem
BASIS
METHOD
SCF_ALGORITHM
SCF_CONVERGENCE
THRESH
$end

=
=
=
=
=

6-31G*
hf
diis_gdm
7
10

0.0
tc3
-tc3
-90.0
90.0
180.0
90.0
-90.0
180.0
180.0

Chapter 4: Self-Consistent Field Ground-State Methods

93

Example 4.16 An example showing how to converge a ROHF calculation on the 3A2 state of DMX. Note the use of
reading in orbitals from a previous closed-shell calculation and the use of MOM to maintain the orbital occupancies.
The 3 B1 is obtained if MOM is not used.
$molecule
+1 1
C
H
C
C
H
H
C
H
C
C
H
C
C
H
H
$end

0.000000
0.000000
-1.233954
-2.444677
-2.464545
-3.400657
-1.175344
-2.151707
0.000000
1.175344
2.151707
1.233954
2.444677
2.464545
3.400657

$rem
UNRESTRICTED
METHOD
BASIS
SCF_GUESS
$end

0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000

false
hf
6-31+G*
core

@@@
$molecule
read
$end
$rem
UNRESTRICTED
METHOD
BASIS
SCF_GUESS
MOM_START
$end

false
hf
6-31+G*
read
1

$occupied
1:26 28
1:26 28
$end
@@@
$molecule
-1 3
read
$end
$rem
UNRESTRICTED
METHOD
BASIS
SCF_GUESS
$end

false
hf
6-31+G*
read

0.990770
2.081970
0.290926
1.001437
2.089088
0.486785
-1.151599
-1.649364
-1.928130
-1.151599
-1.649364
0.290926
1.001437
2.089088
0.486785

94

Chapter 4: Self-Consistent Field Ground-State Methods

Example 4.17 RCA_DIIS algorithm applied a radical
$molecule
0 2
H
1.004123
O
-0.246002
O
-1.312366
$end

-0.180454
0.596152
-0.230256

$rem
UNRESTRICTED
METHOD
BASIS
SCF_GUESS
SCF_ALGORITHM
THRESH
$end

0.000000
0.000000
0.000000

true
hf
cc-pVDZ
gwh
RCA_DIIS
9

Example 4.18 pFON calculation of a metal cluster.
$molecule
0 1
Pt
Pt
Pt
Pt
Pt
$end

-0.20408
2.61132
0.83227
0.95832
-1.66760

$rem
METHOD
ECP
SYMMETRY
OCCUPATIONS
FON_NORB
FON_T_START
FON_T_END
FON_E_THRESH
$end

4.6
4.6.1

1.19210
1.04687
0.03296
-1.05360
-1.07875

0.54029
0.66196
-1.49084
0.92253
-1.02416

pbe
fit-lanl2dz
false
2
! pseudo-fractional occupation numbers
10
300 ! electronic temperature: 300 K
300
5
! freeze occupation numbers once DIIS error is 10^-5

Large Molecules and Linear Scaling Methods
Introduction

Construction of the effective Hamiltonian, or Fock matrix, has traditionally been the rate-determining step in selfconsistent field calculations, due primarily to the cost of two-electron integral evaluation, even with the efficient methods available in Q-C HEM (see Appendix B). However, for large enough molecules, significant speedups are possible
by employing linear-scaling methods for each of the nonlinear terms that can arise. Linear scaling means that if the
molecule size is doubled, then the computational effort likewise only doubles. There are three computationally significant terms:
• Electron-electron Coulomb interactions, for which Q-C HEM incorporates the Continuous Fast Multipole Method
(CFMM) discussed in section 4.6.2
• Exact exchange interactions, which arise in hybrid DFT calculations and Hartree-Fock calculations, for which
Q-C HEM incorporates the LinK method discussed in section 4.6.3 below.

Chapter 4: Self-Consistent Field Ground-State Methods

95

• Numerical integration of the exchange and correlation functionals in DFT calculations, which we have already
discussed in section 5.5.
Q-C HEM supports energies and efficient analytical gradients for all three of these high performance methods to permit
structure optimization of large molecules, as well as relative energy evaluation. Note that analytical second derivatives
of SCF energies do not exploit these methods at present.
For the most part, these methods are switched on automatically by the program based on whether they offer a significant
speedup for the job at hand. Nevertheless it is useful to have a general idea of the key concepts behind each of these
algorithms, and what input options are necessary to control them. That is the primary purpose of this section, in addition
to briefly describing two more conventional methods for reducing computer time in large calculations in Section 4.6.4.
There is one other computationally significant step in SCF calculations, and that is diagonalization of the Fock matrix,
once it has been constructed. This step scales with the cube of molecular size (or basis set size), with a small pre-factor.
So, for large enough SCF calculations (very roughly in the vicinity of 2000 basis functions and larger), diagonalization
becomes the rate-determining step. The cost of cubic scaling with a small pre-factor at this point exceeds the cost of the
linear scaling Fock build, which has a very large pre-factor, and the gap rapidly widens thereafter. This sets an effective
upper limit on the size of SCF calculation for which Q-C HEM is useful at several thousand basis functions.

4.6.2

Continuous Fast Multipole Method (CFMM)

The quantum chemical Coulomb problem, perhaps better known as the DFT bottleneck, has been at the forefront of
many research efforts throughout the 1990s. The quadratic computational scaling behavior conventionally seen in the
construction of the Coulomb matrix in DFT or HF calculations has prevented the application of ab initio methods to
molecules containing many hundreds of atoms. Q-C HEM Inc., in collaboration with White and Head-Gordon at the
University of California at Berkeley, and Gill now at the Australian National University, were the first to develop the
generalization of Greengard’s Fast Multipole Method 27 (FMM) to continuous charged matter distributions in the form
of the CFMM, which is the first linear scaling algorithm for DFT calculations. This initial breakthrough has since lead
to an increasing number of linear scaling alternatives and analogies, but for Coulomb interactions, the CFMM remains
state of the art. There are two computationally intensive contributions to the Coulomb interactions which we discuss in
turn:
• Long-range interactions, which are treated by the CFMM
• Short-range interactions, corresponding to overlapping charge distributions, which are treated by a specialized
“J-matrix engine” together with Q-C HEM’s state-of-the art two-electron integral methods.
The Continuous Fast Multipole Method was the first implemented linear scaling algorithm for the construction of
the J matrix. In collaboration with Q-C HEM Inc., Dr. Chris White began the development of the CFMM by more
efficiently deriving 68 the original Fast Multipole Method before generalizing it to the CFMM. 72 The generalization
applied by White et al. allowed the principles underlying the success of the FMM to be applied to arbitrary (subject
to constraints in evaluating the related integrals) continuous, but localized, matter distributions. White and coworkers
further improved the underlying CFMM algorithm, 69,70 then implemented it efficiently, 73 achieving performance that
is an order of magnitude faster than some competing implementations.
The success of the CFMM follows similarly with that of the FMM, in that the charge system is subdivided into a
hierarchy of boxes. Local charge distributions are then systematically organized into multipole representations so that
each distribution interacts with local expansions of the potential due to all distant charge distributions. Local and distant
distributions are distinguished by a well-separated (WS) index, which is the number of boxes that must separate two
collections of charges before they may be considered distant and can interact through multipole expansions; near-field
interactions must be calculated directly. In the CFMM each distribution is given its own WS index and is sorted on
the basis of the WS index, and the position of their space centers. The implementation in Q-C HEM has allowed the
efficiency gains of contracted basis functions to be maintained.
The CFMM algorithm can be summarized in five steps:

Chapter 4: Self-Consistent Field Ground-State Methods

96

1. Form and translate multipoles.
2. Convert multipoles to local Taylor expansions.
3. Translate Taylor information to the lowest level.
4. Evaluate Taylor expansions to obtain the far-field potential.
5. Perform direct interactions between overlapping distributions.
Accuracy can be carefully controlled by due consideration of tree depth, truncation of the multipole expansion and the
definition of the extent of charge distributions in accordance with a rigorous mathematical error bound. As a rough
guide, 10 poles are adequate for single point energy calculations, while 25 poles yield sufficient accuracy for gradient
calculations. Subdivision of boxes to yield a one-dimensional length of about 8 boxes works quite well for systems
of up to about one hundred atoms. Larger molecular systems, or ones which are extended along one dimension, will
benefit from an increase in this number. The program automatically selects an appropriate number of boxes by default.
For the evaluation of the remaining short-range interactions, Q-C HEM incorporates efficient J-matrix engines, originated by White and Head-Gordon. 71 These are analytically exact methods that are based on standard two-electron
integral methods, but with an interesting twist. If one knows that the two-electron integrals are going to be summed
into a Coulomb matrix, one can ask whether they are in fact the most efficient intermediates for this specific task. Or,
can one instead find a more compact and computationally efficient set of intermediates by folding the density matrix
into the recurrence relations for the two-electron integrals. For integrals that are not highly contracted (i.e., are not
linear combinations of more than a few Gaussians), the answer is a dramatic yes. This is the basis of the J-matrix
approach, and Q-C HEM includes the latest algorithm developed by Yihan Shao working with Martin Head-Gordon at
Berkeley for this purpose. Shao’s J-engine is employed for both energies 49 and forces, 50 and gives substantial speedups
relative to the use of two-electron integrals without any approximation—roughly a factor of 10 for energies and 30 for
forces at the level of an uncontracted dddd shell quartet, and increasing with angular momentum). Its use is automatically selected for integrals with low degrees of contraction, while regular integrals are employed when the degree of
contraction is high, following the state of the art PRISM approach of Gill and coworkers. 5
The CFMM is controlled by the following input parameters:
CFMM_ORDER
Controls the order of the multipole expansions in CFMM calculation.
TYPE:
INTEGER
DEFAULT:
15 For single point SCF accuracy
25 For tighter convergence (optimizations)
OPTIONS:
n Use multipole expansions of order n
RECOMMENDATION:
Use the default.

Chapter 4: Self-Consistent Field Ground-State Methods

97

GRAIN
Controls the number of lowest-level boxes in one dimension for CFMM.
TYPE:
INTEGER
DEFAULT:
-1 Program decides best value, turning on CFMM when useful
OPTIONS:
-1
Program decides best value, turning on CFMM when useful
1
Do not use CFMM
n ≥ 8 Use CFMM with n lowest-level boxes in one dimension
RECOMMENDATION:
This is an expert option; either use the default, or use a value of 1 if CFMM is not desired.

4.6.3

Linear Scaling Exchange (LinK) Matrix Evaluation

Hartree-Fock calculations and the popular hybrid density functionals such as B3LYP also require two-electron integrals
to evaluate the exchange energy associated with a single determinant. There is no useful multipole expansion for the
exchange energy, because the bra and ket of the two-electron integral are coupled by the density matrix, which carries
the effect of exchange. Fortunately, density matrix elements decay exponentially with distance for systems that have
a HOMO/LUMO gap. 47 The better the insulator, the more localized the electronic structure, and the faster the rate of
exponential decay. Therefore, for insulators, there are only a linear number of numerically significant contributions to
the exchange energy. With intelligent numerical thresholding, it is possible to rigorously evaluate the exchange matrix
in linear scaling effort. For this purpose, Q-C HEM contains the linear scaling K (LinK) method 41 to evaluate both
exchange energies and their gradients 40 in linear scaling effort (provided the density matrix is highly sparse). The
LinK method essentially reduces to the conventional direct SCF method for exchange in the small molecule limit (by
adding no significant overhead), while yielding large speedups for (very) large systems where the density matrix is
indeed highly sparse. For full details, we refer the reader to the original papers. 40,41 LinK can be explicitly requested
by the following option (although Q-C HEM automatically switches it on when the program believes it is the preferable
algorithm).
LIN_K
Controls whether linear scaling evaluation of exact exchange (LinK) is used.
TYPE:
LOGICAL
DEFAULT:
Program chooses, switching on LinK whenever CFMM is used.
OPTIONS:
TRUE
Use LinK
FALSE Do not use LinK
RECOMMENDATION:
Use for HF and hybrid DFT calculations with large numbers of atoms.

4.6.4

Incremental and Variable Thresh Fock Matrix Building

The use of a variable integral threshold, operating for the first few cycles of an SCF, is justifiable on the basis that the
MO coefficients are usually of poor quality in these cycles. In Q-C HEM, the integrals in the first iteration are calculated
at a threshold of 10−6 (for an anticipated final integral threshold greater than, or equal to 10−6 ) to ensure the error in
the first iteration is solely sourced from the poor MO guess. Following this, the integral threshold used is computed as
tmp_thresh = varthresh × DIIS_error

(4.43)

Chapter 4: Self-Consistent Field Ground-State Methods

98

where the DIIS_error is that calculated from the previous cycle, varthresh is the variable threshold set by the
program (by default) and tmp_thresh is the temporary threshold used for integral evaluation. Each cycle requires
recalculation of all integrals. The variable integral threshold procedure has the greatest impact in early SCF cycles.
In an incremental Fock matrix build, 46 F is computed recursively as
1
Fm = Fm−1 + ∆Jm−1 − ∆Km−1
2

(4.44)

where m is the SCF cycle, and ∆Jm and ∆Km are computed using the difference density
∆Pm = Pm − Pm−1

(4.45)

Using Schwartz integrals and elements of the difference density, Q-C HEM is able to determine at each iteration which
ERIs are required, and if necessary, recalculated. As the SCF nears convergence, ∆Pm becomes sparse and the number
of ERIs that need to be recalculated declines dramatically, saving the user large amounts of computational time.
Incremental Fock matrix builds and variable thresholds are only used when the SCF is carried out using the direct SCF
algorithm and are clearly complementary algorithms. These options are controlled by the following input parameters,
which are only used with direct SCF calculations.
INCFOCK
Iteration number after which the incremental Fock matrix algorithm is initiated
TYPE:
INTEGER
DEFAULT:
1 Start INCFOCK after iteration number 1
OPTIONS:
User-defined (0 switches INCFOCK off)
RECOMMENDATION:
May be necessary to allow several iterations before switching on INCFOCK.
VARTHRESH
Controls the temporary integral cut-off threshold. tmp_thresh = 10−VARTHRESH ×
DIIS_error
TYPE:
INTEGER
DEFAULT:
0 Turns VARTHRESH off
OPTIONS:
n User-defined threshold
RECOMMENDATION:
3 has been found to be a practical level, and can slightly speed up SCF evaluation.

4.6.5

Fourier Transform Coulomb Method

The Coulomb part of the DFT calculations using ordinary Gaussian representations can be sped up dramatically using
plane waves as a secondary basis set by replacing the most costly analytical electron repulsion integrals with numerical
integration techniques. The main advantages to keeping the Gaussians as the primary basis set is that the diagonalization
step is much faster than using plane waves as the primary basis set, and all electron calculations can be performed
analytically.
The Fourier Transform Coulomb (FTC) technique 24,25 is precise and tunable and all results are practically identical with
the traditional analytical integral calculations. The FTC technique is at least 2–3 orders of magnitude more accurate

Chapter 4: Self-Consistent Field Ground-State Methods

99

then other popular plane wave based methods using the same energy cutoff. It is also at least 2–3 orders of magnitude
more accurate than the density fitting (resolution-of-identity) technique. Recently, an efficient way to implement the
forces of the Coulomb energy was introduced, 22 and a new technique to localize filtered core functions. Both of these
features have been implemented within Q-C HEM and contribute to the efficiency of the method.
The FTC method achieves these spectacular results by replacing the analytical integral calculations, whose computational costs scales as O(N 4 ) (where N is the number of basis function) with procedures that scale as only O(N 2 ). The
asymptotic scaling of computational costs with system size is linear versus the analytical integral evaluation which is
quadratic. Research at Q-C HEM Inc. has yielded a new, general, and very efficient implementation of the FTC method
which work in tandem with the J-engine and the CFMM (Continuous Fast Multipole Method) techniques. 23
In the current implementation the speed-ups arising from the FTC technique are moderate when small or medium
Pople basis sets are used. The reason is that the J-matrix engine and CFMM techniques provide an already highly
efficient solution to the Coulomb problem. However, increasing the number of polarization functions and, particularly,
the number of diffuse functions allows the FTC to come into its own and gives the most significant improvements.
For instance, using the 6-311G+(df,pd) basis set for a medium-to-large size molecule is more affordable today then
before. We found also significant speed ups when non–Pople basis sets are used such as cc-pvTZ. The FTC energy and
gradients calculations are implemented to use up to f -type basis functions.
FTC
Controls the overall use of the FTC.
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
0 Do not use FTC in the Coulomb part
1 Use FTC in the Coulomb part
RECOMMENDATION:
Use FTC when bigger and/or diffuse basis sets are used.
FTC_SMALLMOL
Controls whether or not the operator is evaluated on a large grid and stored in memory to speed
up the calculation.
TYPE:
INTEGER
DEFAULT:
1
OPTIONS:
1 Use a big pre-calculated array to speed up the FTC calculations
0 Use this option to save some memory
RECOMMENDATION:
Use the default if possible and use 0 (or buy some more memory) when needed.

100

Chapter 4: Self-Consistent Field Ground-State Methods
FTC_CLASS_THRESH_ORDER
Together with FTC_CLASS_THRESH_MULT, determines the cutoff threshold for included a shellpair in the dd class, i.e., the class that is expanded in terms of plane waves.
TYPE:
INTEGER
DEFAULT:
5 Logarithmic part of the FTC classification threshold. Corresponds to 10−5
OPTIONS:
n User specified
RECOMMENDATION:
Use the default.
FTC_CLASS_THRESH_MULT
Together with FTC_CLASS_THRESH_ORDER, determines the cutoff threshold for included a
shell-pair in the dd class, i.e., the class that is expanded in terms of plane waves.
TYPE:
INTEGER
DEFAULT:
5 Multiplicative part of the FTC classification threshold. Together with
the default value of the FTC_CLASS_THRESH_ORDER this leads to
the 5 × 10−5 threshold value.
OPTIONS:
n User specified.
RECOMMENDATION:
Use the default. If diffuse basis sets are used and the molecule is relatively big then tighter FTC
classification threshold has to be used. According to our experiments using Pople-type diffuse
basis sets, the default 5 × 10−5 value provides accurate result for an alanine5 molecule while
1 × 10−5 threshold value for alanine10 and 5 × 10−6 value for alanine15 has to be used.

4.6.6

Resolution of the Identity Fock Matrix Methods

Evaluation of the Fock matrix (both Coulomb, J, and exchange, K, pieces) can be sped up by an approximation known
as the resolution-of-the-identity approximation (RI-JK). Essentially, the full complexity in common basis sets required
to describe chemical bonding is not necessary to describe the mean-field Coulomb and exchange interactions between
electrons. That is, ρ in the left side of
X
(µν|ρ) =
(µν|λσ)Pλσ
(4.46)
λσ

is much less complicated than an individual λσ function pair. The same principle applies to the FTC method in
subsection 4.6.5, in which case the slowly varying piece of the electron density is replaced with a plane-wave expansion.
With the RI-JK approximation, the Coulomb interactions of the function pair ρ(r) = λσ(r)Pλσ are fit by a smaller set
of atom-centered basis functions. In terms of J:
XZ
XZ
1
1
d3 r1 Pλσ λσ(r1 )
≈
d3 r1 PK K(r1 )
(4.47)
|r1 − r|
|r1 − r|
λσ

K

The coefficients PK must be determined to accurately represent the potential. This is done by performing a leastsquared minimization of the difference between Pλσ λσ(r1 ) and PK K(r1 ), with differences measured by the Coulomb
metric. This requires a matrix inversion over the space of auxiliary basis functions, which may be done rapidly by
Cholesky decomposition.
The RI method applied to the Fock matrix may be further enhanced by performing local fitting of a density or function
pair element. This is the basis of the atomic-RI method (ARI), which has been developed for both Coulomb (J)

Chapter 4: Self-Consistent Field Ground-State Methods

101

matrix 54 and exchange (K) matrix evaluation. 55 In ARI, only nearby auxiliary functions K(r) are employed to fit the
target function. This reduces the asymptotic scaling of the matrix-inversion step as well as that of many intermediate
steps in the digestion of RI integrals. Briefly, atom-centered auxiliary functions on nearby atoms are only used if they
are within the “outer” radius (R1 ) of the fitting region. Between R1 and the “inner” radius (R0 ), the amplitude of
interacting auxiliary functions is smoothed by a function that goes from zero to one and has continuous derivatives. To
optimize efficiency, the van der Waals radius of the atom is included in the cutoff so that smaller atoms are dropped
from the fitting radius sooner. The values of R0 and R1 are specified as REM variables as described below.
RI_J
Toggles the use of the RI algorithm to compute J.
TYPE:
LOGICAL
DEFAULT:
FALSE RI will not be used to compute J.
OPTIONS:
TRUE Turn on RI for J.
RECOMMENDATION:
For large (especially 1D and 2D) molecules the approximation may yield significant improvements in Fock evaluation time when used with ARI.
RI_K
Toggles the use of the RI algorithm to compute K.
TYPE:
LOGICAL
DEFAULT:
FALSE RI will not be used to compute K.
OPTIONS:
TRUE Turn on RI for K.
RECOMMENDATION:
For large (especially 1D and 2D) molecules the approximation may yield significant improvements in Fock evaluation time when used with ARI.
ARI
Toggles the use of the atomic resolution-of-the-identity (ARI) approximation.
TYPE:
LOGICAL
DEFAULT:
FALSE ARI will not be used by default for an RI-JK calculation.
OPTIONS:
TRUE Turn on ARI.
RECOMMENDATION:
For large (especially 1D and 2D) molecules the approximation may yield significant improvements in Fock evaluation time.
ARI_R0
Determines the value of the inner fitting radius (in Ångstroms)
TYPE:
INTEGER
DEFAULT:
4 A value of 4 Å will be added to the atomic van der Waals radius.
OPTIONS:
n User defined radius.
RECOMMENDATION:
For some systems the default value may be too small and the calculation will become unstable.

Chapter 4: Self-Consistent Field Ground-State Methods

102

ARI_R1
Determines the value of the outer fitting radius (in Ångstroms)
TYPE:
INTEGER
DEFAULT:
5 A value of 5 Å will be added to the atomic van der Waals radius.
OPTIONS:
n User defined radius.
RECOMMENDATION:
For some systems the default value may be too small and the calculation will become unstable.
This value also determines, in part, the smoothness of the potential energy surface.

4.6.7

PARI-K Fast Exchange Algorithm

PARI-K 36 is an algorithm that significantly accelerates the construction of the exchange matrix in Hartree-Fock and
hybrid density functional theory calculations with large basis sets. The speedup is made possible by fitting products of
atomic orbitals using only auxiliary basis functions found on their respective atoms. The PARI-K implementation in
Q-C HEM is an efficient MO-basis formulation similar to the AO-basis formulation of Merlot et al. 38 PARI-K is highly
recommended for calculations using basis sets of size augmented triple-zeta or larger, and should be used in conjunction
with the standard RI-J algorithm for constructing the Coulomb matrix. 67 The exchange fitting basis sets of Weigend 67
(cc-pVTZ-JK and cc-pVQZ-JK) are recommended for use in conjunction with PARI-K. The errors associated with the
PARI-K approximation appear to be only slightly worse than standard RI-HF. 38
PARI_K
Controls the use of the PARI-K approximation in the construction of the exchange matrix
TYPE:
LOGICAL
DEFAULT:
FALSE Do not use PARI-K.
OPTIONS:
TRUE Use PARI-K.
RECOMMENDATION:
Use for basis sets aug-cc-pVTZ and larger.

4.6.8

CASE Approximation

The Coulomb Attenuated Schrödinger Equation (CASE) approximation 6 follows from the KWIK algorithm 19 in which
the Coulomb operator is separated into two pieces using the error function, Eq. (5.12). Whereas in Section 5.6 this partition of the Coulomb operator was used to incorporate long-range Hartree-Fock exchange into DFT, within the CASE
approximation it is used to attenuate all occurrences of the Coulomb operator in Eq. (4.2), by neglecting the long-range
portion of the identity in Eq. (5.12). The parameter ω in Eq. (5.12) is used to tune the level of attenuation. Although
the total energies from Coulomb attenuated calculations are significantly different from non-attenuated energies, it is
found that relative energies, correlation energies and, in particular, wave functions, are not, provided a reasonable value
of ω is chosen.
By virtue of the exponential decay of the attenuated operator, ERIs can be neglected on a proximity basis yielding a
rigorous O(N ) algorithm for single point energies. CASE may also be applied in geometry optimizations and frequency
calculations.

103

Chapter 4: Self-Consistent Field Ground-State Methods
OMEGA
Controls the degree of attenuation of the Coulomb operator.
TYPE:
INTEGER
DEFAULT:
No default
OPTIONS:
n Corresponding to ω = n/1000, in units of bohr−1
RECOMMENDATION:
None
INTEGRAL_2E_OPR
Determines the two-electron operator.
TYPE:
INTEGER
DEFAULT:
-2 Coulomb Operator.
OPTIONS:
-1 Apply the CASE approximation.
-2 Coulomb Operator.
RECOMMENDATION:
Use the default unless the CASE operator is desired.

4.6.9

occ-RI-K Exchange Algorithm

The occupied orbital RI-K (occ-RI-K) algorithm 37 is a new scheme for building the exchange matrix (K) partially in
the MO basis using the RI approximation. occ-RI-K typically matches current alternatives in terms of both the accuracy
(energetics identical to standard RI-K) and convergence (essentially unchanged relative to conventional methods). On
the other hand, this algorithm exhibits significant speedups over conventional integral evaluation (14x) and standard
RI-K (3.3x) for a test system, a graphene fragment (C68 H22 ) using cc-pVQZ basis set (4400 basis functions), whereas
the speedup increases with the size of the AO basis set. Thus occ-RI-K helps to make larger basis set hybrid DFT
calculations more feasible, which is quite desirable for achieving improved accuracy in DFT calculations with modern
functionals.
The idea of the occ-RI-K formalism comes from a simple observation that the exchange energy EK and its gradient
can be evaluated from the diagonal elements of the exchange matrix in the occupied-occupied block Kii , and occupiedvirtual block Kia , respectively, rather than the full matrix in the AO representation, Kµν . Mathematically,
X
X
X
EK = −
Pµν Kµν = −
cµi Kµν cνi = −
Kii
(4.48)
µν

and

µν

i

∂EK
= 2Kai
∂∆ai

(4.49)

where ∆ is a skew-symmetric matrix used to parameterize the unitary transformation U , which represents the variations
of the MO coefficients as follows:
T
U = e(∆−∆ ) .
(4.50)
From Eq. 4.48 and 4.49 it is evident that the exchange energy and gradient need just Kiν rather than Kµν .
In regular RI-K one has to compute two quartic terms, 67 whereas there are three quartic terms for the occ-RI-K algorithm. The speedup of the latter with respect to former can be explained from the following ratio of operations; refer to
Ref. 37 for details.

104

Chapter 4: Self-Consistent Field Ground-State Methods

# of RI-K quartic operations
oN X 2 + oN 2 X
N (X + N )
=
= 2 2
# of occ-RI-K quartic operations
o X + o2 N X + o 2 N X
o(X + 2N )

(4.51)

With a conservative approximation of X ≈ 2N , the speedup is 43 (N/o). The occ-RI-K algorithm also involves some
cubic steps which should be negligible in the very large molecule limit. Tests in the Ref. 37 suggest that occ-RI-K for
small systems with large basis will gain less speed than a large system with small basis, because the cubic terms will
be more dominant for the former than the latter case.
In the course of SCF iteration, the occ-RI-K method does not require us to construct the exact Fock matrix explicitly.
Rather, kiν contributes to the Fock matrix in the mixed MO and AO representations (Fiν ) and yields orbital gradient and
DIIS error vectors for converging SCF. On the other hand, since occ-RI-K does not provide exactly the same unoccupied
eigenvalues, the diagonalization updates can differ from the conventional SCF procedure. In Ref. 37, occ-RI-K was
found to require, on average, the same number of SCF iterations to converge and to yield accurate energies.
OCC_RI_K
Controls the use of the occ-RI-K approximation for constructing the exchange matrix
TYPE:
LOGICAL
DEFAULT:
False Do not use occ-RI-K.
OPTIONS:
True Use occ-RI-K.
RECOMMENDATION:
Larger the system, better the performance

4.6.9.1

occ-RI-K for exchange energy gradient evaluation

A very attractive feature of occ-RI-K framework is that one can compute the exchange energy gradient with respect to
nuclear coordinates with the same leading quartic-scaling operations as the energy calculation.
The occ-RI-K formulation yields the following formula for the gradient of exchange energy in global Coulomb-metric
RI:
x
EK

=
=

(ij|ij)x
XX

P
cµi cνj Cij
(µν|P )x −

µνP ij

XX
RS

R S
Cij
Cij (R|S)x .

(4.52)

ij

The superscript x represents the derivative with respect to a nuclear coordinate. Note that the derivatives of the MO
coefficients cµi are not included here, because they are already included in the total energy derivative calculation by
Q-C HEM via the derivative of the overlap matrix.
P
In Eq. 4.52, the construction of the density fitting coefficients (Cµν
) has the worst scaling of O(M 4 ) because it involves
MO to AO back transformations:
X
P
P
Cµν
=
cµi cνj Cij
(4.53)
ij

where the operation cost is o2 N X + o[NB2]X.

Chapter 4: Self-Consistent Field Ground-State Methods
RI_K_GRAD
Turn on the nuclear gradient calculations
TYPE:
LOGICAL
DEFAULT:
FALSE Do not invoke occ-RI-K based gradient
OPTIONS:
TRUE Use occ-RI-K based gradient
RECOMMENDATION:
Use "RI_J false"

105

Chapter 4: Self-Consistent Field Ground-State Methods

4.6.10

106

Examples

Example 4.19 Q-C HEM input for a large single point energy calculation. The CFMM is switched on automatically
when LinK is requested.
$comment
HF/3-21G single point calculation on a large molecule
read in the molecular coordinates from file
$end
$molecule
read dna.inp
$end
$rem
METHOD
BASIS
LIN_K
$end

HF
3-21G
TRUE

Hartree-Fock
Basis set
Calculate K using LinK

Example 4.20 Q-C HEM input for a large single point energy calculation. This would be appropriate for a mediumsized molecule, but for truly large calculations, the CFMM and LinK algorithms are far more efficient.
$comment
HF/3-21G single point calculation on a large molecule
read in the molecular coordinates from file
$end
$molecule
read dna.inp
$end
$rem
METHOD
BASIS
INCFOCK
VARTHRESH
$end

HF
3-21G
5
3

Hartree-Fock
Basis set
Incremental Fock after 5 cycles
1.0d-03 variable threshold

Example 4.21 Q-C HEM input for a energy and gradient calculations with occ-RI-K method.
$molecule
read C30H62.inp
$end
$rem
JOBTYPE
EXCHANGE
BASIS
AUX_BASIS
OCC_RI_K
RI_K_GRAD
INCFOCK
PURECART
$end

force
HF
cc-pVTZ
cc-pVTZ-JK
1
1
0
1111

Chapter 4: Self-Consistent Field Ground-State Methods

4.7

107

Dual-Basis Self-Consistent Field Calculations

The dual-basis approximation 18,35,56,58–60 to self-consistent field (HF or DFT) energies provides an efficient means for
obtaining large basis set effects at vastly less cost than a full SCF calculation in a large basis set. First, a full SCF
calculation is performed in a chosen small basis (specified by BASIS2). Second, a single SCF-like step in the larger,
target basis (specified, as usual, by BASIS) is used to perturbatively approximate the large basis energy. This correction
amounts to a first-order approximation in the change in density matrix, after the single large-basis step:
Etotal = Esmall basis + tr[(∆P)F]large basis .

(4.54)

Here F (in the large basis) is built from the converged (small basis) density matrix. Thus, only a single Fock build is
required in the large basis set. Currently, HF and DFT energies (SP) as well as analytic first derivatives (FORCE or OPT)
are available.
Note: As of version 4.0, first derivatives of unrestricted dual-basis DFT energies—though correct—require a codeefficiency fix. We do not recommend use of these derivatives until this improvement has been made.
Across the G3 set 10,12,13 of 223 molecules, using cc-pVQZ, dual-basis errors for B3LYP are 0.04 kcal/mol (energy)
and 0.03 kcal/mol (atomization energy per bond) and are at least an order of magnitude less than using a smaller basis
set alone. These errors are obtained at roughly an order of magnitude savings in cost, relative to the full, target-basis
calculation.

4.7.1

Dual-Basis MP2

The dual-basis approximation can also be used for the reference energy of a correlated second-order Møller-Plesset
(MP2) calculation. 58,60 When activated, the dual-basis HF energy is first calculated as described above; subsequently,
the MO coefficients and orbital energies are used to calculate the correlation energy in the large basis. This technique
is particularly effective for RI-MP2 calculations (see Section 6.6), in which the cost of the underlying SCF calculation
often dominates.
Furthermore, efficient analytic gradients of the DB-RI-MP2 energy have been developed 18 and added to Q-C HEM.
These gradients allow for the optimization of molecular structures with RI-MP2 near the basis set limit. Typical
computational savings are on the order of 50% (aug-cc-pVDZ) to 71% (aug-cc-pVTZ). Resulting dual-basis errors are
only 0.001 Å in molecular structures and are, again, significantly less than use of a smaller basis set alone.

4.7.2

Dual-Basis Dynamics

The ability to compute SCF and MP2 energies and forces at reduced cost makes dual-basis calculations attractive for
ab initio molecular dynamics simulations, which are described in Section 10.7. Dual-basis BOMD has demonstrated 61
savings of 58%, even relative to state-of-the-art, Fock-extrapolated BOMD. Savings are further increased to 71% for
dual-basis RI-MP2 dynamics. Notably, these timings outperform estimates of extended Lagrangian (“Car-Parrinello”)
dynamics, without detrimental energy conservation artifacts that are sometimes observed in the latter. 29
Two algorithm improvements make modest but worthwhile improvements to dual-basis dynamics. First, the iterative,
small-basis calculation can benefit from Fock matrix extrapolation. 29 Second, extrapolation of the response equations
(“Z-vector” equations) for nuclear forces further increases efficiency. 57 (See Section 10.7.) Q-C HEM automatically
adjusts to extrapolate in the proper basis set when DUAL_BASIS_ENERGY is activated.

4.7.3

Basis-Set Pairings

We recommend using basis pairings in which the small basis set is a proper subset of the target basis (6-31G into
6-31G*, for example). They not only produce more accurate results; they also lead to more efficient integral screening
in both energies and gradients. Subsets for many standard basis sets (including Dunning-style cc-pVXZ basis sets and

108

Chapter 4: Self-Consistent Field Ground-State Methods

their augmented analogs) have been developed and thoroughly tested for these purposes. A summary of the pairings is
provided in Table 4.7.3; details of these truncations are provided in Figure 4.1.
A new pairing for 6-31G*-type calculations is also available. The 6-4G subset (named r64G in Q-C HEM) is a subset
by primitive functions and provides a smaller, faster alternative for this basis set regime. 56 A case-dependent switch in
the projection code (still OVPROJECTION) properly handles 6-4G. For DB-HF, the calculations proceed as described
above. For DB-DFT, empirical scaling factors (see Ref. 56 for details) are applied to the dual-basis correction. This
scaling is handled automatically by the code and prints accordingly.
As of Q-C HEM version 3.2, the basis set projection code has also been adapted to properly account for linear dependence, 60 which can often be problematic for large, augmented (aug-cc-pVTZ, etc.) basis set calculations. The same
standard keyword (LIN_DEP_THRESH) is used to determine linear dependence in the projection code. Because of the
scheme used to account for linear dependence, only proper-subset pairings are now allowed.
Like single-basis calculations, user-specified general or mixed basis sets may be employed (see Chapter 8) with dualbasis calculations. The target basis specification occurs in the standard $basis section. The smaller, secondary basis
is placed in a similar $basis2 section; the syntax within this section is the same as the syntax for $basis. General and
mixed small basis sets are activated by BASIS2 = BASIS2_GEN and BASIS2 = BASIS2_MIXED, respectively.
BASIS

BASIS2

cc-pVTZ
cc-pVQZ
aug-cc-pVDZ
aug-cc-pVTZ
aug-cc-pVQZ
6-31G*
6-31G**
6-31++G**
6-311++G(3df,3pd)

rcc-pVTZ
rcc-pVQZ
racc-pVDZ
racc-pVTZ
racc-pVQZ
r64G, 6-31G
r64G, 6-31G
6-31G*
6-311G*, 6-311+G*

Table 4.2: Summary and nomenclature of recommended dual-basis pairings

4.7.4

Job Control

Dual-basis calculations are controlled with the following $rem. DUAL_BASIS_ENERGY turns on the dual-basis approximation. Note that use of BASIS2 without DUAL_BASIS_ENERGY only uses basis set projection to generate the initial
guess and does not invoke the dual-basis approximation (see Section 4.4.5). OVPROJECTION is used as the default
projection mechanism for dual-basis calculations; it is not recommended that this be changed. Specification of SCF
variables (e.g., THRESH) will apply to calculations in both basis sets.
DUAL_BASIS_ENERGY
Activates dual-basis SCF (HF or DFT) energy correction.
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
Analytic first derivative available for HF and DFT (see JOBTYPE)
Can be used in conjunction with MP2 or RI-MP2
See BASIS, BASIS2, BASISPROJTYPE
RECOMMENDATION:
Use dual-basis to capture large-basis effects at smaller basis cost. Particularly useful with RIMP2, in which HF often dominates. Use only proper subsets for small-basis calculation.

109

Chapter 4: Self-Consistent Field Ground-State Methods

s
s

s
p

H-He:

p

s

d

−

Li-Ne:

−

s

p

s

s

p

s
p
d

s

p

f

s

s

d

s

p

rcc-pVTZ

s

cc-pVTZ

rcc-pVTZ

s
s

s
−
d

H-He:

f

s

−

Li-Ne:

g

s

d

s
p
s

cc-pVQZ

rcc-pVQZ

s
s

p

p

Li-Ne:

s

s

p
d

s

p

−

p

d

racc-pVDZ

p0
s0

aug-cc-pVDZ

racc-pVDZ

s
s
p
s

H-He:

−

s

p

s

s

s

racc-pVTZ

p

aug-cc-pVTZ

s
p

s

racc-pVTZ

p

H-He:

s

d

−

s

d

s

g

d0

0

racc-pVQZ

aug-cc-pVQZ

−
−
−

d
−

p
−

s
p0
s

s

aug-cc-pVQZ

s

f0

p0

s0

−
d

p
g0

d

s

s

f

p

d
p

f

p

−

s

p0
s0

−

p

s

Li-Ne:

−

s

d0

p
d

p
−

p

f0

p
s

s
−

s
f

s
p

p
d

−
0

s0

s
s

−

p
s

0

s0

aug-cc-pVTZ

−
d

f0
d0

p

d
p

d

s

0

s
f

p

−

0

s

−

s

p0

p
d

p

Li-Ne:

p
d0

s

s

p
d

s
p

s

−

s

p0
s0

aug-cc-pVDZ

d
p

0

s

s0

s0

s
p

s

s

H-He:

d

p

rcc-pVQZ

−
−

p

s

cc-pVQZ

−
d

f

s

s

d
p

d
p

−

p
s

s
f

s

−

s

d
p

p

d

p

s
−

s

p

s
p

p
s

−
d

p

s

cc-pVTZ

d
p

0

racc-pVQZ

Figure 4.1: Structure of the truncated basis set pairings for cc-pV(T,Q)Z and aug-cc-pV(D,T,Q)Z. The most compact
functions are listed at the top. Primed functions depict diffuse function augmentation. Dashes indicate eliminated
functions, relative to the paired standard basis set. In each case, the truncations for hydrogen and heavy atoms are
shown, along with the nomenclature used in Q-C HEM.

Chapter 4: Self-Consistent Field Ground-State Methods

4.7.5

Examples

Example 4.22 Input for a dual-basis B3LYP single-point calculation.
$molecule
0 1
H
H
1
$end

0.75

$rem
JOBTYPE
METHOD
BASIS
BASIS2
DUAL_BASIS_ENERGY
$end

sp
b3lyp
6-311++G(3df,3pd)
6-311G*
true

Example 4.23 Input for a dual-basis B3LYP single-point calculation with a minimal 6-4G small basis.
$molecule
0 1
H
H
1
$end

0.75

$rem
JOBTYPE
METHOD
BASIS
BASIS2
DUAL_BASIS_ENERGY
$end

sp
b3lyp
6-31G*
r64G
true

Example 4.24 Input for a dual-basis RI-MP2 geometry optimization.
$molecule
0 1
H
H
1
$end

0.75

$rem
JOBTYPE
METHOD
AUX_BASIS
BASIS
BASIS2
DUAL_BASIS_ENERGY
$end

opt
rimp2
rimp2-aug-cc-pVDZ
aug-cc-pVDZ
racc-pVDZ
true

110

Chapter 4: Self-Consistent Field Ground-State Methods

Example 4.25 Input for a dual-basis RI-MP2 single-point calculation with mixed basis sets.
$molecule
0 1
H
O
1
H
2
$end

1.1
1.1

1

104.5

$rem
JOBTYPE
METHOD
AUX_BASIS
BASIS
BASIS2
DUAL_BASIS_ENERGY
$end
$basis
H 1
cc-pVTZ
****
O 2
aug-cc-pVTZ
****
H 3
cc-pVTZ
****
$end
$basis2
H 1
rcc-pVTZ
****
O 2
racc-pVTZ
****
H 3
rcc-pVTZ
****
$end
$aux_basis
H 1
rimp2-cc-pVTZ
****
O 2
rimp2-aug-cc-pVTZ
****
H 3
rimp2-cc-pVTZ
****
$end

opt
rimp2
aux_mixed
mixed
basis2_mixed
true

111

Chapter 4: Self-Consistent Field Ground-State Methods

4.8
4.8.1

112

Hartree-Fock and Density-Functional Perturbative Corrections
Theory

Closely related to the dual-basis approach of Section 4.7, but somewhat more general, is the Hartree-Fock perturbative
correction (HFPC) developed by Deng et al.. 15,16 An HFPC calculation consists of an iterative HF calculation in a
small primary basis followed by a single Fock matrix formation, diagonalization, and energy evaluation in a larger,
secondary basis. In the following, we denote a conventional HF calculation by HF/basis, and a HFPC calculation by
HFPC/primary/secondary. Using a primary basis of n functions, the restricted HF matrix elements for a 2m-electron
system are


n
X
1
Fµν = hµν +
Pλσ (µν|λσ) − (µλ|νσ)
(4.55)
2
λσ

Solving the Roothaan-Hall equation in the primary basis results in molecular orbitals and an associated density matrix,
P. In an HFPC calculation, P is subsequently used to build a new Fock matrix, F[1] , in a larger secondary basis of N
functions


n
X
1
[1]
Fab = hab +
Pλσ (ab|λσ) − (aλ|bσ)
(4.56)
2
λσ

where λ, σ indicate primary basis functions and a, b represent secondary basis functions. Diagonalization of F[1]
affords improved molecular orbitals and an associated density matrix P[1] . The HFPC energy is given by
E HFPC =

N
X
ab

[1]

Pab hab +

N

1 X [1] [1] 
Pab Pcd 2(ab|cd) − (ac|bd)
2

(4.57)

abcd

where a, b, c and d represent secondary basis functions. This differs from the DBHF energy evaluation where PP[1] ,
rather than P[1] P[1] , is used. The inclusion of contributions that are quadratic in PP[1] is the key reason for the fact
that HFPC is more accurate than DBHF.
Unlike dual-basis HF, HFPC does not require that the small basis be a proper subset of the large basis, and is therefore
able to jump between any two basis sets. Benchmark study of HFPC on a large and diverse data set of total and reaction
energies demonstrate that, for a range of primary/secondary basis set combinations, the HFPC scheme can reduce the
error of the primary calculation by around two orders of magnitude at a cost of about one third that of the full secondary
calculation. 15,16
A density-functional version of HFPC (“DFPC”) 17 seeks to combine the low cost of pure DFT calculations using small
bases and grids, with the high accuracy of hybrid calculations using large bases and grids. The DFPC approach is motivated by the dual-functional method of Nakajima and Hirao 39 and the dual-grid scheme of Tozer et al. 65 Combining
these features affords a triple perturbation: to the functional, to the grid, and to the basis set. We call this approach
density-functional “triple jumping”.

4.8.2

Job Control

HFPC/DFPC calculations are controlled with the following $rem. HFPT turns on the HFPC/DFPC approximation. Note
that HFPT_BASIS specifies the secondary basis set.

Chapter 4: Self-Consistent Field Ground-State Methods
HFPT
Activates HFPC/DFPC calculation.
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
Single-point energy only
RECOMMENDATION:
Use Dual-Basis to capture large-basis effects at smaller basis cost. See reference for recommended basis set, functional, and grid pairings.
HFPT_BASIS
Specifies the secondary basis in a HFPC/DFPC calculation.
TYPE:
STRING
DEFAULT:
None
OPTIONS:
None
RECOMMENDATION:
See reference for recommended basis set, functional, and grid pairings.
DFPT_XC_GRID
Specifies the secondary grid in a HFPC/DFPC calculation.
TYPE:
STRING
DEFAULT:
None
OPTIONS:
None
RECOMMENDATION:
See reference for recommended basis set, functional, and grid pairings.
DFPT_EXCHANGE
Specifies the secondary functional in a HFPC/DFPC calculation.
TYPE:
STRING
DEFAULT:
None
OPTIONS:
None
RECOMMENDATION:
See reference for recommended basis set, functional, and grid pairings.

113

114

Chapter 4: Self-Consistent Field Ground-State Methods

4.8.3

Examples

Example 4.26 Input for a HFPC single-point calculation.
$molecule
0 1
H
H
1
$end

0.75

$rem
JOBTYPE
EXCHANGE
BASIS
HFPT_BASIS
PURECART
HFPT
$end

sp
hf
cc-pVDZ
cc-pVQZ
1111
true

! primary basis
! secondary basis
! set to purecart of the target basis

Example 4.27 Input for a DFPC single-point calculation.
$molecule
0 1
H
H
1
$end

0.75

$rem
JOBTYPE
METHOD
DFPT_EXCHANGE
DFPT_XC_GRID
XC_GRID
HFPT_BASIS
BASIS
PURECART
HFPT
$end

4.9
4.9.1

sp
blyp
b3lyp
00075000302
0
6-311++G(3df,3pd)
6-311G*
1111
true

!
!
!
!
!
!

primary functional
secondary functional
secondary grid
primary grid
secondary basis
primary basis

Unconventional SCF Calculations
Polarized Atomic Orbital (PAO) Calculations

Polarized atomic orbital (PAO) calculations are an interesting unconventional SCF method, in which the molecular
orbitals and the density matrix are not expanded directly in terms of the basis of atomic orbitals. Instead, an intermediate
molecule-optimized minimal basis of polarized atomic orbitals (PAOs) is used. 33 The polarized atomic orbitals are
defined by an atom-blocked linear transformation from the fixed atomic orbital basis, where the coefficients of the
transformation are optimized to minimize the energy, at the same time as the density matrix is obtained in the PAO
representation. Thus a PAO-SCF calculation is a constrained variational method, whose energy is above that of a
full SCF calculation in the same basis. However, a molecule optimized minimal basis is a very compact and useful
representation for purposes of chemical analysis, and it also has potential computational advantages in the context of
MP2 or local MP2 calculations, as can be done after a PAO-HF calculation is complete to obtain the PAO-MP2 energy.
PAO-SCF calculations tend to systematically underestimate binding energies (since by definition the exact result is
obtained for atoms, but not for molecules). In tests on the G2 database, PAO-B3LYP/6-311+G(2df,p) atomization
energies deviated from full B3LYP/6-311+G(2df,p) atomization energies by roughly 20 kcal/mol, with the error being
essentially extensive with the number of bonds. This deviation can be reduced to only 0.5 kcal/mol with the use of

Chapter 4: Self-Consistent Field Ground-State Methods

115

a simple non-iterative second order correction for “beyond-minimal basis” effects. 34 The second order correction is
evaluated at the end of each PAO-SCF calculation, as it involves negligible computational cost. Analytical gradients
are available using PAOs, to permit structure optimization. For additional discussion of the PAO-SCF method and its
uses, see the references cited above.
Calculations with PAOs are determined controlled by the following $rem variables. PAO_METHOD = PAO invokes
PAO-SCF calculations, while the algorithm used to iterate the PAOs can be controlled with PAO_ALGORITHM.
PAO_ALGORITHM
Algorithm used to optimize polarized atomic orbitals (see PAO_METHOD)
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
0 Use efficient (and riskier) strategy to converge PAOs.
1 Use conservative (and slower) strategy to converge PAOs.
RECOMMENDATION:
None
PAO_METHOD
Controls evaluation of polarized atomic orbitals (PAOs).
TYPE:
STRING
DEFAULT:
EPAO For local MP2 calculations Otherwise no default.
OPTIONS:
PAO
Perform PAO-SCF instead of conventional SCF.
EPAO Obtain EPAOs after a conventional SCF.
RECOMMENDATION:
None

4.9.2

SCF Meta-Dynamics

As the SCF equations are non-linear in the electron density, there are in theory very many solutions, i.e., sets of orbitals
where the energy is stationary with respect to changes in the orbital subset. Most often sought is the solution with
globally minimal energy as this is a variational upper bound to the true eigenfunction in this basis. The SCF methods
available in Q-C HEM allow the user to converge upon an SCF solution, and (using STABILITY_ANALYSIS) ensure it is
a minimum, but there is no known method of ensuring that the found solution is a global minimum; indeed in systems
with many low-lying energy levels the solution converged upon may vary considerably with initial guess.
SCF meta-dynamics 64 is a technique which can be used to locate multiple SCF solutions, and thus gain some confidence
that the calculation has converged upon the global minimum. It works by searching out a solution to the SCF equations.
Once found, the solution is stored, and a biasing potential added so as to avoid re-converging to the same solution. More
formally, the distance between two solutions, w and x, can be expressed as d2wx = hwΨ|wρ̂ − xρ̂|wΨi, where wΨ is a
Slater determinant formed from the orthonormal orbitals, wφi , of solution w, and wρ̂ is the one-particle density operator
for wΨ. This definition is equivalent to d2wx = N − wP µν Sνσ · xP στ Sτ µ . and is easily calculated. The function d2wx is
between zero and the number of electrons, and can be taken as the distance between two solutions. As an example, any
singly-excited determinant (which will not in general be another SCF solution) is a distance 1 away from the reference
(unexcited) determinant.
In a manner analogous to classical meta-dynamics, to bias against the set of previously located solutions, x, we create

116

Chapter 4: Self-Consistent Field Ground-State Methods
a new Lagrangian,
Ẽ = E +

X

2

Nx e−λx d0x

(4.58)

x

where 0 represents the present density. From this we may derive a new effective Fock matrix,
X
2
x
F̃µν = Fµν +
P µν Nx λx e−λx d0x

(4.59)

x

This may be used with very little modification within a standard DIIS procedure to locate multiple solutions. When
close to a new solution, the biasing potential is removed so the location of that solution is not affected by it. If the
calculation ends up re-converging to the same solution, Nx and λx can be modified to avert this. Once a solution is
found it is added to the list of solutions, and the orbitals mixed to provide a new guess for locating a different solution.
This process can be customized by the REM variables below. Both DIIS and GDM methods can be used, but
it is advisable to turn on MOM when using DIIS to maintain the orbital ordering. Post-HF correlation methods
can also be applied. By default they will operate for the last solution located, but this can be changed with the
SCF_MINFIND_RUNCORR variable.
The solutions found through meta-dynamics also appear to be good approximations to diabatic surfaces where the
electronic structure does not significantly change with geometry. In situations where there are such multiple electronic
states close in energy, an adiabatic state may be produced by diagonalizing a matrix of these states, i.e., through a
configuration interaction (CI) procedure. As they are distinct solutions of the SCF equations, these states are nonorthogonal (one cannot be constructed as a single determinant made out of the orbitals of another), and so the CI is a
little more complicated and is a non-orthogonal CI (NOCI). More information on NOCI can be found in Section 7.2.7.
SCF_SAVEMINIMA
Turn on SCF meta-dynamics and specify how many solutions to locate.
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
0 Do not use SCF meta-dynamics
n Attempt to find n distinct SCF solutions.
RECOMMENDATION:
Perform SCF Orbital meta-dynamics and attempt to locate n different SCF solutions. Note that
these may not all be minima. Many saddle points are often located. The last one located will be
the one used in any post-SCF treatments. In systems where there are infinite point groups, this
procedure cannot currently distinguish between spatial rotations of different densities, so will
likely converge on these multiply.

Chapter 4: Self-Consistent Field Ground-State Methods
SCF_READMINIMA
Read in solutions from a previous SCF meta-dynamics calculation
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
n
Read in n previous solutions and attempt to locate them all.
−n Read in n previous solutions, but only attempt to locate solution n.
RECOMMENDATION:
This may not actually locate all solutions required and will probably locate others too. The
SCF will also stop when the number of solutions specified in SCF_SAVEMINIMA are found.
Solutions from other geometries may also be read in and used as starting orbitals. If a solution
is found and matches one that is read in within SCF_MINFIND_READDISTTHRESH, its orbitals
are saved in that position for any future calculations. The algorithm works by restarting from the
orbitals and density of a the minimum it is attempting to find. After 10 failed restarts (defined by
SCF_MINFIND_RESTARTSTEPS), it moves to another previous minimum and attempts to locate
that instead. If there are no minima to find, the restart does random mixing (with 10 times the
normal random mixing parameter).
SCF_MINFIND_WELLTHRESH
Specify what SCF_MINFIND believes is the basin of a solution
TYPE:
INTEGER
DEFAULT:
5
OPTIONS:
n for a threshold of 10−n
RECOMMENDATION:
When the DIIS error is less than 10−n , penalties are switched off to see whether it has converged
to a new solution.
SCF_MINFIND_RESTARTSTEPS
Restart with new orbitals if no minima have been found within this many steps
TYPE:
INTEGER
DEFAULT:
300
OPTIONS:
n Restart after n steps.
RECOMMENDATION:
If the SCF calculation spends many steps not finding a solution, lowering this number may speed
up solution-finding. If the system converges to solutions very slowly, then this number may need
to be raised.

117

Chapter 4: Self-Consistent Field Ground-State Methods
SCF_MINFIND_INCREASEFACTOR
Controls how the height of the penalty function changes when repeatedly trapped at the same
solution
TYPE:
INTEGER
DEFAULT:
10100 meaning 1.01
OPTIONS:
abcde corresponding to a.bcde
RECOMMENDATION:
If the algorithm converges to a solution which corresponds to a previously located solution,
increase both the normalization N and the width lambda of the penalty function there. Then do a
restart.
SCF_MINFIND_INITLAMBDA
Control the initial width of the penalty function.
TYPE:
INTEGER
DEFAULT:
02000 meaning 2.000
OPTIONS:
abcde corresponding to ab.cde
RECOMMENDATION:
The initial inverse-width (i.e., the inverse-variance) of the Gaussian to place to fill solution’s well.
Measured in electrons( − 1). Increasing this will repeatedly converging on the same solution.
SCF_MINFIND_INITNORM
Control the initial height of the penalty function.
TYPE:
INTEGER
DEFAULT:
01000 meaning 1.000
OPTIONS:
abcde corresponding to ab.cde
RECOMMENDATION:
The initial normalization of the Gaussian to place to fill a well. Measured in hartrees.
SCF_MINFIND_RANDOMMIXING
Control how to choose new orbitals after locating a solution
TYPE:
INTEGER
DEFAULT:
00200 meaning .02 radians
OPTIONS:
abcde corresponding to a.bcde radians
RECOMMENDATION:
After locating an SCF solution, the orbitals are mixed randomly to move to a new position in
orbital space. For each occupied and virtual orbital pair picked at random and rotate between
them by a random angle between 0 and this. If this is negative then use exactly this number, e.g.,
−15708 will almost exactly swap orbitals. Any number< −15708 will cause the orbitals to be
swapped exactly.

118

Chapter 4: Self-Consistent Field Ground-State Methods
SCF_MINFIND_NRANDOMMIXES
Control how many random mixes to do to generate new orbitals
TYPE:
INTEGER
DEFAULT:
10
OPTIONS:
n Perform n random mixes.
RECOMMENDATION:
This is the number of occupied/virtual pairs to attempt to mix, per separate density (i.e., for
unrestricted calculations both alpha and beta space will get this many rotations). If this is negative
then only mix the highest 25% occupied and lowest 25% virtuals.
SCF_MINFIND_READDISTTHRESH
The distance threshold at which to consider two solutions the same
TYPE:
INTEGER
DEFAULT:
00100 meaning 0.1
OPTIONS:
abcde corresponding to ab.cde
RECOMMENDATION:
The threshold to regard a minimum as the same as a read in minimum. Measured in electrons. If
two minima are closer together than this, reduce the threshold to distinguish them.
SCF_MINFIND_MIXMETHOD
Specify how to select orbitals for random mixing
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
0 Random mixing: select from any orbital to any orbital.
1 Active mixing: select based on energy, decaying with distance from the Fermi level.
2 Active Alpha space mixing: select based on energy, decaying with distance from the
Fermi level only in the alpha space.
RECOMMENDATION:
Random mixing will often find very high energy solutions. If lower energy solutions are desired,
use 1 or 2.
SCF_MINFIND_MIXENERGY
Specify the active energy range when doing Active mixing
TYPE:
INTEGER
DEFAULT:
00200 meaning 00.200
OPTIONS:
abcde corresponding to ab.cde
RECOMMENDATION:
The standard deviation of the Gaussian distribution used to select the orbitals for mixing (centered on the Fermi level). Measured in Hartree. To find less-excited solutions, decrease this
value

119

Chapter 4: Self-Consistent Field Ground-State Methods

120

SCF_MINFIND_RUNCORR
Run post-SCF correlated methods on multiple SCF solutions
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
If this is set > 0, then run correlation methods for all found SCF solutions.
RECOMMENDATION:
Post-HF correlation methods should function correctly with excited SCF solutions, but their
convergence is often much more difficult owing to intruder states.

4.10

Ground State Method Summary

To summarize the main features of Q-C HEM’s ground state self-consistent field capabilities, the user needs to consider:
• Input a molecular geometry ($molecule keyword)
– Cartesian
– Z-matrix
– Read from prior calculations
• Declare the job specification ($rem keyword)
– JOBTYPE
*
*
*
*

Single point
Optimization
Frequency
See Table 4.1 for further options

– BASIS
* Refer to Chapter 8 (note: $basis keyword for user defined basis sets)
* Effective core potentials (if desired); refer to Chapter 9
– METHOD
* Single method specification for exchange and correlation. Alternatively these can be specified separately.
– EXCHANGE
* Linear scaling algorithms for all methods
* Arsenal of exchange density functionals
* User definable functionals and hybrids
– CORRELATION
*
*
*
*
*

DFT or wave function-based methods
Linear scaling (CPU and memory) incorporation of correlation with DFT
Arsenal of correlation density functionals
User definable functionals and hybrids
See Chapter 6 for wave function-based correlation methods.

• Exploit Q-C HEM’s special features
– CFMM, LinK large molecule options
– SCF rate of convergence increased through improved guesses and alternative minimization algorithms
– Explore novel methods if desired: CASE approximation, PAOs.

121

Chapter 4: Self-Consistent Field Ground-State Methods

References and Further Reading
[1] AOINTS (Appendix B).
[2] Molecular Properties Analysis (Chapter 11).
[3] Basis Sets (Chapter 8) and Effective Core Potentials (Chapter 9).
[4] Molecular Geometry and Critical Points (Chapter 10).
[5] T. R. Adams, R. D. Adamson, and P. M. W. Gill. J. Chem. Phys., 107:124, 1997. DOI: 10.1063/1.474359.
[6] R. D. Adamson, J. P. Dombroski, and P. M. W. Gill. Chem. Phys. Lett., 254:329, 1996. DOI: 10.1016/00092614(96)00280-1.
[7] M. Born and J. R. Oppenheimer. Ann. Phys., 84:457, 1927. DOI: 10.1002/andp.19273892002.
[8] E. Cancès. J. Chem. Phys., 114:10616, 2001. DOI: 10.1063/1.1373430.
[9] E. Cancès and C. Le Bris. Int. J. Quantum Chem., 79:82, 2000. DOI: 10.1002/1097-461X(2000)79:2<82::AIDQUA3>3.0.CO;2-I.
[10] L. A. Curtiss, K. Raghavachari, G. W. Trucks, and J. A. Pople.
10.1063/1.460205.

J. Chem. Phys., 94:7221, 1991.

DOI:

[11] L. A. Curtiss, K. Raghavachari, P. C. Redfern, and J. A. Pople. J. Chem. Phys., 106:1063, 1997. DOI:
10.1063/1.473182.
[12] L. A. Curtiss, K. Raghavachari, P. C. Redfern, V. Rassolov, and J. A. Pople. J. Chem. Phys., 109:7764, 1998.
DOI: 10.1063/1.477422.
[13] L. A. Curtiss, K. Raghavachari, P. C. Redfern, and J. A. Pople. J. Chem. Phys., 112:7374, 2000. DOI:
10.1063/1.481336.
[14] E. R. Davidson. 17:87, 1975. DOI: 10.1016/0021-9991(75)90065-0.
[15] J. Deng, A. T. B. Gilbert, and P. M. W. Gill. J. Chem. Phys., 130:231101, 2009. DOI: 10.1063/1.3152864.
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Chapter 5

Density Functional Theory
5.1

Introduction

DFT 106,112,148,253 has emerged as an accurate, alternative first-principles approach to quantum mechanical molecular
investigations. DFT calculations account for the overwhelming majority of all quantum chemistry calculations, not
only because of its proven chemical accuracy, but also because of its relatively low computational expense, comparable
to Hartree-Fock theory but with treatment of electron correlation that is neglected in a HF calculation. These two
features suggest that DFT is likely to remain a leading method in the quantum chemist’s toolkit well into the future.
Q-C HEM contains fast, efficient and accurate algorithms for all popular density functionals, making calculations on
large molecules possible and practical.
DFT is primarily a theory of electronic ground state structures based on the electron density, ρ(r), as opposed to
the many-electron wave function, Ψ(r1 , . . . , rN ). (Its excited-state extension, time-dependent DFT, is discussed in
Section 7.3.) There are a number of distinct similarities and differences between traditional wave function approaches
and modern DFT methodologies. First, the essential building blocks of the many-electron wave function Ψ are singleelectron orbitals, which are directly analogous to the Kohn-Sham orbitals in the DFT framework. Second, both the
electron density and the many-electron wave function tend to be constructed via a SCF approach that requires the
construction of matrix elements that are conveniently very similar.
However, traditional ab initio approaches using the many-electron wave function as a foundation must resort to a postSCF calculation (Chapter 6) to incorporate correlation effects, whereas DFT approaches incorporate correlation at the
SCF level. Post-SCF methods, such as perturbation theory or coupled-cluster theory are extremely expensive relative
to the SCF procedure. On the other hand, while the DFT approach is exact in principle, in practice it relies on modeling
an unknown exchange-correlation energy functional. While more accurate forms of such functionals are constantly
being developed, there is no systematic way to improve the functional to achieve an arbitrary level of accuracy. Thus,
the traditional approaches offer the possibility of achieving a systematically-improvable level of accuracy, but can be
computationally demanding, whereas DFT approaches offer a practical route, but the theory is currently incomplete.

5.2

Kohn-Sham Density Functional Theory

The density functional theory by Hohenberg, Kohn, and Sham 90,105 stems from earlier work by Dirac, 62 who showed
that the exchange energy of a uniform electron gas can be computed exactly from the charge density along. However,
while this traditional density functional approach, nowadays called “orbital-free” DFT, makes a direct connection to
the density alone, in practice it is constitutes a direct approach where the necessary equations contain only the electron
density, difficult to obtain decent approximations for the kinetic energy functional. Kohn and Sham sidestepped this
difficulty via an indirect approach in which the kinetic energy is computed exactly for a noninteracting reference

125

Chapter 5: Density Functional Theory

system, namely, the Kohn-Sham determinant. 105 It is the Kohn-Sham approach that first made DFT into a practical tool
for calculations.
Within the Kohn-Sham formalism, 105 the ground state electronic energy, E, can be written as
E = ET + EV + EJ + EXC

(5.1)

where ET is the kinetic energy, EV is the electron–nuclear interaction energy, EJ is the Coulomb self-interaction of the
electron density, ρ(r) and EXC is the exchange-correlation energy. Adopting an unrestricted format, the α and β total
electron densities can be written as
ρα (r) =

nα
X

|ψiα |2

i=1

ρβ (r) =

nβ
X

(5.2)
|ψiβ |2

i=1

where nα and nβ are the number of alpha and beta electron respectively, and ψi are the Kohn-Sham orbitals. Thus, the
total electron density is
ρ(r) = ρα (r) + ρβ (r)
(5.3)
Within a finite basis set, the density is represented by 170
X
ρ(r) =
Pµν φµ (r)φν (r) ,

(5.4)

µν

where the Pµν are the elements of the one-electron density matrix; see Eq. (4.23) in the discussion of Hartree-Fock
theory. The various energy components in Eq. (5.1) can now be written
ET

=
=

EV

=
=

EJ

=
=

EXC

=

 X

nβ 
1 ˆ2 β
1 ˆ2 α
ψi +
ψiβ − ∇
ψi
ψiα − ∇
2
2
i=1
i=1


X
1 ˆ2
φν (r)
Pµν φµ (r) − ∇
2
µν
nα 
X

M
X

Z

ρ(r)
dr
|r − RA |
A=1

X
X
ZA
−
φν (r)
Pµν
φµ (r)
|r − RA |
µν
A


1
1
ρ(r2 )
ρ(r1 )
2
|r1 − r2 |
1 XX
Pµν Pλσ (µν|λσ)
2 µν
λσ
Z


ˆ
f ρ(r), ∇ρ(r),
. . . ρ(r) dr .
−

(5.5)

ZA

(5.6)

(5.7)
(5.8)

Minimizing E with respect to the unknown Kohn-Sham orbital coefficients yields a set of matrix equations exactly analogous to Pople-Nesbet equations of the UHF case, Eq. (4.13), but with modified Fock matrix elements [cf. Eq. (4.26)]
α
core
XCα
Fµν
= Hµν
+ Jµν − Fµν
β
core
XCβ
Fµν
= Hµν
+ Jµν − Fµν
.

(5.9)

Here, FXCα and FXCβ are the exchange-correlation parts of the Fock matrices and depend on the exchange-correlation
XCα
α
functional used. UHF theory is recovered as a special case simply by taking Fµν
= Kµν
, and similarly for β. Thus,
the density and energy are obtained in a manner analogous to that for the HF method. Initial guesses are made for the
MO coefficients and an iterative process is applied until self-consistency is achieved.

126

Chapter 5: Density Functional Theory

5.3

Overview of Available Functionals

Q-C HEM currently has more than 30 exchange functionals as well as more than 30 correlation functionals, and in
addition over 150 exchange-correlation (XC) functionals, which refer to functionals that are not separated into exchange
and correlation parts, either because the way in which they were parameterized renders such a separation meaningless
(e.g., B97-D 75 or ωB97X 44 ) or because they are a standard linear combination of exchange and correlation (e.g.,
PBE 155 or B3LYP 20,190 ). User-defined XC functionals can be created as specified linear combinations of any of the
30+ exchange functionals and/or the 30+ correlation functionals.
KS-DFT functionals can be organized onto a ladder with five rungs, in a classification scheme (“Jacob’s Ladder”)
proposed by John Perdew 157 in 2001. The first rung contains a functional that only depends on the (spin-)density ρσ ,
namely, the local spin-density approximation (LSDA). These functionals are exact for the infinite uniform electron gas
(UEG), but are highly inaccurate for molecular properties whose densities exhibit significant inhomogeneity. To improve upon the weaknesses of the LSDA, it is necessary to introduce an ingredient that can account for inhomogeneities
in the density: the density gradient, ∇ρσ . These generalized gradient approximation (GGA) functionals define the second rung of Jacob’s Ladder and tend to improve significantly upon the LSDA. Two additional ingredients that can be
used to further improve the performance of GGA functionals are either the Laplacian of the density ∇2 ρσ , and/or the
kinetic energy density,
nσ
X
2
|∇ψi,σ | .
τσ =
(5.10)
i

While functionals that employ both of these options are available in Q-C HEM, the kinetic energy density is by far the
more popular ingredient and has been used in many modern functionals to add flexibility to the functional form with
respect to both constraint satisfaction (non-empirical functionals) and least-squares fitting (semi-empirical parameterization). Functionals that depend on either of these two ingredients belong to the third rung of the Jacob’s Ladder
and are called meta-GGAs. These meta-GGAs often further improve upon GGAs in areas such as thermochemistry,
kinetics (reaction barrier heights), and even non-covalent interactions.
Functionals on the fourth rung of Jacob’s Ladder are called hybrid density functionals. This rung contains arguably
the most popular density functional of our time, B3LYP, the first functional to see widespread application in chemistry.
“Global” hybrid (GH) functionals such as B3LYP (as distinguished from the “range-separated hybrids" introduced
below) add a constant fraction of “exact” (Hartree-Fock) exchange to any of the functionals from the first three rungs.
Thus, hybrid LSDA, hybrid GGA, and hybrid meta-GGA functionals can be constructed, although the latter two types
are much more common. As an example, the formula for the B3LYP functional, as implemented in Q-C HEM, is
B3LYP
Exc
= cx ExHF + (1 − cx − ax ) ExSlater + ax ExB88 + (1 − ac ) EcVWN1RPA + ac EcLYP

(5.11)

where cx = 0.20, ax = 0.72, and ac = 0.81.
A more recent approach to introducing exact exchange into the functional form is via range separation. Range-separated
hybrid (RSH) functionals split the exact exchange contribution into a short-range (SR) component and a long-range
(LR) component, often by means of the error function (erf) and complementary error function (erfc ≡ 1 − erf):
1
erfc(ωr12 ) erf(ωr12 )
=
+
r12
r12
r12

(5.12)

The first term on the right in Eq. (5.12) is singular but short-range, and decays to zero on a length scale of ∼ 1/ω,
while the second term constitutes a non-singular, long-range background. An RSH XC functional can be expressed
generically as
RSH
HF
HF
DFT
DFT
Exc
= cx,SR Ex,SR
+ cx,LR Ex,LR
+ (1 − cx,SR )Ex,SR
+ (1 − cx,LR )Ex,LR
+ EcDFT ,

(5.13)

HF
where the SR and LR parts of the Coulomb operator are used, respectively, to evaluate the HF exchange energies Ex,SR
HF
and Ex,LR . The corresponding DFT exchange functional is partitioned in the same manner, but the correlation energy
−1
EcDFT is evaluated using the full Coulomb operator, r12
. Of the two linear parameters in Eq. (5.13), cx,LR is usually
either set to 1 to define long-range corrected (LRC) RSH functionals (see Section 5.6) or else set to 0, which defines
screened-exchange (SE) RSH functionals. On the other hand, the fraction of short-range exact exchange (cx,SR ) can

Chapter 5: Density Functional Theory

127

either be determined via least-squares fitting, theoretically justified using the adiabatic connection, or simply set to zero.
As with the global hybrids, RSH functionals can be fashioned using all of the ingredients from the lower three rungs.
The rate at which the local DFT exchange is turned off and the non-local exact exchange is turned on is controlled by
the parameter ω. Large values of ω tend to lead to attenuators that are less smooth (unless the fraction of short-range
exact exchange is very large), while small values of (e.g., ω =0.2–0.3 bohr−1 ) are the most common in semi-empirical
RSH functionals.
The final rung on Jacob’s Ladder contains functionals that use not only occupied orbitals (via exact exchange), but
virtual orbitals as well (via methods such as MP2 or the random phase approximation, RPA). These double hybrids
(DH) are the most expensive density functionals available in Q-C HEM, but can also be very accurate. The most basic
form of a DH functional is
DH
Exc
= cx ExHF + (1 − cx ) ExDFT + cc ExMP2 + (1 − cc ) EcDFT .

(5.14)

As with hybrids, the coefficients can either be theoretically motivated or empirically determined. In addition, double
hybrids can use exact exchange both globally or via range-separation, and their components can be as primitive as
LSDA or as advanced as in meta-GGA functionals. More information on double hybrids can be found in Section 5.9.
Finally, the last major advance in KS-DFT in recent years has been the development of methods that are capable of
accurately describing non-covalent interactions, particularly dispersion. All of the functionals from Jacob’s Ladder
can technically be combined with these dispersion corrections, although in some cases the combination is detrimental,
particularly for semi-empirical functionals that were parameterized in part using data sets of non-covalent interactions,
and already tend to overestimate non-covalent interaction energies. The most popular such methods available in QC HEM are:
• Non-local correlation (NLC) functionals (Section 5.7.1), including those of Vydrov and Van Voorhis 213,215
(VV09 and VV10) and of Lundqvist and Langreth 60,61 (vdW-DF-04 and vdW-DF-10). The revised VV10 NLC
functional of Sabatini and coworkers (rVV10) is also available 182 .
• Damped, atom–atom pairwise empirical dispersion potentials from Grimme and others 45,75,77,78,184,187 [DFTD2, DFT-CHG, DFT-D3(0), DFT-D3(BJ), DFT-D3(CSO), DFT-D3M(0), DFT-D3M(BJ), and DFT-D3(op)]; see
Section 5.7.2.
• The exchange-dipole models (XDM) of Johnson and Becke (XDM6 and XDM10); see Section 5.7.3.
• The Tkatchenko-Scheffler (TS) method for dispersion interactions 199 ; see Section 5.7.4.
• The Many-Body Dispersion (MBD) method for van der Waals interactions 11,200 ; see Section 5.7.5.
Below, we categorize the functionals that are available in Q-C HEM, including exchange functionals (Section 5.3.2),
correlation functionals (Section 5.3.3), and exchange-correlation functionals (Section 5.3.4). Within each category
the functionals will be categorized according to Jacob’s Ladder. Exchange and correlation functionals can be invoked
using the $rem variables EXCHANGE and CORRELATION, while the exchange-correlation functionals can be invoked
either by setting the $rem variable METHOD or alternatively (in most cases, and for backwards compatibility with
earlier versions of Q-C HEM) by using the $rem variable EXCHANGE. Some caution is warranted here. While setting
METHOD to PBE, for example, requests the Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional, 155 which
includes both PBE exchange and PBE correlation, setting EXCHANGE = PBE requests only the exchange component
and setting CORRELATION = PBE requests only the correlation component. Setting both of these values is equivalent
to specifying METHOD = PBE.
Finally, Table 5.1 provides a summary, arranged according to Jacob’s Ladder, of which categories of functionals are
available with analytic first derivatives (for geometry optimizations) or second derivatives (for vibrational frequency
calculations). If analytic derivatives are not available for the requested job type, Q-C HEM will automatically generate
them via finite difference. Tests of the finite-difference procedure, in cases where analytic second derivatives are
available, suggest that finite-difference frequencies are accurate to < 1 cm−1 , except for very low-frequency, nonbonded modes. 128 Also listed in Table 5.1 are which functionals are available for excited-state time-dependent DFT
(TDDFT) calculations, as described in Section 7.3. Lastly, Table 5.1 describes which functionals have been parallelized
with OpenMP and/or MPI.

128

Chapter 5: Density Functional Theory

Ground State

TDDFT

†
?

Single-Point
LSDA†?
GGA†?
meta-GGA†?
GH†?
RSH†?
NLC†?
DFT-D
XDM
LSDA†?
GGA†?
meta-GGA†?
GH†?
RSH†?
—
DFT-D
—

Optimization
LSDA†?
GGA†?
meta-GGA†
GH†?
RSH†?
NLC†?
DFT-D
—
LSDA†?
GGA†?
—
GH†?
RSH†?
—
DFT-D
—

Frequency
LSDA?
GGA?
—
GH?
RSH?
—
DFT-D
—
LSDA
GGA
—
GH
—
—
DFT-D
—

OpenMP parallelization available
MPI parallelization available

Table 5.1: Available analytic properties and parallelization for SCF calculations.

5.3.1

Suggested Density Functionals

Q-C HEM contains over 150 exchange-correlation functionals, not counting those that can be straightforwardly appended with a dispersion correction (such as B3LYP-D3). Therefore, we suggest a few functionals from the second
through fourth rungs of Jacob’s Ladder in order to guide functional selection. Most of these suggestions come from a
benchmark of over 200 density functionals on a vast database of nearly 5000 data points, covering non-covalent interactions, isomerization energies, thermochemistry, and barrier heights. The single recommended method from each
category is indicated in bold.
From the GGAs on Rung 2, we recommend:
• B97-D3(BJ): METHOD B97-D3 and DFT_D D3_BJ
• revPBE-D3(BJ): METHOD revPBE and DFT_D D3_BJ
• BLYP-D3(BJ): METHOD BLYP and DFT_D D3_BJ
• PBE: METHOD PBE
From the meta-GGAs on Rung 3, we recommend:
• B97M-rV: METHOD B97M-rV
• MS1-D3(0): METHOD MS1 and DFT_D D3_ZERO
• MS2-D3(0): METHOD MS2 and DFT_D D3_ZERO
• M06-L-D3(0): METHOD M06-L and DFT_D D3_ZERO
• TPSS-D3(BJ): METHOD TPSS and DFT_D D3_BJ
From the hybrid GGAs on Rung 4, we recommend:

Chapter 5: Density Functional Theory

129

• ωB97X-V: METHOD wB97X-V
• ωB97X-D3: METHOD wB97X-D3
• ωB97X-D: METHOD wB97X-D
• B3LYP-D3(BJ): METHOD B3LYP and DFT_D D3_BJ
• revPBE0-D3(BJ): METHOD revPBE0 and DFT_D D3_BJ
From the hybrid meta-GGAs on Rung 4, we recommend:
• ωB97M-V: METHOD wB97M-V
• ωM05-D: METHOD wM05-D
• M06-2X-D3(0): METHOD M06-2X and DFT_D D3_ZERO
• TPSSh-D3(BJ): METHOD TPSSh and DFT_D D3_BJ

5.3.2

Exchange Functionals

Note: All exchange functionals in this section can be invoked using the $rem variable EXCHANGE. Popular and/or
recommended functionals within each class are listed first and indicated in bold. The rest are in alphabetical
order.
◦ Local Spin-Density Approximation (LSDA)
• Slater: Slater-Dirac exchange functional (Xα method with α = 2/3) 62
• SR_LSDA (BNL): Short-range version of the Slater-Dirac exchange functional 68
◦ Generalized Gradient Approximation (GGA)
• PBE: Perdew, Burke, and Ernzerhof exchange functional 155
• B88: Becke exchange functional from 1988 19
• revPBE: Zhang and Yang one-parameter modification of the PBE exchange functional 240
• AK13: Armiento-Kümmel exchange functional from 2013 12
• B86: Becke exchange functional (Xαβγ) from 1986 16
• G96: Gill exchange functional from 1996 66
• mB86: Becke “modified gradient correction” exchange functional from 1986 17
• mPW91: modified version (Adamo and Barone) of the 1991 Perdew-Wang exchange functional 6
• muB88 (µB88): Short-range version of the B88 exchange functional by Hirao and coworkers 92
• muPBE (µPBE): Short-range version of the PBE exchange functional by Hirao and coworkers 92
• srPBE: Short-range version of the PBE exchange functional by Goll and coworkers 71,72
• optB88: Refit version of the original B88 exchange functional (for use with vdW-DF-04) by Michaelides
and coworkers 104
• OPTX: Two-parameter exchange functional by Handy and Cohen 83
• PBEsol: PBE exchange functional modified for solids 159
• PW86: Perdew-Wang exchange functional from 1986 151
• PW91: Perdew-Wang exchange functional from 1991 154
• RPBE: Hammer, Hansen, and Norskov exchange functional (modification of PBE) 81

Chapter 5: Density Functional Theory

130

• rPW86: Revised version (Murray et al.) of the 1986 Perdew-Wang exchange functional 144
• SOGGA: Second-order GGA functional by Zhao and Truhlar 246
• wPBE (ωPBE): Henderson et al. model for the PBE GGA short-range exchange hole 85
◦ Meta-Generalized Gradient Approximation (meta-GGA)
• TPSS: Tao, Perdew, Staroverov, and Scuseria exchange functional 197
• revTPSS: Revised version of the TPSS exchange functional 161
• BLOC: Minor modification of the TPSS exchange functional that works best with TPSSloc correlation
(both by Della Sala and coworkers) 55
• modTPSS: One-parameter version of the TPSS exchange functional 158
• oTPSS: TPSS exchange functional with 5 refit parameters (for use with oTPSS correlation) by Grimme and
coworkers 69
• PBE-GX: First exchange functional based on a finite uniform electron gas (rather than an infinite UEG) by
Pierre-François Loos 131
• PKZB: Perdew, Kurth, Zupan, and Blaha exchange functional 156
• regTPSS: Regularized (fixed order of limits issue) version of the TPSS exchange functional 181
• SCAN: Strongly Constrained and Appropriately Normed exchange functional 195
• TM: Tao-Mo exchange functional derived via an accurate modeling of the conventional exchange hole 196

5.3.3

Correlation Functionals

Note: All correlation functionals in this section can be invoked using the $rem variable CORRELATION. Popular and/
or recommended functionals within each class are listed first and indicated in bold. The rest are in alphabetical
order.
◦ Local Spin-Density Approximation (LSDA)
• PW92: Perdew-Wang parameterization of the LSDA correlation energy from 1992 152
• VWN5 (VWN): Vosko-Wilk-Nusair parameterization of the LSDA correlation energy #5 211
• srVWN: Short-range version of the VWN correlation functional by Toulouse and coworkers 201
• Liu-Parr: Liu-Parr ρ1/3 model from the functional expansion formulation 129
• PK09: Proynov-Kong parameterization of the LSDA correlation energy from 2009 173
• PW92RPA: Perdew-Wang parameterization of the LSDA correlation energy from 1992 with RPA values 152
• srPW92: Short-range version of the PW92 correlation functional by Paziani and coworkers 149
• PZ81: Perdew-Zunger parameterization of the LSDA correlation energy from 1981 153
• VWN1: Vosko-Wilk-Nusair parameterization of the LSDA correlation energy #1 211
• VWN1RPA: Vosko-Wilk-Nusair parameterization of the LSDA correlation energy #1 with RPA values 211
• VWN2: Vosko-Wilk-Nusair parameterization of the LSDA correlation energy #2 211
• VWN3: Vosko-Wilk-Nusair parameterization of the LSDA correlation energy #3 211
• VWN4: Vosko-Wilk-Nusair parameterization of the LSDA correlation energy #4 211
• Wigner:Wigner correlation functional (simplification of LYP) 191,221
◦ Generalized Gradient Approximation (GGA)
• PBE: Perdew, Burke, and Ernzerhof correlation functional 155
• LYP: Lee-Yang-Parr opposite-spin correlation functional 121

Chapter 5: Density Functional Theory

131

• P86: Perdew-Wang correlation functional from 1986 based on the PZ81 LSDA functional 150
• P86VWN5: Perdew-Wang correlation functional from 1986 based on the VWN5 LSDA functional 150
• PBEloc: PBE correlation functional with a modified beta term by Della Sala and coworkers 54
• PBEsol: PBE correlation functional modified for solids 159
• srPBE: Short-range version of the PBE correlation functional by Goll and coworkers 71,72
• PW91: Perdew-Wang correlation functional from 1991 154
• regTPSS: Slight modification of the PBE correlation functional (also called vPBEc) 181
◦ Meta-Generalized Gradient Approximation (meta-GGA)
• TPSS:Tao, Perdew, Staroverov, and Scuseria correlation functional 197
• revTPSS: Revised version of the TPSS correlation functional 161
• B95: Becke’s two-parameter correlation functional from 1995 22
• oTPSS: TPSS correlation functional with 2 refit parameters (for use with oTPSS exchange) by Grimme and
coworkers 69
• PK06: Proynov-Kong “tLap” functional with τ and Laplacian dependence 171
• PKZB: Perdew, Kurth, Zupan, and Blaha correlation functional 156
• SCAN: Strongly Constrained and Appropriately Normed correlation functional 195
• TM: Tao-Mo correlation functional, representing a minor modification to the TPSS correlation functional 196
• TPSSloc: The TPSS correlation functional with the PBE component replaced by the PBEloc correlation
functional 54

5.3.4

Exchange-Correlation Functionals

Note: All exchange-correlation functionals in this section can be invoked using the $rem variable METHOD. For
backwards compatibility, all of the exchange-correlation functionals except for the ones marked with an asterisk
can be used with the $rem variable EXCHANGE. Popular and/or recommended functionals within each class
are listed first and indicated in bold. The rest are in alphabetical order.
◦ Local Spin-Density Approximation (LSDA)
• SPW92*: Slater LSDA exchange + PW92 LSDA correlation
• LDA: Slater LSDA exchange + VWN5 LSDA correlation
• SVWN5*: Slater LSDA exchange + VWN5 LSDA correlation
◦ Generalized Gradient Approximation (GGA)
• B97-D3(0): B97-D with a fitted DFT-D3(0) tail instead of the original DFT-D2 tail 77
• B97-D: 9-parameter dispersion-corrected (DFT-D2) functional by Grimme 75
• PBE*: PBE GGA exchange + PBE GGA correlation
• BLYP*: B88 GGA exchange + LYP GGA correlation
• revPBE*: revPBE GGA exchange + PBE GGA correlation
• BEEF-vdW: 31-parameter semi-empirical exchange functional developed via a Bayesian error estimation
framework paired with PBE correlation and vdW-DF-10 NLC 218
• BOP: B88 GGA exchange + BOP “one-parameter progressive” GGA correlation 203
• BP86*: B88 GGA exchange + P86 GGA correlation
• BP86VWN*: B88 GGA exchange + P86VWN5 GGA correlation

Chapter 5: Density Functional Theory

132

• BPBE*: B88 GGA exchange + PBE GGA correlation
• EDF1: Modification of BLYP to give good performance in the 6-31+G* basis set 9
• EDF2: Modification of B3LYP to give good performance in the cc-pVTZ basis set for frequencies 124
• GAM: 21-parameter non-separable gradient approximation functional by Truhlar and coworkers 236
• HCTH93 (HCTH/93): 15-parameter functional trained on 93 systems by Handy and coworkers 82
• HCTH120 (HCTH/120): 15-parameter functional trained on 120 systems by Boese et al. 34
• HCTH147 (HCTH/147): 15-parameter functional trained on 147 systems by Boese et al. 34
• HCTH407 (HCTH/407): 15-parameter functional trained on 407 systems by Boese and Handy 31
• HLE16 – HCTH/407 exchange functional enhanced by a factor of 1.25 + HCTH/407 correlation functional
enhanced by a factor of 0.5 210
• KT1: GGA functional designed specifically for shielding constant calculations 100
• KT2: GGA functional designed specifically for shielding constant calculations 100
• KT3: GGA functional with improved results for main-group nuclear magnetic resonance shielding constants 101
• mPW91*: mPW91 GGA exchange + PW91 GGA correlation
• N12: 21-parameter non-separable gradient approximation functional by Peverati and Truhlar 166
• OLYP*: OPTX GGA exchange + LYP GGA correlation
• PBEOP: PBE GGA exchange + PBEOP “one-parameter progressive” GGA correlation 203
• PBEsol*: PBEsol GGA exchange + PBEsol GGA correlation
• PW91*: PW91 GGA exchange + PW91 GGA correlation
• RPBE*: RPBE GGA exchange + PBE GGA correlation
• rVV10*: rPW86 GGA exchange + PBE GGA correlation + rVV10 non-local correlation 182
• SOGGA*: SOGGA GGA exchange + PBE GGA correlation
• SOGGA11: 20-parameter functional by Peverati, Zhao, and Truhlar 169
• VV10: rPW86 GGA exchange + PBE GGA correlation + VV10 non-local correlation 215
◦ Meta-Generalized Gradient Approximation (meta-GGA)
• B97M-V: 12-parameter combinatorially-optimized, dispersion-corrected (VV10) functional by Mardirossian
and Head-Gordon 134
• B97M-rV*: B97M-V density functional with the VV10 NLC functional replaced by the rVV10 NLC
functional 136
• M06-L: 34-parameter functional by Zhao and Truhlar 244
• TPSS*: TPSS meta-GGA exchange + TPSS meta-GGA correlation
• revTPSS*: revTPSS meta-GGA exchange + revTPSS meta-GGA correlation
• BLOC*: BLOC meta-GGA exchange + TPSSloc meta-GGA correlation
• M11-L: 44-parameter dual-range functional by Peverati and Truhlar 165
• mBEEF: 64-parameter exchange functional paired with the PBEsol correlation functional 219
• MGGA_MS0: MGGA_MS0 meta-GGA exchange + regTPSS GGA correlation 192
• MGGA_MS1: MGGA_MS1 meta-GGA exchange + regTPSS GGA correlation 193
• MGGA_MS2: MGGA_MS2 meta-GGA exchange + regTPSS GGA correlation 193
• MGGA_MVS: MGGA_MVS meta-GGA exchange + regTPSS GGA correlation 194
• MN12-L: 58-parameter meta-nonseparable gradient approximation functional by Peverati and Truhlar 167

Chapter 5: Density Functional Theory

133

• MN15-L: 58-parameter meta-nonseparable gradient approximation functional by Yu, He, and Truhlar 238
• oTPSS*: oTPSS meta-GGA exchange + oTPSS meta-GGA correlation
• PKZB*: PKZB meta-GGA exchange + PKZB meta-GGA correlation
• SCAN*: SCAN meta-GGA exchange + SCAN meta-GGA correlation
• t-HCTH (τ -HCTH): 16-parameter functional by Boese and Handy 32
• TM*: TM meta-GGA exchange + TM meta-GGA correlation 196
• VSXC: 21-parameter functional by Voorhis and Scuseria 207
◦ Global Hybrid Generalized Gradient Approximation (GH GGA)
– B3LYP: 20% HF exchange + 8% Slater LSDA exchange + 72% B88 GGA exchange + 19% VWN1RPA
LSDA correlation + 81% LYP GGA correlation 20,190
– PBE0: 25% HF exchange + 75% PBE GGA exchange + PBE GGA correlation 7
– revPBE0: 25% HF exchange + 75% revPBE GGA exchange + PBE GGA correlation
– B97: Becke’s original 10-parameter density functional with 19.43% HF exchange 23
– B1LYP: 25% HF exchange + 75% B88 GGA exchange + LYP GGA correlation 5
– B1PW91: 25% HF exchange + 75% B88 GGA exchange + PW91 GGA correlation 5
– B3LYP5: 20% HF exchange + 8% Slater LSDA exchange + 72% B88 GGA exchange + 19% VWN5 LSDA
correlation + 81% LYP GGA correlation 20,190
– B3P86: 20% HF exchange + 8% Slater LSDA exchange + 72% B88 GGA exchange+ 19% VWN1RPA
LSDA correlation + 81% P86 GGA correlation
– B1LYP: 25% HF exchange + 75% B88 GGA exchange + LYP GGA correlation 5
– B1PW91: 25% HF exchange + 75% B88 GGA exchange + PW91 GGA correlation 5
– B3LYP5: 20% HF exchange + 8% Slater LSDA exchange + 72% B88 GGA exchange + 19% VWN5 LSDA
correlation + 81% LYP GGA correlation 20,190
– B3P86: 20% HF exchange + 8% Slater LSDA exchange + 72% B88 GGA exchange+ 19% VWN1RPA
LSDA correlation + 81% P86 GGA correlation
– B3PW91: 20% HF exchange + 8% Slater LSDA exchange + 72% B88 GGA exchange+ 19% PW92 LSDA
correlation + 81% PW91 GGA correlation 20
– B5050LYP: 50% HF exchange + 8% Slater LSDA exchange + 42% B88 GGA exchange + 19% VWN5
LSDA correlation + 81% LYP GGA correlation 186
– B97-1: Self-consistent parameterization of Becke’s B97 density functional with 21% HF exchange 82
– B97-2: Re-parameterization of B97 by Tozer and coworkers with 21% HF exchange 223
– B97-3: 16-parameter version of B97 by Keal and Tozer with ≈ 26.93% HF exchange 102
– B97-K: Re-parameterization of B97 for kinetics by Boese and Martin with 42% HF exchange 33
– BHHLYP: 50% HF exchange + 50% B88 GGA exchange + LYP GGA correlation
– HFLYP*: 100% HF exchange + LYP GGA correlation
– MPW1K: 42.8% HF exchange + 57.2% mPW91 GGA exchange + PW91 GGA correlation 132
– MPW1LYP: 25% HF exchange + 75% mPW91 GGA exchange + LYP GGA correlation 6
– MPW1PBE: 25% HF exchange + 75% mPW91 GGA exchange + PBE GGA correlation 6
– MPW1PW91: 25% HF exchange + 75% mPW91 GGA exchange + PW91 GGA correlation 6
– O3LYP: 11.61% HF exchange + ≈ 7.1% Slater LSDA exchange + 81.33% OPTX GGA exchange + 19%
VWN5 LSDA correlation + 81% LYP GGA correlation 89
– PBEh-3c: Low-cost composite scheme of Grimme and coworkers for use with the def2-mSVP basis set
only 79

Chapter 5: Density Functional Theory

134

– PBE50: 50% HF exchange + 50% PBE GGA exchange + PBE GGA correlation 28
– SOGGA11-X: 21-parameter functional with 40.15% HF exchange by Peverati and Truhlar 163
– WC04: Hybrid density functional optimized for the computation of 13 C chemical shifts 222
– WP04: Hybrid density functional optimized for the computation of 1 H chemical shifts 222
– X3LYP: 21.8% HF exchange + 7.3% Slater LSDA exchange + ≈ 54.24% B88 GGA exchange + ≈ 16.66%
PW91 GGA exchange + 12.9% VWN1RPA LSDA correlation + 87.1% LYP GGA correlation 234
◦ Global Hybrid Meta-Generalized Gradient Approximation (GH meta-GGA)
• M06-2X: 29-parameter functional with 54% HF exchange by Zhao and Truhlar 248
• M08-HX: 47-parameter functional with 52.23% HF exchange by Zhao and Truhlar 247
• TPSSh: 10% HF exchange + 90% TPSS meta-GGA exchange + TPSS meta-GGA correlation 189
• revTPSSh: 10% HF exchange + 90% revTPSS meta-GGA exchange + revTPSS meta-GGA correlation 58
• B1B95: 28% HF exchange + 72% B88 GGA exchange + B95 meta-GGA correlation 22
• B3TLAP: 17.13% HF exchange + 9.66% Slater LSDA exchange + 72.6% B88 GGA exchange + PK06
meta-GGA correlation 171,172
• BB1K: 42% HF exchange + 58% B88 GGA exchange + B95 meta-GGA correlation 250
• BMK: Boese-Martin functional for kinetics with 42% HF exchange 33
• dlDF: Dispersion-less density functional (based on the M05-2X functional form) by Szalewicz and coworkers 162
• M05: 22-parameter functional with 28% HF exchange by Zhao, Schultz, and Truhlar 251
• M05-2X: 19-parameter functional with 56% HF exchange by Zhao, Schultz, and Truhlar 252
• M06: 33-parameter functional with 27% HF exchange by Zhao and Truhlar 248
• M06-HF: 32-parameter functional with 100% HF exchange by Zhao and Truhlar 245
• M08-SO: 44-parameter functional with 56.79% HF exchange by Zhao and Truhlar 247
• MGGA_MS2h: 9% HF exchange + 91 % MGGA_MS2 meta-GGA exchange + regTPSS GGA correlation 193
• MGGA_MVSh: 25% HF exchange + 75 % MGGA_MVS meta-GGA exchange + regTPSS GGA correlation 194
• MN15: 59-parameter functional with 44% HF exchange by Truhlar and coworkers 237
• MPW1B95: 31% HF exchange + 69% mPW91 GGA exchange + B95 meta-GGA correlation 242
• MPWB1K: 44% HF exchange + 56% mPW91 GGA exchange + B95 meta-GGA correlation 242
• PW6B95: 6-parameter combination of 28 % HF exchange, 72 % optimized PW91 GGA exchange, and
re-optimized B95 meta-GGA correlation by Zhao and Truhlar 243
• PWB6K: 6-parameter combination of 46 % HF exchange, 54 % optimized PW91 GGA exchange, and
re-optimized B95 meta-GGA correlation by Zhao and Truhlar 243
• SCAN0: 25% HF exchange + 75% SCAN meta-GGA exchange + SCAN meta-GGA correlation 91
• t-HCTHh (τ -HCTHh): 17-parameter functional with 15% HF exchange by Boese and Handy 32
• TPSS0: 25% HF exchange + 75% TPSS meta-GGA exchange + TPSS meta-GGA correlation 73
◦ Range-Separated Hybrid Generalized Gradient Approximation (RSH GGA)
• wB97X-V (ωB97X-V): 10-parameter combinatorially-optimized, dispersion-corrected (VV10) functional
with 16.7% SR HF exchange, 100% LR HF exchange, and ω = 0.3 133
• wB97X-D3 (ωB97X-D3): 16-parameter dispersion-corrected (DFT-D3(0)) functional with ≈ 19.57% SR
HF exchange, 100% LR HF exchange, and ω = 0.25 126

Chapter 5: Density Functional Theory

135

• wB97X-D (ωB97X-D): 15-parameter dispersion-corrected (DFT-CHG) functional with ≈ 22.2% SR HF
exchange, 100% LR HF exchange, and ω = 0.2 45
• CAM-B3LYP: Coulomb-attenuating method functional by Handy and coworkers 235
• CAM-QTP00: Re-parameterized CAM-B3LYP designed to satisfy the IP-theorem for all occupied orbitals
of the water molecule 209
• CAM-QTP01: Re-parameterized CAM-B3LYP optimized to satisfy the valence IPs of the water molecule,
34 excitation states, and G2-1 atomization energies 94
• HSE-HJS: Screened-exchange “HSE06” functional with 25% SR HF exchange, 0% LR HF exchange, and
ω=0.11, using the updated HJS PBE exchange hole model 85,110
• LC-rVV10*: LC-VV10 density functional with the VV10 NLC functional replaced by the rVV10 NLC
functional 136
• LC-VV10: 0% SR HF exchange + 100% LR HF exchange + ωPBE GGA exchange + PBE GGA correlation
+ VV10 non-local correlation (ω=0.45) 215
• LC-wPBE08 (LC-ωPBE08): 0% SR HF exchange + 100% LR HF exchange + ωPBE GGA exchange +
PBE GGA correlation (ω=0.45) 217
• LRC-BOP (LRC-µBOP): 0% SR HF exchange + 100% LR HF exchange + muB88 GGA exchange + BOP
GGA correlation (ω=0.47) 188
• LRC-wPBE (LRC-ωPBE): 0% SR HF exchange + 100% LR HF exchange + ωPBE GGA exchange + PBE
GGA correlation (ω=0.3) 178
• LRC-wPBEh (LRC-ωPBEh): 20% SR HF exchange + 100% LR HF exchange + 80% ωPBE GGA exchange + PBE GGA correlation (ω=0.2) 179
• N12-SX: 26-parameter non-separable GGA with 25% SR HF exchange, 0% LR HF exchange, and ω =
0.11 168
• rCAM-B3LYP: Re-fit CAM-B3LYP with the goal of minimizing many-electron self-interaction error 52
• wB97 (ωB97): 13-parameter functional with 0% SR HF exchange, 100% LR HF exchange, and ω = 0.4 44
• wB97X (ωB97X): 14-parameter functional with ≈ 15.77% SR HF exchange, 100% LR HF exchange, and
ω = 0.3 44
• wB97X-rV* (ωB97X-rV): ωB97X-V density functional with the VV10 NLC functional replaced by the
rVV10 NLC functional 136
◦ Range-Separated Hybrid Meta-Generalized Gradient Approximation (RSH meta-GGA)
• wB97M-V (ωB97M-V): 12-parameter combinatorially-optimized, dispersion-corrected (VV10) functional
with 15% SR HF exchange, 100% LR HF exchange, and ω = 0.3 135
• M11: 40-parameter functional with 42.8% SR HF exchange, 100% LR HF exchange, and ω = 0.25 164
• MN12-SX: 58-parameter non-separable meta-GGA with 25% SR HF exchange, 0% LR HF exchange, and
ω = 0.11 168
• wB97M-rV* (ωB97X-rV): ωB97M-V density functional with the VV10 NLC functional replaced by the
rVV10 NLC functional 136
• wM05-D (ωM05-D): 21-parameter dispersion-corrected (DFT-CHG) functional with ≈ 36.96% SR HF
exchange, 100% LR HF exchange, and ω = 0.2 125
• wM06-D3 (ωM06-D3): 25-parameter dispersion-corrected [DFT-D3(0)] functional with ≈ 27.15% SR HF
exchange, 100% LR HF exchange, and ω = 0.3 126
◦ Double Hybrid Generalized Gradient Approximation (DH GGA)
Note: In order to use the resolution-of-the-identity approximation for the MP2 component, specify an auxiliary basis
set with the $rem variable AUX_BASIS

Chapter 5: Density Functional Theory

136

• DSD-PBEPBE-D3: 68% HF exchange + 32% PBE GGA exchange + 49% PBE GGA correlation + 13%
SS MP2 correlation + 55% OS MP2 correlation with DFT-D3(BJ) tail 109
• wB97X-2(LP) (ωB97X-2(LP)): 13-parameter functional with ≈ 67.88% SR HF exchange, 100% LR HF
exchange, ≈ 58.16% SS MP2 correlation, ≈ 47.80% OS MP2 correlation, and ω = 0.3 46
• wB97X-2(TQZ) (ωB97X-2(TQZ)): 13-parameter functional with ≈ 63.62% SR HF exchange, 100% LR
HF exchange, ≈ 52.93% SS MP2 correlation, ≈ 44.71% OS MP2 correlation, and ω = 0.3 46
• XYG3: 80.33% HF exchange - 1.4% Slater LSDA exchange + 21.07% B88 GGA exchange + 67.89% LYP
GGA correlation + 32.11% MP2 correlation (evaluated with B3LYP orbitals) 241
• XYGJ-OS: 77.31% HF exchange + 22.69% Slater LSDA exchange + 23.09% VWN1RPA LSDA correlation + 27.54% LYP GGA correlation + 43.64% OS MP2 correlation (evaluated with B3LYP orbitals) 239
• B2PLYP: 53% HF exchange + 47% B88 GGA exchange + 73% LYP GGA correlation + 27% MP2 correlation 74
• B2GPPLYP: 65% HF exchange + 35% B88 GGA exchange + 64% LYP GGA correlation + 36% MP2
correlation 99
• DSD-PBEP86-D3: 69% HF exchange + 31% PBE GGA exchange + 44% P86 GGA correlation + 22% SS
MP2 correlation + 52% OS MP2 correlation with DFT-D3(BJ) tail 109
• LS1DH-PBE: 75% HF exchange + 25% PBE GGA exchange + 57.8125% PBE GGA correlation + 42.1875%
MP2 correlation 202
• PBE-QIDH: 69.3361% HF exchange + 30.6639% PBE GGA exchange + 66.6667% PBE GGA correlation
+ 33.3333% MP2 correlation 37
• PBE0-2: ≈ 79.37% HF exchange + ≈ 20.63% PBE GGA exchange + 50% PBE GGA correlation + 50%
MP2 correlation 47
• PBE0-DH: 50% HF exchange + 50% PBE GGA exchange + 87.5% PBE GGA correlation + 12.5% MP2
correlation 36
◦ Double Hybrid Meta-Generalized Gradient Approximation (DH MGGA)
• PTPSS-D3: 50% HF exchange + 50% Re-Fit TPSS meta-GGA exchange + 62.5% Re-Fit TPSS meta-GGA
correlation + 37.5% OS MP2 correlation with DFT-D3(0) tail 70
• DSD-PBEB95-D3: 66% HF exchange + 34% PBE GGA exchange + 55% B95 GGA correlation + 9% SS
MP2 correlation + 46% OS MP2 correlation with DFT-D3(BJ) tail 109
• PWPB95-D3: 50% HF exchange + 50% Re-Fit PW91 GGA exchange + 73.1% Re-Fit B95 meta-GGA
correlation + 26.9% OS MP2 correlation with DFT-D3(0) tail 70

5.3.5

Specialized Functionals

• SRC1-R1: TDDFT short-range corrected functional [Eq. (1) in Ref. 29, 1st row atoms]
• SRC1-R2: TDDFT short-range corrected functional [Eq. (1) in Ref. 29, 2nd row atoms]
• SRC2-R1: TDDFT short-range corrected functional [Eq. (2) in Ref. 29, 1st row atoms]
• SRC2-R2: TDDFT short-range corrected functional [Eq. (2) in Ref. 29, 2nd row atoms]
• BR89: Becke-Roussel meta-GGA exchange functional modeled after the hydrogen atom 26
• B94: meta-GGA correlation functional by Becke that uses the BR89 exchange functional to compute the Coulomb
potential 21
• B94hyb: modified version of the B94 correlation functional for use with the BR89B94hyb exchange-correlation
functional 21

Chapter 5: Density Functional Theory

137

• BR89B94h: 15.4% HF exchange + 84.6% BR89 meta-GGA exchange + BR89hyb meta-GGA correlation 21
• BRSC: Exchange component of the original B05 exchange-correlation functional 24
• MB05: Exchange component of the modified B05 (BM05) exchange-correlation functional 175
• B05: A full exact-exchange Kohn-Sham scheme of Becke that uses the exact-exchange energy density (RI) and
accounts for static correlation 24,174,176
• BM05 (XC): Modified B05 hyper-GGA scheme that uses MB05 instead of BRSC as the exchange functional 175
• PSTS: Hyper-GGA (100% HF exchange) exchange-correlation functional of Perdew, Staroverov, Tao, and Scuseria 160
• MCY2: Mori-Sánchez-Cohen-Yang adiabatic connection-based hyper-GGA exchange-correlation functional 51,127,142

5.3.6

User-Defined Density Functionals

Users can also request a customized density functional consisting of any linear combination of exchange and/or correlation functionals available in Q-C HEM. A “general” density functional of this sort is requested by setting EXCHANGE
= GEN and then specifying the functional by means of an $xc_functional input section consisting of one line for each
desired exchange (X) or correlation (C) component of the functional, and having the format shown below.
$xc_functional
X
exchange_symbol
coefficient
X
exchange_symbol
coefficient
...
C
correlation_symbol
coefficient
C
correlation_symbol
coefficient
...
K
coefficient
$end

Each line requires three variables: X or C to designate whether this is an exchange or correlation component; the
symbolic representation of the functional, as would be used for the EXCHANGE or CORRELATION keywords variables
as described above; and a real number coefficient for each component. Note that Hartree-Fock exchange can be

138

Chapter 5: Density Functional Theory
designated either as “X" or as “K". Examples are shown below.
Example 5.1 Q-C HEM input for H2 O with the B3tLap functional.
$molecule
0 1
O
H1 O oh
H2 O oh

H1

hoh

oh =
0.97
hoh = 120.0
$end
$rem
EXCHANGE
CORRELATION
BASIS
THRESH
$end

gen
none
g3large
14

! recommended for high accuracy
! and better convergence

$xc_functional
X
Becke
0.726
X
S
0.0966
C
PK06
1.0
K
0.1713
$end

Example 5.2 Q-C HEM input for H2 O with the BR89B94hyb functional.
$molecule
0 1
O
H1 O oh
H2 O oh

H1

hoh

oh =
0.97
hoh = 120.0
$end
$rem
EXCHANGE
CORRELATION
BASIS
THRESH
$end

gen
none
g3large
14

$xc_functional
X
BR89
C
B94hyb
K
$end

0.846
1.0
0.154

! recommended for high accuracy
! and better convergence

The next two examples illustrate the use of the RI-B05 and RI-PSTS functionals. These are presently available only for
single-point calculations, and convergence is greatly facilitated by obtaining converged SCF orbitals from, e.g., an LDA
or HF calculation first. (LDA is used in the example below but HF can be substituted.) Use of the RI approximation

Chapter 5: Density Functional Theory
(Section 6.6) requires specification of an auxiliary basis set.
Example 5.3 Q-C HEM input of H2 using RI-B05.
$comment
H2, example of SP RI-B05. First do a well-converged LSD, G3LARGE is the
basis of choice for good accuracy. The input lines
PURECART
2222
SCF_GUESS
CORE
are obligatory for the time being here.
$end
$molecule
0 1
H
0. 0.
H
0. 0.
$end

0.0
0.7414

$rem
SCF_GUESS
METHOD
BASIS
PURECART
THRESH
INCDFT
SYM_IGNORE
SYMMETRY
SCF_CONVERGENCE
$end

core
lda
g3large
2222
14
false
true
false
9

@@@
$comment
For the time being the following input lines are obligatory:
PURECART
2222
AUX_BASIS
riB05-cc-pvtz
DFT_CUTOFFS
0
MAX_SCF_CYCLES 0
$end
$molecule
read
$end
$rem
SCF_GUESS
EXCHANGE
PURECART
BASIS
AUX_BASIS
THRESH
PRINT_INPUT
INCDFT
SYM_IGNORE
SYMMETRY
MAX_SCF_CYCLES
DFT_CUTOFFS
$end

read
b05
! or set to psts for ri-psts
2222
g3large
rib05-cc-pvtz ! the aux basis for both RI-B05 and RI-PSTS
4
true
false
true
false
0
0

139

Chapter 5: Density Functional Theory

5.4

140

Basic DFT Job Control

Basic SCF job control was described in Section 4.3 in the context of Hartree-Fock theory and is largely the same for
DFT. The keywords METHOD and BASIS are required, although for DFT the former could be substituted by specifying
EXCHANGE and CORRELATION instead.
METHOD
Specifies the exchange-correlation functional.
TYPE:
STRING
DEFAULT:
No default
OPTIONS:
NAME Use METHOD = NAME, where NAME is either HF for Hartree-Fock theory or
else one of the DFT methods listed in Section 5.3.4.
RECOMMENDATION:
In general, consult the literature to guide your selection. Our recommendations for DFT are
indicated in bold in Section 5.3.4.
EXCHANGE
Specifies the exchange functional (or most exchange-correlation functionals for backwards compatibility).
TYPE:
STRING
DEFAULT:
No default
OPTIONS:
NAME Use EXCHANGE = NAME, where NAME is either:
1) One of the exchange functionals listed in Section 5.3.2
2) One of the XC functionals listed in Section 5.3.4 that is not marked with an
asterisk.
3) GEN, for a user-defined functional (see Section 5.3.6).
RECOMMENDATION:
In general, consult the literature to guide your selection. Our recommendations are indicated in
bold in Sections 5.3.4 and 5.3.2.
CORRELATION
Specifies the correlation functional.
TYPE:
STRING
DEFAULT:
NONE

OPTIONS:
NAME

Use CORRELATION = NAME, where NAME is one of the correlation functionals
listed in Section 5.3.3.
RECOMMENDATION:
In general, consult the literature to guide your selection. Our recommendations are indicated in
bold in Section 5.3.3.

The following $rem variables are related to the choice of the quadrature grid required to integrate the XC part of the
functional, which does not appear in Hartree-Fock theory. DFT quadrature grids are described in Section 5.5.

Chapter 5: Density Functional Theory
FAST_XC
Controls direct variable thresholds to accelerate exchange-correlation (XC) in DFT.
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
TRUE
Turn FAST_XC on.
FALSE Do not use FAST_XC.
RECOMMENDATION:
Caution: FAST_XC improves the speed of a DFT calculation, but may occasionally cause the
SCF calculation to diverge.
XC_GRID
Specifies the type of grid to use for DFT calculations.
TYPE:
INTEGER
DEFAULT:
Functional-dependent; see Table 5.3.
OPTIONS:
0
Use SG-0 for H, C, N, and O; SG-1 for all other atoms.
n
Use SG-n for all atoms, n = 1, 2, or 3
XY
A string of two six-digit integers X and Y , where X is the number of radial points
and Y is the number of angular points where possible numbers of Lebedev angular
points, which must be an allowed value from Table 5.2 in Section 5.5.
−XY Similar format for Gauss-Legendre grids, with the six-digit integer X corresponding
to the number of radial points and the six-digit integer Y providing the number of
Gauss-Legendre angular points, Y = 2N 2 .
RECOMMENDATION:
Use the default unless numerical integration problems arise. Larger grids may be required for
optimization and frequency calculations.
NL_GRID
Specifies the grid to use for non-local correlation.
TYPE:
INTEGER
DEFAULT:
1
OPTIONS:
Same as for XC_GRID
RECOMMENDATION:
Use the default unless computational cost becomes prohibitive, in which case SG-0 may be used.
XC_GRID should generally be finer than NL_GRID.

141

142

Chapter 5: Density Functional Theory
XC_SMART_GRID
Uses SG-0 (where available) for early SCF cycles, and switches to the (larger) target grid specified by XC_GRID for final cycles of the SCF.
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
TRUE (or 1)
Use the smaller grid for the initial cycles.
FALSE (or 0) Use the target grid for all SCF cycles.
RECOMMENDATION:
The use of the smart grid can save some time on initial SCF cycles.

5.5

DFT Numerical Quadrature

In practical DFT calculations, the forms of the approximate exchange-correlation functionals used are quite complicated, such that the required integrals involving the functionals generally cannot be evaluated analytically. Q-C HEM
evaluates these integrals through numerical quadrature directly applied to the exchange-correlation integrand. Several
standard quadrature grids are available (“SG-n”, n = 0, 1, 2, 3), with a default value that is automatically set according
to the complexity of the functional in question.
The quadrature approach in Q-C HEM is generally similar to that found in many DFT programs. The multi-center
XC integrals are first partitioned into “atomic” contributions using a nuclear weight function. Q-C HEM uses the
nuclear partitioning of Becke, 18 though without the “atomic size adjustments” of Ref. 18. The atomic integrals are then
evaluated through standard one-center numerical techniques. Thus, the exchange-correlation energy is obtained as
EXC =

atoms
X points
X
A

wAi f (rAi ) ,

(5.15)

i∈A

where the function f is the aforementioned XC integrand and the quantities wAi are the quadrature weights. The sum
over i runs over grid points belonging to atom A, which are located at positions rAi = RA + ri , so this approach
requires only the choice of a suitable one-center integration grid (to define the ri ), which is independent of nuclear
configuration. These grids are implemented in Q-C HEM in a way that ensures that the EXC is rotationally-invariant,
i.e., that is does not change when the molecule undergoes rigid rotation in space. 95
Quadrature grids are further separated into radial and angular parts. Within Q-C HEM, the radial part is usually treated
by the Euler-Maclaurin scheme proposed by Murray et al., 143 which maps the semi-infinite domain [0, ∞) onto [0, 1)
and applies the extended trapezoid rule to the transformed integrand. Alternatively, Gill and Chien proposed a radial
scheme based on a Gaussian quadrature on the interval [0, 1] with a different weight function. 49 This “MultiExp" radial
quadrature is exact for integrands that are a linear combination of a geometric sequence of exponential functions, and
is therefore well suited to evaluating atomic integrals. However, the task of generating the MultiExp quadrature points
becomes increasingly ill-conditioned as the number of radial points increases, so that a “double exponential" radial
quadrature 137,138 is used for the largest standard grids in Q-C HEM, 137,138 namely SG-2 and SG-3. 59 (See Section 5.5.2.)

5.5.1

Angular Grids

For a fixed value of the radial spherical-polar coordinate r, a function f (r) ≡ f (r, θ, φ) has an exact expansion in
spherical harmonic functions,
∞ X
`
X
f (r, θ, φ) =
c`m Y`m (θ, φ) .
(5.16)
`=0 m=−`

143

Chapter 5: Density Functional Theory

No. Points
6
14
26
38
50
74
86
110
146
170
194

Degree
(`max )
3
5
7
9
11
13
15
17
19
21
23

No. Points
230
266
302
350
434
590
770
974
1202
1454

Degree
(`max )
25
27
29
31
35
41
47
53
59
65

No. Points
1730
2030
2354
2702
3074
3470
3890
4334
4802
5294

Degree
(`max )
71
77
83
89
95
101
107
113
119
125

Table 5.2: Lebedev angular quadrature grids available in Q-C HEM.

Angular quadrature grids are designed to integrate f (r, θ, φ) for fixed r, and are often characterized by their degree,
meaning the maximum value of ` for which the quadrature is exact, as well as by their efficiency, meaning the number
of spherical harmonics exactly integrated per degree of freedom in the formula. Q-C HEM supports the following two
types of angular grids.
• Lebedev grids. These are specially-constructed grids for quadrature on the surface of a sphere, 117–120 based on
the octahedral point group. Lebedev grids available in Q-C HEM are listed in Table 5.2. These grids typically have
near-unit efficiencies, with efficiencies exceeding unity in some cases. A Lebedev grid is selected by specifying
the number of grid points (from Table 5.2) using the $rem keyword XC_GRID, as discussed below.
• Gauss-Legendre grids. These are spherical direct-product grids in the two spherical-polar angles, θ and φ.
Integration in over θ is performed using a Gaussian quadrature derived from the Legendre polynomials, while
integration over φ is performed using equally-spaced points. A Gauss-Legendre grid is selected by specifying
the total number of points, 2N 2 , to be used for the integration, which specifies a grid consisting of 2Nφ points in
φ and Nθ in θ, for a degree of 2N − 1. Gauss-Legendre grids exhibit efficiencies of only 2/3, and are thus lower
in quality than Lebedev grids for the same number of grid points, but have the advantage that they are defined for
arbitrary (and arbitrarily-large) numbers of grid points. This offers a mechanism to achieve arbitrary accuracy in
the angular integration, if desired.
Combining these radial and angular schemes yields an intimidating selection of quadratures, so it is useful to standardize the grids. This is done for the convenience of the user, to facilitate comparisons in the literature, and also
for developers wishing to compared detailed results between different software programs, because the total electronic
energy is sensitive to the details of the grid, just as it is sensitive to details of the basis set. Standard quadrature grids
are discussed next.

5.5.2

Standard Quadrature Grids

Four different “standard grids" are available in Q-C HEM, designated SG-n, for n = 0, 1, 2, or 3; both quality and the
computational cost of these grids increases with n. These grids are constructed starting from a “parent” grid (Nr , NΩ )
consisting of Nr radial spheres with NΩ angular (Lebedev) grid points on each, then systematically pruning the number
of angular points in regions where sophisticated angular quadrature is not necessary, such as near the nuclei where the
charge density is nearly spherically symmetric and at long distance from the nuclei where it varies slowly. A large
number of points is retained in the valence region where angular accuracy is critical. The SG-n grids are summarized
in Table 5.3. While many electronic structure programs use some kind of procedure to delete unnecessary grid points
in the interest of computational efficiency, Q-C HEM’s SG-n grids are notable in that the complete grid specifications

144

Chapter 5: Density Functional Theory
Pruned
Grid
SG-0
SG-1
SG-2
SG-3

Ref.
50
67
59
59

Parent Grid
(Nr , NΩ )
(23, 170)
(50, 194)
(75, 302)
(99, 590)

No. Grid Points
(C atom)a
1,390 (36%)
3,816 (39%)
7,790 (34%)
17,674 (30%)

Default Grid for
Which Functionals?b
None
LDA, most GGAs and hybrids
Meta-GGAs; B95- and B97-based functionals
Minnesota functionals

a
b

Number in parenthesis is the fraction of points retained from the parent grid
Reflects Q-C HEM versions since v. 4.4.2

Table 5.3: Standard quadrature grids available in Q-C HEM, along with the number of grid points for a carbon atom,
showing the reduction in grid points due to pruning.
are available in the peer-reviewed literature, 50,59,67 to facilitate reproduction of Q-C HEM DFT calculations using other
electronic structure programs. Just as computed energies may vary quite strongly with the choice of basis set, so too in
DFT may they vary strongly with the choice of quadrature grid. In publications, users should always specify the grid
that is used, and it is suggested to cite the appropriate literature reference from Table 5.3.
The SG-0 and SG-1 grids are designed for calculations on large molecules using GGA functionals. SG-1 affords
integration errors on the order of ∼0.2 kcal/mol for medium-sized molecules and GGA functionals, including for
demanding test cases such as reaction enthalpies for isomerizations. (Integration errors in total energies are no more
than a few µhartree, or ∼0.01 kcal/mol.) The SG-0 grid was derived in similar fashion, and affords a root-mean-square
error in atomization energies of 72 µhartree with respect to SG-1, while relative energies are reproduced well. 50 In
either case, errors of this magnitude are typically considerably smaller than the intrinsic errors in GGA energies, and
hence acceptable. As seen in Table 5.3, SG-1 retains < 40% of the grid points of its parent grid, which translates
directly into cost savings.
Both SG-0 and SG-1 were optimized so that the integration error in the energy falls below a target threshold, but
derivatives of the energy (including such properties as (hyper)polarizabilities 40 ) are often more sensitive to the quality
of the integration grid. Special care is required, for example, when imaginary vibrational frequencies are encountered,
as low-frequency (but real) vibrational frequencies can manifest as imaginary if the grid is sparse. If imaginary frequencies are found, or if there is some doubt about the frequencies reported by Q-C HEM, the recommended procedure
is to perform the geometry optimization and vibrational frequency calculations again using a higher-quality grid. (The
optimization should converge quite quickly if the previously-optimized geometry is used as an initial guess.)
SG-1 was the default DFT integration grid for all density functionals for Q-C HEM versions 3.2–4.4. Beginning with
Q-C HEM v. 4.4.2, however, the default grid is functional-dependent, as summarized in Table 5.3. This is a reflection
of the fact that although SG-1 is adequate for energy calculations using most GGA and hybrid functionals (although
care must be taken for some other properties, as discussed below), it is not adequate to integrate many functionals
developed since ∼2005. These include meta-GGAs, which are more complicated due to their dependence on the kinetic
energy density (τσ in Eq. (5.10)) and/or the Laplacian of the density (∇2 ρσ ). Functionals based on B97, along with
the Minnesota suite of functionals, 248,249 contain relatively complicated expressions for the exchange inhomogeneity
factor, and are therefore also more sensitive to the quality of the integration grid. 59,133,220 To integrate these modern
density functionals, the SG-2 and SG-3 grids were developed, 59 which are pruned versions of the medium-quality
(75, 302) and high-quality (99, 590) integration grids, respectively. Tests of properties known to be highly sensitive to
the quality of the integration grid, such as vibrational frequencies, hyper-polarizabilities, and potential energy curves
for non-bonded interactions, demonstrate that SG-2 is usually adequate for meta-GGAs and B97-based functionals, and
in many cases is essentially converged with respect to an unpruned (250, 974) grid. 59 The Minnesota functionals are
more sensitive to the grid, and while SG-3 is often adequate, it is not completely converged in the case of non-bonded
interactions. 59

Chapter 5: Density Functional Theory

145

Note: (1) SG-0 was re-optimized for Q-C HEM v. 3.0, so results may differ slightly as compared to older versions of
the program.
(2) The SG-2 and SG-3 grids use a double-exponential radial quadrature, 59 whereas a general grid (selected
by setting XC_GRID = XY , as described in Section 5.4) uses an Euler-MacLaurin radial quadrature. As such,
absolute energies cannot be compared between, e.g., SG-2 and XC_GRID = 000075000302, even though SG-2
uses a pruned (75, 302) grid. However, energy differences should be quite similar between the two.

5.5.3

Consistency Check and Cutoffs

Whenever Q-C HEM calculates numerical density functional integrals, the electron density itself is also integrated numerically as a test of the quality of the numerical quadrature. The extent to which this numerical result differs from
the number of electrons is an indication of the accuracy of the other numerical integrals. A warning message is printed
whenever the relative error in the numerical electron count reaches 0.01%, indicating that the numerical XC results
may not be reliable. If the warning appears on the first SCF cycle it is probably not serious, because the initial-guess
density matrix is sometimes not idempotent. This is the case with the SAD guess discussed in Section 4.4, and also
with a density matrix that is taken from a previous geometry optimization cycle, and in such cases the problem will
likely correct itself in subsequent SCF iterations. If the warning persists, however, then one should consider either
using a finer grid or else selecting an alternative initial guess.
By default, Q-C HEM will estimate the magnitude of various XC contributions on the grid and eliminate those determined to be numerically insignificant. Q-C HEM uses specially-developed cutoff procedures which permits evaluation
of the XC energy and potential in only O(N ) work for large molecules. This is a significant improvement over the
formal O(N 3 ) scaling of the XC cost, and is critical in enabling DFT calculations to be carried out on very large
systems. In rare cases, however, the default cutoff scheme can be too aggressive, eliminating contributions that should
be retained; this is almost always signaled by an inaccurate numerical density integral. An example of when this could
occur is in calculating anions with multiple sets of diffuse functions in the basis. A remedy may be to increase the size
of the quadrature grid.

5.5.4

Multi-resolution Exchange-Correlation (MRXC) Method

The multi-resolution exchange-correlation (MRXC) method is a new approach, courtesy of the Q-C HEM development
team, 48,107,180 for accelerating computation of the exchange-correlation (XC) energy and matrix for any given density
functional. As explained in Section 4.6.5, XC functionals are sufficiently complicated integration of them is usually
performed by numerical quadrature. There are two basic types of quadrature. One is the atom-centered grid (ACG), a
superposition of atomic quadrature described in Section 4.6.5. The ACG has high density of points near the nucleus to
handle the compact core density and low density of points in the valence and non-bonding region where the electron
density is smooth. The other type is even-spaced cubic grid (ESCG), which is typically used together with pseudopotentials and plane-wave basis functions where only the valence and non-bonded electron density is assumed smooth.
In quantum chemistry, an ACG is more often used as it can handle accurately all-electron calculations of molecules.
MRXC combines those two integration schemes seamlessly to achieve an optimal computational efficiency by placing
the calculation of the smooth part of the density and XC matrix onto the ESCG. The computation associated with the
smooth fraction of the electron density is the major bottleneck of the XC part of a DFT calculation and can be done at a
much faster rate on the ESCG due to its low resolution. Fast Fourier transform and B-spline interpolation are employed
for the accurate transformation between the two types of grids such that the final results remain the same as they would
be on the ACG alone, yet a speedup of several times is achieved for the XC matrix. The smooth part of the calculation
with MRXC can also be combined with FTC (see Section 4.6.5) to achieve a further gain in efficiency.

Chapter 5: Density Functional Theory

146

MRXC
Controls the use of MRXC.
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
0 Do not use MRXC
1 Use MRXC in the evaluation of the XC part
RECOMMENDATION:
MRXC is very efficient for medium and large molecules, especially when medium and large
basis sets are used.

The following two keywords control the smoothness precision. The default value is carefully selected to maintain high
accuracy.
MRXC_CLASS_THRESH_MULT
Controls the of smoothness precision
TYPE:
INTEGER
DEFAULT:
1
OPTIONS:
im An integer
RECOMMENDATION:
A prefactor in the threshold for MRXC error control: im × 10−io
MRXC_CLASS_THRESH_ORDER
Controls the of smoothness precision
TYPE:
INTEGER
DEFAULT:
6
OPTIONS:
io An integer
RECOMMENDATION:
The exponent in the threshold of the MRXC error control: im × 10−io

The next keyword controls the order of the B-spline interpolation:
LOCAL_INTERP_ORDER
Controls the order of the B-spline
TYPE:
INTEGER
DEFAULT:
6
OPTIONS:
n An integer
RECOMMENDATION:
The default value is sufficiently accurate

Chapter 5: Density Functional Theory

5.5.5

147

Incremental DFT

Incremental DFT (IncDFT) uses the difference density and functional values to improve the performance of the DFT
quadrature procedure by providing a better screening of negligible values. Using this option will yield improved
efficiency at each successive iteration due to more effective screening.
INCDFT
Toggles the use of the IncDFT procedure for DFT energy calculations.
TYPE:
LOGICAL
DEFAULT:
TRUE
OPTIONS:
FALSE Do not use IncDFT
TRUE
Use IncDFT
RECOMMENDATION:
Turning this option on can lead to faster SCF calculations, particularly towards the end of the
SCF. Please note that for some systems use of this option may lead to convergence problems.
INCDFT_DENDIFF_THRESH
Sets the threshold for screening density matrix values in the IncDFT procedure.
TYPE:
INTEGER
DEFAULT:
SCF_CONVERGENCE + 3
OPTIONS:
n Corresponding to a threshold of 10−n .
RECOMMENDATION:
If the default value causes convergence problems, set this value higher to tighten the threshold.
INCDFT_GRIDDIFF_THRESH
Sets the threshold for screening functional values in the IncDFT procedure
TYPE:
INTEGER
DEFAULT:
SCF_CONVERGENCE + 3
OPTIONS:
n Corresponding to a threshold of 10−n .
RECOMMENDATION:
If the default value causes convergence problems, set this value higher to tighten the threshold.

Chapter 5: Density Functional Theory

148

INCDFT_DENDIFF_VARTHRESH
Sets the lower bound for the variable threshold for screening density matrix values in the IncDFT
procedure. The threshold will begin at this value and then vary depending on the error in the
current SCF iteration until the value specified by INCDFT_DENDIFF_THRESH is reached. This
means this value must be set lower than INCDFT_DENDIFF_THRESH.
TYPE:
INTEGER
DEFAULT:
0 Variable threshold is not used.
OPTIONS:
n Corresponding to a threshold of 10−n .
RECOMMENDATION:
If the default value causes convergence problems, set this value higher to tighten accuracy. If this
fails, set to 0 and use a static threshold.
INCDFT_GRIDDIFF_VARTHRESH
Sets the lower bound for the variable threshold for screening the functional values in the IncDFT
procedure. The threshold will begin at this value and then vary depending on the error in the
current SCF iteration until the value specified by INCDFT_GRIDDIFF_THRESH is reached. This
means that this value must be set lower than INCDFT_GRIDDIFF_THRESH.
TYPE:
INTEGER
DEFAULT:
0 Variable threshold is not used.
OPTIONS:
n Corresponding to a threshold of 10−n .
RECOMMENDATION:
If the default value causes convergence problems, set this value higher to tighten accuracy. If this
fails, set to 0 and use a static threshold.

5.6

Range-Separated Hybrid Density Functionals

Whereas RSH functionals such as LRC-ωPBE are attempts to add 100% LR Hartree-Fock exchange with minimal
perturbation to the original functional (PBE, in this example), other RSH functionals are of a more empirical nature
and their range-separation parameters have been carefully parameterized along with all of the other parameters in the
functional. These cases are functionals are discussed first, in Section 5.6.1, because their range-separation parameters
should be taken as fixed. User-defined values of the range-separation parameter are discussed in Section 5.6.2, and
Section 5.6.3 discusses a procedure for which an optimal, system-specific value of this parameter (ω or µ) can be
chosen for functionals such as LRC-ωPBE or LRC-µPBE.

5.6.1

Semi-Empirical RSH Functionals

Semi-empirical RSH functionals for which the range-separation parameter should be considered fixed include the
ωB97, ωB97X, and ωB97X-D functionals developed by Chai and Head-Gordon; 44,45 ωB97X-V and ωB97M-V from
Mardirossian and Head-Gordon; 133,135 M11 from Peverati and Truhlar; 164 ωB97X-D3, ωM05-D, and ωM06-D3 from
Chai and coworkers; 125,126 and the screened exchange functionals N12-SX and MN12-SX from Truhlar and coworkers. 168 More recently, Mardirossian and Head-Gordon developed two RSH functionals, ωB97X-V and ωB97M-V,
via a combinatorial approach by screening over 100,000 possible functionals in the first case and over 10 billion possible functionals in the second case. Both of the latter functionals use the VV10 non-local correlation functional in
order to improve the description of non-covalent interactions and isomerization energies. ωB97M-V is a 12-parameter

Chapter 5: Density Functional Theory

149

meta-GGA with 15% short-range exact exchange and 100% long-range exact exchange and is one of the most accurate functionals available through rung 4 of Jacob’s Ladder, across a wide variety of applications. This has been
verified by benchmarking the functional on nearly 5000 data points against over 200 alternative functionals available
in Q-C HEM. 135

5.6.2

User-Defined RSH Functionals

As pointed out in Ref. 64 and elsewhere, the description of charge-transfer excited states within density functional
theory (or more precisely, time-dependent DFT, which is discussed in Section 7.3) requires full (100%) non-local HF
exchange, at least in the limit of large donor–acceptor distance. Hybrid functionals such as B3LYP 20,190 and PBE0 8
that are well-established and in widespread use, however, employ only 20% and 25% HF exchange, respectively. While
these functionals provide excellent results for many ground-state properties, they cannot correctly describe the distance
dependence of charge-transfer excitation energies, which are enormously underestimated by most common density
functionals. This is a serious problem in any case, but it is a catastrophic problem in large molecules and in noncovalent clusters, where TDDFT often predicts a near-continuum of spurious, low-lying charge transfer states. 113,114
The problems with TDDFT’s description of charge transfer are not limited to large donor–acceptor distances, but have
been observed at ∼2 Å separation, in systems as small as uracil–(H2 O)4 . 113 Rydberg excitation energies also tend to
be substantially underestimated by standard TDDFT.
One possible avenue by which to correct such problems is to parameterize functionals that contain 100% HF exchange, though few such functionals exist to date. An alternative option is to attempt to preserve the form of common
GGAs and hybrid functionals at short range (i.e., keep the 25% HF exchange in PBE0) while incorporating 100%
HF exchange at long range, which provides a rigorously correct description of the long-range distance dependence
of charge-transfer excitation energies, but aims to avoid contaminating short-range exchange-correlation effects with
additional HF exchange. The separation is accomplished using the range-separation ansatz that was introduced in Section 5.3. In particular, functionals that use 100% HF exchange at long range (cx,LR = 1 in Eq. (5.13)) are known as
“long-range-corrected” (LRC) functionals. An LRC version of PBE0 would, for example, have cx,SR = 0.25.
To fully specify an LRC functional, one must choose a value for the range separation parameter ω in Eq. (5.12). In
the limit ω → 0, the LRC functional in Eq. (5.13) reduces to a non-RSH functional where there is no “SR” or “LR”,
−1
because all exchange and correlation energies are evaluated using the full Coulomb operator, r12
. Meanwhile the
HF
RSH
ω → ∞ limit corresponds to a new functional, Exc = Ec + Ex . Full HF exchange is inappropriate for use with
most contemporary GGA correlation functionals, so the latter limit is expected to perform quite poorly. Values of
ω > 1.0 bohr−1 are likely not worth considering, according to benchmark tests. 115,178
Evaluation of the short- and long-range HF exchange energies is straightforward, 10 so the crux of any RSH functional
is the form of the short-range GGA exchange functional, and several such functionals are available in Q-C HEM. These
include short-range variants of the B88 and PBE exchange described by Hirao and co-workers, 92,188 called µB88 and
µPBE in Q-C HEM, 177 and an alternative formulation of short-range PBE exchange proposed by Scuseria and coworkers, 85 which is known as ωPBE. These functionals are available in Q-C HEM thanks to the efforts of the Herbert
group. 178,179 By way of notation, the terms “µPBE”, “ωPBE”, etc., refer only to the short-range exchange functional,
DFT
Ex,SR
in Eq. (5.13). These functionals could be used in “screened exchange” mode, as described in Section 5.3, as for
example in the HSE03 functional, 87 therefore the designation “LRC-ωPBE”, for example, should only be used when
the short-range exchange functional ωPBE is combined with 100% Hartree-Fock exchange in the long range.
In general, LRC-DFT functionals have been shown to remove the near-continuum of spurious charge-transfer excited
states that appear in large-scale TDDFT calculations. 115 However, certain results depend sensitively upon the value
of the range-separation parameter ω, 114,115,178,179,204 especially in TDDFT calculations (Section 7.3) and therefore the
results of LRC-DFT calculations must therefore be interpreted with caution, and probably for a range of ω values. This
can be accomplished by requesting a functional that contains some short-range GGA exchange functional (ωPBE or
µPBE, in the examples mentioned above), in combination with setting the $rem variable LRC_DFT = TRUE, which
requests the addition of 100% Hartree-Fock exchange in the long-range. Basic job-control variables and an example
can be found below. The value of the range-separation parameter is then controlled by the variable OMEGA, as shown
in the examples below.

Chapter 5: Density Functional Theory

150

LRC_DFT
Controls the application of long-range-corrected DFT
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
FALSE (or 0) Do not apply long-range correction.
TRUE (or 1)
Add 100% long-range Hartree-Fock exchange to the requested functional.
RECOMMENDATION:
The $rem variable OMEGA must also be specified, in order to set the range-separation parameter.
OMEGA
Sets the range-separation parameter, ω, also known as µ, in functionals based on Hirao’s RSH
scheme.
TYPE:
INTEGER
DEFAULT:
No default
OPTIONS:
n Corresponding to ω = n/1000, in units of bohr−1
RECOMMENDATION:
None
COMBINE_K
Controls separate or combined builds for short-range and long-range K
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
FALSE (or 0) Build short-range and long-range K separately (twice as expensive as a global hybrid)
TRUE (or 1)
Build short-range and long-range K together (≈ as expensive as a global hybrid)
RECOMMENDATION:
Most pre-defined range-separated hybrid functionals in Q-C HEM use this feature by default.
However, if a user-specified RSH is desired, it is necessary to manually turn this feature on.

151

Chapter 5: Density Functional Theory

Example 5.4 Application of LRC-BOP to (H2 O)−
2.
$comment
The value of omega is 0.47 by default but can
be overwritten by specifying OMEGA.
$end
$molecule
-1 2
O
H
H
O
H
H
$end

1.347338
1.824285
1.805176
-1.523051
-0.544777
-1.682218

$rem
EXCHANGE
BASIS
LRC_DFT
OMEGA
$end

-0.017773
0.813088
-0.695567
-0.002159
-0.024370
0.174228

-0.071860
0.117645
0.461913
-0.090765
-0.165445
0.849364

LRC-BOP
6-31(1+,3+)G*
TRUE
300
! = 0.300 bohr**(-1)

Rohrdanz et al. 179 published a thorough benchmark study of both ground- and excited-state properties using the LRCωPBEh functional, in which the “h” indicates a short-range hybrid (i.e., the presence of some short-range HF exchange).
Empirically-optimized parameters of cx,SR = 0.2 (see Eq. (5.13)) and ω = 0.2 bohr−1 were obtained, 179 and these
parameters are taken as the defaults for LRC-ωPBEh. Caution is warranted, however, especially in TDDFT calculations
for large systems, as excitation energies for states that exhibit charge-transfer character can be rather sensitive to the
precise value of ω. 114,179 In such cases (and maybe in general), the “tuning” procedure described in Section 5.6.3 is

152

Chapter 5: Density Functional Theory
recommended.
Example 5.5 Application of LRC-ωPBEh to the C2 H4 –C2 F4 dimer at 5 Å separation.
$comment
This example uses the "optimal" parameter set discussed above.
It can also be run by setting METHOD = LRC-wPBEh.
$end
$molecule
0 1
C
C
H
H
H
H
C
C
F
F
F
F
$end

0.670604
-0.670604
1.249222
1.249222
-1.249222
-1.249222
0.669726
-0.669726
1.401152
1.401152
-1.401152
-1.401152

$rem
EXCHANGE
BASIS
LRC_DFT
OMEGA
CIS_N_ROOTS
CIS_TRIPLETS
$end

0.000000
0.000000
0.929447
-0.929447
0.929447
-0.929447
0.000000
0.000000
1.122634
-1.122634
-1.122634
1.122634

GEN
6-31+G*
TRUE
200
4
FALSE

0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
5.000000
5.000000
5.000000
5.000000
5.000000
5.000000

! = 0.2 a.u.

$xc_functional
C PBE
1.00
X wPBE 0.80
X HF
0.20
$end

Both LRC functionals and also the asymptotic corrections that will be discussed in Section 5.10.1 are thought to reduce
self-interaction error in approximate DFT. A convenient way to quantify—or at least depict—this error is by plotting
the DFT energy as a function of the (fractional) number of electrons, N , because E(N ) should in principle consist of
a sequence of line segments with abrupt changes in slope (the so-called derivative discontinuity 53,141 ) at integer values
of N , but in practice these E(N ) plots bow away from straight-line segments. 53 Examination of such plots has been
suggested as a means to adjust the fraction of short-range exchange in an LRC functional, 13 while the range-separation

153

Chapter 5: Density Functional Theory
parameter is tuned as described in Section 5.6.3.

Example 5.6 Example of a DFT job with a fractional number of electrons. Here, we make the −1.x anion of fluoride
by subtracting a fraction of an electron from the HOMO of F2− .
$comment
Subtracting a whole electron recovers the energy of F-.
Adding electrons to the LUMO is possible as well.
$end
$rem
EXCHANGE
BASIS
FRACTIONAL_ELECTRON
GEN_SCFMAN
$end

b3lyp
6-31+G*
-500
! divide by 1000 to get the fraction, -0.5 here.
FALSE ! not yet available in new scf code

$molecule
-2 2
F
$end

FRACTIONAL_ELECTRON
Add or subtract a fraction of an electron.
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
0 Use an integer number of electrons.
n Add n/1000 electrons to the system.
RECOMMENDATION:
Use only if trying to generate E(N ) plots. If n < 0, a fraction of an electron is removed from
the system.

5.6.3

Tuned RSH Functionals

Whereas the range-separation parameters for the functionals described in Section 5.6.1 are wholly empirical in nature
and should not be adjusted, for the functionals described in Section 5.6.2 some adjustment was suggested, especially
for TDDFT calculations and for any properties that require interpretation of orbital energies, e.g., HOMO/LUMO gaps.
This adjustment can be performed in a non-empirical, albeit system-specific way, by “tuning” the value of ω in order
to satisfy certain criteria that ought rigorously to be satisfied by an exact density functional.
System-specific optimization of ω is based on the Koopmans condition that would be satisfied for the exact density
functional, namely, that for an N -electron molecule 14
− εHOMO (N ) = IE(N ) ≡ E(N − 1) − E(N ) .

(5.17)

In other words, the HOMO eigenvalue should be equal to minus the ionization energy (IE), where the latter is defined
by a ∆SCF procedure. 205,231 When an RSH functional is used, all of the quantities in Eq. (5.17) are ω-dependent, so
this parameter is adjusted until the condition in Eq. (5.17) is met, which requires a sequence of SCF calculations on both
the neutral and ionized species, using different values of ω. For proper description of charge-transfer states, Baer and
co-workers 14 suggest finding the value of ω that (to the extent possible) satisfies both Eq. (5.17) for the neutral donor
molecule and its analogue for the anion of the acceptor species. Note that for a given approximate density functional,
there is no guarantee that the IE condition can actually be satisfied for any value of ω, but in practice it usually can
be, and published benchmarks suggest that this system-specific approach affords the most accurate values of IEs and

Chapter 5: Density Functional Theory

154

TDDFT excitation energies. 14,130,183 It should be noted, however, that the optimal value of ω can very dramatically with
− 204
system size, e.g., it is very different for the cluster anion (H2 O)−
6 than it is for cluster (H2 O)70 .
A script that optimizes ω, called OptOmegaIPEA.pl, is located in the $QC/bin directory. The script scans ω over
the range 0.1–0.8 bohr−1 , corresponding to values of the $rem variable OMEGA in the range 100–800. See the script
for the instructions how to modify the script to scan over a wider range. To execute the script, you need to create three
inputs for a BNL job using the same geometry and basis set, for a neutral molecule (N.in), its anion (M.in), and its
cation (P.in), and then run the command
OptOmegaIPEA.pl >& optomega
which both generates the input files (N_*, P_*, M_*) and runs Q-C HEM on them, writing the optimization output into
optomega. This script applies the IE condition to both the neutral molecule and its anion, minimizing the sum of
(IE + εHOMO )2 for these two species. A similar script, OptOmegaIP.pl, uses Eq. (5.17) for the neutral molecule
only.
Note: (i) If the system does not have positive EA, then the tuning should be done according to the IP condition only.
The IP/EA script will yield an incorrect value of ω in such cases.
(ii) In order for the scripts to work, one must specify SCF_FINAL_PRINT = 1 in the inputs. The scripts look for
specific regular expressions and will not work correctly without this keyword.
Although the tuning procedure was originally developed by Baer and co-workers using the BNL functional, 14,130,183 it
has more recently been applied using functionals such as LRC-ωPBE (see, e.g., Ref. 204), and the scripts will work
with functionals other than BNL.

5.6.4

Tuned RSH Functionals Based on the Global Density-Dependent Condition

The value of range-separation parameter based on IP tuning procedure (ωIP ) exhibits a troublesome dependence on system size. 57,65,146,204,208 An alternative method to select ω is the global density-dependent (GDD) tuning procedure, 139
in which the optimal value
ωGDD = Chd2x i−1/2
(5.18)
is related to the average of the distance dx between an electron in the outer regions of a molecule and the exchange
hole in the region of localized valence orbitals. The quantity C is an empirical parameter for a given LRC functional,
which was determined for LRC-ωPBE (C = 0.90) and LRC-ωPBEh (C = 0.75) using the def2-TZVPP basis set. 139
(A slightly different value, C = 0.885, was determined for Q-C HEM’s implementation of LRC-ωPBE. 116 ) Since LRCωPBE(ωGDD ) provides a better description of polarizabilities in polyacetylene as compared to ωIP 84 , it is anticipated
that using ωGDD in place of ωIP may afford more accurate molecular properties, especially in conjugated systems.
GDD tuning of an RSH functional is involving by setting the $rem variable OMEGA_GDD = TRUE. The electron
density is obviously needed to compute ωGDD in Eq. (5.18) and this is accomplished using the converged SCF density
computed using the RSH functional with the value of ω given by the $rem variable OMEGA. The value of ωGDD
therefore depends, in principle, upon the value of OMEGA, although in practice it is not very sensitive to this value.
OMEGA_GDD
Controls the application of ωGDD tuning for long-range-corrected DFT
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
FALSE (or 0) Do not apply ωGDD tuning.
TRUE (or 1)
Use ωGDD tuning.
RECOMMENDATION:
The $rem variable OMEGA must also be specified, in order to set the initial range-separation
parameter.

Chapter 5: Density Functional Theory

155

OMEGA_GDD_SCALING
Sets the empirical constant C in ωGDD tuning procedure.
TYPE:
INTEGER
DEFAULT:
885
OPTIONS:
n Corresponding to C = n/1000.
RECOMMENDATION:
The quantity n = 885 was determined by Lao and Herbert in Ref. 116 using LRC-ωPBE and def2TZVPP augmented with diffuse functions on non-hydrogen atoms that are taken from Dunning’s
aug-cc-pVTZ basis set.

Example 5.7 Sample input illustrating a calculation to determine the ω value for LRC-ωPBE based on the ωGDD
tuning procedure.
$comment
The initial omega value has to set.
$end
$rem
exchange
basis
lrc_dft
omega
omega_gdd
$end

gen
aug-cc-pvdz
true
300
true

$xc_functional
x
wPBE
1.0
c
PBE
1.0
$end
$molecule
0 1
O -0.042500 0.091700 0.110000
H 0.749000 0.556800 0.438700
H -0.825800 0.574700 0.432500
$end

5.7

DFT Methods for van der Waals Interactions

This section describes five different procedures for obtaining a better description of dispersion (van der Waals) interactions in DFT calculations: non-local correlation functionals (Section 5.7.1), empirical atom–atom dispersion potentials
(“DFT-D”, Section 5.7.2), the Becke-Johnson exchange-dipole model (XDM, Section 5.7.3), the Tkatchenko-Scheffler
van der Waals method (TS-vdW, Section 5.7.4), and finally the many-body dispersion method (MBD, Section 5.7.5).

5.7.1

Non-Local Correlation (NLC) Functionals

From the standpoint of the electron density, the vdW interaction is a non-local one: even for two non-overlapping,
spherically-symmetric charge densities (two argon atoms, say), the presence of molecule B in the non-covalent A· · · B
complex induces ripples in the tail of A’s charge distribution, which are the hallmarks of non-covalent interactions. 56
(This is the fundamental idea behind the non-covalent interaction plots described in Section 11.5.5; the vdW interaction
manifests as large density gradients in regions of space where the density itself is small.) Semi-local GGAs that depend

Chapter 5: Density Functional Theory

156

only on the density and its gradient cannot describe this long-range, correlation-induced interaction, and meta-GGAs at
best describe it at middle-range via the Laplacian of the density and/or the kinetic energy density. A proper description
of long-range electron correlation requires a non-local functional, i.e., an exchange-correlation potential having the
form
Z
vcnl (r) = f (r, r0 ) dr0 .
(5.19)
In this way, a perturbation at a point r0 (due to B, say) then induces an exchange-correlation potential at a (possibly
far-removed) point r (on A).
Q-C HEM includes four such functionals that can describe dispersion interactions:
• vdW-DF-04, developed by Langreth, Lundqvist, and coworkers, 60,61 implemented as described in Ref. 216.
• vdW-DF-10 (also known as vdW-DF2), which is a re-parameterization of vdW-DF-04. 122
• VV09, developed 213 and implemented 214 by Vydrov and Van Voorhis.
• VV10 by Vydrov and Van Voorhis. 215
• rVV10 by Sabatini and coworkers. 182
Each of these functionals is implemented in a self-consistent manner, and analytic gradients with respect to nuclear
displacements are available. 214–216 The non-local correlation is governed by the $rem variable NL_CORRELATION,
which can be set to one of the four values: vdW-DF-04, vdW-DF-10, VV09, or VV10. The vdW-DF-04, vdW-DF-10, and
VV09 functionals are used in combination with LSDA correlation, which must be specified explicitly. For instance,
vdW-DF-10 is invoked by the following keyword combination:
CORRELATION
NL_CORRELATION

PW92
vdW-DF-10

VV10 is used in combination with PBE correlation, which must be added explicitly. In addition, the values of two
parameters, C and b (see Ref. 216), must be specified for VV10. These parameters are controlled by the $rem variables
NL_VV_C and NL_VV_B, respectively. For instance, to invoke VV10 with C = 0.0093 and b = 5.9, the following
input is used:
CORRELATION
NL_CORRELATION
NL_VV_C
NL_VV_B

PBE
VV10
93
590

The variable NL_VV_C may also be specified for VV09, where it has the same meaning. By default, C = 0.0089 is
used in VV09 (i.e. NL_VV_C is set to 89). However, in VV10 neither C nor b are assigned a default value and must
always be provided in the input.
Unlike local (LSDA) and semi-local (GGA and meta-GGA) functionals, for non-local functionals evaluation of the
correlation energy requires a double integral over the spatial variables, as compared to the single integral [Eq. (5.8)]
required for semi-local functionals:
Z
Z
Ecnl = vcnl (r) dr = f (r, r0 ) ρ(r) dr dr0 .
(5.20)
In practice, this double integration is performed numerically on a quadrature grid. 214–216 By default, the SG-1 quadrature (described in Section 5.5.2 below) is used to evaluate Ecnl , but a different grid can be requested via the $rem
variable NL_GRID. The non-local energy is rather insensitive to the fineness of the grid such that SG-1 or even SG-0
grids can be used in most cases, but a finer grid may be required to integrate other components of the functional. This
is controlled by the XC_GRID variable discussed in Section 5.5.2.

157

Chapter 5: Density Functional Theory

The two functionals originally developed by Vydrov and Van Voorhis can be requested by specifying METHOD =
VV10 or METHOD LC-VV10. In addition, the combinatorially-optimized functionals of Mardirossian and Head-Gordon
(ωB97X-V, B97M-V, and ωB97M-V) make use of non-local correlation and can be invoked by setting METHOD to
wB97X-V, B97M-V, or wB97M-V.
Example 5.8 Geometry optimization of the methane dimer using VV10 with rPW86 exchange.
$molecule
0 1
C
0.000000
H -0.888551
H
0.888551
H
0.000000
H
0.000000
C
0.000000
H
0.000000
H -0.888551
H
0.888551
H
0.000000
$end
$rem
JOBTYPE
BASIS
EXCHANGE
CORRELATION
XC_GRID
NL_CORRELATION
NL_GRID
NL_VV_C
NL_VV_B
$end

-0.000140
0.513060
0.513060
-1.026339
0.000089
0.000140
-0.000089
-0.513060
-0.513060
1.026339

1.859161
1.494685
1.494685
1.494868
2.948284
-1.859161
-2.948284
-1.494685
-1.494685
-1.494868

opt
aug-cc-pVTZ
rPW86
PBE
2
VV10
1
93
590

In the above example, the SG-2 grid is used to evaluate the rPW86 exchange and PBE correlation, but a coarser SG-1
grid is used for the non-local part of VV10. Furthermore, the above example is identical to specifying METHOD = VV10.
NL_CORRELATION
Specifies a non-local correlation functional that includes non-empirical dispersion.
TYPE:
STRING
DEFAULT:
None No non-local correlation.
OPTIONS:
None
No non-local correlation
vdW-DF-04 the non-local part of vdW-DF-04
vdW-DF-10 the non-local part of vdW-DF-10 (also known as vdW-DF2)
VV09
the non-local part of VV09
VV10
the non-local part of VV10
RECOMMENDATION:
Do not forget to add the LSDA correlation (PW92 is recommended) when using vdW-DF-04,
vdW-DF-10, or VV09. VV10 should be used with PBE correlation. Choose exchange functionals carefully: HF, rPW86, revPBE, and some of the LRC exchange functionals are among the
recommended choices.

Chapter 5: Density Functional Theory

158

NL_VV_C
Sets the parameter C in VV09 and VV10. This parameter is fitted to asymptotic van der Waals
C6 coefficients.
TYPE:
INTEGER
DEFAULT:
89
for VV09
No default for VV10
OPTIONS:
n Corresponding to C = n/10000
RECOMMENDATION:
C = 0.0093 is recommended when a semi-local exchange functional is used. C = 0.0089 is
recommended when a long-range corrected (LRC) hybrid functional is used. For further details
see Ref. 215.
NL_VV_B
Sets the parameter b in VV10. This parameter controls the short range behavior of the non-local
correlation energy.
TYPE:
INTEGER
DEFAULT:
No default
OPTIONS:
n Corresponding to b = n/100
RECOMMENDATION:
The optimal value depends strongly on the exchange functional used. b = 5.9 is recommended
for rPW86. For further details see Ref. 215.
USE_RVV10
Used to turn on the rVV10 NLC functional
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
FALSE Use VV10 NLC (the default for NL_CORRELATION)
TRUE
Use rVV10 NLC
RECOMMENDATION:
Set to TRUE if the rVV10 NLC is desired.

5.7.2

Empirical Dispersion Corrections: DFT-D

A major development in DFT during the mid-2000s was the recognition that, first of all, semi-local density functionals
do not properly capture dispersion (van der Waals) interactions, a problem that has been addressed only much more
recently by the non-local correlation functionals discussed in Section 5.7.1; and second, that a cheap and simple solution
to this problem is to incorporate empirical potentials of the form −C6 /R6 , where the C6 coefficients are pairwise
atomic parameters. This approach, which is an alternative to the use of a non-local correlation functional, is known as
dispersion-corrected DFT (DFT-D). 76,80
There are currently three unique DFT-D methods in Q-C HEM. These are requested via the $rem variable DFT_D and
are discussed below.

159

Chapter 5: Density Functional Theory
DFT_D
Controls the empirical dispersion correction to be added to a DFT calculation.
TYPE:
LOGICAL
DEFAULT:
None
OPTIONS:
FALSE
(or 0) Do not apply the DFT-D2, DFT-CHG, or DFT-D3 scheme
EMPIRICAL_GRIMME
DFT-D2 dispersion correction from Grimme 75
EMPIRICAL_CHG
DFT-CHG dispersion correction from Chai and Head-Gordon 45
EMPIRICAL_GRIMME3 DFT-D3(0) dispersion correction from Grimme (deprecated as
of Q-C HEM 5.0)
D3_ZERO
DFT-D3(0) dispersion correction from Grimme et al. 77
D3_BJ
DFT-D3(BJ) dispersion correction from Grimme et al. 78
D3_CSO
DFT-D3(CSO) dispersion correction from Schröder et al. 184
D3_ZEROM
DFT-D3M(0) dispersion correction from Smith et al. 187
D3_BJM
DFT-D3M(BJ) dispersion correction from Smith et al. 187
D3_OP
DFT-D3(op) dispersion correction from Witte et al. 224
D3
Automatically select the "best" available D3 dispersion correction
RECOMMENDATION:
Use the D3 option, which selects the empirical potential based on the density functional specified
by the user.

The oldest of these approaches is DFT-D2, 75 in which the empirical dispersion potential has the aforementioned form,
namely, pairwise atomic −C/R6 terms:
atoms
X atoms
X  C6,AB 
D2
D2
fdamp
(RAB ) .
(5.21)
Edisp = −s6
6
RAB
A

This function is damped at short range, where

B3.0.CO;2-K.
[11] A. Ambrosetti, A. M. Reilly, R. A. DiStasio, Jr., and A. Tkatchenko. J. Chem. Phys., 140:18A508, 2014. DOI:
10.1063/1.4865104.
[12] R. Armiento and S. Kümmel. Phys. Rev. Lett., 111:036402, 2013. DOI: 10.1103/PhysRevLett.111.036402.
[13] J. Autschbach and M. Srebro. Acc. Chem. Res., 47:2592, 2014. DOI: 10.1021/ar500171t.
[14] R. Baer, E. Livshits, and U. Salzner.
nurev.physchem.012809.103321.

Annu. Rev. Phys. Chem., 61:85, 2010.

DOI: 10.1146/an-

[15] D. Barton, K. U. Lao, R. A. DiStasio, Jr., and A. Tkatchenko. Tkatchenko-Scheffler and many-body dispersion
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[20] A. D. Becke. J. Chem. Phys., 98:5648, 1993. DOI: 10.1063/1.464913.
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Chapter 6

Wave Function-Based Correlation Methods
6.1

Introduction

The Hartree-Fock procedure, while often qualitatively correct, is frequently quantitatively deficient. The deficiency
is due to the underlying assumption of the Hartree-Fock approximation: that electrons move independently within
molecular orbitals subject to an averaged field imposed by the remaining electrons. The error that this introduces is
called the correlation energy and a wide variety of procedures exist for estimating its magnitude. The purpose of this
Chapter is to introduce the main wave function-based methods available in Q-C HEM to describe electron correlation.
Wave function-based electron correlation methods concentrate on the design of corrections to the wave function beyond
the mean-field Hartree-Fock description. This is to be contrasted with the density functional theory methods discussed
in the previous Chapter. While density functional methods yield a description of electronic structure that accounts
for electron correlation subject only to the limitations of present-day functionals (which, for example, omit dispersion
interactions), DFT cannot be systematically improved if the results are deficient. Wave function-based approaches for
describing electron correlation 4,5 offer this main advantage. Their main disadvantage is relatively high computational
cost, particularly for the higher-level theories.
There are four broad classes of models for describing electron correlation that are supported within Q-C HEM. The first
three directly approximate the full time-independent Schrödinger equation. In order of increasing accuracy, and also
increasing cost, they are:
1. Perturbative treatment of pair correlations between electrons, typically capable of recovering 80% or so of the
correlation energy in stable molecules.
2. Self-consistent treatment of pair correlations between electrons (most often based on coupled-cluster theory),
capable of recovering on the order of 95% or so of the correlation energy.
3. Non-iterative corrections for higher than double substitutions, which can account for more than 99% of the
correlation energy. They are the basis of many modern methods that are capable of yielding chemical accuracy
for ground state reaction energies, as exemplified by the G2 17 and G3 methods. 18
These methods are discussed in the following subsections.
There is also a fourth class of methods supported in Q-C HEM, which have a different objective. These active space
methods aim to obtain a balanced description of electron correlation in highly correlated systems, such as diradicals,
or along bond-breaking coordinates. Active space methods are discussed in Section 6.10. Finally, equation-of-motion
(EOM) methods provide tools for describing open-shell and electronically excited species. Selected configuration
interaction (CI) models are also available.
In order to carry out a wave function-based electron correlation calculation using Q-C HEM, three $rem variables need
to be set:

Chapter 6: Wave Function-Based Correlation Methods

206

• BASIS to specify the basis set (see Chapter 8)
• METHOD for treating correlation
• N_FROZEN_CORE frozen core electrons (FC default, optionally FC, or n)
For wave function-based correlation methods, the default option for exchange is Hartree-Fock. If desired, correlated
calculations can employ DFT orbitals, which should be set up using a pair of EXCHANGE and CORRELATION keywords. EXCHANGE should be set to a specific DFT method (see Section 6.12).
Additionally, for EOM or CI calculations the number of target states of each type (excited, spin-flipped, ionized,
attached, etc.) in each irreducible representation (irrep) should be specified (see Section 7.7.13). The level of correlation
of the target EOM states may be different from that used for the reference, and can be specified by EOM_CORR keyword.
The full range of ground and excited state wave function-based correlation methods available (i.e. the recognized
options to the METHOD keyword) are as follows. Ground-state methods are also a valid option for the CORRELATION
keyword.
METHOD
Specifies the level of theory, either DFT or wave function-based.
TYPE:
STRING
DEFAULT:
HF No correlation, Hartree-Fock exchange
OPTIONS:
MP2
Sections 6.3 and 6.4
RI-MP2
Section 6.6
Local_MP2 Section 6.5
RILMP2
Section 6.6.1
ATTMP2
Section 6.7
ATTRIMP2 Section 6.7
ZAPT2
A more efficient restricted open-shell MP2 method. 51
MP3
Section 6.3
MP4SDQ
Section 6.3
MP4
Section 6.3
CCD
Section 6.8
CCD(2)
Section 6.9
CCSD
Section 6.8
CCSD(T)
Section 6.9
CCSD(2)
Section 6.9
CCSD(fT)
Section 6.9.3
CCSD(dT)
Section 6.9.3
QCISD
Section 6.8
QCISD(T)
Section 6.9
OD
Section 6.8
OD(T)
Section 6.9
OD(2)
Section 6.9
VOD
Section 6.10
VOD(2)
Section 6.10
QCCD
Section 6.8
QCCD(T)
QCCD(2)
VQCCD
Section 6.10
RECOMMENDATION:
Consult the literature for guidance.

207

Chapter 6: Wave Function-Based Correlation Methods

6.2

Treatment and the Definition of Core Electrons

Treatment of core electrons is controlled by N_FROZEN_CORE. Starting from version Q-C HEM 5.0, the core electrons
are frozen by default in most post-Hartree–Fock calculations. Selected virtual orbitals can also be frozen by using
N_FROZEN_VIRTUAL keyword (the default for this is zero).
The number of core electrons in an atom is relatively well-defined, and consists of certain atomic shells. (Note that
ECPs are available in both “small-core” and “large-core” varieties; see Chapter 9.) For example, in phosphorus the
core consists of 1s, 2s, and 2p shells, for a total of ten electrons. In molecular systems, the core electrons are usually
chosen as those occupying the n/2 lowest energy orbitals, where n is the number of core electrons in the constituent
atoms. In some cases, particularly in the lower parts of the periodic table, this definition is inappropriate and can lead to
significant errors in the correlation energy. Vitaly Rassolov has implemented an alternative definition of core electrons
within Q-C HEM which is based on a Mulliken population analysis, and which addresses this problem. 84
The current implementation is restricted to n-kl type basis sets such as 3-21 or 6-31, and related bases such as 631+G(d). There are essentially two cases to consider, the outermost 6G functions (or 3G in the case of the 3-21G basis
set) for Na, Mg, K and Ca, and the 3d functions for the elements Ga—Kr. Whether or not these are treated as core or
valence is determined by the CORE_CHARACTER $rem, as summarized in Table 6.2.
CORE_CHARACTER

1
2
3
4

Outermost 6G (3G)
for Na, Mg, K, Ca
valence
valence
core
core

3d (Ga–Kr)
valence
core
core
valence

Table 6.1: A summary of the effects of different core definitions

N_FROZEN_CORE
Sets the number of frozen core orbitals in a post-Hartree–Fock calculation.
TYPE:
INTEGER
DEFAULT:
FC
OPTIONS:
FC Frozen Core approximation (all core orbitals frozen).
n
Freeze n core orbitals (if set to 0, all electrons will be active).
RECOMMENDATION:
Correlated calculations calculations are more efficient with frozen core orbitals. Use default if
possible.
N_FROZEN_VIRTUAL
Sets the number of frozen virtual orbitals in a post-Hartree–Fock calculation.
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
n Freeze n virtual orbitals.
RECOMMENDATION:
None

Chapter 6: Wave Function-Based Correlation Methods

208

CORE_CHARACTER
Selects how the core orbitals are determined in the frozen-core approximation.
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
0
Use energy-based definition.
1-4 Use Mulliken-based definition (see Table 6.2 for details).
RECOMMENDATION:
Use the default, unless performing calculations on molecules with heavy elements.
PRINT_CORE_CHARACTER
Determines the print level for the CORE_CHARACTER option.
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
0 No additional output is printed.
1 Prints core characters of occupied MOs.
2 Print level 1, plus prints the core character of AOs.
RECOMMENDATION:
Use the default, unless you are uncertain about what the core character is.

6.3
6.3.1

Møller-Plesset Perturbation Theory
Introduction

Møller-Plesset Perturbation Theory 74 is a widely used method for approximating the correlation energy of molecules.
In particular, second-order Møller-Plesset perturbation theory (MP2) is one of the simplest and most useful levels of
theory beyond the Hartree-Fock approximation. Conventional and local MP2 methods available in Q-C HEM are discussed in detail in Sections 6.4 and 6.5 respectively. The MP3 method is still occasionally used, while MP4 calculations
are quite commonly employed as part of the G2 and G3 thermochemical methods. 17,18 In the remainder of this section,
the theoretical basis of Møller-Plesset theory is reviewed.

6.3.2

Theoretical Background

The Hartree-Fock wave function Ψ0 and energy E0 are approximate solutions (eigenfunction and eigenvalue) to the
exact Hamiltonian eigenvalue problem or Schrödinger’s electronic wave equation, Eq. (4.5). The HF wave function
and energy are, however, exact solutions for the Hartree-Fock Hamiltonian H0 eigenvalue problem. If we assume that
the Hartree-Fock wave function Ψ0 and energy E0 lie near the exact wave function Ψ and energy E, we can now write
the exact Hamiltonian operator as
H = H0 + λV
(6.1)
where V is the small perturbation and λ is a dimensionless parameter. Expanding the exact wave function and energy
in terms of the HF wave function and energy yields
E = E (0) + λE (1) + λ2 E (2) + λ3 E (3) + . . .

(6.2)

Ψ = Ψ0 + λΨ(1) + λ2 Ψ(2) + λ3 Ψ(3) + . . .

(6.3)

and

209

Chapter 6: Wave Function-Based Correlation Methods
Substituting these expansions into the Schrödinger equation and collecting terms according to powers of λ yields
H0 Ψ0 = E (0) Ψ0

(6.4)

H0 Ψ(1) + V Ψ0 = E (0) Ψ(1) + E (1) Ψ0

(6.5)

H0 Ψ(2) + V Ψ(1) = E (0) Ψ(2) + E (1) Ψ(1) + E (2) Ψ0

(6.6)

and so forth. Multiplying each of the above equations by Ψ0 and integrating over all space yields the following
expression for the nth-order (MPn) energy:
E (0) = hΨ0 |H0 |Ψ0 i
(6.7)
E (1) = hΨ0 |V |Ψ0 i
E
D
E (2) = Ψ0 |V |Ψ(1)

(6.8)

E0 = hΨ0 | H0 + V |Ψ0 i

(6.10)

(6.9)

Thus, the Hartree-Fock energy
is simply the sum of the zeroth- and first- order energies
E0 = E (0) + E (1)

(6.11)

The correlation energy can then be written
(2)

(3)

(4)

Ecorr = E0 + E0 + E0 + . . .

(6.12)

of which the first term is the MP2 energy.
It can be shown that the MP2 energy can be written (in terms of spin-orbitals) as
virt occ

2

1 XX
|hab| |iji|
4
εa + εb − εi − εj
ij

(6.13)

hab kij i = hab|abijiji − hab|abjijii

(6.14)

(2)

E0

=−

ab

where
and

Z
hab|abcdcdi =




1
ψa (r1 )ψc (r1 )
ψb (r2 )ψd (r2 )dr1 dr2
r12

which can be written in terms of the two-electron repulsion integrals
XXXX
hab|abcdcdi =
Cµa Cνc Cλb Cσd (µν|λσ)
µ

ν

λ

(6.15)

(6.16)

σ

Expressions for higher order terms follow similarly, although with much greater algebraic and computational complexity. MP3 and particularly MP4 (the third and fourth order contributions to the correlation energy) are both occasionally
used, although they are increasingly supplanted by the coupled-cluster methods described in the following sections.
The disk and memory requirements for MP3 are similar to the self-consistent pair correlation methods discussed in
Section 6.8 while the computational cost of MP4 is similar to the (T) corrections discussed in Section 6.9.

6.4
6.4.1

Exact MP2 Methods
Algorithm

Second-order Møller-Plesset theory 74 (MP2) is probably the simplest useful wave function-based electron correlation method. Revived in the mid-1970s, it remains highly popular today, because it offers systematic improvement
in optimized geometries and other molecular properties relative to Hartree-Fock (HF) theory. 47 Indeed, in a recent

Chapter 6: Wave Function-Based Correlation Methods

210

comparative study of small closed-shell molecules, 48 MP2 outperformed much more expensive singles and doubles
coupled-cluster theory for such properties! Relative to state-of-the-art Kohn-Sham density functional theory (DFT)
methods, which are the most economical methods to account for electron correlation effects, MP2 has the advantage
of properly incorporating long-range dispersion forces. The principal weaknesses of MP2 theory are for open shell
systems, and other cases where the HF determinant is a poor starting point.
Q-C HEM contains an efficient conventional semi-direct method to evaluate the MP2 energy and gradient. 44 These
methods require OV N memory (O, V , N are the numbers of occupied, virtual and total orbitals, respectively), and
disk space which is bounded from above by OV N 2 /2. The latter can be reduced to IV N 2 /2 by treating the occupied
orbitals in batches of size I, and re-evaluating the two-electron integrals O/I times. This approach is tractable on
modern workstations for energy and gradient calculations of at least 500 basis functions or so, or molecules of between
15 and 30 first row atoms, depending on the basis set size. The computational cost increases between the 3rd and 5th
power of the size of the molecule, depending on which part of the calculation is time-dominant.
The algorithm and implementation in Q-C HEM is improved over earlier methods, 34,45 particularly in the following
areas:
• Uses pure functions, as opposed to Cartesians, for all fifth-order steps. This leads to large computational savings
for basis sets containing pure functions.
• Customized loop unrolling for improved efficiency.
• The sort-less semi-direct method avoids a read and write operation resulting in a large I/O savings.
• Reduction in disk and memory usage.
• No extra integral evaluation for gradient calculations.
• Full exploitation of frozen core approximation.
The implementation offers the user the following alternatives:
• Direct algorithm (energies only).
• Disk-based sort-less semi-direct algorithm (energies and gradients).
• Local occupied orbital method (energies only).
The semi-direct algorithm is the only choice for gradient calculations. It is also normally the most efficient choice for
energy calculations. There are two classes of exceptions:
• If the amount of disk space available is not significantly larger than the amount of memory available, then the
direct algorithm is preferred.
• If the calculation involves a very large basis set, then the local orbital method may be faster, because it performs
the transformation in a different order. It does not have the large memory requirement (no OV N array needed),
and always evaluates the integrals four times. The AO2MO_DISK option is also ignored in this algorithm, which
requires up to O2 V N megabytes of disk space.
There are three important options that should be wisely chosen by the user in order to exploit the full efficiency of
Q-C HEM’s direct and semi-direct MP2 methods (as discussed above, the LOCAL_OCCUPIED method has different
requirements).
• MEM_STATIC: The value specified for this $rem variable must be sufficient to permit efficient integral evaluation
(10-80Mb) and to hold a large temporary array whose size is OV N , the product of the number of occupied,
virtual and total numbers of orbitals.

Chapter 6: Wave Function-Based Correlation Methods

211

• AO2MO_DISK: The value specified for this $rem variable should be as large as possible (i.e., perhaps 80% of the
free space on your $QCSCRATCH partition where temporary job files are held). The value of this variable will
determine how many times the two-electron integrals in the atomic orbital basis must be re-evaluated, which is a
major computational step in MP2 calculations.
• N_FROZEN_CORE: The computational requirements for MP2 are proportional to the number of occupied orbitals
for some steps, and the square of that number for other steps. Therefore the CPU time can be significantly reduced
if your job employs the frozen core approximation. Additionally the memory and disk requirements are reduced
when the frozen core approximation is employed.

6.4.2

Algorithm Control and Customization

The direct and semi-direct integral transformation algorithms used by Q-C HEM (e.g., MP2, CIS(D)) are limited by
available disk space, D, and memory, C, the number of basis functions, N , the number of virtual orbitals, V and the
number of occupied orbitals, O, as discussed above. The generic description of the key $rem variables are:
MEM_STATIC
Sets the memory for Fortran AO integral calculation and transformation modules.
TYPE:
INTEGER
DEFAULT:
64 corresponding to 64 Mb.
OPTIONS:
n User-defined number of megabytes.
RECOMMENDATION:
For direct and semi-direct MP2 calculations, this must exceed OVN + requirements for AO
integral evaluation (32–160 Mb), as discussed above.
MEM_TOTAL
Sets the total memory available to Q-C HEM, in megabytes.
TYPE:
INTEGER
DEFAULT:
2000 Corresponding to 2000 Mb.
OPTIONS:
n User-defined number of megabytes.
RECOMMENDATION:
Use the default, or set equal to the physical memory of your machine. Note that if the memory
allocation total more than 1 Gb for a CCMAN job, the memory is allocated as follows
12% MEM_STATIC
50% CC_MEMORY
35% Other memory requirements:
AO2MO_DISK
Sets the amount of disk space (in megabytes) available for MP2 calculations.
TYPE:
INTEGER
DEFAULT:
2000 Corresponding to 2000 Mb.
OPTIONS:
n User-defined number of megabytes.
RECOMMENDATION:
Should be set as large as possible, discussed in Section 6.4.1.

Chapter 6: Wave Function-Based Correlation Methods

212

CD_ALGORITHM
Determines the algorithm for MP2 integral transformations.
TYPE:
STRING
DEFAULT:
Program determined.
OPTIONS:
DIRECT
Uses fully direct algorithm (energies only).
SEMI_DIRECT
Uses disk-based semi-direct algorithm.
LOCAL_OCCUPIED Alternative energy algorithm (see 6.4.1).
RECOMMENDATION:
Semi-direct is usually most efficient, and will normally be chosen by default.

6.4.3

Example

Example 6.1 Example of an MP2/6-31G* calculation employing the frozen core approximation. Note that the
EXCHANGE $rem variable will default to HF
$molecule
0 1
O
H1 O oh
H2 O oh

H1

hoh

oh = 1.01
hoh = 105
$end
$rem
METHOD
BASIS
N_FROZEN_CORE
$end

6.5
6.5.1

mp2
6-31g*
fc

Local MP2 Methods
Local Triatomics in Molecules (TRIM) Model

The development of what may be called “fast methods” for evaluating electron correlation is a problem of both fundamental and practical importance, because of the unphysical increases in computational complexity with molecular
size which afflict “exact” implementations of electron correlation methods. Ideally, the development of fast methods
for treating electron correlation should not impact either model errors or numerical errors associated with the original
electron correlation models. Unfortunately this is not possible at present, as may be appreciated from the following
rough argument. Spatial locality is what permits re-formulations of electronic structure methods that yield the same
answer as traditional methods, but faster. The one-particle density matrix decays exponentially with a rate that relates
to the HOMO-LUMO gap in periodic systems. When length scales longer than this characteristic decay length are
examined, sparsity will emerge in both the one-particle density matrix and also pair correlation amplitudes expressed
in terms of localized functions. Very roughly, such a length scale is about 5 to 10 atoms in a line, for good insulators
such as alkanes. Hence sparsity emerges beyond this number of atoms in 1-D, beyond this number of atoms squared
in 2-D, and this number of atoms cubed in 3-D. Thus for three-dimensional systems, locality only begins to emerge for
systems of between hundreds and thousands of atoms.

213

Chapter 6: Wave Function-Based Correlation Methods

If we wish to accelerate calculations on systems below this size regime, we must therefore introduce additional errors
into the calculation, either as numerical noise through looser tolerances, or by modifying the theoretical model, or
perhaps both. Q-C HEM’s approach to local electron correlation is based on modifying the theoretical models describing
correlation with an additional well-defined local approximation. We do not attempt to accelerate the calculations by
introducing more numerical error because of the difficulties of controlling the error as a function of molecule size, and
the difficulty of achieving reproducible significant results. From this perspective, local correlation becomes an integral
part of specifying the electron correlation treatment. This means that the considerations necessary for a correlation
treatment to qualify as a well-defined theoretical model chemistry apply equally to local correlation modeling. The
local approximations should be
• Size-consistent: meaning that the energy of a super-system of two non-interacting molecules should be the sum
of the energy obtained from individual calculations on each molecule.
• Uniquely defined: Require no input beyond nuclei, electrons, and an atomic orbital basis set. In other words, the
model should be uniquely specified without customization for each molecule.
• Yield continuous potential energy surfaces: The model approximations should be smooth, and not yield energies
that exhibit jumps as nuclear geometries are varied.
To ensure that these model chemistry criteria are met, Q-C HEM’s local MP2 methods 46,65 express the double substitutions (i.e., the pair correlations) in a redundant basis of atom-labeled functions. The advantage of doing this is that local
models satisfying model chemistry criteria can be defined by performing an atomic truncation of the double substitutions. A general substitution in this representation will then involve the replacement of occupied functions associated
with two given atoms by empty (or virtual) functions on two other atoms, coupling together four different atoms. We
can force one occupied to virtual substitution (of the two that comprise a double substitution) to occur only between
functions on the same atom, so that only three different atoms are involved in the double substitution. This defines
the triatomics in molecules (TRIM) local model for double substitutions. The TRIM model offers the potential for
reducing the computational requirements of exact MP2 theory by a factor proportional to the number of atoms. We
could also force each occupied to virtual substitution to be on a given atom, thereby defining a more drastic diatomics
in molecules (DIM) local correlation model.
The simplest atom-centered basis that is capable of spanning the occupied space is a minimal basis of core and valence
atomic orbitals on each atom. Such a basis is necessarily redundant because it also contains sufficient flexibility to
describe the empty valence anti-bonding orbitals necessary to correctly account for non-dynamical electron correlation
effects such as bond-breaking. This redundancy is actually important for the success of the atomic truncations because
occupied functions on adjacent atoms to some extent describe the same part of the occupied space. The minimal
functions we use to span the occupied space are obtained at the end of a large basis set calculation, and are called
extracted polarized atomic orbitals (EPAOs). 64 We discuss them briefly below. It is even possible to explicitly perform
an SCF calculation in terms of a molecule-optimized minimal basis of polarized atomic orbitals (PAOs) (see Chapter 4).
To span the virtual space, we use the full set of atomic orbitals, appropriately projected into the virtual space.
We summarize the situation. The number of functions spanning the occupied subspace will be the minimal basis set
dimension, M , which is greater than the number of occupied orbitals, O, by a factor of up to about two. The virtual
space is spanned by the set of projected atomic orbitals whose number is the atomic orbital basis set size N , which
is fractionally greater than the number of virtuals V N O. The number of double substitutions in such a redundant
representation will be typically three to five times larger than the usual total. This will be more than compensated by
reducing the number of retained substitutions by a factor of the number of atoms, A, in the local triatomics in molecules
model, or a factor of A2 in the diatomics in molecules model.
The local MP2 energy in the TRIM and DIM models are given by the following expressions, which can be compared
against the full MP2 expression given earlier in Eq. (6.13). First, for the DIM model:
EDIM MP2 = −

1 X (P̄ |Q̄)(P̄ ||Q̄)
2
∆P̄ + ∆Q̄
P̄ ,Q̄

(6.17)

Chapter 6: Wave Function-Based Correlation Methods

214

The sums run over the linear number of atomic single excitations after they have been canonicalized. Each term in the
denominator is thus an energy difference between occupied and virtual levels in this local basis. Similarly, the TRIM
model corresponds to the following local MP2 energy:
ETRIM MP2 = −

X (P̄ |jb)(P̄ ||jb)
− EDIM MP2
∆P̄ + εb − εj

(6.18)

P̄ ,jb

where the sum is now mixed between atomic substitutions P̄ , and non-local occupied j to virtual b substitutions. See
Refs. 46,65 for a full derivation and discussion.
The accuracy of the local TRIM and DIM models has been tested in a series of calculations. 46,65 In particular, the TRIM
model has been shown to be quite faithful to full MP2 theory via the following tests:
• The TRIM model recovers around 99.7% of the MP2 correlation energy for covalent bonding. This is significantly higher than the roughly 98–99% correlation energy recovery typically exhibited by the Saebo-Pulay local
correlation method. 89 The DIM model recovers around 95% of the correlation energy.
• The performance of the TRIM model for relative energies is very robust, as shown in Ref. 65 for the challenging case of torsional barriers in conjugated molecules. The RMS error in these relative energies is only
0.031 kcal/mol, as compared to around 1 kcal/mol when electron correlation effects are completely neglected.
• For the water dimer with the aug-cc-pVTZ basis, 96% of the MP2 contribution to the binding energy is recovered
with the TRIM model, as compared to 62% with the Saebo-Pulay local correlation method.
• For calculations of the MP2 contribution to the G3 and G3(MP2) energies with the larger molecules in the G3-99
database, 19 introduction of the TRIM approximation results in an RMS error relative to full MP2 theory of only
0.3 kcal/mol, even though the absolute magnitude of these quantities is on the order of tens of kcal/mol.

6.5.2

EPAO Evaluation Options

When a local MP2 job (requested by the LOCAL_MP2 option for CORRELATION) is performed, the first new step
after the SCF calculation is converged is to extract a minimal basis of polarized atomic orbitals (EPAOs) that spans
the occupied space. There are three valid choices for this basis, controlled by the PAO_METHOD and EPAO_ITERATE
keywords described below.
• Non-iterated EPAOs: The initial guess EPAOs are the default for local MP2 calculations, and are defined
as follows. For each atom, the covariant density matrix (SPS) is diagonalized, giving eigenvalues which are
approximate natural orbital occupancies, and eigenvectors which are corresponding atomic orbitals. The m
eigenvectors with largest populations are retained (where m is the minimal basis dimension for the current
atom). This non-orthogonal minimal basis is symmetrically orthogonalized, and then modified as discussed in
Ref. 64 to ensure that these functions rigorously span the occupied space of the full SCF calculation that has just
been performed. These orbitals may be denoted as EPAO(0) to indicate that no iterations have been performed
after the guess. In general, the quality of the local MP2 results obtained with this option is very similar to the
EPAO option below, but it is much faster and fully robust. For the example of the torsional barrier calculations
discussed above, 65 the TRIM RMS deviations of 0.03 kcal/mol from full MP2 calculations are increased to only
0.04 kcal/mol when EPAO(0) orbitals are employed rather than EPAOs.
• EPAOs: EPAOs are defined by minimizing a localization functional as described in Ref. 64. These functions
were designed to be suitable for local MP2 calculations, and have yielded excellent results in all tests performed
so far. Unfortunately the functional is difficult to converge for large molecules, at least with the algorithms that
have been developed to this stage. Therefore it is not the default, but is switched on by specifying a (large) value
for EPAO_ITERATE, as discussed below.

Chapter 6: Wave Function-Based Correlation Methods

215

• PAO: If the SCF calculation is performed in terms of a molecule-optimized minimal basis, as described in
Chapter 4, then the resulting PAO-SCF calculation can be corrected with either conventional or local MP2 for
electron correlation. PAO-SCF calculations alter the SCF energy, and are therefore not the default. This can be
enabled by specifying PAO_METHOD as PAO, in a job which also requests CORRELATION as LOCAL_MP2.
PAO_METHOD
Controls the type of PAO calculations requested.
TYPE:
STRING
DEFAULT:
EPAO For local MP2, EPAOs are chosen by default.
OPTIONS:
EPAO Find EPAOs by minimizing delocalization function.
PAO
Do SCF in a molecule-optimized minimal basis.
RECOMMENDATION:
None
EPAO_ITERATE
Controls iterations for EPAO calculations (see PAO_METHOD).
TYPE:
INTEGER
DEFAULT:
0 Use non-iterated EPAOs based on atomic blocks of SPS.
OPTIONS:
n Optimize the EPAOs for up to n iterations.
RECOMMENDATION:
Use the default. For molecules that are not too large, one can test the sensitivity of the results to
the type of minimal functions by the use of optimized EPAOs in which case a value of n = 500
is reasonable.
EPAO_WEIGHTS
Controls algorithm and weights for EPAO calculations (see PAO_METHOD).
TYPE:
INTEGER
DEFAULT:
115 Standard weights, use 1st and 2nd order optimization
OPTIONS:
15 Standard weights, with 1st order optimization only.
RECOMMENDATION:
Use the default, unless convergence failure is encountered.

6.5.3

Algorithm Control and Customization

A local MP2 calculation (requested by the LOCAL_MP2 option for CORRELATION) consists of the following steps:
• After the SCF is converged, a minimal basis of EPAOs are obtained.
• The TRIM (and DIM) local MP2 energies are then evaluated (gradients are not yet available).
Details of the efficient implementation of the local MP2 method described above are reported in the recent thesis of Dr.
Michael Lee. 63 Here we simply summarize the capabilities of the program. The computational advantage associated
with these local MP2 methods varies depending upon the size of molecule and the basis set. As a rough general estimate,

Chapter 6: Wave Function-Based Correlation Methods

216

TRIM MP2 calculations are feasible on molecule sizes about twice as large as those for which conventional MP2
calculations are feasible on a given computer, and this is their primary advantage. Our implementation is well suited
for large basis set calculations. The AO basis two-electron integrals are evaluated four times. DIM MP2 calculations
are performed as a by-product of TRIM MP2 but no separately optimized DIM algorithm has been implemented.
The resource requirements for local MP2 calculations are as follows:
• Memory: The memory requirement for the integral transformation does not exceed OON , and is thresholded so
that it asymptotically grows linearly with molecule size. Additional memory of approximately 32N 2 is required
to complete the local MP2 energy evaluation.
• Disk: The disk space requirement is only about 8OV N , but is not governed by a threshold. This is a very large
reduction from the case of a full MP2 calculation, where, in the case of four integral evaluations, OV N 2 /4 disk
space is required. As the local MP2 disk space requirement is not adjustable, the AO2MO_DISK keyword is
ignored for LOCAL_MP2 calculations.
The evaluation of the local MP2 energy does not require any further customization. An adequate amount of MEM_STATIC
(80 to 160 Mb) should be specified to permit efficient AO basis two-electron integral evaluation, but all large scratch
arrays are allocated from MEM_TOTAL.

217

Chapter 6: Wave Function-Based Correlation Methods

6.5.4

Examples

Example 6.2 A relative energy evaluation using the local TRIM model for MP2 with the 6-311G** basis set. The
energy difference is the internal rotation barrier in propenal, with the first geometry being planar trans, and the second
the transition structure.
$molecule
0 1
C
C 1 1.32095
C 2 1.47845
O 3 1.18974
H 1 1.07686
H 1 1.07450
H 2 1.07549
H 3 1.09486
$end
$rem
METHOD
BASIS
$end

1
2
2
2
1
2

121.19
123.83
121.50
122.09
122.34
115.27

1
3
3
3
4

180.00
0.00
180.00
180.00
180.00

local_mp2
6-311g**

@@@
$molecule
0 1
C
C 1 1.31656
C 2 1.49838
O 3 1.18747
H 1 1.07631
H 1 1.07484
H 2 1.07813
H 3 1.09387
$end
$rem
CORRELATION
BASIS
$end

6.6

1
2
2
2
1
2

123.44
123.81
122.03
121.43
120.96
115.87

1
3
3
3
4

92.28
-0.31
180.28
180.34
179.07

local_mp2
6-311g**

Auxiliary Basis (Resolution of the Identity) MP2 Methods

For a molecule of fixed size, increasing the number of basis functions per atom, n, leads to O(n4 ) growth in the number
of significant four-center two-electron integrals, since the number of non-negligible product charge distributions, |µνi,
grows as O(n2 ). As a result, the use of large (high-quality) basis expansions is computationally costly. Perhaps the
most practical way around this “basis set quality” bottleneck is the use of auxiliary basis expansions. 25,32,55 The ability
to use auxiliary basis sets to accelerate a variety of electron correlation methods, including both energies and analytical
gradients, is a major feature of Q-C HEM.
The auxiliary basis {|Ki} is used to approximate products of Gaussian basis functions:
X
K
|µνi ≈ |f
µνi =
|KiCµν

(6.19)

K

Auxiliary basis expansions were introduced long ago, and are now widely recognized as an effective and powerful
approach, which is sometimes synonymously called resolution of the identity (RI) or density fitting (DF). When using

218

Chapter 6: Wave Function-Based Correlation Methods

auxiliary basis expansions, the rate of growth of computational cost of large-scale electronic structure calculations with
n is reduced to approximately n3 .
If n is fixed and molecule size increases, auxiliary basis expansions reduce the pre-factor associated with the computation, while not altering the scaling. The important point is that the pre-factor can be reduced by 5 or 10 times or more.
Such large speedups are possible because the number of auxiliary functions required to obtain reasonable accuracy, X,
has been shown to be only about 3 or 4 times larger than N .
The auxiliary basis expansion coefficients, C, are determined by minimizing the deviation between the fitted distribution and the actual distribution, hµν − µ
fν|µν − µ
fνi, which leads to the following set of linear equations:
X
L
hK |L iCµν
= hK |µν i
(6.20)
L

Evidently solution of the fit equations requires only two- and three-center integrals, and as a result the (four-center)
two-electron integrals can be approximated as the following optimal expression for a given choice of auxiliary basis
set:
X
L
K
f =
hµν|λσi ≈ hf
µν|λσi
Cµν
hL|KiCλσ
(6.21)
K,L

In the limit where the auxiliary basis is complete (i.e. all products of AOs are included), the fitting procedure described
above will be exact. However, the auxiliary basis is invariably incomplete (as mentioned above, X ≈ 3N ) because this
is essential for obtaining increased computational efficiency.
Standardized auxiliary basis sets have been developed by the Karlsruhe group for second-order perturbation (MP2)
calculations of the correlation energy. 109,110 Using these basis sets, absolute errors in the correlation energy are small
(e.g., below 60 µHartree per atom), and errors in relative energies are smaller still At the same time, speedups of 3–
30× are realized. This development has made the routine use of auxiliary basis sets for electron correlation calculations
possible.
Correlation calculations that can take advantage of auxiliary basis expansions are described in the remainder of this
section (MP2, and MP2-like methods) and in Section 6.15 (simplified active space coupled cluster methods such as PP,
PP(2), IP, RP). These methods automatically employ auxiliary basis expansions when a valid choice of auxiliary basis
set is supplied using the AUX_BASIS keyword which is used in the same way as the BASIS keyword. The PURECART
$rem is no longer needed here, even if using a auxiliary basis that does not have a predefined value. There is a built-in
automatic procedure that provides the effect of the PURECART $rem in these cases by default.

6.6.1

RI-MP2 Energies and Gradients.

Following common convention, the MP2 energy evaluated approximately using an auxiliary basis is referred to as
“resolution of the identity” MP2, or RI-MP2 for short. RI-MP2 energy and gradient calculations are enabled simply
by specifying the AUX_BASIS keyword discussed above. As discussed above, RI-MP2 energies 32 and gradients 23,108
are significantly faster than the best conventional MP2 energies and gradients, and cause negligible loss of accuracy,
when an appropriate standardized auxiliary basis set is employed. Therefore they are recommended for jobs where
turnaround time is an issue. Disk requirements are very modest; one merely needs to hold various 3-index arrays.
Memory requirements grow more slowly than our conventional MP2 algorithms—only quadratically with molecular
size. The minimum memory requirement is approximately 3X 2 , where X is the number of auxiliary basis functions, for
both energy and analytical gradient evaluations, with some additional memory being necessary for integral evaluation
and other small arrays.
In fact, for molecules that are not too large (perhaps no more than 20 or 30 heavy atoms) the RI-MP2 treatment of
electron correlation is so efficient that the computation is dominated by the initial Hartree-Fock calculation. This is
despite the fact that as a function of molecule size, the cost of the RI-MP2 treatment still scales more steeply with
molecule size (it is just that the pre-factor is so much smaller with the RI approach). Its scaling remains 5th order with
the size of the molecule, which only dominates the initial SCF calculation for larger molecules. Thus, for RI-MP2

Chapter 6: Wave Function-Based Correlation Methods

219

energy evaluation on moderate size molecules (particularly in large basis sets), it is desirable to use the dual basis HF
method to further improve execution times (see Section 4.7).

6.6.2

Example

Example 6.3 Q-C HEM input for an RI-MP2 geometry optimization.
$molecule
0 1
O
H 1 0.9
F 1 1.4
$end
$rem
JOBTYPE
METHOD
BASIS
AUX_BASIS
SYMMETRY
$end

2

100.

opt
rimp2
cc-pvtz
rimp2-cc-pvtz
false

For the size of required memory, the followings need to be considered.
MEM_STATIC
Sets the memory for AO-integral evaluations and their transformations.
TYPE:
INTEGER
DEFAULT:
64 corresponding to 64 Mb.
OPTIONS:
n User-defined number of megabytes.
RECOMMENDATION:
For RI-MP2 calculations, 150(ON + V ) of MEM_STATIC is required. Because a number of
matrices with N 2 size also need to be stored, 32–160 Mb of additional MEM_STATIC is needed.
MEM_TOTAL
Sets the total memory available to Q-C HEM, in megabytes.
TYPE:
INTEGER
DEFAULT:
2000 2 Gb
OPTIONS:
n User-defined number of megabytes.
RECOMMENDATION:
Use the default, or set to the physical memory of your machine. The minimum requirement is
3X 2 .

6.6.3

OpenMP Implementation of RI-MP2

An OpenMP RI-MP2 energy algorithm is used by default in Q-C HEM 4.1 onward. This can be invoked by using
CORR=primp2 for older versions, but note that in 4.01 and below, only RHF/RI-MP2 was supported. Now UHF/RIMP2 is supported, and the formation of the ‘B’ matrices as well as three center integrals are parallelized. This algorithm

Chapter 6: Wave Function-Based Correlation Methods

220

uses the remaining memory from the MEM_TOTAL allocation for all computation, which can drastically reduce hard
drive reads in the formation of t-amplitudes.
Example 6.4 Example of OpenMP-parallel RI-MP2 job.
$molecule
0 1
C1
H1 C1
H2 C1
$end

1.077260
1.077260

$rem
JOBTYPE
EXCHANGE
CORRELATION
BASIS
AUX_BASIS
PURECART
SYMMETRY
THRESH
SCF_CONVERGENCE
MAX_SUB_FILE_NUM
!TIME_MP2
$end

H1

131.608240

SP
HF
pRIMP2
cc-pVTZ
rimp2-cc-pVTZ
11111
false
12
8
128
true

6.6.4

GPU Implementation of RI-MP2

6.6.4.1

Requirements

Q-C HEM currently offers the possibility of accelerating RI-MP2 calculations using graphics processing units (GPUs).
Currently, this is implemented for CUDA-enabled NVIDIA graphics cards only, such as (in historical order from 2008)
the GeForce, Quadro, Tesla and Fermi cards. More information about CUDA-enabled cards is available at
http://www.nvidia.com/object/cuda_gpus.html
It should be noted that these GPUs have specific power and motherboard requirements.
Software requirements include the installation of the appropriate NVIDIA CUDA driver (at least version 1.0, currently
3.2) and linear algebra library, CUBLAS (at least version 1.0, currently 2.0). These can be downloaded jointly in
NVIDIA’s developer website:
http://developer.nvidia.com/object/cuda_3_2_downloads.html
We have implemented a mixed-precision algorithm in order to get better than single precision when users only have
single-precision GPUs. This is accomplished by noting that RI-MP2 matrices have a large fraction of numerically
“small” elements and a small fraction of numerically “large” ones. The latter can greatly affect the accuracy of the
calculation in single-precision only calculations, but calculation involves a relatively small number of compute cycles.
So, given a threshold value δ, we perform a separation between “small” and “large” elements and accelerate the former
compute-intensive operations using the GPU (in single-precision) and compute the latter on the CPU (using doubleprecision). We are thus able to determine how much double-precision we desire by tuning the δ parameter, and tailoring
the balance between computational speed and accuracy.

Chapter 6: Wave Function-Based Correlation Methods
6.6.4.2

Options
CUDA_RI-MP2
Enables GPU implementation of RI-MP2
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
FALSE GPU-enabled MGEMM off
TRUE
GPU-enabled MGEMM on
RECOMMENDATION:
Necessary to set to 1 in order to run GPU-enabled RI-MP2
USECUBLAS_THRESH
Sets threshold of matrix size sent to GPU (smaller size not worth sending to GPU).
TYPE:
INTEGER
DEFAULT:
250
OPTIONS:
n user-defined threshold
RECOMMENDATION:
Use the default value. Anything less can seriously hinder the GPU acceleration
USE_MGEMM
Use the mixed-precision matrix scheme (MGEMM) if you want to make calculations in your
card in single-precision (or if you have a single-precision-only GPU), but leave some parts of the
RI-MP2 calculation in double precision)
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
FALSE MGEMM disabled
TRUE
MGEMM enabled
RECOMMENDATION:
Use when having single-precision cards
MGEMM_THRESH
Sets MGEMM threshold to determine the separation between “large” and “small” matrix elements. A larger threshold value will result in a value closer to the single-precision result. Note
that the desired factor should be multiplied by 10000 to ensure an integer value.
TYPE:
INTEGER
DEFAULT:
10000 (corresponds to 1)
OPTIONS:
n User-specified threshold
RECOMMENDATION:
For small molecules and basis sets up to triple-ζ, the default value suffices to not deviate too
much from the double-precision values. Care should be taken to reduce this number for larger
molecules and also larger basis-sets.

221

222

Chapter 6: Wave Function-Based Correlation Methods
6.6.4.3

Input examples

Example 6.5 RI-MP2 double-precision calculation
$molecule
0 1
c
h1 c
h2 c
h3 c
h4 c
$end

1.089665
1.089665
1.089665
1.089665

$rem
JOBTYPE
EXCHANGE
METHOD
BASIS
AUX_BASIS
CUDA_RIMP2
$end

h1
h1
h1

109.47122063
109.47122063
109.47122063

h2
h2

120.
-120.

h2
h2

120.
-120.

sp
hf
rimp2
cc-pvdz
rimp2-cc-pvdz
1

Example 6.6 RI-MP2 calculation with MGEMM
$molecule
0 1
c
h1 c
h2 c
h3 c
h4 c
$end

1.089665
1.089665
1.089665
1.089665

$rem
JOBTYPE
EXCHANGE
METHOD
BASIS
AUX_BASIS
CUDA_RIMP2
USE_MGEMM
MGEMM_THRESH
$end

6.6.5

h1
h1
h1

109.47122063
109.47122063
109.47122063

sp
hf
rimp2
cc-pvdz
rimp2-cc-pvdz
1
1
10000

Spin-Biased MP2 Methods (SCS-MP2, SOS-MP2, MOS-MP2, and O2)

The accuracy of MP2 calculations can be significantly improved by semi-empirically scaling the opposite-spin (OS)
and same-spin (SS) correlation components with separate scaling factors, as shown by Grimme. 41 Scaling with 1.2
and 0.33 (or OS and SS components) defines the SCS-MP2 method, but other parameterizations are desirable for
systems involving intermolecular interactions, as in the SCS-MI-MP2 method, which uses 0.40 and 1.29 (for OS and
SS components). 20
Results of similar quality for thermochemistry can be obtained by only retaining and scaling the opposite spin correlation (by 1.3), as was recently demonstrated. 54 Furthermore, the SOS-MP2 energy can be evaluated using the RI
approximation together with a Laplace transform technique, in effort that scales only with the 4th power of molecular
size. Efficient algorithms for the energy 54 and the analytical gradient 69 of this method are available since Q-C HEM
v. 3.0, and offer advantages in speed over MP2 for larger molecules, as well as statistically significant improvements in
accuracy.

Chapter 6: Wave Function-Based Correlation Methods

223

However, we note that the SOS-MP2 method does systematically underestimate long-range dispersion (for which the
appropriate scaling factor is 2 rather than 1.3) but this can be accounted for by making the scaling factor distancedependent, which is done in the modified opposite spin variant (MOS-MP2) that has recently been proposed and
tested. 68 The MOS-MP2 energy and analytical gradient are also available in Q-C HEM 3.0 at a cost that is essentially
identical with SOS-MP2. Timings show that the 4th-order implementation of SOS-MP2 and MOS-MP2 yields substantial speedups over RI-MP2 for molecules in the 40 heavy atom regime and larger. It is also possible to customize
the scale factors for particular applications, such as weak interactions, if required.
A fourth order scaling SOS-MP2/MOS-MP2 energy calculation can be invoked by setting the CORRELATION keyword
to either SOSMP2 or MOSMP2. MOS-MP2 further requires the specification of the $rem variable OMEGA, which tunes
the level of attenuation of the MOS operator: 68
gω (r12 ) =

1
erf (ωr12 )
+ cMOS
r12
r12

(6.22)

The recommended OMEGA value is ω = 0.6 bohr−1 . 68 The fast algorithm makes use of auxiliary basis expansions and
therefore, the keyword AUX_BASIS should be set consistently with the user’s choice of BASIS. Fourth-order scaling
analytical gradient for both SOS-MP2 and MOS-MP2 are also available and is automatically invoked when JOBTYPE is
set to OPT or FORCE. The minimum memory requirement is 3X 2 , where X = the number of auxiliary basis functions,
for both energy and analytical gradient evaluations. Disk space requirement for closed shell calculations is ∼ 2OV X
for energy evaluation and ∼ 4OV X for analytical gradient evaluation.
More recently, Brueckner orbitals (BO) are introduced into SOSMP2 and MOSMP2 methods to resolve the problems
of symmetry breaking and spin contamination that are often associated with Hartree-Fock orbitals. So the molecular
orbitals are optimized with the mean-field energy plus a correlation energy taken as the opposite-spin component of the
second-order many-body correlation energy, scaled by an empirically chosen parameter. This “optimized second-order
opposite-spin” (O2) method 67 requires fourth-order computation on each orbital iteration. O2 is shown to yield predictions of structure and frequencies for closed-shell molecules that are very similar to scaled MP2 methods. However,
it yields substantial improvements for open-shell molecules, where problems with spin contamination and symmetry
breaking are shown to be greatly reduced.
Summary of key $rem variables to be specified:
CORRELATION

JOBTYPE

BASIS
AUX_BASIS
OMEGA
N_FROZEN_CORE
N_FROZEN_VIRTUAL
SCS

RIMP2
SOSMP2
MOSMP2
sp (default) single point energy evaluation
opt geometry optimization with analytical gradient
force evaluation with analytical gradient
user’s choice (standard or user-defined: GENERAL or MIXED)
corresponding auxiliary basis (standard or user-defined:
AUX_GENERAL or AUX_MIXED
no default n; use ω = n/1000. The recommended value is
n = 600 (ω = 0.6 bohr−1 )
Optional
Optional
Turns on spin-component scaling with SCS-MP2(1),
SOS-MP2(2), and arbitrary SCS-MP2(3)

Chapter 6: Wave Function-Based Correlation Methods

224

225

Chapter 6: Wave Function-Based Correlation Methods

6.6.6

Examples

Example 6.7 Example of SCS-MP2 geometry optimization
$molecule
0 1
C
H 1 1.0986
H 1 1.0986
H 1 1.0986
H 1 1.0986
$end

2
2
2

$rem
JOBTYPE
EXCHANGE
CORRELATION
BASIS
AUX_BASIS
BASIS2
THRESH
SCF_CONVERGENCE
MAX_SUB_FILE_NUM
SCS
DUAL_BASIS_ENERGY
N_FROZEN_CORE
SYMMETRY
SYM_IGNORE
$end

109.5
109.5
109.5

3
3

120.0 0
-120.0 0

opt
hf
rimp2
aug-cc-pvdz
rimp2-aug-cc-pvdz
racc-pvdz
12
8
128
1
true
fc
false
true

Optional Secondary basis

Turn on spin-component scaling
Optional dual-basis approximation

Example 6.8 Example of SCS-MI-MP2 energy calculation
$molecule
0 1
C
0.000000
H
-0.888551
H
0.888551
H
0.000000
H
0.000000
C
0.000000
H
0.000000
H
-0.888551
H
0.888551
H
0.000000
$end
$rem
EXCHANGE
CORRELATION
BASIS
AUX_BASIS
BASIS2
THRESH
SCF_CONVERGENCE
MAX_SUB_FILE_NUM
SCS
SOS_FACTOR
SSS_FACTOR
DUAL_BASIS_ENERGY
N_FROZEN_CORE
SYMMETRY
SYM_IGNORE
$end

-0.000140
0.513060
0.513060
-1.026339
0.000089
0.000140
-0.000089
-0.513060
-0.513060
1.026339

1.859161
1.494685
1.494685
1.494868
2.948284
-1.859161
-2.948284
-1.494685
-1.494685
-1.494868

hf
rimp2
aug-cc-pvtz
rimp2-aug-cc-pvtz
racc-pvtz
Optional Secondary basis
12
8
128
3
Spin-component scale arbitrarily
0400000
Specify OS parameter
1290000
Specify SS parameter
true
Optional dual-basis approximation
fc
false
true

226

Chapter 6: Wave Function-Based Correlation Methods

Example 6.9 Example of SOS-MP2 geometry optimization
$molecule
0 3
C1
H1
C1
H2
C1
$end

1.07726
1.07726

$rem
JOBTYPE
METHOD
BASIS
AUX_BASIS
UNRESTRICTED
SYMMETRY
$end

H1

131.60824

opt
sosmp2
cc-pvdz
rimp2-cc-pvdz
true
false

Example 6.10 Example of MOS-MP2 energy evaluation with frozen core approximation
$molecule
0 1
Cl
Cl 1 2.05
$end
$rem
JOBTYPE
METHOD
OMEGA
BASIS
AUX_BASIS
N_FROZEN_CORE
THRESH
SCF_CONVERGENCE
$end

sp
mosmp2
600
cc-pVTZ
rimp2-cc-pVTZ
fc
12
8

Example 6.11 Example of O2 methodology applied to O(N 4 ) SOSMP2
$molecule
1 2
F
H 1 1.001
$end
$rem
UNRESTRICTED
JOBTYPE
EXCHANGE
DO_O2
SOS_FACTOR
SCF_ALGORITHM
SCF_GUESS
BASIS
AUX_BASIS
SCF_CONVERGENCE
THRESH
SYMMETRY
PURECART
$end

TRUE
FORCE
HF
1
1000000
DIIS_GDM
GWH
sto-3g
rimp2-vdz
8
14
FALSE
1111

Options are SP/FORCE/OPT
O2 with O(N^4) SOS-MP2 algorithm
Opposite Spin scaling factor = 1.0

227

Chapter 6: Wave Function-Based Correlation Methods

Example 6.12 Example of O2 methodology applied to O(N 4 ) MOSMP2
$molecule
1 2
F
H 1 1.001
$end
$rem
UNRESTRICTED
JOBTYPE
EXCHANGE
DO_O2
OMEGA
SCF_ALGORITHM
SCF_GUESS
BASIS
AUX_BASIS
SCF_CONVERGENCE
THRESH
SYMMETRY
PURECART
$end

6.6.7

TRUE
FORCE
HF
2
600
DIIS_GDM
GWH
sto-3g
rimp2-vdz
8
14
FALSE
1111

Options are SP/FORCE/OPT
O2 with O(N^4) MOS-MP2 algorithm
Omega = 600/1000 = 0.6 a.u.

RI-TRIM MP2 Energies

The triatomics in molecules (TRIM) local correlation approximation to MP2 theory 65 was described in detail in Section 6.5.1 which also discussed our implementation of this approach based on conventional four-center two-electron
integrals. Starting from Q-C HEM v. 3.0, an auxiliary basis implementation of the TRIM model is available. The new
RI-TRIM MP2 energy algorithm 21 greatly accelerates these local correlation calculations (often by an order of magnitude or more for the correlation part), which scale with the 4th power of molecule size. The electron correlation part
of the calculation is speeded up over normal RI-MP2 by a factor proportional to the number of atoms in the molecule.
For a hexadecapeptide, for instance, the speedup is approximately a factor of 4. 21 The TRIM model can also be applied
to the scaled opposite spin models discussed above. As for the other RI-based models discussed in this section, we
recommend using RI-TRIM MP2 instead of the conventional TRIM MP2 code whenever run-time of the job is a significant issue. As for RI-MP2 itself, TRIM MP2 is invoked by adding AUX_BASIS $rems to the input deck, in addition
to requesting CORRELATION = RILMP2.
Example 6.13 Example of RI-TRIM MP2 energy evaluation
$molecule
0 3
C1
H1
C1
H2
C1
$end

1.07726
1.07726

$rem
METHOD
BASIS
AUX_BASIS
PURECART
UNRESTRICTED
SYMMETRY
$end

H1

131.60824

rilmp2
cc-pVDZ
rimp2-cc-pVDZ
1111
true
false

Chapter 6: Wave Function-Based Correlation Methods

6.6.8

228

Dual-Basis MP2

The successful computational cost speedups of the previous sections often leave the cost of the underlying SCF calculation dominant. The dual-basis method provides a means of accelerating the SCF by roughly an order of magnitude,
with minimal associated error (see Section 4.7). This dual-basis reference energy may be combined with RI-MP2
calculations for both energies 98,99 and analytic first derivatives. 22 In the latter case, further savings (beyond the SCF
alone) are demonstrated in the gradient due to the ability to solve the response (Z-vector) equations in the smaller basis
set. Refer to Section 4.7 for details and job control options.

6.7

Attenuated MP2

MP2(attenuator, basis) approximates MP2 by splitting the Coulomb operator in two pieces and preserving only shortrange two-electron interactions, akin to the CASE approximation, 6,24 but without modification of the underlying SCF
calculation. While MP2 is a comparatively efficient method for estimating the correlation energy, it converges slowly
with basis set size — and, even in the complete basis limit, contains fundamentally inaccurate physics for long-range
interactions. Basis set superposition error and the MP2-level treatment of long-range interactions both typically artificially increase correlation energies for non-covalent interactions. Attenuated MP2 improves upon MP2 for interand intramolecular interactions, with significantly better performance for relative and binding energies of non-covalent
complexes, frequently outperforming complete basis set estimates of MP2. 39,40
Attenuated MP2, denoted MP2(attenuator, basis) is implemented in Q-C HEM based on the complementary terf function, below:
1
s(r) = terfc(r, r0 ) = {erfc [ω(r − r0 )] + erfc [ω(r + r0 )]}
(6.23)
2
By choosing the terfc short-range operator, we optimally preserve the short-range behavior of the Coulomb operator
while smoothly and rapidly switching off around the distance r0 . Since this directly addresses basis set superposition
error, parameterization must be done for specific basis sets. This has been performed for the basis sets, aug-cc-pVDZ 39
and aug-cc-pVTZ. 40 Other basis sets are not recommended for general use until further testing has been done.
Energies and gradients are functional with and without the resolution of the identity approximation using correlation

Chapter 6: Wave Function-Based Correlation Methods

229

keywords ATTMP2 and ATTRIMP2.
Example 6.14 Example of RI-MP2(terfc, aug-cc-pVDZ) energy evaluation
$molecule
0 1
O
-1.551007
H
-1.934259
H
-0.599677
$end
$rem
JOBTYPE
METHOD
BASIS
AUX_BASIS
N_FROZEN_CORE
$end

-0.114520
0.762503
0.040712

0.000000
0.000000
0.000000

sp
attrimp2
aug-cc-pvdz
rimp2-aug-cc-pvdz
fc

Example 6.15 Example of MP2(terfc, aug-cc-pVTZ) geometry optimization
$molecule
0 1
H
0.0
H
0.0
$end

0.0
0.0

$rem
JOBTYPE
METHOD
BASIS
N_FROZEN_CORE
$end

6.8

0.0
0.9

opt
attmp2
aug-cc-pvtz
fc

Coupled-Cluster Methods

The following sections give short summaries of the various coupled-cluster based methods available in Q-C HEM, most
of which are variants of coupled-cluster theory. The basic object-oriented tools necessary to permit the implementation
of these methods in Q-C HEM was accomplished by Profs. Anna Krylov and David Sherrill, working at Berkeley with
Martin Head-Gordon, and then continuing independently at the University of Southern California and Georgia Tech,
respectively. While at Berkeley, Krylov and Sherrill also developed the optimized orbital coupled-cluster method, with
additional assistance from Ed Byrd. The extension of this code to MP3, MP4, CCSD and QCISD is the work of Prof.
Steve Gwaltney at Berkeley, while the extensions to QCCD were implemented by Ed Byrd at Berkeley. The original
tensor library and CC/EOM suite of methods are handled by the CCMAN module of Q-C HEM. Recently, a new code
(termed CCMAN2) has been developed in Krylov group by Evgeny Epifanovsky and others, and a gradual transition
from CCMAN to CCMAN2 has begun. During the transition time, both codes will be available for users via the
CCMAN2 keyword.

Chapter 6: Wave Function-Based Correlation Methods

230

CORRELATION
Specifies the correlation level of theory handled by CCMAN/CCMAN2.
TYPE:
STRING
DEFAULT:
None No Correlation
OPTIONS:
CCMP2
Regular MP2 handled by CCMAN/CCMAN2
MP3
CCMAN and CCMAN2
MP4SDQ
CCMAN
MP4
CCMAN
CCD
CCMAN and CCMAN2
CCD(2)
CCMAN
CCSD
CCMAN and CCMAN2
CCSD(T)
CCMAN and CCMAN2
CCSD(2)
CCMAN
CCSD(fT)
CCMAN and CCMAN2
CCSD(dT)
CCMAN
CCVB-SD
CCMAN2
QCISD
CCMAN and CCMAN2
QCISD(T)
CCMAN and CCMAN2
OD
CCMAN
OD(T)
CCMAN
OD(2)
CCMAN
VOD
CCMAN
VOD(2)
CCMAN
QCCD
CCMAN
QCCD(T)
CCMAN
QCCD(2)
CCMAN
VQCCD
CCMAN
VQCCD(T) CCMAN
VQCCD(2) CCMAN
RECOMMENDATION:
Consult the literature for guidance.

Note: All methods implemented in CCMAN2 can be executed in combination with the C-PCM implicit solvent model
(section 7.7.11) and with the EFP method (section 12.5). Only energies and unrelaxed properties are available
(no gradient).

6.8.1

Coupled Cluster Singles and Doubles (CCSD)

The standard approach for treating pair correlations self-consistently are coupled-cluster methods where the cluster
operator contains all single and double substitutions, 81 abbreviated as CCSD. CCSD yields results that are only slightly
superior to MP2 for structures and frequencies of stable closed-shell molecules. However, it is far superior for reactive
species, such as transition structures and radicals, for which the performance of MP2 is quite erratic.
A full textbook presentation of CCSD is beyond the scope of this manual, and several comprehensive references are
available. However, it may be useful to briefly summarize the main equations. The CCSD wave function is:


|ΨCCSD i = exp T̂1 + T̂2 |Φ0 i

(6.24)

231

Chapter 6: Wave Function-Based Correlation Methods

where the single and double excitation operators may be defined by their actions on the reference single determinant
(which is normally taken as the Hartree-Fock determinant in CCSD):

T̂1 |Φ0 i =

occ X
virt
X
i

tai |Φai i

(6.25)

a

occ virt

T̂2 |Φ0 i =

1 X X ab ab
tij Φij
4 ij

(6.26)

ab

It is not feasible to determine the CCSD energy by variational minimization of hEiCCSD with respect to the singles and
doubles amplitudes because the expressions terminate at the same level of complexity as full configuration interaction
(!). So, instead, the Schrödinger equation is satisfied in the subspace spanned by the reference determinant, all single
substitutions, and all double substitutions. Projection with these functions and integration over all space provides
sufficient equations to determine the energy, the singles and doubles amplitudes as the solutions of sets of nonlinear
equations. These equations may be symbolically written as follows:

ECCSD

0

0

= hΦ0 |Ĥ|ΨCCSD i


 
1
=
Φ0 Ĥ 1 + T̂1 + T̂12 + T̂2 Φ0
2
C
D
E
a
=
Φi Ĥ − ECCSD ΨCCSD

 

1 2
1 3
a
=
Φi Ĥ 1 + T̂1 + T̂1 + T̂2 + T̂1 T̂2 + T̂1 Φ0
2
3!
C
D
E
ab
=
Φij Ĥ − ECCSD ΨCCSD


1
1
=
Φab
Ĥ
1 + T̂1 + T̂12 + T̂2 + T̂1 T̂2 + T̂13
ij
2
3!
 
1
1
1
+ T̂22 + T̂12 T̂2 + T̂14 Φ0
2
2
4!
C

(6.27)

(6.28)

(6.29)

The result is a set of equations which yield an energy that is not necessarily variational (i.e., may not be above the true
energy), although it is strictly size-consistent. The equations are also exact for a pair of electrons, and, to the extent
that molecules are a collection of interacting electron pairs, this is the basis for expecting that CCSD results will be of
useful accuracy.
The computational effort necessary to solve the CCSD equations can be shown to scale with the 6th power of the
molecular size, for fixed choice of basis set. Disk storage scales with the 4th power of molecular size, and involves a
number of sets of doubles amplitudes, as well as two-electron integrals in the molecular orbital basis. Therefore the
improved accuracy relative to MP2 theory comes at a steep computational cost. Given these scalings it is relatively
straightforward to estimate the feasibility (or non feasibility) of a CCSD calculation on a larger molecule (or with a
larger basis set) given that a smaller trial calculation is first performed. Q-C HEM supports both energies and analytic
gradients for CCSD for RHF and UHF references (including frozen-core). For ROHF, only energies and unrelaxed
properties are available.

6.8.2

Quadratic Configuration Interaction (QCISD)

Quadratic configuration interaction with singles and doubles (QCISD) 77 is a widely used alternative to CCSD, that
shares its main desirable properties of being size-consistent, exact for pairs of electrons, as well as being also non
variational. Its computational cost also scales in the same way with molecule size and basis set as CCSD, although
with slightly smaller constants. While originally proposed independently of CCSD based on correcting configuration

232

Chapter 6: Wave Function-Based Correlation Methods

interaction equations to be size-consistent, QCISD is probably best viewed as approximation to CCSD. The defining
equations are given below (under the assumption of Hartree-Fock orbitals, which should always be used in QCISD).
The QCISD equations can clearly be viewed as the CCSD equations with a large number of terms omitted, which are
evidently not very numerically significant:

 E
D
(6.30)
EQCISD = Φ0 Ĥ 1 + T̂2 Φ0
C


 E
D
0 = Φai Ĥ T̂1 + T̂2 + T̂1 T̂2 Φ0
C


 
1
0 = Φab
1 + T̂1 + T̂2 + T̂22 Φ0
ij Ĥ
2
C

(6.31)
(6.32)

QCISD energies are available in Q-C HEM, and are requested with the QCISD keyword. As discussed in Section 6.9,
the non iterative QCISD(T) correction to the QCISD solution is also available to approximately incorporate the effect
of higher substitutions.

6.8.3

Optimized Orbital Coupled Cluster Doubles (OD)

It is possible to greatly simplify the CCSD equations by omitting the single substitutions (i.e., setting the T1 operator
to zero). If the same single determinant reference is used (specifically the Hartree-Fock determinant), then this defines
the coupled-cluster doubles (CCD) method, by the following equations:
D

 E
ECCD =
Φ0 Ĥ 1 + T̂2 Φ0
(6.33)
C


 
1
1 + T̂2 + T̂22 Φ0
0 =
Φab
(6.34)
ij Ĥ
2
C
The CCD method cannot itself usually be recommended because while pair correlations are all correctly included, the
neglect of single substitutions causes calculated energies and properties to be significantly less reliable than for CCSD.
Single substitutions play a role very similar to orbital optimization, in that they effectively alter the reference determinant to be more appropriate for the description of electron correlation (the Hartree-Fock determinant is optimized in
the absence of electron correlation).
This suggests an alternative to CCSD and QCISD that has some additional advantages. This is the optimized orbital
CCD method (OO-CCD), which we normally refer to as simply optimized doubles (OD). 91 The OD method is defined
by the CCD equations above, plus the additional set of conditions that the cluster energy is minimized with respect to
orbital variations. This may be mathematically expressed by
∂ECCD
=0
∂θia

(6.35)

where the rotation angle θia mixes the ith occupied orbital with the ath virtual (empty) orbital. Thus the orbitals that
define the single determinant reference are optimized to minimize the coupled-cluster energy, and are variationally best
for this purpose. The resulting orbitals are approximate Brueckner orbitals.
The OD method has the advantage of formal simplicity (orbital variations and single substitutions are essentially redundant variables). In cases where Hartree-Fock theory performs poorly (for example artificial symmetry breaking, or
non-convergence), it is also practically advantageous to use the OD method, where the HF orbitals are not required,
rather than CCSD or QCISD. Q-C HEM supports both energies and analytical gradients using the OD method. The
computational cost for the OD energy is more than twice that of the CCSD or QCISD method, but the total cost of
energy plus gradient is roughly similar, although OD remains more expensive. An additional advantage of the OD
method is that it can be performed in an active space, as discussed later, in Section 6.10.

233

Chapter 6: Wave Function-Based Correlation Methods

6.8.4

Quadratic Coupled Cluster Doubles (QCCD)

The non variational determination of the energy in the CCSD, QCISD, and OD methods discussed in the above subsections is not normally a practical problem. However, there are some cases where these methods perform poorly. One such
example are potential curves for homolytic bond dissociation, using closed shell orbitals, where the calculated energies
near dissociation go significantly below the true energies, giving potential curves with unphysical barriers to formation
of the molecule from the separated fragments. 104 The Quadratic Coupled Cluster Doubles (QCCD) method 105 recently
proposed by Troy Van Voorhis at Berkeley uses a different energy functional to yield improved behavior in problem
cases of this type. Specifically, the QCCD energy functional is defined as
 

  
1
(6.36)
EQCCD = Φ0 1 + Λ̂2 + Λ̂22 Ĥ exp T̂2 Φ0
2
C
where the amplitudes of both the T̂2 and Λ̂2 operators are determined by minimizing the QCCD energy functional.
Additionally, the optimal orbitals are determined by minimizing the QCCD energy functional with respect to orbital
rotations mixing occupied and virtual orbitals.
To see why the QCCD energy should be an improvement on the OD energy, we first write the latter in a different way
than before. Namely, we can write a CCD energy functional which when minimized with respect to the T̂2 and Λ̂2
operators, gives back the same CCD equations defined earlier. This energy functional is
D 

  E
ECCD = Φ0 1 + Λ̂2 Ĥ exp T̂2 Φ0
(6.37)
C

Minimization with respect to the Λ̂2 operator gives the equations for the T̂2 operator presented previously, and, if those
equations are satisfied then it is clear that we do not require knowledge of the Λ̂2 operator itself to evaluate the energy.
Comparing the two energy functionals, Eqs. (6.36) and (6.37), we see that the QCCD functional includes up through
quadratic terms of the Maclaurin expansion of exp(Λ̂2 ) while the conventional CCD functional includes only linear
terms. Thus the bra wave function and the ket wave function in the energy expression are treated more equivalently in
QCCD than in CCD. This makes QCCD closer to a true variational treatment 104 where the bra and ket wave functions
are treated precisely equivalently, but without the exponential cost of the variational method.
In practice QCCD is a dramatic improvement relative to any of the conventional pair correlation methods for processes
involving more than two active electrons (i.e., the breaking of at least a double bond, or, two spatially close single
bonds). For example calculations, we refer to the original paper, 105 and the follow-up paper describing the full implementation. 15 We note that these improvements carry a computational price. While QCCD scales formally with the
6th power of molecule size like CCSD, QCISD, and OD, the coefficient is substantially larger. For this reason, QCCD
calculations are by default performed as OD calculations until they are partly converged. Q-C HEM also contains some
configuration interaction models (CISD and CISDT). The CI methods are inferior to CC due to size-consistency issues,
however, these models may be useful for benchmarking and development purposes.

6.8.5

Resolution of the Identity with CC (RI-CC)

The RI approximation (see Section 6.6) can be used in coupled-cluster calculations, which substantially reduces the
cost of integral transformation and disk storage requirements. The RI approximations may be used for integrals only
such that integrals are generated in conventional MO form and canonical CC/EOM calculations are performed, or
in a more complete version when modified CC/EOM equations are used such that the integrals are used in their RI
representation. The latter version allows for more substantial savings in storage and in computational speed-up.
The RI for integrals is invoked when AUX_BASIS is specified. All two-electron integrals are used in RI decomposed
form in CC when AUX_BASIS is specified.
By default, the integrals will be stored in the RI form and special CC/EOM code will be invoked. Keyword CC_DIRECT_RI
allows one to use RI generated integrals in conventional form (by transforming RI integrals back to the standard format)
invoking conventional CC procedures.

234

Chapter 6: Wave Function-Based Correlation Methods

Note: RI for integrals is available for all CCMAN/CCMAN2 methods. CCMAN requires that the unrestricted reference be used, CCMAN2 does not have this limitation. In addition, while RI is available for jobs that need
analytical gradients, only energies and properties are computed using RI. Energy derivatives are calculated
using regular electron repulsion integral derivatives. Full RI implementation (with integrals used in decomposed form) is only available for CCMAN2. For maximum computational efficiency, combine with FNO (see
Sections 6.11 and 7.7.8) when appropriate.

6.8.6

Cholesky decomposition with CC (CD-CC)

Two-electron integrals can be decomposed using Cholesky decomposition 27 giving rise to the same representation as
in RI and substantially reducing the cost of integral transformation, disk storage requirements, and improving parallel
performance:
M
X
P
P
(µν|λσ) ≈
Bµν
Bλσ
,
(6.38)
P =1

The rank of Cholesky decomposition, M , is typically 3-10 times larger than the number of basis functions N (Ref. 7);
it depends on the decomposition threshold δ and is considerably smaller than the full rank of the matrix, N (N + 1)/2
(Refs. 7,10,111). Cholesky decomposition removes linear dependencies in product densities (µν|, 7 allowing one to
obtain compact approximation to the original matrix with accuracy, in principle, up to machine precision.
Decomposition threshold δ is the only parameter that controls accuracy and the rank of the decomposition. Cholesky
decomposition is invoked by specifying CHOLESKY_TOL that defines the accuracy with which decomposition should
be performed. For most calculations tolerance of δ = 10−3 gives a good balance between accuracy and compactness
of the rank. Tolerance of δ = 10−2 can be used for exploratory calculations and δ = 10−4 for high-accuracy calculations. Similar to RI, Cholesky-decomposed integrals can be transformed back, into the canonical MO form, using
CC_DIRECT_RI keyword.
Note: Cholesky decomposition is available for all CCMAN2 methods. Analytic gradients are not yet available; only
energies and properties are computed using CD. For maximum computational efficiency, combine with FNO
(see Sections 6.11 and 7.7.8) when appropriate.

6.8.7

Job Control Options

There are a large number of options for the coupled-cluster singles and doubles methods. They are documented in
Appendix C, and, as the reader will find upon following this link, it is an extensive list indeed. Fortunately, many of
them are not necessary for routine jobs. Most of the options for non-routine jobs concern altering the default iterative
procedure, which is most often necessary for optimized orbital calculations (OD, QCCD), as well as the active space
and EOM methods discussed later in Section 6.10. The more common options relating to convergence control are
discussed there, in Section 6.10.6. Below we list the options that one should be aware of for routine calculations.
For memory options and parallel execution, see Section 6.14.

235

Chapter 6: Wave Function-Based Correlation Methods
CC_CONVERGENCE
Overall convergence criterion for the coupled-cluster codes. This is designed to ensure at least n
significant digits in the calculated energy, and automatically sets the other convergence-related
variables (CC_E_CONV, CC_T_CONV, CC_THETA_CONV, CC_THETA_GRAD_CONV) [10−n ].
TYPE:
INTEGER
DEFAULT:
6 Energies.
7 Gradients.
OPTIONS:
n Corresponding to 10−n convergence criterion. Amplitude convergence is set
automatically to match energy convergence.
RECOMMENDATION:
Use the default

Note: For single point calculations, CC_E_CONV = 6 and CC_T_CONV = 4.
(CC_T_CONV = 5) is used for gradients and EOM calculations.

Tighter amplitude convergence

CC_DOV_THRESH
Specifies minimum allowed values for the coupled-cluster energy denominators. Smaller values
are replaced by this constant during early iterations only, so the final results are unaffected, but
initial convergence is improved when the HOMO-LUMO gap is small or when non-conventional
references are used.
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
abcde Integer code is mapped to abc × 10−de , e.g., 2502 corresponds to 0.25
RECOMMENDATION:
Increase to 0.25, 0.5 or 0.75 for non convergent coupled-cluster calculations.
CC_SCALE_AMP
If not 0, scales down the step for updating coupled-cluster amplitudes in cases of problematic
convergence.
TYPE:
INTEGER
DEFAULT:
0 no scaling
OPTIONS:
abcd Integer code is mapped to abcd × 10−2 , e.g., 90 corresponds to 0.9
RECOMMENDATION:
Use 0.9 or 0.8 for non convergent coupled-cluster calculations.

Chapter 6: Wave Function-Based Correlation Methods
CC_MAX_ITER
Maximum number of iterations to optimize the coupled-cluster energy.
TYPE:
INTEGER
DEFAULT:
200
OPTIONS:
n up to n iterations to achieve convergence.
RECOMMENDATION:
None
CC_PRINT
Controls the output from post-MP2 coupled-cluster module of Q-C HEM
TYPE:
INTEGER
DEFAULT:
1
OPTIONS:
0 − 7 higher values can lead to deforestation. . .
RECOMMENDATION:
Increase if you need more output and don’t like trees
CHOLESKY_TOL
Tolerance of Cholesky decomposition of two-electron integrals
TYPE:
INTEGER
DEFAULT:
3
OPTIONS:
n Corresponds to a tolerance of 10−n
RECOMMENDATION:
2 - qualitative calculations, 3 - appropriate for most cases, 4 - quantitative (error in total energy
typically less than 1 µhartree)
CC_DIRECT_RI
Controls use of RI and Cholesky integrals in conventional (undecomposed) form
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
FALSE use all integrals in decomposed format
TRUE
transform all RI or Cholesky integral back to conventional format
RECOMMENDATION:
By default all integrals are used in decomposed format allowing significant reduction of memory use. If all integrals are transformed back (TRUE option) no memory reduction is achieved
and decomposition error is introduced, however, the integral transformation is performed significantly faster and conventional CC/EOM algorithms are used.

236

Chapter 6: Wave Function-Based Correlation Methods

6.8.8

237

Examples

Example 6.16 A series of jobs evaluating the correlation energy (with core orbitals frozen) of the ground state of the
NH2 radical with three methods of coupled-cluster singles and doubles type: CCSD itself, OD, and QCCD.
$molecule
0 2
N
H1 N 1.02805
H2 N 1.02805
$end
$rem
METHOD
BASIS
N_FROZEN_CORE
$end

H1

103.34

ccsd
6-31g*
fc

@@@
$molecule
read
$end
$rem
METHOD
BASIS
N_FROZEN_CORE
$end

od
6-31g*
fc

@@@
$molecule
read
$end
$rem
METHOD
BASIS
N_FROZEN_CORE
$end

qccd
6-31g*
fc

Example 6.17 A job evaluating CCSD energy of water using RI-CCSD
$molecule
0 1
O
H1 O OH
H2 O OH H1 HOH
OH = 0.947
HOH = 105.5
$end
$rem
METHOD
BASIS
AUX_BASIS
$end

ccsd
aug-cc-pvdz
rimp2-aug-cc-pvdz

Chapter 6: Wave Function-Based Correlation Methods

238

Example 6.18 A job evaluating CCSD energy of water using CD-CCSD (tolerance = 10−3 )
$molecule
0 1
O
H1 O OH
H2 O OH H1 HOH
OH = 0.947
HOH = 105.5
$end
$rem
METHOD
BASIS
CHOLESKY_TOL
$end

ccsd
aug-cc-pvdz
3

Example 6.19 A job evaluating CCSD energy of water using CD-CCSD (tolerance = 10−3 ) with FNO
$molecule
0 1
O
H1 O OH
H2 O OH H1 HOH
OH = 0.947
HOH = 105.5
$end
$rem
METHOD
BASIS
CHOLESKY_TOL
CC_FNO_THRESH
$end

6.9
6.9.1

ccsd
aug-cc-pvdz
3
9950

Non-Iterative Corrections to Coupled Cluster Energies
(T) Triples Corrections

To approach chemical accuracy in reaction energies and related properties, it is necessary to account for electron
correlation effects that involve three electrons simultaneously, as represented by triple substitutions relative to the mean
field single determinant reference, which arise in MP4. The best standard methods for including triple substitutions are
the CCSD(T) 82 and QCISD(T) methods. 77 The accuracy of these methods is well-documented for many cases, 66 and
in general is a very significant improvement relative to the starting point (either CCSD or QCISD). The cost of these
corrections scales with the 7th power of molecule size (or the 4th power of the number of basis functions, for a fixed
molecule size), although no additional disk resources are required relative to the starting coupled-cluster calculation.
Q-C HEM supports the evaluation of CCSD(T) and QCISD(T) energies, as well as the corresponding OD(T) correction
to the optimized doubles method discussed in the previous subsection. Gradients and properties are not yet available
for any of these (T) corrections.

Chapter 6: Wave Function-Based Correlation Methods

6.9.2

239

(2) Triples and Quadruples Corrections

While the (T) corrections discussed above have been extraordinarily successful, there is nonetheless still room for
further improvements in accuracy, for at least some important classes of problems. They contain judiciously chosen
terms from 4th- and 5th-order Møller-Plesset perturbation theory, as well as higher order terms that result from the fact
that the converged cluster amplitudes are employed to evaluate the 4th- and 5th-order order terms. The (T) correction
therefore depends upon the bare reference orbitals and orbital energies, and in this way its effectiveness still depends
on the quality of the reference determinant. Since we are correcting a coupled-cluster solution rather than a single
determinant, this is an aspect of the (T) corrections that can be improved. Deficiencies of the (T) corrections show up
computationally in cases where there are near-degeneracies between orbitals, such as stretched bonds, some transition
states, open shell radicals, and diradicals.
Prof. Steve Gwaltney, while working at Berkeley with Martin Head-Gordon, has suggested a new class of non iterative
correction that offers the prospect of improved accuracy in problem cases of the types identified above. 42 Q-C HEM
contains Gwaltney’s implementation of this new method, for energies only. The new correction is a true second-order
correction to a coupled-cluster starting point, and is therefore denoted as (2). It is available for two of the cluster
methods discussed above, as OD(2) and CCSD(2). 42,43 Only energies are available at present.
The basis of the (2) method is to partition not the regular Hamiltonian into perturbed and unperturbed parts, but rather
to partition a similarity-transformed Hamiltonian, defined as H̃ = e−T̂ ĤeT̂ . In the truncated space (call it the p-space)
within which the cluster problem is solved (e.g., singles and doubles for CCSD), the coupled-cluster wave function is a
true eigenvalue of H̃. Therefore we take the zero order Hamiltonian, H̃ (0) , to be the full H̃ in the p-space, while in the
space of excluded substitutions (the q-space) we take only the one-body part of H̃ (which can be made diagonal). The
fluctuation potential describing electron correlations in the q-space is H̃ − H̃ (0) , and the (2) correction then follows
from second-order perturbation theory.
The new partitioning of terms between the perturbed and unperturbed Hamiltonians inherent in the (2) correction leads
to a correction that shows both similarities and differences relative to the existing (T) corrections. There are two types
of higher correlations that enter at second-order: not only triple substitutions, but also quadruple substitutions. The
quadruples are treated with a factorization ansatz, that is exact in 5th order Møller-Plesset theory, 57 to reduce their
computational cost from N 9 to N 6 . For large basis sets this can still be larger than the cost of the triples terms, which
scale as the 7th power of molecule size, with a factor twice as large as the usual (T) corrections.
These corrections are feasible for molecules containing between four and ten first row atoms, depending on computer
resources, and the size of the basis set chosen. There is early evidence that the (2) corrections are superior to the (T)
corrections for highly correlated systems. 42 This shows up in improved potential curves, particularly at long range
and may also extend to improved energetic and structural properties at equilibrium in problematical cases. It will be
some time before sufficient testing on the new (2) corrections has been done to permit a general assessment of the
performance of these methods. However, they are clearly very promising, and for this reason they are available in
Q-C HEM.

6.9.3

(dT) and (fT) corrections

Alternative inclusion of non-iterative N 7 triples corrections is described in Section 7.7.21. These methods called
(dT) and (fT) are of similar accuracy to other triples corrections. CCSD(dT) and CCSD(fT) are equivalent to the
CR-CCSD(T)L and CR-CCSD(T)2 methods of Piecuch and coworkers. 76
Note: Due to a violation of orbital invariance, the (dT) correction can sometimes lead to spurious results. Therefore,
its use is discouraged. Use (fT) instead!

6.9.4

Job Control Options

The evaluation of a non-iterative (T) or (2) correction after a coupled-cluster singles and doubles level calculation
(either CCSD, QCISD or OD) is controlled by the correlation keyword, and the specification of any frozen orbitals via

Chapter 6: Wave Function-Based Correlation Methods

240

N_FROZEN_CORE (and possibly N_FROZEN_VIRTUAL).

For the (2) correction, it is possible to apply the frozen core approximation in the reference coupled cluster calculation,
and then correlate all orbitals in the (2) correction. This is controlled by CC_INCL_CORE_CORR, described below.
The default is to include core and core-valence correlation automatically in the CCSD(2) or OD(2) correction, if the
reference CCSD or OD calculation was performed with frozen core orbitals. The reason for this choice is that core
correlation is economical to include via this method (the main cost increase is only linear in the number of core orbitals),
and such effects are important to account for in accurate calculations. This option should be made false if a job with
explicitly frozen core orbitals is desired. One good reason for freezing core orbitals in the correction is if the basis set
is physically inappropriate for describing core correlation (e.g., standard Pople basis sets, and Dunning cc-pVxZ basis
sets are designed to describe valence-only correlation effects). Another good reason is if a direct comparison is desired
against another method such as CCSD(T) which is always used in the same orbital window as the CCSD reference.
There are several implementations of non-iterative triples available in Q-C HEM. In the original CCMAN suite, (T), (2),
and (dT)/(fT) corrections can be computed. The parallel scaling of this code is very modest (4 cores max). CCMAN2
currently allows only the calculation of (T) correction for CCSD wave fucntions. By default, the CCMAN2 code is
used for (T). The CCMAN code CCMAN2 is set to false. There are two versions of (T) in CCMAN2: The default
version (native CCMAN2) and a new version using libpt. The implementation based on libpt is in-core MPI/OpenMP
distributed-parallel. It is significantly faster in most realistic calculations (but it does not use point group symmetry, so
it might show slower performance for small jobs with high symmetry). The libpt code is enabled by setting USE_LIBPT
to true.
Note: For the best performance of libpt (T) code, parallel execution should be requested, see Section 2.8.
USE_LIBPT
Enable libpt for CCSD(T) calculations in CCMAN2.
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
TRUE FALSE
RECOMMENDATION:
libpt is now used by default in all real-valued CC/EOM-CC calculations
CC_INCL_CORE_CORR
Whether to include the correlation contribution from frozen core orbitals in non iterative (2)
corrections, such as OD(2) and CCSD(2).
TYPE:
LOGICAL
DEFAULT:
TRUE
OPTIONS:
TRUE FALSE
RECOMMENDATION:
Use the default unless no core-valence or core correlation is desired (e.g., for comparison with
other methods or because the basis used cannot describe core correlation).

Chapter 6: Wave Function-Based Correlation Methods

241

Chapter 6: Wave Function-Based Correlation Methods

6.9.5

242

Examples

Example 6.20 Two jobs that compare the correlation energy calculated via the standard CCSD(T) method with the
new CCSD(2) approximation, both using the frozen core approximation. This requires that CC_INCL_CORE_CORR
must be specified as FALSE in the CCSD(2) input.
$molecule
0 2
O
H O 0.97907
$end
$rem
METHOD
BASIS
N_FROZEN_CORE
$end

ccsd(t)
cc-pvtz
fc

@@@
$molecule
read
$end
$rem
METHOD
BASIS
N_FROZEN_CORE
CC_INCL_CORE_CORR
$end

ccsd(2)
cc-pvtz
fc
false

Example 6.21 Using libpt for a standard CCSD(T) calculation
$molecule
0 2
O
H O 0.97907
$end
$rem
METHOD
BASIS
N_FROZEN_CORE
USE_LIBPT
$end

ccsd(t)
cc-pvtz
fc
true

Example 6.22 Water: Ground state CCSD(dT) calculation using RI
$molecule
0 1
O
H1 O OH
H2 O OH

H1

HOH

OH = 0.957
HOH = 104.5
$end
$rem
JOBTYPE
BASIS
AUX_BASIS
METHOD
$end

SP
cc-pvtz
rimp2-cc-pvtz
CCSD(dT)

Chapter 6: Wave Function-Based Correlation Methods

6.10

Coupled Cluster Active Space Methods

6.10.1

Introduction

243

Electron correlation effects can be qualitatively divided into two classes. The first class is static or non-dynamical
correlation: long wavelength low-energy correlations associated with other electron configurations that are nearly
as low in energy as the lowest energy configuration. These correlation effects are important for problems such as
homolytic bond breaking, and are the hardest to describe because by definition the single configuration Hartree-Fock
description is not a good starting point. The second class is dynamical correlation: short wavelength high-energy
correlations associated with atomic-like effects. Dynamical correlation is essential for quantitative accuracy, but a
reasonable description of static correlation is a prerequisite for a calculation being qualitatively correct.
In the methods discussed in the previous several subsections, the objective was to approximate the total correlation
energy. However, in some cases, it is useful to model directly the non-dynamical and dynamical correlation energies
separately. The reasons for this are pragmatic: with approximate methods, such a separation can give a more balanced
treatment of electron correlation along bond-breaking coordinates, or reaction coordinates that involve diradicaloid intermediates. The non-dynamical correlation energy is conveniently defined as the solution of the Schrödinger equation
within a small basis set composed of valence bonding, anti-bonding and lone pair orbitals: the so-called full valence
active space. Solved exactly, this is the so-called full valence complete active space SCF (CASSCF), 86 or equivalently,
the fully optimized reaction space (FORS) method. 88
Full valence CASSCF and FORS involve computational complexity which increases exponentially with the number
of atoms, and is thus unfeasible beyond systems of only a few atoms, unless the active space is further restricted on a
case-by-case basis. Q-C HEM includes two relatively economical methods that directly approximate these theories using a truncated coupled-cluster doubles wave function with optimized orbitals. 56 They are active space generalizations
of the OD and QCCD methods discussed previously in Sections 6.8.3 and 6.8.4, and are discussed in the following two
subsections. By contrast with the exponential growth of computational cost with problem size associated with exact solution of the full valence CASSCF problem, these cluster approximations have only 6th-order growth of computational
cost with problem size, while often providing useful accuracy.
The full valence space is a well-defined theoretical chemical model. For these active space coupled-cluster doubles
methods, it consists of the union of valence levels that are occupied in the single determinant reference, and those that
are empty. The occupied levels that are to be replaced can only be the occupied valence and lone pair orbitals, whose
number is defined by the sum of the valence electron counts for each atom (i.e., 1 for H, 2 for He, 1 for Li, etc..). At
the same time, the empty virtual orbitals to which the double substitutions occur are restricted to be empty (usually
anti-bonding) valence orbitals. Their number is the difference between the number of valence atomic orbitals, and the
number of occupied valence orbitals given above. This definition (the full valence space) is the default when either of
the “valence” active space methods are invoked (VOD or VQCCD)
There is also a second useful definition of a valence active space, which we shall call the 1:1 or perfect pairing active
space. In this definition, the number of occupied valence orbitals remains the same as above. The number of empty
correlating orbitals in the active space is defined as being exactly the same number, so that each occupied orbital may
be regarded as being associated 1:1 with a correlating virtual orbital. In the water molecule, for example, this means
that the lone pair electrons as well as the bond-orbitals are correlated. Generally the 1:1 active space recovers more
correlation for molecules dominated by elements on the right of the periodic table, while the full valence active space
recovers more correlation for molecules dominated by atoms to the left of the periodic table.
If you wish to specify either the 1:1 active space as described above, or some other choice of active space based on
your particular chemical problem, then you must specify the numbers of active occupied and virtual orbitals. This is
done via the standard “window options”, documented earlier in this Chapter.
Finally we note that the entire discussion of active spaces here leads only to specific numbers of active occupied and
virtual orbitals. The orbitals that are contained within these spaces are optimized by minimizing the trial energy with
respect to all the degrees of freedom previously discussed: the substitution amplitudes, and the orbital rotation angles
mixing occupied and virtual levels. In addition, there are new orbital degrees of freedom to be optimized to obtain the

244

Chapter 6: Wave Function-Based Correlation Methods

best active space of the chosen size, in the sense of yielding the lowest coupled-cluster energy. Thus rotation angles
mixing active and inactive occupied orbitals must be varied until the energy is stationary. Denoting inactive orbitals by
primes and active orbitals without primes, this corresponds to satisfying
∂ECCD
∂θij

0

=0

(6.39)

Likewise, the rotation angles mixing active and inactive virtual orbitals must also be varied until the coupled-cluster
energy is minimized with respect to these degrees of freedom:
∂ECCD
=0
∂θab0

6.10.2

(6.40)

VOD and VOD(2) Methods

The VOD method is the active space version of the OD method described earlier in Section 6.8.3. Both energies and
gradients are available for VOD, so structure optimization is possible. There are a few important comments to make
about the usefulness of VOD. First, it is a method that is capable of accurately treating problems that fundamentally
involve 2 active electrons in a given local region of the molecule. It is therefore a good alternative for describing single
bond-breaking, or torsion around a double bond, or some classes of diradicals. However it often performs poorly for
problems where there is more than one bond being broken in a local region, with the non variational solutions being
quite possible. For such problems the newer VQCCD method is substantially more reliable.
Assuming that VOD is a valid zero order description for the electronic structure, then a second-order correction,
VOD(2), is available for energies only. VOD(2) is a version of OD(2) generalized to valence active spaces. It permits more accurate calculations of relative energies by accounting for dynamical correlation.

6.10.3

VQCCD

The VQCCD method is the active space version of the QCCD method described earlier in Section 6.8.3. Both energies and gradients are available for VQCCD, so that structure optimization is possible. VQCCD is applicable to a
substantially wider range of problems than the VOD method, because the modified energy functional is not vulnerable
to non variational collapse. Testing to date suggests that it is capable of describing double bond breaking to similar
accuracy as full valence CASSCF, and that potential curves for triple bond-breaking are qualitatively correct, although
quantitatively in error by a few tens of kcal/mol. The computational cost scales in the same manner with system size
as the VOD method, albeit with a significantly larger prefactor.

6.10.4

CCVB-SD

Working with Prof. Head-Gordon at Berkeley, Dr. D. W. Small and Joonho Lee have developed and implemented a
novel single-reference coupled-cluster method with singles and doubles, called CCVB-SD. 50,94 CCVB-SD improves
upon a more crude model CCVB (Section 6.15.2) and can be considered a simple modification to restricted CCSD
(RCCSD). CCVB-SD inherits good properties from CCVB and RCCSD; it is spin-pure, size-extensive, and capable
of breaking multiple bonds as long as only the valence space is correlated. It is a full doubles model and thus scales
O(N 6 ). However, its energy is invariant under rotations in occupied space and virtual space, which makes it much
more black-box than CCVB. Its energy function follows
*
! +


 
Q̂2
ECCVB−SD = Φ0 1 + Λ̂ Ĥ exp T̂ − ÎS
Φ0
(6.41)
2
C

where ÎS is a singlet projection operator and Q̂ is a quintet doubles operator. Unlike QCCD, CCVB-SD improves
the right eigenfunction while leaving the left eigenfunction unchanged. The quintet term in Eq. (6.41) represents
approximate connected quadruples which are responsible for describing strong correlation. The cost of CCVB-SD is

Chapter 6: Wave Function-Based Correlation Methods

245

only twice as expensive as RCCSD, and it is better suited for strong correlation than QCCD/VQCCD in the sense that
the method becomes exact at the dissociation limits of most multiple bond breaking whereas QCCD does not except
special cases.
Although CCVB-SD can be used without the active space constraints, we recommend that users use it with the valence
active space in general. For benchmarking purposes, using a minimal basis will automatically provide the valence
space correctly with frozen cores. Both the energy and nuclear gradients of CCVB-SD are available through CCMAN2.
It should be noted that there is no orbital optimization implemented for CCVB-SD at the moment. This means that
using basis sets larger than minimal basis requires choosing right valence orbitals to use. Therefore, we recommend
that users run GVB-PP (or CCVB) to obtain orbitals to begin with. Orbital optimization (i.e. CCVB-OD) will soon
be implemented and running CCVB-OD will be much more black-box than CCVB-SD as it does not require selecting
proper valence space orbitals.
Furthermore, CCVB-SD can be applied to only closed-shell molecules at the moment. The extension to open-shell
molecules is under development.
Example 6.23 A CCVB-SD force calculation of benzene in a minimal basis.
$comment
CCVB-SD job for benzene
It will compute energy+gradients.
It will also print out natural orbital occupation numbers (NOONs)
$end
$molecule
0 1
C
0.000000
C
0.000000
C
1.209318
C
1.209318
C
2.418636
C
2.418636
H
-0.931410
H
-0.931410
H
1.209318
H
1.209318
H
3.350046
H
3.350046
$end
$rem
JOBTYPE
BASIS
METHOD
THRESH
SCF_ALGORITHM
SCF_CONVERGENCE
CC_REF_PROP
SYMMETRY
SYM_IGNORE
$end

6.10.5

0.698200
-0.698200
1.396400
-1.396400
0.698200
-0.698200
1.235950
-1.235950
2.471900
-2.471900
1.235950
-1.235950

=
=
=
=
=
=
=
=
=

0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000

force
sto-3g
ccvbsd
14
gdm
10
true
false
true

Local Pair Models for Valence Correlations Beyond Doubles

Working with Prof. Head-Gordon at Berkeley, John Parkhill has developed implementations for pair models which
couple 4 and 6 electrons together quantitatively. Because these truncate the coupled cluster equations at quadruples and
hextuples respectively they have been termed the “Perfect Quadruples” and “Perfect Hextuples” models. These can be
viewed as local approximations to CASSCF. The PQ and PH models are executed through an extension of Q-C HEM’s

Chapter 6: Wave Function-Based Correlation Methods

246

coupled cluster code, and several options defined for those models will have the same effects although the mechanism
may be different (CC_DIIS_START, CC_DIIS_SIZE, CC_DOV_THRESH, CC_CONV, etc.).
In the course of implementation, the non-local coupled cluster models were also implemented up to T̂6 . Because the
algorithms are explicitly sparse their costs relative to the existing implementations of CCSD are much higher (and
should never be used in lieu of an existing CCMAN code), but this capability may be useful for development purposes,
and when computable, models above CCSDTQ are highly accurate. To use PQ, PH, their dynamically correlated “+SD”
versions or this machine generated cluster code set: METHOD = MGC.
MGC_AMODEL
Choice of approximate cluster model.
TYPE:
INTEGER
DEFAULT:
Determines how the CC equations are approximated:
OPTIONS:
0 Local Active-Space Amplitude iterations (pre-calculate GVB orbitals with your method of choice
(RPP is good)).
7 Optimize-Orbitals using the VOD 2-step solver.
(Experimental-only use with MGC_AMPS = 2, 24 ,246)
8 Traditional Coupled Cluster up to CCSDTQPH.
9 MR-CC version of the Pair-Models. (Experimental)
RECOMMENDATION:
None
MGC_NLPAIRS
Number of local pairs on an amplitude.
TYPE:
INTEGER
DEFAULT:
None
OPTIONS:
Must be greater than 1, which corresponds to the PP model. 2 for PQ, and 3 for PH.
RECOMMENDATION:
None
MGC_AMPS
Choice of Amplitude Truncation
TYPE:
INTEGER
DEFAULT:
None
OPTIONS:
2≤ n ≤ 123456, a sorted list of integers for every amplitude
which will be iterated. Choose 1234 for PQ and 123456 for PH
RECOMMENDATION:
None

Chapter 6: Wave Function-Based Correlation Methods

247

MGC_LOCALINTS
Pair filter on an integrals.
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
Enforces a pair filter on the 2-electron integrals, significantly
reducing computational cost. Generally useful. for more than 1 pair locality.
RECOMMENDATION:
None
MGC_LOCALINTER
Pair filter on an intermediate.
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
Any nonzero value enforces the pair constraint on intermediates,
significantly reducing computational cost. Not recommended for ≤ 2 pair locality
RECOMMENDATION:
None

6.10.6

Convergence Strategies and More Advanced Options

These optimized orbital coupled-cluster active space methods enable the use of the full valence space for larger systems than is possible with conventional complete active space codes. However, we should note at the outset that often
there are substantial challenges in converging valence active space calculations (and even sometimes optimized orbital
coupled cluster calculations without an active space). Active space calculations cannot be regarded as “routine” calculations in the same way as SCF calculations, and often require a considerable amount of computational trial and error
to persuade them to converge. These difficulties are largely because of strong coupling between the orbital degrees of
freedom and the amplitude degrees of freedom, as well as the fact that the energy surface is often quite flat with respect
to the orbital variations defining the active space.
Being aware of this at the outset, and realizing that the program has nothing against you personally is useful information
for the uninitiated user of these methods. What the program does have, to assist in the struggle to achieve a converged
solution, are accordingly many convergence options, fully documented in Appendix C. In this section, we describe the
basic options and the ideas behind using them as a starting point. Experience plays a critical role, however, and so
we encourage you to experiment with toy jobs that give rapid feedback in order to become proficient at diagnosing
problems.
If the default procedure fails to converge, the first useful option to employ is CC_PRECONV_T2Z, with a value of
between 10 and 50. This is useful for jobs in which the MP2 amplitudes are very poor guesses for the converged cluster
amplitudes, and therefore initial iterations varying only the amplitudes will be beneficial:

Chapter 6: Wave Function-Based Correlation Methods

248

CC_PRECONV_T2Z
Whether to pre-converge the cluster amplitudes before beginning orbital optimization in optimized orbital cluster methods.
TYPE:
INTEGER
DEFAULT:
0
(FALSE)
10 If CC_RESTART, CC_RESTART_NO_SCF or CC_MP2NO_GUESS are TRUE
OPTIONS:
0 No pre-convergence before orbital optimization.
n Up to n iterations in this pre-convergence procedure.
RECOMMENDATION:
Experiment with this option in cases of convergence failure.

Other options that are useful include those that permit some damping of step sizes, and modify or disable the standard
DIIS procedure. The main choices are as follows.
CC_DIIS
Specify the version of Pulay’s Direct Inversion of the Iterative Subspace (DIIS) convergence
accelerator to be used in the coupled-cluster code.
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
0 Activates procedure 2 initially, and procedure 1 when gradients are smaller
than DIIS12_SWITCH.
1 Uses error vectors defined as differences between parameter vectors from
successive iterations. Most efficient near convergence.
2 Error vectors are defined as gradients scaled by square root of the
approximate diagonal Hessian. Most efficient far from convergence.
RECOMMENDATION:
DIIS1 can be more stable. If DIIS problems are encountered in the early stages of a calculation
(when gradients are large) try DIIS1.
CC_DIIS_START
Iteration number when DIIS is turned on. Set to a large number to disable DIIS.
TYPE:
INTEGER
DEFAULT:
3
OPTIONS:
n User-defined
RECOMMENDATION:
Occasionally DIIS can cause optimized orbital coupled-cluster calculations to diverge through
large orbital changes. If this is seen, DIIS should be disabled.

Chapter 6: Wave Function-Based Correlation Methods

249

CC_DOV_THRESH
Specifies minimum allowed values for the coupled-cluster energy denominators. Smaller values
are replaced by this constant during early iterations only, so the final results are unaffected, but
initial convergence is improved when the guess is poor.
TYPE:
INTEGER
DEFAULT:
2502 Corresponding to 0.25
OPTIONS:
abcde Integer code is mapped to abc × 10−de
RECOMMENDATION:
Increase to 0.5 or 0.75 for non convergent coupled-cluster calculations.
CC_THETA_STEPSIZE
Scale factor for the orbital rotation step size. The optimal rotation steps should be approximately
equal to the gradient vector.
TYPE:
INTEGER
DEFAULT:
100 Corresponding to 1.0
OPTIONS:
abcde Integer code is mapped to abc × 10−de
If the initial step is smaller than 0.5, the program will increase step
when gradients are smaller than the value of THETA_GRAD_THRESH,
up to a limit of 0.5.
RECOMMENDATION:
Try a smaller value in cases of poor convergence and very large orbital gradients. For example,
a value of 01001 translates to 0.1

An even stronger—and more-or-less last resort—option permits iteration of the cluster amplitudes without changing
the orbitals:
CC_PRECONV_T2Z_EACH
Whether to pre-converge the cluster amplitudes before each change of the orbitals in optimized
orbital coupled-cluster methods. The maximum number of iterations in this pre-convergence
procedure is given by the value of this parameter.
TYPE:
INTEGER
DEFAULT:
0 (FALSE)
OPTIONS:
0 No pre-convergence before orbital optimization.
n Up to n iterations in this pre-convergence procedure.
RECOMMENDATION:
A very slow last resort option for jobs that do not converge.

Chapter 6: Wave Function-Based Correlation Methods

6.10.7

250

Examples

Example 6.24 Two jobs that compare the correlation energy of the water molecule with partially stretched bonds,
calculated via the two coupled-cluster active space methods, VOD, and VQCCD. These are relatively “easy” jobs to
converge, and may be contrasted with the next example, which is not easy to converge. The orbitals are restricted.
$molecule
0 1
O
H 1 r
H 1 r

a

r =
1.5
a = 104.5
$end
$rem
METHOD
BASIS
$end

vod
6-31G

@@@
$molecule
read
$end
$rem
METHOD
BASIS
$end

vqccd
6-31G

Chapter 6: Wave Function-Based Correlation Methods

251

Example 6.25 The water molecule with highly stretched bonds, calculated via the two coupled-cluster active space
methods, VOD, and VQCCD. These are “difficult” jobs to converge. The convergence options shown permitted the job
to converge after some experimentation (thanks due to Ed Byrd for this!). The difficulty of converging this job should
be contrasted with the previous example where the bonds were less stretched. In this case, the VQCCD method yields
far better results than VOD!.
$molecule
0 1
O
H 1 r
H 1 r

a

r =
3.0
a = 104.5
$end
$rem
METHOD
BASIS
SCF_CONVERGENCE
THRESH
CC_PRECONV_T2Z
CC_PRECONV_T2Z_EACH
CC_DOV_THRESH
CC_THETA_STEPSIZE
CC_DIIS_START
$end

vod
6-31G
9
12
50
50
7500
3200
75

@@@
$molecule
read
$end
$rem
METHOD
BASIS
SCF_CONVERGENCE
THRESH
CC_PRECONV_T2Z
CC_PRECONV_T2Z_EACH
CC_DOV_THRESH
CC_THETA_STEPSIZE
CC_DIIS_START
$end

6.11

vqccd
6-31G
9
12
50
50
7500
3200
75

Frozen Natural Orbitals in CCD, CCSD, OD, QCCD, and QCISD Calculations

Large computational savings are possible if the virtual space is truncated using the frozen natural orbital (FNO) approach. For example, using a fraction f of the full virtual space results in a 1/(1 − f )4 -fold speed up for each CCSD
iteration (CCSD scales with the forth power of the virtual space size). FNO-based truncation for ground-states CC
methods was introduced by Bartlett and coworkers. 97,101,102 Extension of the FNO approach to ionized states within
EOM-CC formalism was recently introduced and benchmarked; 58 see Section 7.7.8.
The FNOs are computed as the eigenstates of the virtual-virtual block of the MP2 density matrix [O(N 5 ) scaling], and
the eigenvalues are the occupation numbers associated with the respective FNOs. By using a user-specified threshold,

Chapter 6: Wave Function-Based Correlation Methods

252

the FNOs with the smallest occupations are frozen in CC calculations. This could be done in CCSD, CCSD(T),
CCSD(2), CCSD(dT), CCSD(fT) as well as CCD, OD,QCCD, VQCCD, and all possible triples corrections for these
wave functions.
The truncation can be performed using two different schemes. The first approach is to simply specify the total number
of virtual orbitals to retain, e.g., as the percentage of total virtual orbitals, as was done in Refs. 101,102. The second
approach is to specify the percentage of total natural occupation (in the virtual space) that needs to be recovered in the
truncated space. These two criteria are referred to as the POVO (percentage of virtual orbitals) and OCCT (occupation
threshold) cutoffs, respectively. 58
Since the OCCT criterion is based on the correlation in a specific molecule, it yields more consistent results than POVO.
For ionization energy calculations employing 99–99.5% natural occupation threshold should yields errors (relative
to the full virtual space values) below 1 kcal/mol. 58 The errors decrease linearly as a function of the total natural
occupation recovered, which can be exploited by extrapolating truncated calculations to the full virtual space values.
This extrapolation scheme is called the extrapolated FNO (XFNO) procedure. 58 The linear behavior is exhibited by
the total energies of the ground and the ionized states as a function of OCCT. Therefore, the XFNO scheme can be
employed even when the two states are not calculated on the same level, e.g., in adiabatic energy differences and
EOM-IP-CC(2,3) calculations (more on this in Ref. 58).
The FNO truncation often causes slower convergence of the CCSD and EOM procedures. Nevertheless, despite larger
number of iterations, the FNO-based truncation of orbital space reduces computational cost considerably, with a negligible decline in accuracy. 58

6.11.1

Job Control Options
CC_FNO_THRESH
Initialize the FNO truncation and sets the threshold to be used for both cutoffs (OCCT and
POVO)
TYPE:
INTEGER
DEFAULT:
None
OPTIONS:
range 0000-10000
abcd
Corresponding to ab.cd%
RECOMMENDATION:
None
CC_FNO_USEPOP
Selection of the truncation scheme
TYPE:
INTEGER
DEFAULT:
1 OCCT
OPTIONS:
0 POVO
RECOMMENDATION:
None

Chapter 6: Wave Function-Based Correlation Methods

6.11.2

253

Example

Example 6.26 CCSD(T) calculation using FNO with POVO=65%
$molecule
0 1
O
H 1 1.0
H 1 1.0 2 100.
$end
$rem
METHOD
BASIS
CC_FNO_THRESH
CC_FNO_USEPOP
$end

6.12

=
=
=
=

CCSD(T)
6-311+G(2df,2pd)
6500
65% of the virtual space
0

Non-Hartree-Fock Orbitals in Correlated Calculations

In cases of problematic open-shell references, e.g., strongly spin-contaminated doublet radicals, one may choose to use
DFT orbitals, which can yield significantly improved results. 11 This can be achieved by first doing DFT calculation
and then reading the orbitals and turning the Hartree-Fock procedure off. A more convenient way is just to specify EXCHANGE, e.g., EXCHANGE = B3LYP means that B3LYP orbitals will be computed and used in the CCMAN/
CCMAN2 module, as in the following example.
Example 6.27 CCSD calculation of triplet methylene using B3LYP orbitals
$molecule
0 3
C
H 1 CH
H 1 CH

2

HCH

CH = 1.07
HCH = 111.0
$end
$rem
JOBTYPE
EXCHANGE
CORRELATION
BASIS
N_FROZEN_CORE
$end

6.13

sp
single point
b3lyp
ccsd
cc-pvdz
1

Analytic Gradients and Properties for Coupled-Cluster Methods

Analytic gradients are available for CCSD, OO-CCD/VOD, CCD, and QCCD/VQCCD methods for both closed- and
open-shell references (UHF and RHF only), including frozen core and/or virtual functionality. Analytic gradients are
available for CCVB-SD for only closed-shell references (RHF). In addition, gradients for selected GVB models are
available.
For the CCSD and OO-CCD wave functions, Q-C HEM can also calculate dipole moments, hR2 i (as well as XX, YY
and ZZ components separately, which is useful for assigning different Rydberg states, e.g., 3px vs. 3s, etc.), and the

Chapter 6: Wave Function-Based Correlation Methods

254

hS 2 i values. Interface of the CCSD and (V)OO-CCD codes with the NBO 5.0 package is also available. This code is
closely related to EOM-CCSD properties/gradient calculations (Section 7.7.16). Solvent models available for CCSD
are described in Chapter 12.2.
Limitations: Gradients and fully relaxed properties for ROHF and non-HF (e.g., B3LYP) orbitals as well as RI approximation are not yet available.
Note: If gradients or properties are computed with frozen core/virtual, the algorithm will replace frozen orbitals to
restricted. This will not affect the energies, but will change the orbital numbering in the CCMAN printout.

6.13.1

Job Control Options
CC_REF_PROP
Whether or not the non-relaxed (expectation value) or full response (including orbital relaxation
terms) one-particle CCSD properties will be calculated. The properties currently include permanent dipole moment, the second moments hX 2 i, hY 2 i, and hZ 2 i of electron density, and the total
hR2 i = hX 2 i+hY 2 i+hZ 2 i (in atomic units). Incompatible with JOBTYPE=FORCE, OPT, FREQ.
TYPE:
LOGICAL
DEFAULT:
FALSE (no one-particle properties will be calculated)
OPTIONS:
FALSE, TRUE
RECOMMENDATION:
Additional equations need to be solved (lambda CCSD equations) for properties with the cost
approximately the same as CCSD equations. Use the default if you do not need properties. The
cost of the properties calculation itself is low. The CCSD one-particle density can be analyzed
with NBO package by specifying NBO=TRUE, CC_REF_PROP=TRUE and JOBTYPE=FORCE.
CC_REF_PROP_TE
Request for calculation of non-relaxed two-particle CCSD properties. The two-particle properties currently include hS 2 i. The one-particle properties also will be calculated, since the additional cost of the one-particle properties calculation is inferior compared to the cost of hS 2 i. The
variable CC_REF_PROP must be also set to TRUE.
TYPE:
LOGICAL
DEFAULT:
FALSE (no two-particle properties will be calculated)
OPTIONS:
FALSE, TRUE
RECOMMENDATION:
The two-particle properties are computationally expensive, since they require calculation and use
of the two-particle density matrix (the cost is approximately the same as the cost of an analytic
gradient calculation). Do not request the two-particle properties unless you really need them.

Chapter 6: Wave Function-Based Correlation Methods

255

CC_FULLRESPONSE
Fully relaxed properties (including orbital relaxation terms) will be computed. The variable
CC_REF_PROP must be also set to TRUE.
TYPE:
LOGICAL
DEFAULT:
FALSE (no orbital response will be calculated)
OPTIONS:
FALSE, TRUE
RECOMMENDATION:
Not available for non UHF/RHF references and for the methods that do not have analytic gradients (e.g., QCISD).

6.13.2

Examples

Example 6.28 CCSD geometry optimization of HHeF followed up by properties calculations
$molecule
0 1
H
0.000000
He
0.000000
F
0.000000
$end

0.000000
0.000000
0.000000

$rem
JOBTYPE
METHOD
BASIS
GEOM_OPT_TOL_GRADIENT
GEOM_OPT_TOL_DISPLACEMENT
GEOM_OPT_TOL_ENERGY
$end

-1.886789
-1.093834
0.333122

OPT
CCSD
aug-cc-pVDZ
1
1
1

@@@
$molecule
read
$end
$rem
JOBTYPE
METHOD
BASIS
SCF_GUESS
CC_REF_PROP
CC_FULLRESPONSE
$end

6.14

SP
CCSD
aug-cc-pVDZ
READ
1
1

Memory Options and Parallelization of Coupled-Cluster Calculations

The coupled-cluster suite of methods, which includes ground-state methods mentioned earlier in this Chapter and
excited-state methods in the next Chapter, has been parallelized to take advantage of distributed memory and multicore architectures. The code is parallelized at the level of the underlying tensor algebra library. 26

Chapter 6: Wave Function-Based Correlation Methods

6.14.1

256

Serial and Shared Memory Parallel Jobs

Parallelization on multiple CPUs or CPU cores is achieved by breaking down tensor operations into batches and running
each batch in a separate thread. Because each thread occupies one CPU core entirely, the maximum number of threads
must not exceed the total available number of CPU cores. If multiple computations are performed simultaneously, they
together should not run more threads than available cores. For example, an eight-core node can accommodate one
eight-thread calculation, two four-thread calculations, and so on.
The number of threads to be used in a calculation is specified as a command line option (-nt nthreads). Here nthreads
should be given a positive integer value. If this option is not specified, the job will run in the serial mode.
Both CCMAN (old version of the couple-cluster codes) and CCMAN2 (default) have shared-memory parallel capabilities. However, they have different memory requirements as described below.
Setting the memory limit correctly is very important for attaining high performance when running large jobs. To
roughly estimate the amount of memory required for a coupled-cluster calculation use the following formula:
Memory =

(Number of basis set functions)4
Mb
131072

(6.42)

If CCMAN2 is used and the calculation is based on a RHF reference, the amount of memory needed is a half of that
given by the formula. If forces or excited states are calculated, the amount should be multiplied by a factor of two.
Because the size of data increases steeply with the size of the molecule computed, both CCMAN and CCMAN2 are
able to use disk space to supplement physical RAM if so required. The strategies of memory management in CCMAN
and CCMAN2 slightly differ, and that should be taken into account when specifying memory-related keywords in the
input file.
The MEM_STATIC keyword specifies the amount of memory in megabytes to be made available to routines that run prior
to coupled-clusters calculations: Hartree-Fock and electronic repulsion integrals evaluation. A safe recommended value
is 500 Mb. The value of MEM_STATIC should not exceed 2000 Mb even for very large jobs.
The memory limit for coupled-clusters calculations is set by CC_MEMORY. When running CCMAN, CC_MEMORY
value is used as the recommended amount of memory, and the calculation can in fact use less or run over the limit. If
the job is to run exclusively on a node, CC_MEMORY should be given 50% of all RAM. If the calculation runs out of
memory, the amount of CC_MEMORY should be reduced forcing CCMAN to use memory-saving algorithms.
CCMAN2 uses a different strategy. It allocates the entire amount of RAM given by CC_MEMORY before the calculation
and treats that as a strict memory limit. While that significantly improves the stability of larger jobs, it also requires the
user to set the correct value of CC_MEMORY to ensure high performance. The default value is computed automatically
based on the job size, but may not always be appropriate for large calculations, especially if the node has more resources
available. When running CCMAN2 exclusively on a node, CC_MEMORY should be set to 75–80% of the total available
RAM.
Note: When running small jobs, using too large CC_MEMORY in CCMAN2 is not recommended because Q-C HEM
will allocate more resources than needed for the calculation, which may affect other jobs that you may wish to
run on the same node.
For large disk-based coupled cluster calculations it is recommended to use a new tensor contraction code available in
CCMAN2 via libxm, which can significantly speed up calculations on Linux nodes. Use the CC_BACKEND variable
to switch on libxm. The new algorithm represents tensor contractions as multiplications of large matrices, which
are performed using efficient BLAS routines. Tensor data is stored on disk and is asynchronously prefetched to fast
memory before evaluating contractions. The performance of the code is not affected by the amount of RAM after about
128 GB if fast disks (such as SAS array in RAID0) are available on the system.

6.14.2

Distributed Memory Parallel Jobs

CCMAN2 has capabilities to run ground and excited state energy and property calculations on computer clusters and
supercomputers using the Cyclops Tensor Framework 96 (CTF) as a computational back-end. To switch on the use

Chapter 6: Wave Function-Based Correlation Methods

257

of CTF, use the CC_BACKEND keyword. In addition, Q-C HEM should be invoked with the -np nproc command line
option to specify the number of processors for a distributed calculation as nproc. Consult Section 2.8 for more details
about running Q-C HEM in parallel.

6.14.3

Summary of Keywords
MEM_STATIC
Sets the memory for individual Fortran program modules
TYPE:
INTEGER
DEFAULT:
240 corresponding to 240 Mb or 12% of MEM_TOTAL
OPTIONS:
n User-defined number of megabytes.
RECOMMENDATION:
For direct and semi-direct MP2 calculations, this must exceed OVN + requirements for AO
integral evaluation (32–160 Mb). Up to 2000 Mb for large coupled-clusters calculations.

CC_MEMORY
Specifies the maximum size, in Mb, of the buffers for in-core storage of block-tensors in CCMAN
and CCMAN2.
TYPE:
INTEGER
DEFAULT:
50% of MEM_TOTAL. If MEM_TOTAL is not set, use 1.5 Gb. A minimum of
192 Mb is hard-coded.
OPTIONS:
n Integer number of Mb
RECOMMENDATION:
Larger values can give better I/O performance and are recommended for systems with large memory (add to your .qchemrc file. When running CCMAN2 exclusively on a node, CC_MEMORY
should be set to 75–80% of the total available RAM. )
CC_BACKEND
Used to specify the computational back-end of CCMAN2.
TYPE:
STRING
DEFAULT:
VM Default shared-memory disk-based back-end
OPTIONS:
XM libxm shared-memory disk-based back-end
CTF Distributed-memory back-end for MPI jobs
RECOMMENDATION:
Use XM for large jobs with limited memory or when the performance of the default disk-based
back-end is not satisfactory, CTF for MPI jobs

6.15

Simplified Coupled-Cluster Methods Based on a Perfect-Pairing Active
Space

The methods described below are related to valence bond theory and are handled by the GVBMAN module. The
following models are available:

Chapter 6: Wave Function-Based Correlation Methods

258

CORRELATION
Specifies the correlation level in GVB models handled by GVBMAN.
TYPE:
STRING
DEFAULT:
None No Correlation
OPTIONS:
PP
CCVB
GVB_IP
GVB_SIP
GVB_DIP
OP
NP
2P
RECOMMENDATION:
As a rough guide, use PP for biradicaloids, and CCVB for polyradicaloids involving strong spin
correlations. Consult the literature for further guidance.

Molecules where electron correlation is strong are characterized by small energy gaps between the nominally occupied
orbitals (that would comprise the Hartree-Fock wave function, for example) and nominally empty orbitals. Examples
include so-called diradicaloid molecules, 53 or molecules with partly broken chemical bonds (as in some transition-state
structures). Because the energy gap is small, electron configurations other than the reference determinant contribute to
the molecular wave function with considerable amplitude, and omitting them leads to a significant error.
Including all possible configurations however, is a vast overkill. It is common to restrict the configurations that one
generates to be constructed not from all molecular orbitals, but just from orbitals that are either “core” or “active”. In
this section, we consider just one type of active space, which is composed of two orbitals to represent each electron
pair: one nominally occupied (bonding or lone pair in character) and the other nominally empty, or correlating (it is
typically anti-bonding in character). This is usually called the perfect pairing active space, and it clearly is well-suited
to represent the bonding/anti-bonding correlations that are associated with bond-breaking.
The quantum chemistry within this (or any other) active space is given by a Complete Active Space SCF (CASSCF)
calculation, whose exponential cost growth with molecule size makes it prohibitive for systems with more than about
14 active orbitals. One well-defined coupled cluster (CC) approximation based on CASSCF is to include only double
substitutions in the valence space whose orbitals are then optimized. In the framework of conventional CC theory, this
defines the valence optimized doubles (VOD) model, 56 which scales as O(N 6 ) (see Section 6.10.2). This is still too
expensive to be readily applied to large molecules.
The methods described in this section bridge the gap between sophisticated but expensive coupled cluster methods and
inexpensive methods such as DFT, HF and MP2 theory that may be (and indeed often are) inadequate for describing
molecules that exhibit strong electron correlations such as diradicals. The coupled cluster perfect pairing (PP), 12,16
imperfect pairing 103 (IP) and restricted coupled cluster 106 (RCC) models are local approximations to VOD that include
only a linear and quadratic number of double substitution amplitudes respectively. They are close in spirit to generalized
valence bond (GVB)-type wave functions, 38 because in fact they are all coupled cluster models for GVB that share the
same perfect pairing active space. The most powerful method in the family, the Coupled Cluster Valence Bond (CCVB)
method, 92–94 is a valence bond approach that goes well beyond the power of GVB-PP and related methods, as discussed
below in Sec. 6.15.2.

259

Chapter 6: Wave Function-Based Correlation Methods

6.15.1

Perfect pairing (PP)

To be more specific, the coupled cluster PP wave function is written as
|Ψi = exp

nX
active

!
ti â†i∗ â†ī∗ âī âi

|Φi

(6.43)

i=1

where nactive is the number of active electrons, and the ti are the linear number of unknown cluster amplitudes,
corresponding to exciting the two electrons in the ith electron pair from their bonding orbital pair to their anti-bonding
orbital pair. In addition to ti , the core and the active orbitals are optimized as well to minimize the PP energy. The
algorithm used for this is a slight modification of the GDM method, described for SCF calculations in Section 4.5.4.
Despite the simplicity of the PP wave function, with only a linear number of correlation amplitudes, it is still a useful
theoretical model chemistry for exploring strongly correlated systems. This is because it is exact for a single electron
pair in the PP active space, and it is also exact for a collection of non-interacting electron pairs in this active space.
Molecules, after all, are in a sense a collection of interacting electron pairs! In practice, PP on molecules recovers
between 60% and 80% of the correlation energy in its active space.
If the calculation is perfect pairing (CORRELATION = PP), it is possible to look for unrestricted solutions in addition to
restricted ones. Unrestricted orbitals are the default for molecules with odd numbers of electrons, but can also be specified for molecules with even numbers of electrons. This is accomplished by setting GVB_UNRESTRICTED = TRUE.
Given a restricted guess, this will, however usually converge to a restricted solution anyway, so additional REM variables should be specified to ensure an initial guess that has broken spin symmetry. This can be accomplished by using
an unrestricted SCF solution as the initial guess, using the techniques described in Chapter 4. Alternatively a restricted
set of guess orbitals can be explicitly symmetry broken just before the calculation starts by using GVB_GUESS_MIX,
which is described below. There is also the implementation of Unrestricted-in-Active Pairs (UAP), 62 which is the default unrestricted implementation for GVB methods. This method simplifies the process of unrestriction by optimizing
only one set of ROHF MO coefficients and a single rotation angle for each occupied-virtual pair. These angles are used
to construct a series of 2x2 Given’s rotation matrices which are applied to the ROHF coefficients to determine the α
spin MO coefficients and their transpose is applied to the ROHF coefficients to determine the β spin MO coefficients.
This algorithm is fast and eliminates many of the pathologies of the unrestricted GVB methods near the dissociation
limit. To generate a full potential curve we find it is best to start at the desired UHF dissociation solution as a guess for
GVB and follow it inwards to the equilibrium bond distance.
GVB_UNRESTRICTED
Controls restricted versus unrestricted PP jobs. Usually handled automatically.
TYPE:
LOGICAL
DEFAULT:
same value as UNRESTRICTED
OPTIONS:
TRUE/FALSE
RECOMMENDATION:
Set this variable explicitly only to do a UPP job from an RHF or ROHF initial guess. Leave this
variable alone and specify UNRESTRICTED = TRUE to access the new Unrestricted-in-ActivePairs GVB code which can return an RHF or ROHF solution if used with GVB_DO_ROHF

Chapter 6: Wave Function-Based Correlation Methods

260

GVB_DO_ROHF
Sets the number of Unrestricted-in-Active Pairs to be kept restricted.
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
n User-Defined
RECOMMENDATION:
If n is the same value as GVB_N_PAIRS returns the ROHF solution for GVB, only works with
the UNRESTRICTED = TRUE implementation of GVB with GVB_OLD_UPP = 0 (its default value)
GVB_OLD_UPP
Which unrestricted algorithm to use for GVB.
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
0 Use Unrestricted-in-Active Pairs described in Ref. 62
1 Use Unrestricted Implementation described in Ref. 12
RECOMMENDATION:
Only works for Unrestricted PP and no other GVB model.
GVB_GUESS_MIX
Similar to SCF_GUESS_MIX, it breaks alpha/beta symmetry for UPP by mixing the alpha HOMO
and LUMO orbitals according to the user-defined fraction of LUMO to add the HOMO. 100
corresponds to a 1:1 ratio of HOMO and LUMO in the mixed orbitals.
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
n User-defined, 0 ≤ n ≤ 100
RECOMMENDATION:
25 often works well to break symmetry without overly impeding convergence.

Whilst all of the description in this section refers to PP solved via projection, it is also possible, as described in Sec.
6.15.2 below, to solve variationally for the PP energy. This variational PP solution is the reference wave function for the
CCVB method. In most cases use of spin-pure CCVB is preferable to attempting to improve restricted PP by permitting
the orbitals to spin polarize.

6.15.2

Coupled Cluster Valence Bond (CCVB)

Cases where PP needs improvement include molecules with several strongly correlated electron pairs that are all localized in the same region of space, and therefore involve significant inter-pair, as well as intra-pair correlations. For
some systems of this type, Coupled Cluster Valence Bond (CCVB) is an appropriate method. 92,93 CCVB is designed to
qualitatively treat the breaking of covalent bonds. At the most basic theoretical level, as a molecular system dissociates
into a collection of open-shell fragments, the energy should approach the sum of the ROHF energies of the fragments.
CCVB is able to reproduce this for a wide class of problems, while maintaining proper spin symmetry. Along with
this, CCVB’s main strength, come many of the spatial symmetry breaking issues common to the GVB-CC methods.

261

Chapter 6: Wave Function-Based Correlation Methods

Like the other methods discussed in this section, the leading contribution to the CCVB wave function is the perfect
pairing wave function, which is shown in Eq. (6.43). One important difference is that CCVB uses the PP wave function
as a reference in the same way that other GVBMAN methods use a reference determinant.
The PP wave function is a product of simple, strongly orthogonal singlet geminals. Ignoring normalization, two equivalent ways of displaying these geminals are
(φi φi + ti φ∗i φ∗i )(αβ − βα) (Natural-orbital form)
χi χ0i (αβ − βα) (Valence-bond form),

(6.44)

where on the left and right we have the spatial part (involving φ and χ orbitals) and the spin coupling, respectively. The
VB-form orbitals are non-orthogonal within a pair and are generally AO-like. The VB form is used in CCVB and the
NO form is used in the other GVBMAN methods. It turns out that occupied UHF orbitals can also be rotated (without
affecting the energy) into the VB form (here the spin part would be just αβ), and as such we store the CCVB orbital
coefficients in the same way as is done in UHF (even though no one spin is assigned to an orbital in CCVB).
These geminals are uncorrelated in the same way that molecular orbitals are uncorrelated in a HF calculation. Hence,
they are able to describe uncoupled, or independent, single-bond-breaking processes, like that found in C2 H6 → 2
CH3 , but not coupled multiple-bond-breaking processes, such as the dissociation of N2 . In the latter system the three
bonds may be described by three singlet geminals, but this picture must somehow translate into the coupling of two
spin-quartet N atoms into an overall singlet, as found at dissociation. To achieve this sort of thing in a GVB context,
it is necessary to correlate the geminals. The part of this correlation that is essential to bond breaking is obtained by
replacing clusters of singlet geminals with triplet geminals, and re-coupling the triplets to an overall singlet. A triplet
geminal is obtained from a singlet by simply modifying the spin component accordingly. We thus obtain the CCVB
wave function:

|Ψi = |Φ0 i +

X
k0
RECOMMENDATION:
Increase for computations that are difficult to converge.
RDM_CG_CONVERGENCE
The minimum threshold for the conjugate gradient solver.
TYPE:
INTEGER
DEFAULT:
12
OPTIONS:
N for a threshold of 10−N
RECOMMENDATION:
Should be at least (RDM_EPS_CONVERGENCE+2).

278

Chapter 6: Wave Function-Based Correlation Methods
RDM_CG_MAXITER
Maximum number of iterations for each conjugate gradient computations in the BPSDP algorithm.
TYPE:
INTEGER
DEFAULT:
1000
OPTIONS:
N >0
RECOMMENDATION:
Use default unless problems arise.
RDM_TAU
Step-length parameter used in the BPSDP solver.
TYPE:
INTEGER
DEFAULT:
10
OPTIONS:
N for a value of 0.1 * N
RECOMMENDATION:
RDM_TAU should range between 10 and 16 for 1.0 ≤ τ ≤ 1.6.
RDM_MU_UPDATE_FREQUENCY
The number of v2RDM iterations after which the penalty parameter µ is updated.
TYPE:
INTEGER
DEFAULT:
200
OPTIONS:
N >0
RECOMMENDATION:
Change if convergence problems arise.
RDM_SOLVER
Indicates which solver to use for the v2RDM optimization.
TYPE:
STRING
DEFAULT:
VECTOR
OPTIONS:
VECTOR
Picks the hand-tuned loop-based code.
BLOCK_TENSOR Picks the libtensor-based code.
RECOMMENDATION:
Use the default.

279

Chapter 6: Wave Function-Based Correlation Methods
RDM_CONSTRAIN_SPIN
Indicates if the spin-constraints are enforced.
TYPE:
BOOLEAN
DEFAULT:
TRUE
OPTIONS:
TRUE
Enforce spin-constraints.
FALSE Do not enforce spin-constraints.
RECOMMENDATION:
Use default.
RDM_OPTIMIZE_ORBITALS
Indicates if the molecular orbitals will be optimized.
TYPE:
BOOLEAN
DEFAULT:
TRUE
OPTIONS:
TRUE
Optimize orbitals.
FALSE Do not optimize orbitals.
RECOMMENDATION:
Use default unless all orbitals are active.
RDM_ORBOPT_FREQUENCY
The number of v2RDM iterations after which the orbital optimization routine is called.
TYPE:
INTEGER
DEFAULT:
500
OPTIONS:
N >0
RECOMMENDATION:
Use default unless convergence problems arise.
RDM_ORBOPT_GRADIENT_CONVERGENCE
The threshold for the orbital gradient during orbital optimization.
TYPE:
INTEGER
DEFAULT:
4
OPTIONS:
N for threshold of 10−N
RECOMMENDATION:
Tighten for gradient computations.

280

Chapter 6: Wave Function-Based Correlation Methods
RDM_ORBOPT_ENERGY_CONVERGENCE
The threshold for energy convergence during orbital optimization.
TYPE:
INTEGER
DEFAULT:
8
OPTIONS:
N for threshold of 10−N
RECOMMENDATION:
Tighten for gradient computations.
RDM_ORBOPT_MAXITER
The maximum number of orbital optimization steps each time the orbital optimization routine is
called.
TYPE:
INTEGER
DEFAULT:
20
OPTIONS:
N >0
RECOMMENDATION:
Use default unless convergence problems arise.
RDM_PRINT
Controls the amount of printing.
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
0 Print minimal information.
1 Print information about all iterations.
RECOMMENDATION:
Use 1 to analyze convergence issues.

281

282

Chapter 6: Wave Function-Based Correlation Methods

References and Further Reading
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Chapter 7

Open-Shell and Excited-State Methods
7.1

General Excited-State Features

As for ground state calculations, performing an adequate excited-state calculation involves making an appropriate
choice of method and basis set. The development of effective approaches to modeling electronic excited states has
historically lagged behind advances in treating the ground state. In part this is because of the much greater diversity in
the character of the wave functions for excited states, making it more difficult to develop broadly applicable methods
without molecule-specific or even state-specific specification of the form of the wave function. Recently, however,
a hierarchy of single-reference ab initio methods has begun to emerge for the treatment of excited states. Broadly
speaking, Q-C HEM contains methods that are capable of giving qualitative agreement, and in many cases quantitative
agreement with experiment for lower optically allowed states. The situation is less satisfactory for states that involve
two-electron excitations, although even here reasonable results can sometimes be obtained. Moreover, some of the
excited state methods can treat open-shell wave functions, e.g. diradicals, ionized and electron attachment states and
more. 63
In excited-state calculations, as for ground state calculations, the user must strike a compromise between cost and
accuracy. This chapter summarizes Q-C HEM’s capabilities in four general classes of excited state methods:
• Single-electron wave function-based methods (Section 7.2). These are excited state treatments of roughly the
same level of sophistication as the Hartree-Fock ground state method, in the sense that electron correlation is
essentially ignored. Single excitation configuration interaction (CIS) is the workhorse method of this type. The
spin-flip variant of CIS extends it to diradicals.
• Time-dependent density functional theory (TDDFT, Section 7.3). TDDFT is a widely used extension of DFT
to excited states. For a cost that is only a little larger than that of a CIS calculation, TDDFT typically affords
significantly greater accuracy due to a treatment of electron correlation. It, too, has a spin-flip variant that can be
used to study di- and tri-radicals as well as bond breaking.
• The Maximum Overlap Method (MOM) for excited SCF states (Section 7.4). This method overcomes some of
the deficiencies of TDDFT and, in particular, can be used for modeling charge-transfer and Rydberg transitions
as well as core-excited states.
• Restricted open-shell Kohn-Sham (ROKS) method is another ∆-SCF approach for excited states (Section 7.5).
• Wave function-based electron correlation treatments (Sections 7.6, 7.8, 7.9 and 7.7). Roughly speaking, these
are excited state analogues of the ground state wave function-based electron correlation methods discussed in
Chapter 6. They are more accurate than the methods of Section 7.2, but also significantly more computationally
expensive. These methods can also describe certain multi-configurational wave functions, for example, problematic doublet radicals, diradicals, triradicals, and more.

Chapter 7: Open-Shell and Excited-State Methods

288

Note: Core electrons are frozen by default in most correlated excited-state calculations (see Section 6.2).
In general, a basis set appropriate for a ground state density functional theory or a Hartree-Fock calculation will be
appropriate for describing valence excited states. However, many excited states involve significant contributions from
diffuse Rydberg orbitals, and, therefore, it is often advisable to use basis sets that include additional diffuse functions.
The 6-31+G* basis set is a reasonable compromise for the low-lying valence excited states of many organic molecules.
To describe true Rydberg excited states, Q-C HEM allows the user to add two or more sets of diffuse functions (see
Chapter 8). For example the 6-311(2+)G* basis includes two sets of diffuse functions on heavy atoms and is generally
adequate for description of both valence and Rydberg excited states.
Q-C HEM supports four main types of excited state calculation:
• Vertical absorption spectrum
This is the calculation of the excited states of the molecule at the ground state geometry, as appropriate for
absorption spectroscopy. The methods supported for performing a vertical absorption calculation are: CIS, RPA,
XCIS, SF-XCIS, CIS(D), ADC(2)-s, ADC(2)-x, ADC(3), RAS-SF, EOM-CCSD and EOM-OD, each of which
will be discussed in turn. The calculation of core-excited states for the simulation of X-ray absorption spectra
can be performed with TDDFT as well as EOM-CCSD and ADC within the CVS approximation (Section 7.10.
All ADC- and EOM-based methods can be combined with the polarizable continuum model (PCM) to model the
absorption spectrum in solution following state-specific non-equilibrium approach. Most EOM methods can be
combined with explicit solvent treatments using classical (QM/MM) and polarizable (QM/EFP) embedding.
• Visualization
It is possible to visualize the excited states either by attachment/detachment density analysis (available for CIS,
RPA, TDDFT, ADC, EOM-CC) or by plotting the transition density (see $plots descriptions in Chapters 3 and
11). Transition densities can be calculated for CIS, EOM-CCSD, and ADC methods. The theoretical basis
of the attachment/detachment density analysis is discussed in Section 7.12.1 of this Chapter (more details are
given in Section 11.2.6. In addition Dyson orbitals can be calculated and plotted for ionization from the ground
and electronically excited states or detachment from electron-attached states for CCSD and EOM-CCSD wave
functions. For the RAS-SF method (Section 7.9), one can plot the natural orbitals of a computed electronic state.
• Excited-state optimization
Optimization of the geometry of stationary points on excited state potential energy surfaces is valuable for understanding the geometric relaxation that occurs between the ground and excited state. Analytic first derivatives are
available for UCIS, RCIS, TDDFT and EOM-CCSD. Excited state optimizations may also be performed using
finite difference methods, however, these can be very time-consuming to perform.
• Optimization of the crossings between potential energy surfaces
Seams between potential energy surfaces can be located and optimized by using analytic gradients within EOMCCSD, CIS, and TD-DFT formalisms.
• Properties
Properties such as dipole moments, spatial extent of electron densities and hS 2 i values can be computed for ADC,
EOM-CCSD, EOM-MP2, EOM-OD, RAS-SF and CIS wave functions. Static polarizabilities are available for
CCSD, EOM-EE-CCSD, and EOM-SF-CCSD methods.
• Transition properties and state interactions
Transition dipole moments and oscillator strengths can be computed with practically all excited-state methods.
Matrix elements and cross-sections for two-photon absorption are available for EOM-EE-CCSD and ADC methods. Spin-orbit couplings can be computed for EOM-CCSD, CIS, and TDDFT wave functions. Dyson orbitals
are available for EOM-CC wave functions. Transition properties can be computed between the reference and
target states (e.g., HF-CIS) or between different target states (e.g., CIS-CIS).
• Excited-state vibrational analysis
Given an optimized excited state geometry, Q-C HEM can calculate the force constants at the stationary point to
predict excited state vibrational frequencies. Stationary points can also be characterized as minima, transition

Chapter 7: Open-Shell and Excited-State Methods

289

structures or nth-order saddle points. Analytic excited state vibrational analysis can only be performed using
the UCIS, RCIS, and TDDFT methods, for which efficient analytical second derivatives are available. EOMCCSD frequencies are also available using analytic first derivatives and second derivatives obtained from finite
difference methods. EOM-OD frequencies are only available through finite difference calculations.
Note: EOM-CC and most of the CI codes are part of CCMAN and CCMAN2. CCMAN is a legacy code which is
being phased out. All new developments and performance-enhancing features are implemented in CCMAN2.
METHOD
Specifies the level of theory.
TYPE:
STRING
DEFAULT:
None No Correlation
OPTIONS:
CIS
Section 7.2.1
CIS(D)
Section 7.6.1
RI-CIS(D)
Section 7.6.2
SOS-CIS(D)
Section 7.6.3
SOS-CIS(D0)
Section 7.6.4
CISD
Section 7.7.2
CISDT
Section 7.7.2
EOM-OD
Section 7.7.2
EOM-CCSD
Section 7.7.2
EOM-MP2
Section 7.7.9
EOM-MP2T
Section 7.7.9
EOM-CCSD-S(D) Section 7.7.10
EOM-MP2-S(D)
Section 7.7.10
EOM-CCSD(dT)
Section 7.7.21
EOM-CCSD(fT)
Section 7.7.21
EOM-CC(2,3)
Section 7.7.18
ADC(0)
Section 7.8
ADC(1)
Section 7.8
ADC(2)
Section 7.8
ADC(2)-X
Section 7.8
ADC(3)
Section 7.8
SOS-ADC(2)
Section 7.8
SOS-ADC(2)-X
Section 7.8
CVS-ADC(1)
Section 7.8
CVS-ADC(2)
Section 7.8
CVS-ADC(2)-X
Section 7.8
CVS-ADC(3)
Section 7.8
RAS-CI
Section 7.9
RAS-CI-2
Section 7.9
RECOMMENDATION:
Consult the literature for guidance.

7.2

Uncorrelated Wave Function Methods

Q-C HEM includes several excited state methods which do not incorporate correlation: CIS, XCIS and RPA. These
methods are sufficiently inexpensive that calculations on large molecules are possible, and are roughly comparable to

290

Chapter 7: Open-Shell and Excited-State Methods

the HF treatment of the ground state in terms of performance. They tend to yield qualitative rather than quantitative
insight. Excitation energies tend to exhibit errors on the order of an electron volt, consistent with the neglect of electron
correlation effects, which are generally different in the ground state and the excited state.

7.2.1

Single Excitation Configuration Interaction (CIS)

The derivation of the CI-singles energy and wave function 30,36 begins by selecting the HF single-determinant wave
function as reference for the ground state of the system:
1
ΨHF = √ det {χ1 χ2 · · · χi χj · · · χn }
n!

(7.1)

where n is the number of electrons, and the spin orbitals
χi =

N
X

cµi φµ

(7.2)

µ

are expanded in a finite basis of N atomic orbital basis functions. Molecular orbital coefficients {cµi } are usually found
by SCF procedures which solve the Hartree-Fock equations
FC = εSC ,

(7.3)

where S is the overlap matrix, C is the matrix of molecular orbital coefficients, ε is a diagonal matrix of orbital
eigenvalues and F is the Fock matrix with elements
XX
Fµυ = Hµυ +
cµi cυi (µλ || υσ)
(7.4)
λσ

i

involving the core Hamiltonian and the anti-symmetrized two-electron integrals


Z Z

1 
φλ (r1 )φσ (r2 ) − φσ (r1 )φλ (r2 ) dr1 dr2
(µµ||λσ) =
φµ (r1 )φν (r2 )
r12

(7.5)

On solving Eq. (7.3), the total energy of the ground state single determinant can be expressed as
EHF =

X
µυ

HF
Pµυ
Hµυ +

1 X HF HF
Pµυ Pλσ (µλ || υσ) + Vnuc
2

(7.6)

µυλσ

where P HF is the HF density matrix and Vnuc is the nuclear repulsion energy.
Equation (7.1) represents only one of many possible determinants made from orbitals of the system; there are in fact
n(N − n) possible singly substituted determinants constructed by replacing an orbital occupied in the ground state (i,
j, k, . . .) with an orbital unoccupied in the ground state (a, b, c, . . .). Such wave functions and energies can be written
1
Ψai = √ det {χ1 χ2 · · · χa χj · · · χn }
n!

(7.7)

Eia = EHF + εa − εi − (ia || ia)

(7.8)

where we have introduced the anti-symmetrized two-electron integrals in the molecular orbital basis
X
(pq || rs) =
cµp cυq cλr cσs (µλ || υσ)

(7.9)

µυλσ

These singly excited wave functions and energies could be considered crude approximations to the excited states of the
system. However, determinants of the form Eq. (7.7) are deficient in that they:
• do not yield pure spin states

291

Chapter 7: Open-Shell and Excited-State Methods
• resemble more closely ionization rather than excitation
• are not appropriate for excitation into degenerate states

These deficiencies can be partially overcome by representing the excited state wave function as a linear combination of
all possible singly excited determinants,
X
ΨCIS =
aai Ψai
(7.10)
ia

where the coefficients {aia } can be obtained by diagonalizing the many-electron Hamiltonian, A, in the space of all
single substitutions. The appropriate matrix elements are:
Aia,jb = hΨai | H Ψbj = (εa − εj )δij δab − (ja || ib)

(7.11)

According to Brillouin’s, theorem single substitutions do not interact directly with a reference HF determinant, so the
resulting eigenvectors from the CIS excited state represent a treatment roughly comparable to that of the HF ground
state. The excitation energy is simply the difference between HF ground state energy and CIS excited state energies,
and the eigenvectors of A correspond to the amplitudes of the single-electron promotions.
CIS calculations can be performed in Q-C HEM using restricted (RCIS), 30,36 unrestricted (UCIS), or restricted openshell 85 (ROCIS) spin orbitals.

7.2.2

Random Phase Approximation (RPA)

The Random Phase Approximation (RPA), 14,41 also known as time-dependent Hartree-Fock (TD-HF) theory, is an
alternative to CIS for uncorrelated calculations of excited states. It offers some advantages for computing oscillator
strengths, e.g., exact satisfaction of the Thomas-Reike-Kuhn sum rule, 90 and is roughly comparable in accuracy to CIS
for singlet excitation energies, but is inferior for triplet states. RPA energies are non-variational, and in moving around
on excited-state potential energy surfaces, this method can occasionally encounter singularities that prevent numerical
solution of the underlying equations, 28 whereas such singularities are mathematically impossible in CIS calculations.

7.2.3

Extended CIS (XCIS)

The motivation for the extended CIS procedure 86 (XCIS) stems from the fact that ROCIS and UCIS are less effective
for radicals that CIS is for closed shell molecules. Using the attachment/detachment density analysis procedure, 44 the
failing of ROCIS and UCIS methodologies for the nitromethyl radical was traced to the neglect of a particular class of
double substitution which involves the simultaneous promotion of an α spin electron from the singly occupied orbital
and the promotion of a β spin electron into the singly occupied orbital. The spin-adapted configurations
E


1
2
Ψ̃ai (1) = √ |Ψāī i − |Ψai i + √ |Ψpaīp̄ i
6
6

(7.12)

are of crucial importance. (Here, a, b, c, . . . are virtual orbitals; i, j, k, . . . are occupied orbitals; and p, q, r, . . . are
singly-occupied orbitals.) It is quite likely that similar excitations are also very significant in other radicals of interest.
The XCIS proposal, a more satisfactory generalization of CIS to open shell molecules, is to simultaneously include a
restricted class of double substitutions similar to those in Eq. (7.12). To illustrate this, consider the resulting orbital
spaces of an ROHF calculation: doubly occupied (d), singly occupied (s) and virtual (v). From this starting point we
can distinguish three types of single excitations of the same multiplicity as the ground state: d → s, s → v and d → v.
Thus, the spin-adapted ROCIS wave function is
dv
sv
ds
X

 X
1 X a
|ΨROCIS i = √
ai |Ψai i + |Ψāī i +
aap |Ψap i +
ap̄ī |Ψp̄ī i
2 ia
pa
ip

(7.13)

292

Chapter 7: Open-Shell and Excited-State Methods

The extension of CIS theory to incorporate higher excitations maintains the ROHF as the ground state reference and
adds terms to the ROCIS wave function similar to that of Eq. (7.13), as well as those where the double excitation occurs
through different orbitals in the α and β space:
|ΨXCIS i =

√1
2

dv
X

sv
ds
X

 X
aai |Ψai i + |Ψāī i +
aap |Ψap i +
ap̄ī |Ψp̄ī i
pa

ia

+

dvs
X
iap

ãai (p)|Ψ̃ai (p)i

ip

+

dv,ss
X

(7.14)

aaq̄
|Ψaq̄
i
pī
pī

ia,p6=q

XCIS is defined only from a restricted open shell Hartree-Fock ground state reference, as it would be difficult to
uniquely define singly occupied orbitals in a UHF wave function. In addition, β unoccupied orbitals, through which the
spin-flip double excitation proceeds, may not match the half-occupied α orbitals in either character or even symmetry.
For molecules with closed shell ground states, both the HF ground and CIS excited states emerge from diagonalization
of the Hamiltonian in the space of the HF reference and singly excited substituted configuration state functions. The
XCIS case is different because the restricted class of double excitations included could mix with the ground state and
lower its energy. This mixing is avoided to maintain the size consistency of the ground state energy.
With the inclusion of the restricted set of doubles excitations in the excited states, but not in the ground state, it could be
expected that some fraction of the correlation energy be recovered, resulting in anomalously low excited state energies.
However, the fraction of the total number of doubles excitations included in the XCIS wave function is very small and
those introduced cannot account for the pair correlation of any pair of electrons. Thus, the XCIS procedure can be
considered one that neglects electron correlation.
The computational cost of XCIS is approximately four times greater than CIS and ROCIS, and its accuracy for open
shell molecules is generally comparable to that of the CIS method for closed shell molecules. In general, it achieves
qualitative agreement with experiment. XCIS is available for doublet and quartet excited states beginning from a
doublet ROHF treatment of the ground state, for excitation energies only.

7.2.4

Spin-Flip Extended CIS (SF-XCIS)

Spin-flip extended CIS (SF-XCIS) 20 is a spin-complete extension of the spin-flip single excitation configuration interaction (SF-CIS) method. 59 The method includes all configurations in which no more than one virtual level of the high
spin triplet reference becomes occupied and no more than one doubly occupied level becomes vacant.
SF-XCIS is defined only from a restricted open shell Hartree-Fock triplet ground state reference. The final SF-XCIS
wave functions correspond to spin-pure MS = 0 (singlet or triplet) states. The fully balanced treatment of the halfoccupied reference orbitals makes it very suitable for applications with two strongly correlated electrons, such as single
bond dissociation, systems with important diradical character or the study of excited states with significant double
excitation character.
The computational cost of SF-XCIS scales in the same way with molecule size as CIS itself, with a pre-factor 13 times
larger.

7.2.5

Spin-Adapted Spin-Flip CIS

Spin-Adapted Spin-Flip CIS (SA-SF-CIS) 140 is a spin-complete extension of the spin-flip single excitation configuration interaction (SF-CIS) method. 59 Unlike SF-XCIS, SA-SF-CIS only includes the minimal set of necessary electronic
configurations to remove the spin contamination in the conventional SF-CIS method. The target SA-SF-CIS states have
spin eigenvalues one less than the reference ROHF state. Based on a tensor equation-of-motion formalism, 140 the dimension of the CI vectors in SA-SF-CIS remains exactly the same as that in conventional SF-CIS. Currently, the DFT
correction to SA-SF-CIS is added in an ad hoc way. 140

Chapter 7: Open-Shell and Excited-State Methods

7.2.6

293

CIS Analytical Derivatives

While CIS excitation energies are relatively inaccurate, with errors of the order of 1 eV, CIS excited state properties,
such as structures and frequencies, are much more useful. This is very similar to the manner in which ground state
Hartree-Fock (HF) structures and frequencies are much more accurate than HF relative energies. Generally speaking,
for low-lying excited states, it is expected that CIS vibrational frequencies will be systematically 10% higher or so relative to experiment. 38,120,142 If the excited states are of pure valence character, then basis set requirements are generally
similar to the ground state. Excited states with partial Rydberg character require the addition of one or preferably two
sets of diffuse functions.
Q-C HEM includes efficient analytical first and second derivatives of the CIS energy, 84,87 to yield analytical gradients,
excited state vibrational frequencies, force constants, polarizabilities, and infrared intensities. Analytical gradients
can be evaluated for any job where the CIS excitation energy calculation itself is feasible, so that efficient excitedstate geometry optimizations and vibrational frequency calculations are possible at the CIS level. In such cases, it is
necessary to specify on which Born-Oppenheimer potential energy surface the optimization should proceed, and care
must be taken to ensure that the optimization remains on the excited state of interest, as state crossings may occur. (A
“state-tracking” algorithm, as discussed in Section 10.6.5, can aid with this.)
Sometimes it is precisely the crossings between Born-Oppenheimer potential energy surfaces (i.e., conical intersections) that are of interest, as these intersections provide pathways for non-adiabatic transitions between electronic
states. 48,83 A feature of Q-C HEM that is not otherwise widely available in an analytic implementation (for both CIS
and TDDFT) of the non-adiabatic couplings that define the topology around conical intersections. 34,99,138,139 Due to
the analytic implementation, these couplings can be evaluated at a cost that is not significantly greater than the cost of
a CIS or TDDFT analytic gradient calculation, and the availability of these couplings allows for much more efficient
optimization of minimum-energy crossing points along seams of conical intersection, as compared to when only analytic gradients are available. 138 These features, including a brief overview of the theory of conical intersections, can be
found in Section 10.6.1.
For CIS vibrational frequencies, a semi-direct algorithm similar to that used for ground-state Hartree-Fock frequencies is available, whose computer time scales as approximately O(N 3 ) for large molecules. 86 The main complication
associated with analytical CIS frequency calculations is ensuring that Q-C HEM has sufficient memory to perform the
calculations. Default settings are adequate for many purposes but if a large calculation fails due to a memory limitation,
then the following additional information may be useful.
The memory requirements for CIS (and HF) analytic frequencies primarily come from dynamic memory, defined as
dynamic memory = MEM_TOTAL − MEM_STATIC .
2
This quantity must be large enough to contain several arrays whose size is 3Natoms Nbasis
. Meanwhile the value of
the $rem variable MEM_STATIC, which obviously reduces the available dynamic memory, must be sufficiently large to
permit integral evaluation, else the job may fail. For most purposes, setting MEM_STATIC to about 80 Mb is sufficient,
and by default MEM_TOTAL is set to a larger value that what is available on most computers, so that the user need not
guess or experiment about an appropriate value of MEM_TOTAL for low-memory jobs. However, a memory allocation
error will occur if the calculation demands more memory than available.

Note: Unlike Q-C HEM’s MP2 frequency code, the analytic CIS second derivative code currently does not support
frozen core or virtual orbitals. These approximations do not lead to large savings at the CIS level, as all
computationally-expensive steps are performed in the atomic orbital basis.

7.2.7

Non-Orthogonal Configuration Interaction

Systems such as transition metals, open-shell species, and molecules with highly-stretched bonds often exhibit multiple, near-degenerate solutions to the SCF equations. Multiple solutions can be located using SCF meta-dynamics
(Section 4.9.2), but given the approximate nature of the SCF calculation in the first place, there is in such cases no clear
reason to choose one of these solutions over another. These SCF solutions are not subject to any non-crossing rule, and

Chapter 7: Open-Shell and Excited-State Methods

294

often do cross (i.e., switch energetic order) as the geometry is changed, so the lowest energy state may switch abruptly
with consequent discontinuities in the energy gradients. It is therefore desirable to have a method that treats all of these
near-degenerate SCF solutions on an equal footing an might yield a smoother, qualitatively correct potential energy
surface. This can be achieved by using multiple SCF solutions (obtained, e.g., via SCF meta-dynamics) as a basis for a
configuration interaction (CI) calculation. Since the various SCF solutions are not orthogonal to one another—meaning
that one solution cannot be constructed as a single determinant composed of orbitals from another solution—this CI is
a bit more complicated and is denoted as a non-orthogonal CI (NOCI). 123
NOCI can be viewed as an alternative to CASSCF within an “active space” consisting of the SCF states of interest,
and has the advantage that the SCF states, and thus the NOCI wave functions, are size-consistent. In common with
CASSCF, it is able to describe complicated phenomena such as avoided crossings (where states mix instead of passing
through each other) as well as conical intersections (whereby via symmetry or else accidental reasons, there is no
coupling between the states, and they pass cleanly through each other at a degeneracy).
Another use for a NOCI calculation is that of symmetry restoration. At some geometries, the SCF states break spatial
or spin symmetry to achieve a lower energy single determinant than if these symmetries were conserved. As these
symmetries still exist within the proper electron Hamiltonian, its exact eigenfunctions should preserve them. In the case
of spin this manifests as spin contamination and for spatial symmetries it usually manifests as artefactual localization.
To recover a (yet lower energy) wave function retaining the correct symmetries, one can include these broken-symmetry
states (with all relevant symmetry permutations) in a NOCI calculation; the resultant eigenfunction will have the true
symmetries restored, as a linear combination of the broken-symmetry states.
A common example occurs in the case of a spin-contaminated UHF reference state. Performing a NOCI calculation
in a basis consisting of this state, plus a second state in which all α and β orbitals have been switched, often reduces
spin contamination in the same way as the half-projected Hartree-Fock method, 89 although there is no guarantee that
the resulting wave function is an eigenfunction of Ŝ 2 . Another example consists in using a UHF wave function with
MS = 0 along with its spin-exchanged version (wherein all α ↔ β orbitals are switched), which two new NOCI
eigenfunctions, one with even S (a mixture of S = 0, 2, . . .), and one with odd S (mixing S = 1, 3, . . .). These may be
used to approximate singlet and triplet wave functions.
NOCI can be enabled by specifying CORRELATION_NOCI, and will automatically use all of the states located with
SCF meta-dynamics. Two spin-exchanged versions of a UHF wave function can be requested simply by not turning on
meta-dynamics. For more customization, a $noci input section can be included having, e.g., the following format:
$noci
1 2 -2 4
2
$end
In this particular case, the first line specifies that states 1, 2, and 4 are to be included in the NOCI calculation, along
with state “−2”, which indicates the spin-exchanged version of state 2. The second (optional) line indicates which
eigenvalue is to be returned to Q-C HEM, with the convention that 0 indicates the lowest state so the $noci input section
above is requesting the third state.
Analytic gradients are not available for NOCI but geometry optimizations can be performed automatically using finitedifference gradients.

Chapter 7: Open-Shell and Excited-State Methods

295

NOCI_PRINT
Specify the debug print level of NOCI.
TYPE:
INTEGER
DEFAULT:
1
OPTIONS:
n Positive integer
RECOMMENDATION:
Increase this for additional debug information.

7.2.8

Basic CIS Job Control Options

CIS-type jobs are requested by setting the $rem variable EXCHANGE = HF and CORRELATION = NONE, as in a
ground-state Hartree-Fock calculation, but then also specifying a number of excited-state roots using the $rem keyword
CIS_N_ROOTS.
Note: For RHF case, n singlets and n triplets will be computed, unless specified otherwise by using CIS_TRIPLETS
and CIS_SINGLETS.
CIS_N_ROOTS
Sets the number of CI-Singles (CIS) excited state roots to find.
TYPE:
INTEGER
DEFAULT:
0 Do not look for any excited states.
OPTIONS:
n n > 0 Looks for n CIS excited states.
RECOMMENDATION:
None
CIS_SINGLETS
Solve for singlet excited states in RCIS calculations (ignored for UCIS).
TYPE:
LOGICAL
DEFAULT:
TRUE
OPTIONS:
TRUE
Solve for singlet states.
FALSE Do not solve for singlet states.
RECOMMENDATION:
None

Chapter 7: Open-Shell and Excited-State Methods
CIS_TRIPLETS
Solve for triplet excited states in RCIS calculations (ignored for UCIS).
TYPE:
LOGICAL
DEFAULT:
TRUE
OPTIONS:
TRUE
Solve for triplet states.
FALSE Do not solve for triplet states.
RECOMMENDATION:
None
RPA
Do an RPA calculation in addition to a CIS or TDDFT/TDA calculation.
TYPE:
LOGICAL/INTEGER
DEFAULT:
FALSE
OPTIONS:
FALSE Do not do an RPA calculation.
TRUE
Do an RPA calculation.
2
Do an RPA calculation without running CIS or TDDFT/TDA first.
RECOMMENDATION:
None
CIS_STATE_DERIV
Sets CIS state for excited state optimizations and vibrational analysis.
TYPE:
INTEGER
DEFAULT:
0 Does not select any of the excited states.
OPTIONS:
n Select the nth state.
RECOMMENDATION:
Check to see that the states do not change order during an optimization, due to state crossings.
SPIN_FLIP
Selects whether to perform a standard excited state calculation, or a spin-flip calculation. Spin
multiplicity should be set to 3 for systems with an even number of electrons, and 4 for systems
with an odd number of electrons.
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
TRUE/FALSE
RECOMMENDATION:
None

296

Chapter 7: Open-Shell and Excited-State Methods
SPIN_FLIP_XCIS
Do a SF-XCIS calculation.
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
FALSE Do not do an SF-XCIS calculation.
TRUE
Do an SF-XCIS calculation (requires ROHF triplet ground state).
RECOMMENDATION:
None
SFX_AMP_OCC_A
Defines a customer amplitude guess vector in SF-XCIS method.
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
n builds a guess amplitude with an α-hole in the nth orbital (requires SFX_AMP_VIR_B).
RECOMMENDATION:
Only use when default guess is not satisfactory.
SFX_AMP_VIR_B
Defines a user-specified amplitude guess vector in SF-XCIS method.
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
n builds a guess amplitude with a β-particle in the nth orbital (requires SFX_AMP_OCC_A).
RECOMMENDATION:
Only use when default guess is not satisfactory.
XCIS
Do an XCIS calculation in addition to a CIS calculation.
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
FALSE Do not do an XCIS calculation.
TRUE
Do an XCIS calculation (requires ROHF ground state).
RECOMMENDATION:
None

297

Chapter 7: Open-Shell and Excited-State Methods
SASF_RPA
Do an SA-SF-CIS/DFT calculation.
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
FALSE Do not do an SA-SF-CIS/DFT calculation.
TRUE
Do an SA-SF-CIS/DFT calculation (requires ROHF ground state).
RECOMMENDATION:
None

7.2.9

CIS Job Customization
N_FROZEN_CORE
Controls the number of frozen core orbitals.
TYPE:
INTEGER/STRING
DEFAULT:
0 No frozen core orbitals.
OPTIONS:
FC Frozen core approximation.
n
Freeze n core orbitals.
RECOMMENDATION:
There is no computational advantage to using frozen core for CIS, and analytical derivatives are
only available when no orbitals are frozen. It is helpful when calculating CIS(D) corrections (see
Sec. 7.6).
N_FROZEN_VIRTUAL
Controls the number of frozen virtual orbitals.
TYPE:
INTEGER
DEFAULT:
0 No frozen virtual orbitals.
OPTIONS:
n Freeze n virtual orbitals.
RECOMMENDATION:
There is no computational advantage to using frozen virtuals for CIS, and analytical derivatives
are only available when no orbitals are frozen.
MAX_CIS_CYCLES
Maximum number of CIS iterative cycles allowed.
TYPE:
INTEGER
DEFAULT:
30
OPTIONS:
n User-defined number of cycles.
RECOMMENDATION:
Default is usually sufficient.

298

Chapter 7: Open-Shell and Excited-State Methods
MAX_CIS_SUBSPACE
Maximum number of subspace vectors allowed in the CIS iterations
TYPE:
INTEGER
DEFAULT:
As many as required to converge all roots
OPTIONS:
n User-defined number of subspace vectors
RECOMMENDATION:
The default is usually appropriate, unless a large number of states are requested for a large
molecule. The total memory required to store the subspace vectors is bounded above by 2nOV ,
where O and V represent the number of occupied and virtual orbitals, respectively. n can be
reduced to save memory, at the cost of a larger number of CIS iterations. Convergence may be
impaired if n is not much larger than CIS_N_ROOTS.
CIS_CONVERGENCE
CIS is considered converged when error is less than 10−CIS_CONVERGENCE
TYPE:
INTEGER
DEFAULT:
6 CIS convergence threshold 10−6
OPTIONS:
n Corresponding to 10−n
RECOMMENDATION:
None
CIS_DYNAMIC_MEM
Controls whether to use static or dynamic memory in CIS and TDDFT calculations.
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
FALSE Partly use static memory
TRUE
Fully use dynamic memory
RECOMMENDATION:
The default control requires static memory (MEM_STATIC) to hold a temporary array whose
minimum size is OV × CIS_N_ROOTS. For a large calculation, one has to specify a large value
for MEM_STATIC, which is not recommended (see Chapter 2). Therefore, it is recommended to
use dynamic memory for large calculations.
CIS_RELAXED_DENSITY
Use the relaxed CIS density for attachment/detachment density analysis.
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
FALSE Do not use the relaxed CIS density in analysis.
TRUE
Use the relaxed CIS density in analysis.
RECOMMENDATION:
None

299

Chapter 7: Open-Shell and Excited-State Methods
CIS_GUESS_DISK
Read the CIS guess from disk (previous calculation).
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
FALSE Create a new guess.
TRUE
Read the guess from disk.
RECOMMENDATION:
Requires a guess from previous calculation.
CIS_GUESS_DISK_TYPE
Determines the type of guesses to be read from disk
TYPE:
INTEGER
DEFAULT:
Nil
OPTIONS:
0 Read triplets only
1 Read triplets and singlets
2 Read singlets only
RECOMMENDATION:
Must be specified if CIS_GUESS_DISK is TRUE.
STS_MOM
Control calculation of the transition moments between excited states in the CIS and TDDFT
calculations (including SF-CIS and SF-DFT).
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
FALSE Do not calculate state-to-state transition moments.
TRUE
Do calculate state-to-state transition moments.
RECOMMENDATION:
When set to true requests the state-to-state dipole transition moments for all pairs of excited
states and for each excited state with the ground state.

Note: This option is not available for SF-XCIS.
CIS_MOMENTS
Controls calculation of excited-state (CIS or TDDFT) multipole moments.
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
FALSE Do not calculate excited-state moments.
TRUE
Calculate moments for each excited state.
RECOMMENDATION:
Set to TRUE if excited-state moments are desired. (This is a trivial additional calculation.) The
MULTIPOLE_ORDER controls how many multipole moments are printed.

300

301

Chapter 7: Open-Shell and Excited-State Methods

7.2.10

Examples

Example 7.1 A basic CIS excitation energy calculation on formaldehyde at the HF/6-31G* optimized ground state
geometry, which is obtained in the first part of the job. Above the first singlet excited state, the states have Rydberg
character, and therefore a basis with two sets of diffuse functions is used.
$molecule
0 1
C
O 1 CO
H 1 CH
H 1 CH
CO
CH
A
D
$end

2
2

A
A

3

D

=
1.2
=
1.0
= 120.0
= 180.0

$rem
JOBTYPE
EXCHANGE
BASIS
$end

=
=
=

opt
hf
6-31G*

@@@
$molecule
read
$end
$rem
EXCHANGE
BASIS
CIS_N_ROOTS
CIS_SINGLETS
CIS_TRIPLETS
$end

=
=
=
=
=

hf
6-311(2+)G*
15
true
false

Do 15 states
Do do singlets
Don’t do Triplets

Chapter 7: Open-Shell and Excited-State Methods

302

Example 7.2 An XCIS calculation of excited states of an unsaturated radical, the phenyl radical, for which double
substitutions make considerable contributions to low-lying excited states.
$comment
C6H5 phenyl radical C2v symmetry MP2(full)/6-31G* = -230.7777459
$end
$molecule
0 2
c1
x1 c1
c2 c1
x2 c2
c3 c1
c4 c1
c5 c3
c6 c4
h1 c2
h2 c3
h3 c4
h4 c5
h5 c6
rh1
rh2
rc2
rc3
rh4
rc5
tc3
ah2
ah4
ac5
$end

=
=
=
=
=
=
=
=
=
=

1.0
rc2
1.0
rc3
rc3
rc5
rc5
rh1
rh2
rh2
rh4
rh4

x1
c1
x1
x1
c1
c1
x2
c1
c1
c3
c4

90.0
90.0
90.0
90.0
ac5
ac5
90.0
ah2
ah2
ah4
ah4

x1
c2
c2
x1
x1
c1
x1
x1
c1
c1

0.0
tc3
-tc3
-90.0
90.0
180.0
90.0
-90.0
180.0
180.0

1.08574
1.08534
2.67299
1.35450
1.08722
1.37290
62.85
122.16
119.52
116.45

$rem
BASIS
EXCHANGE
MEM_STATIC
INTSBUFFERSIZE
SCF_CONVERGENCE
CIS_N_ROOTS
XCIS
$end

=
=
=
=
=
=
=

6-31+G*
hf
80
15000000
8
5
true

Chapter 7: Open-Shell and Excited-State Methods

303

Example 7.3 A SF-XCIS calculation of ground and excited states of trimethylenemethane (TMM) diradical, for which
double substitutions make considerable contributions to low-lying excited states.
$molecule
0 3
C
C 1 CC1
C 1 CC2
C 1 CC2
H 2 C2H
H 2 C2H
H 3 C3Hu
H 3 C3Hd
H 4 C3Hu
H 4 C3Hd
CC1
CC2
C2H
C3Hu
C3Hd
C2CH
C3CHu
C3CHd
A2
$end

=
=
=
=
=
=
=
=
=

2
2
1
1
1
1
1
1

A2
A2
C2CH
C2CH
C3CHu
C3CHd
C3CHu
C3CHd

3
3
4
2
4
2
3

180.0
0.0
0.0
0.0
0.0
0.0
0.0

1.35
1.47
1.083
1.08
1.08
121.2
120.3
121.3
121.0

$rem
UNRESTRICTED
EXCHANGE
BASIS
SCF_CONVERGENCE
SCF_ALGORITHM
MAX_SCF_CYCLES
SPIN_FLIP_XCIS
CIS_N_ROOTS
CIS_SINGLETS
CIS_TRIPLETS
$end

=
=
=
=
=
=
=
=
=
=

false
HF
6-31G*
10
DM
100
true
3
true
true

SF-XCIS runs from ROHF triplet reference

Do SF-XCIS
Do singlets
Do triplets

304

Chapter 7: Open-Shell and Excited-State Methods

Example 7.4 An SA-SF-DFT calculation of singlet ground and excited states of ethylene.
$molecule
0 3
C
C
H
H
H
H
B1
B2
B3
B4
B5
A1
A2
A3
A4
D1
D2
D3
$end

1
1
1
2
2
1.32808942
1.08687297
1.08687297
1.08687297
1.08687297
121.62604150
121.62604150
121.62604150
121.62604150
180.00000000
0.00000000
180.00000000

$rem
unrestricted 0
jobtype sp
basis
cc-pvtz
basis2 sto-3g
exchange bhhlyp
cis_n_roots 5
sasf_rpa 1
cis_triplets false
$end

B1
B2
B3
B4
B5

2
2
1
1

A1
A2
A3
A4

3
3
3

D1
D2
D3

Chapter 7: Open-Shell and Excited-State Methods

305

Example 7.5 This example illustrates a CIS geometry optimization followed by a vibrational frequency analysis on
the lowest singlet excited state of formaldehyde. This n → π ∗ excited state is non-planar, unlike the ground state. The
optimization converges to a non-planar structure with zero forces, and all frequencies real.
$molecule
0 1
C
O 1 CO
H 1 CH
H 1 CH
CO
CH
A
D
$end

=
=
=
=

2
2

A
A

3

D

1.2
1.0
120.0
150.0

$rem
JOBTYPE
EXCHANGE
BASIS
CIS_STATE_DERIV
CIS_N_ROOTS
CIS_SINGLETS
CIS_TRIPLETS
$end

=
=
=
=
=
=
=

opt
hf
6-31+G*
1
3
true
false

Optimize state 1
Do 3 states
Do do singlets
Don’t do Triplets

=
=
=
=
=
=
=

freq
hf
6-31+G*
1
3
true
false

Focus on state 1
Do 3 states
Do do singlets
Don’t do Triplets

@@@
$molecule
read
$end
$rem
JOBTYPE
EXCHANGE
BASIS
CIS_STATE_DERIV
CIS_N_ROOTS
CIS_SINGLETS
CIS_TRIPLETS
$end

7.3
7.3.1

Time-Dependent Density Functional Theory (TDDFT)
Brief Introduction to TDDFT

Excited states may be obtained from density functional theory by time-dependent density functional theory, 25,31 which
calculates poles in the response of the ground state density to a time-varying applied electric field. These poles are Bohr
frequencies, or in other words the excitation energies. Operationally, this involves solution of an eigenvalue equation






A B
x
−1 0
x
=
ω
(7.15)
B† A†
y
0 1
y
where the elements of the matrix A similar to those used at the CIS level, Eq. (7.11), but with an exchange-correlation
correction. 49 Elements of B are similar. Equation (7.15) is solved iteratively for the lowest few excitation energies, ω.

Chapter 7: Open-Shell and Excited-State Methods

306

Alternatively, one can make a CIS-like Tamm-Dancoff approximation (TDA), 50 in which the “de-excitation” amplitudes Y are neglected, the B matrix is not required, and Eq. (7.15) reduces to Ax = ωx.
TDDFT is popular because its computational cost is roughly similar to that of the simple CIS method, but a description
of differential electron correlation effects is implicit in the method. It is advisable to only employ TDDFT for low-lying
valence excited states that are below the first ionization potential of the molecule, 25 or more conservatively, below the
first Rydberg state, and in such cases the valence excitation energies are often remarkably improved relative to CIS,
with an accuracy of ∼0.3 eV for many functionals. 69 The calculation of the nuclear gradients of full TDDFT and within
the TDA is implemented. 74
On the other hand, standard density functionals do not yield a potential with the correct long-range Coulomb tail, owing
to the so-called self-interaction problem, and therefore excitation energies corresponding to states that sample this tail
(e.g., diffuse Rydberg states and some charge transfer excited states) are not given accurately. 26,67,124 The extent to
which a particular excited state is characterized by charge transfer can be assessed using an a spatial overlap metric
proposed by Peach, Benfield, Helgaker, and Tozer (PBHT). 101 (However, see Ref. 108 for a cautionary note regarding
this metric.)
Standard TDDFT also does not yield a good description of static correlation effects (see Section 6.10), because it is
based on a single reference configuration of Kohn-Sham orbitals. Recently, a new variation of TDDFT called spin-flip
(SF) DFT was developed by Yihan Shao, Martin Head-Gordon and Anna Krylov to address this issue. 115 SF-DFT is
different from standard TDDFT in two ways:
• The reference is a high-spin triplet (quartet) for a system with an even (odd) number of electrons;
• One electron is spin-flipped from an alpha Kohn-Sham orbital to a beta orbital during the excitation.
SF-DFT can describe the ground state as well as a few low-lying excited states, and has been applied to bond-breaking
processes, and di- and tri-radicals with degenerate or near-degenerate frontier orbitals. Recently, we also implemented 7
a SF-DFT method with a non-collinear exchange-correlation potential from Tom Ziegler et al., 114,129 which is in many
case an improvement over collinear SF-DFT. 115 Recommended functionals for SF-DFT calculations are 5050 and
PBE50 (see Ref. 7 for extensive benchmarks). See also Section 7.7.3 for details on wave function-based spin-flip
models.

7.3.2

TDDFT within a Reduced Single-Excitation Space

Much of chemistry and biology occurs in solution or on surfaces. The molecular environment can have a large effect
on electronic structure and may change chemical behavior. Q-C HEM is able to compute excited states within a local
region of a system through performing the TDDFT (or CIS) calculation with a reduced single excitation subspace, 8
in which some of the amplitudes x in Eq. (7.15) are excluded. (This is implemented within the TDA, so y ≡ 0.)
This allows the excited states of a solute molecule to be studied with a large number of solvent molecules reducing
the rapid rise in computational cost. The success of this approach relies on there being only weak mixing between the
electronic excitations of interest and those omitted from the single excitation space. For systems in which there are
strong hydrogen bonds between solute and solvent, it is advisable to include excitations associated with the neighboring
solvent molecule(s) within the reduced excitation space.
The reduced single excitation space is constructed from excitations between a subset of occupied and virtual orbitals.
These can be selected from an analysis based on Mulliken populations and molecular orbital coefficients. For this
approach the atoms that constitute the solvent needs to be defined. Alternatively, the orbitals can be defined directly.
The atoms or orbitals are specified within a $solute block. These approach is implemented within the TDA and has
been used to study the excited states of formamide in solution, 11 CO on the Pt(111) surface, 9 and the tryptophan
chromophore within proteins. 109

Chapter 7: Open-Shell and Excited-State Methods

7.3.3

307

Job Control for TDDFT

Input for time-dependent density functional theory calculations follows very closely the input already described for the
uncorrelated excited state methods described in the previous section (in particular, see Section 7.2.8). There are several
points to be aware of:
• The exchange and correlation functionals are specified exactly as for a ground state DFT calculation, through
EXCHANGE and CORRELATION.
• If RPA is set to TRUE, a “full” TDDFT calculation will be performed, however the default value is RPA = FALSE,
which invokes the TDA, 50 in which the de-excitation amplitudes Y in Eq. (7.15) are neglected, which is usually
a good approximation for excitation energies, although oscillator strengths within the TDA no longer formally
satisfy the Thomas-Reiche-Kuhn sum rule. 25 For RPA = TRUE, a TDA calculation is performed first and used as
the initial guess for the full TDDFT calculation. The TDA calculation can be skipped altogether using RPA = 2
• If SPIN_FLIP is set to TRUE when performing a TDDFT calculation, a SF-DFT calculation will also be performed.
At present, SF-DFT is only implemented within TDDFT/TDA so RPA must be set to FALSE. Remember to set
the spin multiplicity to 3 for systems with an even-number of electrons (e.g., diradicals), and 4 for odd-number
electron systems (e.g., triradicals).
TRNSS
Controls whether reduced single excitation space is used.
TYPE:
LOGICAL
DEFAULT:
FALSE Use full excitation space.
OPTIONS:
TRUE Use reduced excitation space.
RECOMMENDATION:
None
TRTYPE
Controls how reduced subspace is specified.
TYPE:
INTEGER
DEFAULT:
1
OPTIONS:
1 Select orbitals localized on a set of atoms.
2 Specify a set of orbitals.
3 Specify a set of occupied orbitals, include excitations to all virtual orbitals.
RECOMMENDATION:
None

Chapter 7: Open-Shell and Excited-State Methods
N_SOL
Specifies number of atoms or orbitals in the $solute section.
TYPE:
INTEGER
DEFAULT:
No default.
OPTIONS:
User defined.
RECOMMENDATION:
None
CISTR_PRINT
Controls level of output.
TYPE:
LOGICAL
DEFAULT:
FALSE Minimal output.
OPTIONS:
TRUE Increase output level.
RECOMMENDATION:
None
CUTOCC
Specifies occupied orbital cutoff.
TYPE:
INTEGER
DEFAULT:
50
OPTIONS:
0-200 CUTOFF = CUTOCC/100
RECOMMENDATION:
None
CUTVIR
Specifies virtual orbital cutoff.
TYPE:
INTEGER
DEFAULT:
0 No truncation
OPTIONS:
0-100 CUTOFF = CUTVIR/100
RECOMMENDATION:
None

308

Chapter 7: Open-Shell and Excited-State Methods
PBHT_ANALYSIS
Controls whether overlap analysis of electronic excitations is performed.
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
FALSE Do not perform overlap analysis.
TRUE
Perform overlap analysis.
RECOMMENDATION:
None
PBHT_FINE
Increases accuracy of overlap analysis.
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
FALSE
TRUE
Increase accuracy of overlap analysis.
RECOMMENDATION:
None
SRC_DFT
Selects form of the short-range corrected functional.
TYPE:
INTEGER
DEFAULT:
No default
OPTIONS:
1 SRC1 functional.
2 SRC2 functional.
RECOMMENDATION:
None
OMEGA
Sets the Coulomb attenuation parameter for the short-range component.
TYPE:
INTEGER
DEFAULT:
No default
OPTIONS:
n Corresponding to ω = n/1000, in units of bohr−1
RECOMMENDATION:
None

309

Chapter 7: Open-Shell and Excited-State Methods

310

OMEGA2
Sets the Coulomb attenuation parameter for the long-range component.
TYPE:
INTEGER
DEFAULT:
No default
OPTIONS:
n Corresponding to ω2 = n/1000, in units of bohr−1
RECOMMENDATION:
None
HF_SR
Sets the fraction of Hartree-Fock exchange at r12 = 0.
TYPE:
INTEGER
DEFAULT:
No default
OPTIONS:
n Corresponding to HF_SR = n/1000
RECOMMENDATION:
None
HF_LR
Sets the fraction of Hartree-Fock exchange at r12 = ∞.
TYPE:
INTEGER
DEFAULT:
No default
OPTIONS:
n Corresponding to HF_LR = n/1000
RECOMMENDATION:
None
WANG_ZIEGLER_KERNEL
Controls whether to use the Wang-Ziegler non-collinear exchange-correlation kernel in a SFDFT calculation.
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
FALSE Do not use non-collinear kernel.
TRUE
Use non-collinear kernel.
RECOMMENDATION:
None

7.3.4

TDDFT Coupled with C-PCM for Excitation Energies and Properties Calculations

As described in Section 12.2 (and especially Section 12.2.2), continuum solvent models such as C-PCM allow one to
include solvent effect in the calculations. TDDFT/C-PCM allows excited-state modeling in solution. Q-C HEM also

Chapter 7: Open-Shell and Excited-State Methods

311

features TDDFT coupled with C-PCM which extends TDDFT to calculations of properties of electronically-excited
molecules in solution. In particular, TDDFT/C-PCM allows one to perform geometry optimization and vibrational
analysis. 77
When TDDFT/C-PCM is applied to calculate vertical excitation energies, the solvent around vertically excited solute is
out of equilibrium. While the solvent electron density equilibrates fast to the density of the solute (electronic response),
the relaxation of nuclear degrees of freedom (e.g., orientational polarization) takes place on a slower timescale. To describe this situation, an optical dielectric constant is employed. To distinguish between equilibrium and non-equilibrium
calculations, two dielectric constants are used in these calculations: a static constant (ε0 ), equal to the equilibrium bulk
value, and a fast constant (εf ast ) related to the response of the medium to high frequency perturbations. For vertical
excitation energy calculations (corresponding to the unrelaxed solvent nuclear degrees of freedom), it is recommended
to use the optical dielectric constant for εf ast ), whereas for the geometry optimization and vibrational frequency calculations, the static dielectric constant should be used. 77
The example below illustrates TDDFT/C-PCM calculations of vertical excitation energies. More information concerning the C-PCM and the various PCM job control options can be found in Section 12.2.
Example 7.6 TDDFT/C-PCM low-lying vertical excitation energy
$molecule
0 1
C
0.0
O
0.0
$end

0.0
0.0

$rem
EXCHANGE
CIS_N_ROOTS
CIS_SINGLETS
CIS_TRIPLETS
RPA
BASIS
XC_GRID
SOLVENT_METHOD
$end

0.0
1.21

$pcm
Theory
Method
Solver
Radii
$end

B3lyp
10
true
true
TRUE
6-31+G*
1
pcm

CPCM
SWIG
Inversion
Bondi

$solvent
Dielectric
OpticalDielectric
$end

7.3.5

78.39
1.777849

Analytical Excited-State Hessian in TDDFT

To carry out vibrational frequency analysis of an excited state with TDDFT, 75,76 an optimization of the excited-state
geometry is always necessary. Like the vibrational frequency analysis of the ground state, the frequency analysis of the
excited state should be also performed at a stationary point on the excited state potential surface. The $rem variable
CIS_STATE_DERIV should be set to the excited state for which an optimization and frequency analysis is needed, in
addition to the $rem keywords used for an excitation energy calculation.
Compared to the numerical differentiation method, the analytical calculation of geometrical second derivatives of the
excitation energy needs much less time but much more memory. The computational cost is mainly consumed by the

Chapter 7: Open-Shell and Excited-State Methods

312

steps to solve both the CPSCF equations for the derivatives of molecular orbital coefficients Cx and the CP-TDDFT
equations for the derivatives of the transition vectors, as well as to build the Hessian matrix. The memory usages for
these steps scale as O(3mN 2 ), where N is the number of basis functions and m is the number of atoms. For large
systems, it is thus essential to solve all the coupled-perturbed equations in segments. In this case, the $rem variable
CPSCF_NSEG is always needed.
In the calculation of the analytical TDDFT excited-state Hessian, one has to evaluate a large number of energyfunctional derivatives: the first-order to fourth-order functional derivatives with respect to the density variables as well
as their derivatives with respect to the nuclear coordinates. Therefore, a very fine integration grid for DFT calculation
should be adapted to guarantee the accuracy of the results.
Analytical TDDFT/C-PCM Hessian has been implemented in Q-C HEM. Normal mode analysis for a system in solution
can be performed with the frequency calculation by TDDFT/C-PCM method. The $rem and $pcm variables for the
excited state calculation with TDDFT/C-PCM included in the vertical excitation energy example above are needed.
When the properties of large systems are calculated, you must pay attention to the memory limit. At present, only a
few exchange correlation functionals, including Slater+VWN, BLYP, B3LYP, are available for the analytical Hessian
calculation.
Example 7.7 A B3LYP/6-31G* optimization in gas phase, followed by a frequency analysis for the first excited state
of the peroxy radical
$molecule
0 2
C 1.004123
O -0.246002
O -1.312366
H 1.810765
H 1.036648
H 1.036648
$end

-0.180454
0.596152
-0.230256
0.567203
-0.805445
-0.805445

$rem
JOBTYPE
EXCHANGE
CIS_STATE_DERIV
BASIS
CIS_N_ROOTS
CIS_SINGLETS
CIS_TRIPLETS
XC_GRID
RPA
$end

0.000000
0.000000
0.000000
0.000000
-0.904798
0.904798

opt
b3lyp
1
6-31G*
10
true
false
000075000302
0

@@@
$molecule
read
$end
$rem
JOBTYPE
EXCHANGE
CIS_STATE_DERIV
BASIS
CIS_N_ROOTS
CIS_SINGLETS
CIS_TRIPLETS
XC_GRID
RPA
$end

freq
b3lyp
1
6-31G*
10
true
false
000075000302
0

Chapter 7: Open-Shell and Excited-State Methods

313

Example 7.8 The optimization and Hessian calculation for low-lying excited state of 9-Fluorenone + 2 methanol in
methanol solution using TDDFT/C-PCM
$molecule
0 1
6
-1.987249
6
-1.987187
6
-0.598049
6
0.282546
6
-0.598139
6
-0.319285
6
-1.386049
6
-2.743097
6
-3.049918
6
-3.050098
6
-2.743409
6
-1.386397
6
-0.319531
8
1.560568
1
0.703016
1
-1.184909
1
-3.533126
1
-4.079363
1
0.702729
1
-1.185378
1
-3.533492
1
-4.079503
8
3.323150
1
2.669309
6
3.666902
1
4.397551
1
4.116282
1
2.795088
1
2.669205
8
3.322989
6
3.666412
1
4.396966
1
4.115800
1
2.794432
$end

0.699711
-0.699537
-1.148932
0.000160
1.149219
-2.505397
-3.395376
-2.962480
-1.628487
1.628566
2.962563
3.395575
2.505713
0.000159
-2.862338
-4.453877
-3.698795
-1.292006
2.862769
4.454097
3.698831
1.291985
2.119222
1.389642
2.489396
3.298444
1.654650
2.849337
-1.389382
-2.119006
-2.489898
-3.299023
-1.655485
-2.850001

$rem
JOBTYPE
EXCHANGE
CIS_N_ROOTS
CIS_SINGLETS
CIS_TRIPLETS
CIS_STATE_DERIV
RPA
BASIS
XC_GRID
SOLVENT_METHOD
$end
$pcm
Theory
Method
Solver
Radii
$end

@@@

OPT
B3lyp
10
true
true
1
Lowest TDDFT state
TRUE
6-311G**
000075000302
pcm

CPCM
SWIG
Inversion
Bondi

$solvent
Dielectric 32.613
$end

0.080583
-0.080519
-0.131299
0.000137
0.131479
-0.285378
-0.388447
-0.339290
-0.186285
0.186246
0.339341
0.388596
0.285633
0.000209
-0.324093
-0.510447
-0.423022
-0.147755
0.324437
0.510608
0.422983
0.147594
0.125454
0.084386
-1.208239
-1.151310
-1.759486
-1.768206
-0.084343
-0.125620
1.207974
1.150789
1.759730
1.767593

314

Chapter 7: Open-Shell and Excited-State Methods

7.3.6

Calculations of Spin-Orbit Couplings Between TDDFT States

Calculations of spin-orbit couplings (SOCs) for TDDFT states within the Tamm-Dancoff approximation or RPA (including TDHF and CIS states) is available. We employ the one-electron Breit Pauli Hamiltonian to calculate the SOC
constant between TDDFT states.
α 2 X ZA
ĤSO = − 0
(7.16)
3 (riA × pi ) · si
2
riA
i,A

where i denotes electrons, A denotes nuclei, α0 = 137.037−1 is the fine structure constant. ZA is the bare positive
charge on nucleus A. In the second quantization representation, the spin-orbit Hamiltonian in different directions can
be expressed as

α2 X ˜
~ †
ap aq̄ + a†p̄ aq
(7.17)
ĤSOx = − 0
Lxpq ·
2 pq
2

α2 X ˜
~  †
ap aq̄ − a†p̄ aq
(7.18)
ĤSOy = − 0
Ly pq ·
2 pq
2i

α2 X ˜
~ †
(7.19)
ĤSOz = − 0
ap aq − a†p̄ aq̄
Lz pq ·
2 pq
2
where L˜α = Lα /r3 (α = x, y, z). The single-reference ab initio excited states (within the Tamm-Dancoff approximation) are given by


X
†
†
|ΦIsinglet i =
sIa
a
a
+
a
a
(7.20)
ā ī |ΦHF i
i
a i
i,a
s =0
|ΦI,m
i
triplet

=

X



tIa
a†a ai − a†ā aī |ΦHF i
i

(7.21)

i,a

s =1
|ΦI,m
i =
triplet

X√

†
2tIa
i aa aī |ΦHF i

(7.22)

†
2tIa
i aā ai |ΦHF i

(7.23)

i,a

I,ms =−1
|Φtriplet
i =

X√
i,a

where sIa
tIa
and triplet excitation coefficients of the I th singlet or triplet state respectively, with the
i andP
i are singlet
P
2
2
normalization sIa
=
tIa
= 21 ; |ΦHF i refers to the Hartree-Fock ground state. Thus the SOC constant from the
i
i
ia

ia

singlet state to different triplet manifolds can be obtained as follows,


2
X
X
α
~
0 
Jb
Ja 
s =0
hΦIsinglet |ĤSO |ΦJ,m
i =
L˜z ab sIa
L˜z ij sIa
i ti −
i tj
triplet
2
i,j,a
i,a,b


2
X
X
α
~
J,ms =±1
0 
Jb
Ja 
hΦIsinglet |ĤSO |Φtriplet
L˜xab sIa
L˜xij sIa
i = ∓ √
i ti −
i tj
2 2 i,a,b
i,j,a


X
α02 ~  X ˜
Jb
Ja 
+ √
Ly sIa
L˜y ij sIa
i ti −
i tj
2 2i i,a,b ab
i,j,a
The SOC constant between different triplet manifolds can be obtained


X
α02 ~  X ˜
I,ms =0
J,ms =±1
Jb
Ja 
hΦtriplet |ĤSO |Φtriplet
i = ∓ √
Lxab tIa
L˜xij tIa
i ti +
i tj
2 2 i,a,b
i,j,a


X
α02 ~  X ˜
Jb
Ja 
+ √
Ly tIa
L˜y ij tIa
i ti +
i tj
2 2i i,a,b ab
i,j,a


X
α02 ~  X ˜
I,ms =±1
J,ms =±1
Ia Jb
Ia
Ja
hΦtriplet
|ĤSO |Φtriplet
i = ±
Lz ab ti ti +
L˜z ij ti tj 
2
i,j,a
i,a,b

(7.24)

(7.25)

(7.26)

(7.27)

315

Chapter 7: Open-Shell and Excited-State Methods

I,ms =±1
J,ms =∓1
s =0
s =0
Note that hΦI,m
|ĤSO |ΦJ,m
i = hΦtriplet
|ĤSO |Φtriplet
i = 0. The total (root-mean-square) spin-orbit coutriplet
triplet
pling is given by
s X
s
2
khΦIsinglet |ĤSO |ΦJ,m
(7.28)
hΦIsinglet |ĤSO |ΦJtriplet i =
triplet ik
ms =0,±1

hΦItriplet |ĤSO |ΦJtriplet i =

s X

J,ms 2
s
khΦI,m
triplet |ĤSO |Φtriplet ik

(7.29)

ms =0,±1
Ia
Jb
Ia
Jb
Jb Ia Jb
For RPA states, the SOC constant can simply be obtained by replacing sIa
i tj (ti tj ) with Xi,singlet Xj,triplet +Yi,singlet Yj,triplet
Ia
Jb
Ia
Jb
(Xi,triplet Xj,triplet + Yi,triplet Yj,triplet ) Setting the $rem variable CALC_SOC = TRUE will enable the SOC calculation for all
calculated TDDFT states.

CALC_SOC
Controls whether to calculate the SOC constants for EOM-CC, ADC, TDDFT/TDA and TDDFT.
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
FALSE Do not perform the SOC calculation.
TRUE
Perform the SOC calculation.
RECOMMENDATION:
None

316

Chapter 7: Open-Shell and Excited-State Methods

Example 7.9 Calculation of SOCs for water molecule using TDDFT/B3LYP functional within the TDA.
$comment
This sample input calculates the spin-orbit coupling constants for water
between its ground state and its TDDFT/TDA excited triplets as well as the
coupling between its TDDFT/TDA singlets and triplets. Results are given in
cm-1.
$end
$molecule
0 1
H
H
O
$end

0.000000
0.000000
0.000000

$rem
JOBTYPE
EXCHANGE
BASIS
CIS_N_ROOTS
CIS_CONVERGENCE
CORRELATION
MAX_SCF_CYCLES
MAX_CIS_CYCLES
SCF_ALGORITHM
MEM_STATIC
MEM_TOTAL
SYMMETRY
SYM_IGNORE
UNRESTRICTED
CIS_SINGLETS
CIS_TRIPLETS
CALC_SOC
SET_ITER
$end

-0.115747
1.109931
0.005817

sp
b3lyp
6-31G
4
8
none
600
50
diis
300
2000
false
true
false
true
true
true
300

1.133769
-0.113383
-0.020386

Chapter 7: Open-Shell and Excited-State Methods

7.3.7

317

Various TDDFT-Based Examples

Example 7.10 This example shows two jobs which request variants of time-dependent density functional theory calculations. The first job, using the default value of RPA = FALSE, performs TDDFT in the Tamm-Dancoff approximation
(TDA). The second job, with RPA = TRUE performs a both TDA and full TDDFT calculations.
$comment
methyl peroxy radical
TDDFT/TDA and full TDDFT with 6-31+G*
$end
$molecule
0 2
C 1.00412
O -0.24600
O -1.31237
H 1.81077
H 1.03665
H 1.03665
$end

-0.18045
0.59615
-0.23026
0.56720
-0.80545
-0.80545

$rem
EXCHANGE
CORRELATION
CIS_N_ROOTS
BASIS
SCF_CONVERGENCE
$end

0.00000
0.00000
0.00000
0.00000
-0.90480
0.90480

b
lyp
5
6-31+G*
7

@@@
$molecule
read
$end
$rem
EXCHANGE
CORRELATION
CIS_N_ROOTS
RPA
BASIS
SCF_CONVERGENCE
$end

b
lyp
5
true
6-31+G*
7

Chapter 7: Open-Shell and Excited-State Methods

318

Example 7.11 This example shows a calculation of the excited states of a formamide-water complex within a reduced
excitation space of the orbitals located on formamide
$comment
formamide-water TDDFT/TDA in reduced excitation space
$end
$molecule
0 1
H 1.13 0.49 -0.75
C 0.31 0.50 -0.03
N -0.28 -0.71 0.08
H -1.09 -0.75 0.67
H 0.23 -1.62 -0.22
O -0.21 1.51 0.47
O -2.69 1.94 -0.59
H -2.59 2.08 -1.53
H -1.83 1.63 -0.30
$end
$rem
EXCHANGE
CIS_N_ROOTS
BASIS
TRNSS
TRTYPE
CUTOCC
CUTVIR
CISTR_PRINT
$end
$solute
1
2
3
4
5
6
$end

b3lyp
10
6-31++G**
TRUE
1
60
40
TRUE

Chapter 7: Open-Shell and Excited-State Methods

319

Example 7.12 This example shows a calculation of the core-excited states at the oxygen K-edge of CO with a shortrange corrected functional.
$comment
TDDFT with short-range corrected (SRC1) functional for the
oxygen K-edge of CO
$end
$molecule
0 1
C
0.000000
O
0.000000
$end
$rem
EXCHANGE
BASIS
CIS_N_ROOTS
CIS_TRIPLETS
TRNSS
TRTYPE
N_SOL
$end

0.000000
0.000000

-0.648906
0.486357

SRC1-R1
6-311(2+,2+)G**
6
false
true
3
1

$solute
1
$end

Example 7.13 This example shows a calculation of the core-excited states at the phosphorus K-edge with a short-range
corrected functional.
$comment
TDDFT with short-range corrected (SRC2) functional for the
phosphorus K-edge of PH3
$end
$molecule
0 1
H
1.196206
P
0.000000
H
-0.598103
H
-0.598103
$end
$rem
EXCHANGE
BASIS
CIS_N_ROOTS
CIS_TRIPLETS
TRNSS
TRTYPE
N_SOL
$end
$solute
1
$end

0.000000
0.000000
-1.035945
1.035945

-0.469131
0.303157
-0.469131
-0.469131

SRC2-R2
6-311(2+,2+)G**
6
false
true
3
1

Chapter 7: Open-Shell and Excited-State Methods

320

Example 7.14 SF-TDDFT SP calculation of the 6 lowest states of the TMM diradical using recommended 50-50
functional
$molecule
0 3
C
C 1 CC1
C 1 CC2
C 1 CC2
H 2 C2H
H 2 C2H
H 3 C3Hu
H 3 C3Hd
H 4 C3Hu
H 4 C3Hd

2
2
1
1
1
1
1
1

A2
A2
C2CH
C2CH
C3CHu
C3CHd
C3CHu
C3CHd

3
3
4
2
4
2
3

CC1
= 1.35
CC2
= 1.47
C2H
= 1.083
C3Hu
= 1.08
C3Hd
= 1.08
C2CH
= 121.2
C3CHu = 120.3
C3CHd = 121.3
A2
= 121.0
$end
$rem
EXCHANGE
BASIS
SCF_GUESS
SCF_CONVERGENCE
MAX_SCF_CYCLES
SPIN_FLIP
CIS_N_ROOTS
CIS_CONVERGENCE
MAX_CIS_CYCLES
$end
$xc_functional
X HF
0.50
X S
0.08
X B
0.42
C VWN
0.19
C LYP
0.81
$end

gen
6-31G*
core
10
100
1
6
10
100

180.0
0.0
0.0
0.0
0.0
0.0
0.0

Chapter 7: Open-Shell and Excited-State Methods

321

Example 7.15 SF-DFT with non-collinear exchange-correlation functional for low-lying states of CH2
$comment
Non-collinear SF-DFT calculation for CH2 at 3B1 state geometry from
EOM-CCSD(fT) calculation
$end
$molecule
0 3
C
H 1 rCH
H 1 rCH

2 HCH

rCH = 1.0775
HCH = 133.29
$end
$rem
EXCHANGE
BASIS
SPIN_FLIP
WANG_ZIEGLER_KERNEL
SCF_CONVERGENCE
CIS_N_ROOTS
CIS_CONVERGENCE
$end

7.4

PBE0
cc-pVTZ
1
TRUE
10
6
10

Maximum Overlap Method (MOM) for SCF Excited States

The Maximum Overlap Method (MOM) is a useful alternative to CIS and TDDFT for obtaining low-cost excited
states. 37 It works by modifying the orbital selection step in the SCF procedure. By choosing orbitals that most resemble
those from the previous cycle, rather than those with the lowest eigenvalues, excited SCF determinants are able to be
obtained. The MOM has several advantages over existing low-cost excited state methods. Current implementations
of TDDFT usually struggle to accurately model charge-transfer and Rydberg transitions, both of which can be wellmodeled using the MOM. The MOM also allows the user to target very high energy states, such as those involving
excitation of core electrons, 12 which are hard to capture using other excited state methods.
In order to calculate an excited state using MOM, the user must correctly identify the orbitals involved in the transition.
For example, in a π → π ∗ transition, the π and π ∗ orbitals must be identified and this usually requires a preliminary
calculation. The user then manipulates the orbital occupancies using the $occupied section, removing an electron from
the π and placing it in the π ∗ . The MOM is then invoked to preserve this orbital occupancy. The success of the MOM
relies on the quality of the initial guess for the calculation. If the virtual orbitals are of poor quality then the calculation
may ‘fall down’ to a lower energy state of the same symmetry. Often the virtual orbitals of the corresponding cation
are more appropriate for using as initial guess orbitals for the excited state.
Because the MOM states are single determinants, all of Q-C HEM’s existing single determinant properties and derivatives are available. This allows, for example, analytic harmonic frequencies to be computed on excited states. The
orbitals from a Hartree-Fock MOM calculation can also be used in an MP2 calculation. For all excited state calculations, it is important to add diffuse functions to the basis set. This is particularly true if Rydberg transitions are being
sought. For DFT based methods, it is also advisable to increase the size of the quadrature grid so that the more diffuse

Chapter 7: Open-Shell and Excited-State Methods
densities are accurately integrated.
Example 7.16 Calculation of the lowest singlet state of CO.
$comment
CO spin-purified calculation
$end
$molecule
0 1
C
O C 1.05
$end
$rem
METHOD
BASIS
$end

B3LYP
6-31G*

@@@
$molecule
read
$end
$rem
METHOD
BASIS
SCF_GUESS
MOM_START
UNRESTRICTED
OPSING
$end

B3LYP
6-31G*
read
1
true
true

$occupied
1 2 3 4 5 6 7
1 2 3 4 5 6 8
$end

The following $rem is used to invoke the MOM:
MOM_START
Determines when MOM is switched on to preserve orbital occupancies.
TYPE:
INTEGER
DEFAULT:
0 (FALSE)
OPTIONS:
0 (FALSE) MOM is not used
n
MOM begins on cycle n.
RECOMMENDATION:
For calculations on excited states, an initial calculation without MOM is usually required to
get satisfactory starting orbitals. These orbitals should be read in using SCF_GUESS = true and
MOM_START = 1.

322

Chapter 7: Open-Shell and Excited-State Methods

323

Example 7.17 Input for obtaining the 2 A0 excited state of formamide corresponding to the π → π ∗ transition. The 1 A0
ground state is obtained if MOM is not used in the second calculation. Note the use of diffuse functions and a larger
quadrature grid to accurately model the larger excited state.
$molecule
1 2
C
H 1 1.091480
O 1 1.214713
N 1 1.359042
H 4 0.996369
H 4 0.998965
$end
$rem
METHOD
BASIS
XC_GRID
$end

2
2
1
1

123.10
111.98
121.06
119.25

3
2
2

-180.00
-0.00
-180.00

B3LYP
6-311(2+,2+)G(d,p)
000100000194

@@@
$molecule
0 1
C
H 1 1.091480
O 1 1.214713
N 1 1.359042
H 4 0.996369
H 4 0.998965
$end
$rem
METHOD
BASIS
XC_GRID
MOM_START
SCF_GUESS
UNRESTRICTED
$end

2
2
1
1

123.10
111.98
121.06
119.25

3
2
2

-180.00
-0.00
-180.00

B3LYP
6-311(2+,2+)G(d,p)
000100000194
1
read
true

$occupied
1:12
1:11 13
$end

Additionally, it is possible to perform a CIS/TDDFT calculation on top of the MOM excitation. This capability can
be useful when modeling pump-probe spectra. In order to run MOM followed by CIS/TDDFT, the $rem variable

Chapter 7: Open-Shell and Excited-State Methods

324

cis_n_roots must be specified. Truncated subspaces may also be used, see Section 7.3.2.
Example 7.18 MOM valence excitation followed by core-state TDDFT using a restricted subspace
$molecule
0 1
O
0.0000
H
0.0000
H
0.0000
$end
$rem
METHOD
BASIS
SYMMETRY
SYM_IGNORE
$end

0.0000
0.7629
-0.7629

0.1168
-0.4672
-0.4672

B3LYP
aug-cc-pvdz
false
true

@@@
$molecule
read
$end
$rem
METHOD
BASIS
SCF_GUESS
MOM_START
UNRESTRICTED
SYMMETRY
SYM_IGNORE
CIS_N_ROOTS
TRNSS
TRTYPE
CUTVIR
N_SOL
$end

B3LYP
aug-cc-pvdz
read
1
true
false
true
5
true ! use truncated subspace for TDDFT
3
! specify occupied orbitals
15
! truncate high energy virtual orbitals
1
! number core orbitals, specified in $solute section

$solute
1
$end
$occupied
1 2 3 4 5
1 2 3 4 6
$end

If the MOM excitation corresponds to a core hole, a reduced subspace must be used to avoid de-excitations to the core
hole. The $rem variable CORE_IONIZE allows only the hole to be specified so that not all occupied orbitals need to be
entered in the $solute section.

Chapter 7: Open-Shell and Excited-State Methods

325

CORE_IONIZE
Indicates how orbitals are specified for reduced excitation spaces.
TYPE:
INTEGER
DEFAULT:
1
OPTIONS:
1 all valence orbitals are listed in $solute section
2 only hole(s) are specified all other occupations same as ground state
RECOMMENDATION:
For MOM + TDDFT this specifies the input form of the $solute section. If set to 1 all occupied
orbitals must be specified, 2 only the empty orbitals to ignore must be specified.

Example 7.19 O(1s) core excited state using MOM followed by excitations among valence orbitals. Note that a
reduced excitation subspace must be used to avoid “excitations” into the empty core hole
$molecule
0 1
O
0.0000
H
0.0000
H
0.0000
$end
$rem
METHOD
BASIS
SYMMETRY
SYM_IGNORE
$end

0.0000
0.7629
-0.7629

0.1168
-0.4672
-0.4672

B3LYP
aug-cc-pvdz
false
true

@@@
$molecule
read
$end
$rem
METHOD
BASIS
SCF_GUESS
MOM_START
UNRESTRICTED
SYMMETRY
SYMMETRY_IGNORE
CIS_N_ROOTS
TRNSS
TRTYPE
N_SOL
CORE_IONIZE
$end
$solute
6
$end
$occupied
1 2 3 4 5
2 3 4 5 6
$end

B3LYP
aug-cc-pvdz
read
1
true
false
true
5
true ! use truncated subspace for TDDFT
3
! specify occupied orbitals
1
! number core holes, specified in $solute section
2
! hole orbital specified

Chapter 7: Open-Shell and Excited-State Methods

7.5

326

Restricted Open-Shell Kohn-Sham Method for ∆-SCF Calculations of
Excited States

Q-C HEM provides access to certain singlet excited states – namely, those well-described by a single-electron HOMOLUMO transition – via restricted open-shell Kohn-Sham (ROKS) theory. In contrast to the MOM approach (see Section 7.4), which requires separate SCF calculations of the non-aufbau and triplet energies, the ROKS approach attempts
to combine the properties of both determinants at the level of the Fock matrix in one SCF calculation. ROKS thus
presents as a single SCF loop, but the structure of the Fock matrix differs from the ground-state case. Note that this
excited-state method is distinct from ROKS theory for open-shell ground states.
The implementation of ROKS excited states in Q-C HEM largely follows the theoretical framework established by
Filatov and Shaik 35 and is described in detail in Ref. 56. Singlet excited state energies and gradients are available,
enabling single-point, geometry optimization and molecular dynamics.
To perform an ROKS excited state calculation, simply set the keywords ROKS = TRUE and UNRESTRICTED = FALSE.
An additional keyword ROKS_LEVEL_SHIFT is included to assist in cases of convergence difficulties with a standard
level-shift technique. It is recommended to perform a preliminary ground-state calculation on the system first, and then
use the ground-state orbitals to construct the initial guess using SCF_GUESS = READ.
ROKS
Controls whether ROKS calculation will be performed.
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
FALSE ROKS is not performed.
TRUE
ROKS will be performed.
RECOMMENDATION:
Set to TRUE if ROKS calculation is desired. You should also set UNRESTRICTED = FALSE
ROKS_LEVEL_SHIFT
Introduce a level shift of N/100 hartree to aid convergence.
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
0 No shift
N level shift of N/100 hartree.
RECOMMENDATION:
Use in cases of problematic convergence.

327

Chapter 7: Open-Shell and Excited-State Methods

Example 7.20 RO-PBE0/6-311+G* excited state gradient of formaldehyde, using the ground state orbitals as an initial
guess.
$comment
ROKS excited state gradient of formaldehyde
Use orbitals from ground state for initial guess
$end
$rem
EXCHANGE
BASIS
SCF_CONVERGENCE
SYM_IGNORE
$end
$molecule
0 1
H
-0.940372
H
0.940372
C
0.000000
O
0.000000
$end

pbe0
6-311+G*
9
true

0.000000
0.000000
0.000000
0.000000

1.268098
1.268098
0.682557
-0.518752

@@@
$molecule
read
$end
$rem
ROKS
UNRESTRICTED
EXCHANGE
BASIS
JOBTYPE
SCF_CONVERGENCE
SYM_IGNORE
SCF_GUESS
$end

7.6

true
false
pbe0
6-311+G*
force
9
true
read

Correlated Excited State Methods: The CIS(D) Family

CIS(D) is a simple, size-consistent doubles correction to CIS which has a computational cost scaling as the fifth power
of the basis set for each excited state. 43,45 In this sense, CIS(D) can be considered as an excited state analog of the
ground state MP2 method. CIS(D) yields useful improvements in the accuracy of excitation energies relative to CIS,
and yet can be applied to relatively large molecules using Q-C HEM’s efficient integrals transformation package. In
addition, as in the case of MP2 method, the efficiency can be significantly improved through the use of the auxiliary
basis expansions (Section 6.6). 107

7.6.1

CIS(D) Theory

The CIS(D) excited state procedure is a second-order perturbative approximation to the computationally expensive
CCSD, based on a single excitation configuration interaction (CIS) reference. The coupled-cluster wave function,
truncated at single and double excitations, is the exponential of the single and double substitution operators acting on

328

Chapter 7: Open-Shell and Excited-State Methods
the Hartree-Fock determinant:
|Ψi = exp (T1 + T2 ) |Ψ0 i
Determination of the singles and doubles amplitudes requires solving the two equations

 
1 2
1 3
a
hΨi | H − E 1 + T1 + T2 + T1 + T1 T2 + T1 Ψ0 = 0
2
3!
and
Ψab
ij


 
1 2
1 3 1 2 1 2
1 4
H − E 1 + T1 + T2 + T1 + T1 T2 + T1 + T2 + T1 T2 + T1 Ψ0 = 0
2
3!
2
2
4!

which lead to the CCSD excited state equations. These can be written

 
1
hΨai | H − E U1 + U2 + T1 U1 + T1 U2 + U1 T2 + T12 U1 Ψ0 = ωbai
2
and

hΨai | H − E

U1 + U2 + T1 U1 + T1 U2 + U1 T2 + 21 T12 U1 + T2 U2
1 3
+ 21 T12 U2 + T1 T2 U1 + 3!
T1 U1 Ψ0 i = ωbab
ij

(7.30)

(7.31)

(7.32)

(7.33)

(7.34)

This is an eigenvalue equation Ab = ωb for the transition amplitudes (b vectors), which are also contained in the U
operators.
The second-order approximation to the CCSD eigenvalue equation yields a second-order contribution to the excitation
energy which can be written in the form
t

t

ω (2) = b(0) A(1) b(1) + b(0) A(2) b(0)

(7.35)

ω (2) = ω CIS(D) = E CIS(D) − E MP2

(7.36)

E CIS(D) = ΨCIS V U2 ΨHF + ΨCIS V T2 U1 ΨHF

(7.37)

E MP2 = ΨHF V T2 ΨHF

(7.38)

or in the alternative form
where
and

The output of a CIS(D) calculation contains useful information beyond the CIS(D) corrected excitation energies themselves. The stability of the CIS(D) energies is tested by evaluating a diagnostic, termed the “theta diagnostic”. 100 The
theta diagnostic calculates a mixing angle that measures the extent to which electron correlation causes each pair of
calculated CIS states to couple. Clearly the most extreme case would be a mixing angle of 45◦ , which would indicate
breakdown of the validity of the initial CIS states and any subsequent corrections. On the other hand, small mixing
angles on the order of only a degree or so are an indication that the calculated results are reliable. The code can report
the largest mixing angle for each state to all others that have been calculated.

7.6.2

Resolution of the Identity CIS(D) Methods

Because of algorithmic similarity with MP2 calculation, the “resolution of the identity” approximation can also be used
in CIS(D). In fact, RI-CIS(D) is orders of magnitudes more efficient than previously explained CIS(D) algorithms for
effectively all molecules with more than a few atoms. Like in MP2, this is achieved by reducing the prefactor of the
computational load. In fact, the overall cost still scales with the fifth power of the system size.
Presently in Q-C HEM, RI approximation is supported for closed-shell restricted CIS(D) and open-shell unrestricted
UCIS(D) energy calculations. The theta diagnostic is not implemented for RI-CIS(D).

Chapter 7: Open-Shell and Excited-State Methods

7.6.3

329

SOS-CIS(D) Model

As in MP2 case, the accuracy of CIS(D) calculations can be improved by semi-empirically scaling the opposite-spin
components of CIS(D) expression:
E SOS−CIS(D) = cU ΨCIS V U2OS ΨHF + cT ΨCIS V T2OS U1 ΨHF

(7.39)

with the corresponding ground state energy
E SOS−MP2 = cT ΨHF V T2OS ΨHF

(7.40)

More importantly, this SOS-CIS(D) energy can be evaluated with the 4th power of the molecular size by adopting
Laplace transform technique. 107 Accordingly, SOS-CIS(D) can be applied to the calculations of excitation energies for
relatively large molecules.

7.6.4

SOS-CIS(D0 ) Model

CIS(D) and its cousins explained in the above are all based on a second-order non-degenerate perturbative correction
scheme on the CIS energy (“diagonalize-and-then-perturb” scheme). Therefore, they may fail when multiple excited
states come close in terms of their energies. In this case, the system can be handled by applying quasi-degenerate
perturbative correction scheme (“perturb-and-then-diagonalize” scheme). The working expression can be obtained by
slightly modifying CIS(D) expression shown in Section 7.6.1. 46
First, starting from Eq. (7.35), one can be explicitly write the CIS(D) energy as 23,46

−1
t
t
t
(0)
(2)
(1)
(0)
(1)
ω CIS + ω (2) = b(0) ASS b(0) + b(0) ASS b(0) − b(0) ASD DDD − ω CIS
ADS b(0)

(7.41)

To avoid the failures of the perturbation theory near degeneracies, the entire single and double blocks of the response
matrix should be diagonalized. Because such a diagonalization is a non-trivial non-linear problem, an additional

−1
(0)
approximation from the binomial expansion of the DDD − ω CIS
is further applied: 46


−1 
−1 

−1

−2
(0)
(0)
(0)
(0)
DDD − ω CIS
= DDD
1 + ω DDD
+ ω 2 DDD
+ ...

(7.42)

The CIS(D0 ) energy ω is defined as the eigen-solution of the response matrix with the zero-th order expansion of this
equation. Namely,


(0)
(2)
(1)
(0)
(1)
ASS + ASS − ASD (DDD )−1 ADS b = ωb
(7.43)
Similar to SOS-CIS(D), SOS-CIS(D0 ) theory is defined by taking the opposite-spin portions of this equation and then
scaling them with two semi-empirical parameters: 23


(0)
OS(2)
OS(1)
(0)
OS(1)
ASS + cT ASS − cU ASD (DDD )−1 ADS
b = ωb
(7.44)
Using the Laplace transform and the auxiliary basis expansion techniques, this can also be handled with a 4th-order
scaling computational effort. In Q-C HEM, an efficient 4th-order scaling analytical gradient of SOS-CIS(D0 ) is also
available. This can be used to perform excited state geometry optimizations on the electronically excited state surfaces.

7.6.5

CIS(D) Job Control and Examples

The legacy CIS(D) algorithm in Q-C HEM is handled by the CCMAN/CCMAN2 modules of Q-C HEM’s and shares
many of the $rem options. RI-CIS(D), SOS-CIS(D), and SOS-CIS(D0 ) do not depend on the coupled-cluster routines.
Users who will not use this legacy CIS(D) method may skip to Section 7.6.6.

Chapter 7: Open-Shell and Excited-State Methods

330

As with all post-HF calculations, it is important to ensure there are sufficient resources available for the necessary integral calculations and transformations. For CIS(D), these resources are controlled using the $rem variables
CC_MEMORY, MEM_STATIC and MEM_TOTAL (see Section 6.8.7).
To request a CIS(D) calculation the METHOD $rem should be set to CIS(D) and the number of excited states to calculate
should be specified by EE_STATES (or EE_SINGLETS and EE_TRIPLETS when appropriate). Alternatively, CIS(D) will
be performed when EXCHANGE = HF, CORRELATION = CI and EOM_CORR = CIS(D). The SF-CIS(D) is invoked by
using SF_STATES.
EE_STATES
Sets the number of excited state roots to find. For closed-shell reference, defaults into
EE_SINGLETS. For open-shell references, specifies all low-lying states.
TYPE:
INTEGER/INTEGER ARRAY
DEFAULT:
0 Do not look for any excited states.
OPTIONS:
[i, j, k . . .] Find i excited states in the first irrep, j states in the second irrep etc.
RECOMMENDATION:
None
EE_SINGLETS
Sets the number of singlet excited state roots to find. Valid only for closed-shell references.
TYPE:
INTEGER/INTEGER ARRAY
DEFAULT:
0 Do not look for any excited states.
OPTIONS:
[i, j, k . . .] Find i excited states in the first irrep, j states in the second irrep etc.
RECOMMENDATION:
None
EE_TRIPLETS
Sets the number of triplet excited state roots to find. Valid only for closed-shell references.
TYPE:
INTEGER/INTEGER ARRAY
DEFAULT:
0 Do not look for any excited states.
OPTIONS:
[i, j, k . . .] Find i excited states in the first irrep, j states in the second irrep etc.
RECOMMENDATION:
None

Chapter 7: Open-Shell and Excited-State Methods

331

SF_STATES
Sets the number of spin-flip target states roots to find.
TYPE:
INTEGER/INTEGER ARRAY
DEFAULT:
0 Do not look for any spin-flip states.
OPTIONS:
[i, j, k . . .] Find i SF states in the first irrep, j states in the second irrep etc.
RECOMMENDATION:
None

Note: It is a symmetry of a transition rather than that of a target state that is specified in excited state calculations.
The symmetry of the target state is a product of the symmetry of the reference state and the transition. For
closed-shell molecules, the former is fully symmetric and the symmetry of the target state is the same as that
of transition, however, for open-shell references this is not so.
CC_STATE_TO_OPT
Specifies which state to optimize.
TYPE:
INTEGER ARRAY
DEFAULT:
None
OPTIONS:
[i,j] optimize the jth state of the ith irrep.
RECOMMENDATION:
None

332

Chapter 7: Open-Shell and Excited-State Methods

Note: Since there are no analytic gradients for CIS(D), the symmetry should be turned off for geometry optimization
and frequency calculations, and CC_STATE_TO_OPT should be specified assuming C1 symmetry, i.e., as [1,N]
where N is the number of state to optimize (the states are numbered from 1).
Example 7.21 CIS(D) excitation energy calculation for ozone at the experimental ground state geometry C2v
$molecule
0 1
O
O 1 RE
O 2 RE

1

A

RE = 1.272
A = 116.8
$end
$rem
JOBTYPE
METHOD
BASIS
N_FROZEN_CORE
EE_SINGLETS
EE_TRIPLETS
$end

SP
CIS(D)
6-31G*
3
[2,2,2,2]
[2,2,2,2]

use frozen core
find 2 lowest singlets in each irrep.
find two lowest triplets in each irrep.

Example 7.22 CIS(D) geometry optimization for the lowest triplet state of water. The symmetry is automatically
turned off for finite difference calculations
$molecule
0 1
o
h 1 r
h 1 r
r
a
$end

2

a

0.95
104.0

$rem
JOBTYPE
BASIS
METHOD
EE_TRIPLETS
CC_STATE_TO_OPT
$end

opt
3-21g
cis(d)
1 calculate one lowest triplet
[1,1] optimize the lowest state (1st state in 1st irrep)

333

Chapter 7: Open-Shell and Excited-State Methods

Example 7.23 CIS(D) excitation energy and transition property calculation (between all states) for ozone at the
experimental ground state geometry C2v
$molecule
0 1
O
O 1 RE
O 2 RE

1

A

RE = 1.272
A = 116.8
$end
$rem
JOBTYPE
BASIS
PURCAR
METHOD
EE_SINGLETS
EE_TRIPLETS
CC_TRANS_PROP
$end

7.6.6

SP
6-31G*
2
CIS(D)
[2,2,2,2]
[2,2,2,2]
1

Non-spherical (6D)

RI-CIS(D), SOS-CIS(D), and SOS-CIS(D0 ): Job Control

These methods are activated by setting the $rem keyword METHOD to RICIS(D), SOSCIS(D), and SOSCIS(D0), respectively. Other keywords are the same as in CIS method explained in Section 7.2.1. As these methods rely on the RI
approximation, AUX_BASIS needs to be set by following the same guide as in RI-MP2 (Section 6.6).
METHOD
Excited state method of choice
TYPE:
STRING
DEFAULT:
None
OPTIONS:
RICIS(D)
Activate RI-CIS(D)
SOSCIS(D)
Activate SOS-CIS(D)
SOSCIS(D0) Activate SOS-CIS(D0 )
RECOMMENDATION:
None
CIS_N_ROOTS
Sets the number of excited state roots to find
TYPE:
INTEGER
DEFAULT:
0 Do not look for any excited states
OPTIONS:
n n > 0 Looks for n excited states
RECOMMENDATION:
None

Chapter 7: Open-Shell and Excited-State Methods
CIS_SINGLETS
Solve for singlet excited states (ignored for spin unrestricted systems)
TYPE:
LOGICAL
DEFAULT:
TRUE
OPTIONS:
TRUE
Solve for singlet states
FALSE Do not solve for singlet states.
RECOMMENDATION:
None
CIS_TRIPLETS
Solve for triplet excited states (ignored for spin unrestricted systems)
TYPE:
LOGICAL
DEFAULT:
TRUE
OPTIONS:
TRUE
Solve for triplet states
FALSE Do not solve for triplet states.
RECOMMENDATION:
None
SET_STATE_DERIV
Sets the excited state index for analytical gradient calculation for geometry optimizations and
vibrational analysis with SOS-CIS(D0 )
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
n Select the nth state.
RECOMMENDATION:
Check to see that the states do no change order during an optimization. For closed-shell systems,
either CIS_SINGLETS or CIS_TRIPLETS must be set to false.
MEM_STATIC
Sets the memory for individual program modules
TYPE:
INTEGER
DEFAULT:
64 corresponding to 64 Mb
OPTIONS:
n User-defined number of megabytes.
RECOMMENDATION:
At least 150(N 2 + N )D of MEM_STATIC is required (N : number of basis functions, D: size of
a double precision storage, usually 8). Because a number of matrices with N 2 size also need to
be stored, 32–160 Mb of additional MEM_STATIC is needed.

334

Chapter 7: Open-Shell and Excited-State Methods
MEM_TOTAL
Sets the total memory available to Q-C HEM
TYPE:
INTEGER
DEFAULT:
2000 2 Gb
OPTIONS:
n User-defined number of megabytes
RECOMMENDATION:
The minimum memory requirement of RI-CIS(D) is approximately MEM_STATIC +
max(3SV XD, 3X 2 D) (S: number of excited states, X: number of auxiliary basis functions,
D: size of a double precision storage, usually 8). However, because RI-CIS(D) uses a batching
scheme for efficient evaluations of electron repulsion integrals, specifying more memory will
significantly speed up the calculation. Put as much memory as possible if you are not sure what
to use, but never put any more than what is available. The minimum memory requirement of
SOS-CIS(D) and SOS-CIS(D0 ) is approximately MEM_STATIC + 20X 2 D. SOS-CIS(D0 ) gradient calculation becomes more efficient when 30X 2 D more memory space is given. Like in
RI-CIS(D), put as much memory as possible if you are not sure what to use. The actual memory
size used in these calculations will be printed out in the output file to give a guide about the
required memory.
AO2MO_DISK
Sets the scratch space size for individual program modules
TYPE:
INTEGER
DEFAULT:
2000 2 Gb
OPTIONS:
n User-defined number of megabytes.
RECOMMENDATION:
The minimum disk requirement of RI-CIS(D) is approximately 3SOV XD. Again, the batching
scheme will become more efficient with more available disk space. There is no simple formula
for SOS-CIS(D) and SOS-CIS(D0 ) disk requirement. However, because the disk space is abundant in modern computers, this should not pose any problem. Just put the available disk space
size in this case. The actual disk usage information will also be printed in the output file.
SOS_FACTOR
Sets the scaling parameter cT
TYPE:
INTEGER
DEFAULT:
1300000 corresponding to 1.30
OPTIONS:
n cT = n/1000000
RECOMMENDATION:
Use the default

335

Chapter 7: Open-Shell and Excited-State Methods
SOS_UFACTOR
Sets the scaling parameter cU
TYPE:
INTEGER
DEFAULT:
151 For SOS-CIS(D), corresponding to 1.51
140 For SOS-CIS(D0 ), corresponding to 1.40
OPTIONS:
n cU = n/100
RECOMMENDATION:
Use the default

336

337

Chapter 7: Open-Shell and Excited-State Methods

7.6.7

Examples

Example 7.24 Input for an RI-CIS(D) calculation.
$molecule
0 1
C
0.667472
C
-0.667472
H
1.237553
H
1.237553
H
-1.237553
H
-1.237553
$end
$rem
METHOD
BASIS
MEM_TOTAL
MEM_STATIC
AO2MO_DISK
AUX_BASIS
PURECART
CIS_N_ROOTS
CIS_SINGLETS
CIS_TRIPLETS
$end

0.000000
0.000000
0.922911
-0.922911
0.922911
-0.922911

0.000000
0.000000
0.000000
0.000000
0.000000
0.000000

ricis(d)
aug-cc-pVDZ
1000
100
1000
rimp2-aug-cc-pVDZ
1111
10
true
false

Example 7.25 Input for an SOS-CIS(D) calculation.
$molecule
0 1
C
-0.627782
O
0.730618
H
-1.133677
H
-1.133677
$end
$rem
METHOD
BASIS
MEM_TOTAL
MEM_STATIC
AO2MO_DISK
AUX_BASIS
PURECART
CIS_N_ROOTS
CIS_SINGLETS
CIS_TRIPLETS
$end

0.141553
-0.073475
-0.033018
-0.033018

0.000000
0.000000
-0.942848
0.942848

soscis(d)
aug-cc-pVDZ
1000
100
500000
! 0.5 Terabyte of disk space available
rimp2-aug-cc-pVDZ
1111
5
true
true

Chapter 7: Open-Shell and Excited-State Methods

338

Example 7.26 Input for an SOS-CIS(D0 ) geometry optimization on S2 surface.
$molecule
0 1
o
h 1 r
h 1 r
r
a
$end

2

a

0.95
104.0

$rem
JOBTYPE
METHOD
BASIS
AUX_BASIS
PURECART
SET_STATE_DERIV
CIS_N_ROOTS
CIS_SINGLETS
CIS_TRIPLETS
$end

7.7

=
=
=
=
=
=
=
=
=

opt
soscis(d0)
6-31G**
rimp2-VDZ
1112
2
5
true
false

Coupled-Cluster Excited-State and Open-Shell Methods

EOM-CC and most of the CI codes are part of CCMAN and CCMAN2. CCMAN is a legacy code which is being
phased out. All new developments and performance-enhancing features are implemented in CCMAN2. Some options
behave differently in the two modules. Below we make an effort to mark which features are available in legacy code
only.

7.7.1

Excited States via EOM-EE-CCSD

One can describe electronically excited states at a level of theory similar to that associated with coupled-cluster theory
for the ground state by applying either linear response theory 54 or equation-of-motion methods. 117 A number of groups
have demonstrated that excitation energies based on a coupled-cluster singles and doubles ground state are generally
very accurate for states that are primarily single electron promotions. The error observed in calculated excitation
energies to such states is typically 0.1–0.2 eV, with 0.3 eV as a conservative estimate, including both valence and
Rydberg excited states. This, of course, assumes that a basis set large and flexible enough to describe the valence and
Rydberg states is employed. The accuracy of excited state coupled-cluster methods is much lower for excited states that
involve a substantial double excitation character, where errors may be 1 eV or even more. Such errors arise because
the description of electron correlation of an excited state with substantial double excitation character requires higher
truncation of the excitation operator. The description of these states can be improved by including triple excitations, as
in EOM(2,3).
Q-C HEM includes coupled-cluster methods for excited states based on the coupled cluster singles and doubles (CCSD)
method described earlier. CCMAN also includes the optimized orbital coupled-cluster doubles (OD) variant. OD
excitation energies have been shown to be essentially identical in numerical performance to CCSD excited states. 62
These methods, while far more computationally expensive than TDDFT, are nevertheless useful as proven high accuracy
methods for the study of excited states of small molecules. Moreover, they are capable of describing both valence and
Rydberg excited states, as well as states of a charge-transfer character. Also, when studying a series of related molecules
it can be very useful to compare the performance of TDDFT and coupled-cluster theory for at least a small example

339

Chapter 7: Open-Shell and Excited-State Methods

to understand its performance. Along similar lines, the CIS(D) method described earlier as an economical correlation
energy correction to CIS excitation energies is in fact an approximation to EOM-CCSD. It is useful to assess the
performance of CIS(D) for a class of problems by benchmarking against the full coupled-cluster treatment. Finally,
Q-C HEM includes extensions of EOM methods to treat ionized or electron attachment systems, as well as di- and
triradicals.
EOM-EE

Ψ(MS = 0) = R(MS = 0)Ψ0 (MS = 0)





|



{z



}
Φai

EOM-IP

Φab
ij

Ψ(N ) = R(−1)Ψ0 (N + 1)







|



{z






}

|

{z

}

Ψ(N ) = R(+1)Ψ0 (N − 1)






|





{z





}

|




{z

Φa
EOM-SF





Φaij

Φi
EOM-EA












}
Φab
i

Ψ(MS = 0) = R(MS = −1)Ψ0 (MS = 1)






{z

|






}

Φai
Figure 7.1: In the EOM formalism, target states Ψ are described as excitations from a reference state Ψ0 : Ψ = RΨ0 ,
where R is a general excitation operator. Different EOM models are defined by choosing the reference and the form of
the operator R. In the EOM models for electronically excited states (EOM-EE, upper panel), the reference is the closedshell ground state Hartree-Fock determinant, and the operator R conserves the number of α and β electrons. Note that
two-configurational open-shell singlets can be correctly described by EOM-EE since both leading determinants appear
as single electron excitations. The second and third panels present the EOM-IP/EA models. The reference states for
EOM-IP/EA are determinants for N + 1/N − 1 electron states, and the excitation operator R is ionizing or electronattaching, respectively. Note that both the EOM-IP and EOM-EA sets of determinants are spin-complete and balanced
with respect to the target multi-configurational ground and excited states of doublet radicals. Finally, the EOM-SF
method (the lowest panel) employs the high-spin triplet state as a reference, and the operator R includes spin-flip, i.e.,
does not conserve the number of α and β electrons. All the determinants present in the target low-spin states appear
as single excitations, which ensures their balanced treatment both in the limit of large and small HOMO/LUMO gaps.
Other EOM methods available in Q-C HEM are EOM-2SF and EOM-DIP.

340

Chapter 7: Open-Shell and Excited-State Methods

7.7.2

EOM-XX-CCSD and CI Suite of Methods

Q-C HEM features the most complete set of EOM-CCSD models, 61 enabling accurate, robust, and efficient calculations
of electronically excited states (EOM-EE-CCSD or EOM-EE-OD); 55,62,71,113,117 ; ground and excited states of diradicals
and triradicals (EOM-SF-CCSD and EOM-SF-OD); 58,71 ionization potentials and electron attachment energies, as well
as problematic doublet radicals and cation or anion radicals (EOM-IP/EA-CCSD). 96,116,118 The EOM-DIP-CCSD and
EOM-2SF-CCSD methods are available as well. Conceptually, EOM is very similar to configuration interaction (CI):
target EOM states are found by diagonalizing the similarity transformed Hamiltonian H̄ = e−T HeT ,
H̄R = ER,

(7.45)

where T and R are general excitation operators with respect to the reference determinant |Φ0 i. In the EOM-CCSD
models, T and R are truncated at single and double excitations, and the amplitudes T satisfy the CC equations for the
reference state |Φ0 i:
hΦai |H̄|Φ0 i =
hΦab
ij |H̄|Φ0 i

=

0

(7.46)

0

(7.47)

The computational scaling of EOM-CCSD and CISD methods is identical, i.e., O(N 6 ), however EOM-CCSD is numerically superior to CISD because correlation effects are “folded in” in the transformed Hamiltonian, and because
EOM-CCSD is rigorously size-intensive.
By combining different types of excitation operators and references |Φ0 i, different groups of target states can be
accessed as explained in Fig. 7.1. For example, electronically excited states can be described when the reference |Φ0 i
corresponds to the ground state wave function, and operators R conserve the number of electrons and a total spin. 117 In
the ionized/electron attached EOM models, 96,118 operators R are not electron conserving (i.e., include different number
of creation and annihilation operators)—these models can accurately treat ground and excited states of doublet radicals
and some other open-shell systems. For example, singly ionized EOM methods, i.e., EOM-IP-CCSD and EOM-EACCSD, have proven very useful for doublet radicals whose theoretical treatment is often plagued by symmetry breaking.
Finally, the EOM-SF method 58,71 in which the excitation operators include spin-flip allows one to access diradicals,
triradicals, and bond-breaking. 63
Q-C HEM features EOM-EE/SF/IP/EA/DIP/DSF-CCSD methods for both closed and open-shell references (RHF/UHF/
ROHF), including frozen core/virtual options. For EE, SF, IP, and EA, a more economical flavor of EOM-CCSD is
available (EOM-MP2 family of methods). All EOM models take full advantage of molecular point group symmetry.
Analytic gradients are available for RHF and UHF references, for the full orbital space, and with frozen core/virtual
orbitals. 72 Properties calculations (permanent and transition dipole moments, hS 2 i, hR2 i, etc.) are also available.
The current implementation of the EOM-XX-CCSD methods enables calculations of medium-size molecules, e.g.,
up to 15–20 heavy atoms. Using RI approximation 6.8.5 or Cholesky decomposition 6.8.6 helps to reduce integral
transformation time and disk usage enabling calculations on much larger systems. EOM-MP2 and EOM-MP2t variants
are also less computationally demanding. The computational cost of EOM-IP calculations can be considerably reduced
(with negligible decline in accuracy) by truncating virtual orbital space using FNO scheme (see Section 7.7.8).
Legacy features available in CCMAN. The CCMAN module of Q-C HEM includes two implementations of EOM-IPCCSD. The proper implementation 103 is used by default is more efficient and robust. The EOM_FAKE_IPEA keyword
invokes is a pilot implementation in which EOM-IP-CCSD calculation is set up by adding a very diffuse orbital to a
requested basis set, and by solving EOM-EE-CCSD equations for the target states that include excitations of an electron
to this diffuse orbital. The implementation of EOM-EA-CCSD in CCMAN also uses this trick. Fake IP/EA calculations
are only recommended for Dyson orbital calculations and debug purposes. (CCMAN2 features proper implementations
of EOM-IP and EOM-EA (including Dyson orbitals)).
A more economical CI variant of EOM-IP-CCSD, IP-CISD is also available in CCMAN. This is an N5 approximation
of IP-CCSD, and can be used for geometry optimizations of problematic doublet states. 39

341

Chapter 7: Open-Shell and Excited-State Methods

7.7.3

Spin-Flip Methods for Di- and Triradicals

The spin-flip method 58–60 addresses the bond-breaking problem associated with a single-determinant description of
the wave function. Both closed and open shell singlet states are described within a single reference as spin-flipping,
(e.g., α → β excitations from the triplet reference state, for which both dynamical and non-dynamical correlation
effects are smaller than for the corresponding singlet state. This is because the exchange hole, which arises from the
Pauli exclusion between same-spin electrons, partially compensates for the poor description of the coulomb hole by the
mean-field Hartree-Fock model. Furthermore, because two α electrons cannot form a bond, no bond breaking occurs
as the internuclear distance is stretched, and the triplet wave function remains essentially single-reference in character.
The spin-flip approach has also proved useful in the description of di- and tri-radicals as well as some problematic
doublet states.
The spin-flip method is available for the CIS, CIS(D), CISD, CISDT, OD, CCSD, and EOM-(2,3) levels of theory and
the spin complete SF-XCIS (see Section 7.2.4). An N7 non-iterative triples corrections are also available. For the OD
and CCSD models, the following non-relaxed properties are also available: dipoles, transition dipoles, eigenvalues of
the spin-squared operator (hS 2 i), and densities. Analytic gradients are also for SF-CIS and EOM-SF-CCSD methods.
To invoke a spin-flip calculation the SF_STATES $rem should be used, along with the associated $rem settings for
the chosen level of correlation by using METHOD (recommended) or using older keywords (CORRELATION, and,
optionally, EOM_CORR). Note that the high multiplicity triplet or quartet reference states should be used.
Several double SF methods have also been implemented. 24 To invoke these methods, use DSF_STATES.

7.7.4

EOM-DIP-CCSD

Double-ionization potential (DIP) is another non-electron-conserving variant of EOM-CCSD. 64,65,136 In DIP, target
states are reached by detaching two electrons from the reference state:
Ψk = RN −2 Ψ0 (N + 2),

(7.48)

and the excitation operator R has the following form:
R
R1

= R1 + R2 ,
X
= 1/2
rij ji,

(7.49)
(7.50)

ij

R2

=

1/6

X

a
rijk
a† kji.

(7.51)

ijka

As a reference state in the EOM-DIP calculations one usually takes a well-behaved closed-shell state. EOM-DIP is
a useful tool for describing molecules with electronic degeneracies of the type “2n − 2 electrons on n degenerate
orbitals”. The simplest examples of such systems are diradicals with two-electrons-on-two-orbitals pattern. Moreover,
DIP is a preferred method for four-electrons-on-three-orbitals wave functions.
Accuracy of the EOM-DIP-CCSD method is similar to accuracy of other EOM-CCSD models, i.e., 0.1–0.3 eV. The
scaling of EOM-DIP-CCSD is O(N 6 ), analogous to that of other EOM-CCSD methods. However, its computational
cost is less compared to, e.g., EOM-EE-CCSD, and it increases more slowly with the basis set size. An EOM-DIP
calculation is invoked by using DIP_STATES, or DIP_SINGLETS and DIP_TRIPLETS.

7.7.5

EOM-CC Calculations of Core-Level States: Core-Valence Separation within EOMCCSD

The core-valence separation (CVS) approximation 27 allows one to extend standard methods for excited and ionized
states to the core-level states. In this approach, the excitations involving core electrons are decoupled from the rest
of the configurational space. This allows one to reduce computational costs and decouple the highly excited core

Chapter 7: Open-Shell and Excited-State Methods

342

states from the continuum. Currently, CVS is implemented within EOM-EE/IP-CCSD for energies and transition
properties (oscillator strengths, NTOs, Dyson orbitals, exciton descriptors). CVS-EOM-EE-CCSD can be used to
model NEXAFS.
In Q-C HEM, a slightly different version of CVS-EOM-EE-CCSD than the original theory by Coriani and Koch 29 is
implemented: the reference coupled-cluster amplitudes do not include core electrons 128 . To distinguish this method
from the original 29 , below we refer to the Q-C HEM implementation as frozen-core-ground-state/core-valence-separated
EOM (FC-CVS-EOM) approach. 128
In the FC-CVS-EOM approach the ground-state parameters (amplitudes and Lagrangian multipliers) are computed
within the frozen-core approximation, whereas the core-excitation energies and strengths are obtained imposing that at
least one index in the EOM excitation (and ionization) operators refer to a core occupied orbital.
To ensure the best convergence of EOM equations, the calculation is edge-specific with respect to the highest lying
edges (or deepest lying core orbitals): the frozen-core and CVS spaces are selected for each edge such that the core
orbitals we are addressing in the excited state calculations are explicitly frozen in the ground state calculation and
specifically included in the EOM calculation. Examples 27 and 28 below illustrate this point.
Although the convergence of FC-CVS-EOM is much more robust that that of regular EOM-CCSD, sometimes calculations would collapse to low-lying artificial states. If this happens, rerun the calculation using EOM_SHIFT to specify
an approximate onset of the edge.
Note: Using EOM_SHIFT will only work correctly when only CVS states are requested.
To invoke the CVS approximation, use METHOD = CCSD and CVS_EE_STATES instead of EE_STATES to specify the
desired target states (likewise, CVS_EE_SINGLETS and CVS_EE_TRIPLETS can be used in exactly the same way as
in regular EOM calculations). For ionized states, use CVS_IP_STATES or CVS_IP_ALPHA/CVS_IP_BETA. Transition
properties and Dyson orbitals can be computed either within CVS manifold or between CVS and valence manifolds
(see Section 7.7.23 for definition of Dyson orbitals). CVS-EOM-CCSD is only available with CCMAN2.
Note: Core electrons must be frozen in CVS-EOM calculations. The exact definition of the core depends on the edge,
so using default values may be not appropriate.
7.7.5.1

Examples

In example 27, the 1s orbital on the oxygen atom is frozen in the CCSD calculation (N_FROZEN_CORE = FC). In
the EOM calculation, the CVS approximation is invoked (CVS_EE_SINGLETS), so that the core-excitation energies are
obtained as the lowest excitations. The calculation of the oscillator strengths is activated by selecting CC_TRANS_PROP
= 1 and the LIBWFA analysis is invoked by STATE_ANALYSIS = TRUE (see Section 11.2.6).
Example 28 illustrates CVS-EOM-EE-CCSD calculations in a two-edge molecule (CO). In the present implementation,
the calculation should be done separately for each edge. The first job computes carbon-edge states. Since the carbon
1s orbital is the highest in energy (among the core 1s orbitals of the molecule), the input for the C-edge is similar
to example 27. Both the oxygen’s and the carbon’s 1s orbitals are frozen in the reference CCSD calculation. In the
EOM part, the carbon core-excited states are automatically selected. In this case, using default frozen core settings
(N_FROZEN_CORE = FC) is equivalent to specifying N_FROZEN_CORE = 2. In the second input, the oxygen edge is
computed. As the core-orbitals of oxygen lie deeper, the frozen core and CVS selection specifically targets the oxygen
edge by using a smaller core. The 1s orbital of the oxygen atom is selected by N_FROZEN_CORE = 1. If the molecule
has other edges, the deepest lying core orbitals, up to and including those of the edge of interest, should be selected by
an appropriate value of N_FROZEN_CORE.
Examples 29 and 30 illustrate calculations of Dyson orbitals between core-excited and core-ionized states and between

Chapter 7: Open-Shell and Excited-State Methods

343

core-excited and valence-ionized states.
Example 7.27 FC-CVS-EOM-CCSD calculation of the first six dipole allowed core excitation energies and their
intensities at the oxygen edge of water. Wave-function analysis is also performed.
$molecule
0 1
O 0.0000 0.0000 0.1173
H 0.0000 0.7572 -0.4692
H 0.0000 -0.7572 -0.4692
$end
$rem
method
= eom-ccsd
cvs_ee_singlets = [3,0,2,1]
basis
= aug-cc-pVDZ
n_frozen_core
= fc
CC_TRANS_PROP
= true
eom_preconv_singles = true
state_analysis = true !invoke libwa to compute NTOs and exciton descriptors
! libwa controls below
molden_format = true
nto_pairs
= 3
pop_mulliken = true
$end

Example 7.28 FC-CVS-EOM-EE-CCSD calculations of the first two dipole allowed core excitation energies per
irreducible representation and their intensities at the carbon and oxygen edges of carbon monoxide.
$comment
CO, carbon edge
$end
$molecule
0 1
O 0.0000
C 0.0000
$end

0.0000
0.0000

0.913973
-1.218243

$rem
input_bohr = true
method
= eom-ccsd
cvs_ee_singlets = [2,0,2,2]
basis
= aug-cc-pVDZ
n_frozen_core
= fc
eom_preconv_singles = true
CC_TRANS_PROP
= true
$end
@@@
$comment
CO, oxygen edge
$end
$molecule
read
$end
$rem
method
= eom-ccsd
cvs_ee_singlets = [2,0,2,2]
basis
= aug-cc-pVDZ
n_frozen_core
= 1
eom_preconv_singles = true
CC_TRANS_PROP = true
$end

Example 7.29 Calculation of Dyson orbitals between FC-CVS-EOM-EE-CCSD and FC-CVS-EOM-IP-CCSD mani-

Chapter 7: Open-Shell and Excited-State Methods

7.7.6

344

EOM-CC Calculations of Metastable States: Super-Excited Electronic States, Temporary Anions, and More

While conventional coupled-cluster and equation-of-motion methods allow one to tackle electronic structure ranging
from well-behaved closed shell molecules to various open-shell and electronically excited species, 61 meta-stable electronic states, so-called resonances, present a difficult case for theory. By using complex scaling and complex absorbing
potential techniques, we extended these powerful methods to describe auto-ionizing states, such as transient anions,
highly excited electronic states, and core-ionized species. 15,52,53 In addition, users can employ stabilization techniques
using charged sphere and scaled atomic charges options. 65 These methods are only available within CCMAN2. The
complex CC/EOM code is engaged by COMPLEX_CCMAN; the specific parameters should be specified in the $complex_ccman section.
COMPLEX_CCMAN
Requests complex-scaled or CAP-augmented CC/EOM calculations.
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
TRUE Engage complex CC/EOM code.
RECOMMENDATION:
Not available in CCMAN. Need to specify CAP strength or complex-scaling parameter in $complex_ccman section.
The $complex_ccman section is used to specify the details of the complex-scaled/CAP calculations, as illustrated below.
If user specifies CS_THETA, complex scaling calculation is performed.
$complex_ccman
CS_THETA 10
Complex-scaling parameter theta=0.01, r->r exp(-i*theta)
CS_ALPHA 10
Real part of the scaling parameter alpha=0.01,
!
r->alpha r exp(-itheta)
$end
Alternatively, for CAP calculations, the CAP parameters need to be specified.
$complex_ccman
CAP_ETA 1000 CAP strength in 10-5 a.u. (0.01)
CAP_X 2760 CAP onset along X in 10^-3 bohr (2.76 bohr)
CAP_Y 2760 CAP onset along Y in 10^-3 bohr (2.76 bohr)
CAP_Z 4880 CAP onset along Z in 10^-3 bohr (4.88 bohr)
CAP_TYPE 1 Use cuboid cap (CAP_TYPE=0 will use spherical CAP)
$end
CS_THETA is specified in radian× 10−3 . CS_ALPHA, CAP_X/Y/Z are specified in a.u.× 10−3 , i.e., CS_THETA = 10
means θ=0.01; CAP_ETA is specified in a.u.× 10−5 . The CAP is calculated by numerical integration, the default grid

is 000099000590. For testing the accuracy of numerical integration, the numerical overlap matrix is calculated and
compared to the analytical one. If the performance of the default grid is poor, the grid type can be changed using the
keyword XC_GRID (see Section 5.5 for further details). When CAP calculations are performed, CC_EOM_PROP = 1 by
default; this is necessary for calculating first-order perturbative correction.
Advanced users may find the following options useful. Several ways of conducing complex calculations are possible, i.e., complex scaling/CAPs can be either engaged at all levels (HF, CCSD, EOM), or not. By default, if
COMPLEX_CCMAN is specified, the EOM calculations are conducted using complex code. Other parameters are set up
as follows:

Chapter 7: Open-Shell and Excited-State Methods

345

$complex_ccman
CS_HF=true
CS_CCSD=true
$end
Alternatively, the user can disable complex HF. These options are experimental and should only be used by advanced
users. For CAP-EOM-CC, only CS_HF = TRUE and CS_CCSD = TRUE is implemented.
Non-iterative triples corrections are available for all complex scaled and CAP-augmented CC/EOM-CC models and
requested in analogy to regular CC/EOM-CC (see Section 7.7.21 for details).
Molecular properties and transition moments are requested for complex scaled or CAP-augmented CC/EOM-CC calculations in analogy to regular CC/EOM-CC (see Section 7.7.16 for details). Natural orbitals and natural transition
orbitals can be computed and the exciton wave-functions can be analyzed, similarly to real-valued EOM-CCSD (same
keywords are used to invoke the analysis). Analytic gradients are available for complex CC/EOM-CC only for cuboid
CAPs (CAP_TYPE = 1) introduced at the HF level (CS_HF = TRUE), as described in Ref. 6. The frozen core approximation is disabled for CAP-CC/EOM-CC gradient calculations. Geometry optimization can be requested in analogy to
regular CC/EOM-CC (see Section 7.7.16 for details).

7.7.7

Charge Stabilization for EOM-DIP and Other Methods

The performance of EOM-DIP deteriorates when the reference state is unstable with respect to electron-detachment, 64,65
which is usually the case for dianion references employed to describe neutral diradicals by EOM-DIP. Similar problems are encountered by all excited-state methods when dealing with excited states lying above ionization or electrondetachment thresholds.
To remedy this problem, one can employ charge stabilization methods, as described in Refs. 64,65. In this approach
(which can also be used with any other electronic structure method implemented in Q-C HEM), an additional Coulomb
potential is introduced to stabilize unstable wave functions. The following keywords invoke stabilization potentials:
SCALE_NUCLEAR_CHARGE and ADD_CHARGED_CAGE. In the former case, the potential is generated by increasing
nuclear charges by a specified amount. In the latter, the potential is generated by a cage built out of point charges
comprising the molecule. There are two cages available: dodecahedral and spherical. The shape, radius, number of
points, and the total charge of the cage are set by the user.
Note: (1) A perturbative correction estimating the effect of the external Coulomb potential on EOM energy will be
computed when target state densities are calculated, e.g., when CC_EOM_PROP = TRUE.
(2) Charge stabilization techniques can be used with other methods such as EOM-EE, CIS, and TDDFT to
improve the description of resonances. It can also be employed to describe meta-stable ground states.

7.7.8

Frozen Natural Orbitals in CC and IP-CC Calculations

Large computational savings are possible if the virtual space is truncated using the frozen natural orbital (FNO) approach (see Section 6.11). Extension of the FNO approach to ionized states within EOM-CC formalism was recently
introduced and benchmarked. 66 In addition to ground-state coupled-cluster calculations, FNOs can also be used in
EOM-IP-CCSD, EOM-IP-CCSD(dT/fT) and EOM-IP-CC(2,3). In IP-CC the FNOs are computed for the reference
(neutral) state and then are used to describe several target (ionized) states of interest. Different truncation scheme are
described in Section 6.11.

7.7.9

Approximate EOM-CC Methods: EOM-MP2 and EOM-MP2T

Approximate EOM-CCSD models with T -amplitudes obtained at the MP2 level offer reduced computational cost compared to the full EOM-CCSD since the computationally demanding O(N 6 ) CCSD step is eliminated from the calculation. Two methods of this type are implemented in Q-C HEM. The first is invoked with the keyword METHOD = EOM-MP2.

Chapter 7: Open-Shell and Excited-State Methods

346

Its formulation and implementation follow the original EOM-CCSD(2) approach developed by Stanton and coworkers. 119 The second method can be requested with the METHOD = EOM-MP2T keyword and is similar to EOM-MP2, but
it accounts for the additional terms in H̄ that appear because the MP2 T −amplitudes do not satisfy the CCSD equations.
EOM-MP2 ansatz is implemented for IP/EA/EE/SF energies, state properties, and interstate properties (EOM-EOM,
but not REF-EOM). EOM-MP2t is available for the IP/EE/EA energy calculations only.

7.7.10

Approximate EOM-CC Methods: EOM-CCSD-S(D) and EOM-MP2-S(D)

These are very light-weight EOM methods in which the EOM problem is solved in the singles block and the effect
of doubles is evaluated perturbatively. The H̄ is evaluated by using either CCSD or MP2 amplitudes, just as in the
regular EOM calculations. The EOM-MP2-S(D) method, which is similar in level of correlation treatment to SOSCIS(D), is particularly fast. These methods are implemented for IP and EE states. For valence states, the errors for
absolute ionization or excitation energies against regular EOM-CCSD are about 0.4 eV and appear to be systematically
blue-shifted; the EOM-EOM energy gaps look better. The calculations are set as in regular EOM-EE/IP, but using
method = EOM-CCSD-SD(D) or method = EOM-MP2-SD(D). State properties and EOM-EOM transition properties can be
computed using these methods (reference-EOM properties are not yet implemented). These methods are designed for
treating core-level states. 110
Note: These methods are still in the experimental stage.

7.7.11

Implicit solvent models in EOM-CC/MP2 calculations.

Vertical excitation/ionization/attachment energies can be computed for all EOM-CC/MP2 methods using a non-equilibrium
C-PCM model. To perform a PCM-EOM calculation, one has to invoke the PCM (SOLVENT_METHOD to PCM in the
$rem block) and specify the solvent parameters, i.e. the dielectric constant  and the squared refractive index n2
(DIELECTRIC and DIELECTRIC_INFI in the $solvent block). If nothing is given, the parameters for water will be used
by default. For EOM methods, only the simplest model, C-PCM, is implemented. More sophisticated flavors of PCM
are available for ADC methods (see Section 7.8.7). For a detailed description of PCM theory, see Sections 7.8.7, 12.2.2
and 12.2.3.
Note: Only energies and unrelaxed properties can be computed (no gradient).
Note: Symmetry is turned off for C-CPM calculations.

7.7.12

EOM-CC Jobs: Controlling Guess Formation and Iterative Diagonalizers

An EOM-CC eigen-problem is solved by an iterative diagonalization procedure that avoids full diagonalization and
only looks for several eigen-states, as specified by the XX_STATES keywords.
The default procedure is based on the modified Davidson diagonalization algorithm, as explained in Ref. 71. In addition
to several keywords that control the convergence of algorithm, memory usage, and fine details of its execution, there
are several important keywords that allow user to specify how the target state selection will be performed.
By default, the diagonalization looks for several lowest eigenstates, as specified by XX_STATES. The guess vectors
are generated as singly excited determinants selected by using Koopmans’ theorem; the number of guess vectors is
equal to the number of target states. If necessary, the user can increase the number of singly excited guess vectors
(EOM_NGUESS_SINGLES) and include doubly excited guess vectors (EOM_NGUESS_DOUBLES).
Note: In CCMAN2, if there is not enough singly excited guess vectors, the algorithm adds doubly excited guess
vectors. In CCMAN, doubly excited guess vectors are generated only if EOM_NGUESS_DOUBLES is invoked.
The user can request to pre-converge singles (solve the equations in singles-only block of the Hamiltonian. This is done
by using EOM_PRECONV_SINGLES.

Chapter 7: Open-Shell and Excited-State Methods

347

Note: In CCMAN, the user can pre-converge both singles and doubles blocks (EOM_PRECONV_SINGLES and.
EOM_PRECONV_DOUBLES)
If a state (or several states) of a particular character is desired (e.g., HOMO → LUMO + 10 excitation or HOMO − 10
ionization), the user can specify this by using EOM_USER_GUESS keyword and $eom_user_guess section, as illustrated
by an example below. The algorithm will attempt to find an eigenstate that has the maximum overlap with this guess
vector. The multiplicity of the state is determined as in the regular calculations, by using the XX_SINGLETS and
EE_TRIPLETS keywords. This option is useful for looking for high-lying states such as core-ionized or core-excited
states. It is only available with CCMAN2.
The examples below illustrate how to use user-specified guess in EOM calculations:
$eom_user_guess
4 Corresponds to 4(OCC)->5(VIRT) transition.
5
$end
or
$eom_user_guess
1 5
Ex. states corresponding to 1(OCC)->5(VIRT) and 1(OCC)->6(VIRT)
1 6
$end
In IP/EA calculations, only one set of orbitals is specified:
$eom_user_guess
4 5 6
$end
If IP_STATES is specified, this will invoke calculation of the EOM-IP states corresponding to the ionization from 4th,
5th, and 6th occupied MOs. If EA_STATES is requested, then EOM-EA equations will be solved for a root corresponding to electron-attachment to the 4th, 5th, and 6th virtual MOs.
For these options to work correctly, user should make sure that XX_STATES requests a sufficient number of states. In
case of symmetry, one can request several states in each irrep, but the algorithm will only compute those states which
are consistent with the user guess orbitals.
Alternatively, the user can specify an energy shift by EOM_SHIFT. In this case, the solver looks for the XX_STATES
eigenstates that are closest to this energy; the guess vectors are generated accordingly, using Koopmans’ theorem. This
option is useful when highly excited states (i.e., interior eigenstates) are desired.

7.7.13

Equation-of-Motion Coupled-Cluster Job Control

It is important to ensure there are sufficient resources available for the necessary integral calculations and transformations. For CCMAN/CCMAN2 algorithms, these resources are controlled using the $rem variables CC_MEMORY,
MEM_STATIC and MEM_TOTAL (see Section 6.14).
The exact flavor of correlation treatment within equation-of-motion methods is defined by METHOD (see Section 7.1).
For EOM-CCSD, once should set METHOD to EOM-CCSD, for EOM-MP2, METHOD = EOM-CCSD, etc.. In addition, a
specification of the number of target states is required through XX_STATES (XX designates the type of the target states,
e.g., EE, SF, IP, EA, DIP, DSF, etc.). Users must be aware of the point group symmetry of the system being studied and
also the symmetry of the initial and target states of interest, as well as symmetry of transition. It is possible to turn off
the use of symmetry by CC_SYMMETRY. If set to FALSE the molecule will be treated as having C1 symmetry and all
states will be of A symmetry.

348

Chapter 7: Open-Shell and Excited-State Methods

Note: (1) In finite-difference calculations, the symmetry is turned off automatically, and the user must ensure that
XX_STATES is adjusted accordingly.
(2) In CCMAN, mixing different EOM models in a single calculation is only allowed in Dyson orbitals calculations. In CCMAN2, different types of target states can be requested in a single calculation.
7.7.13.1

Alternative way to set up EOM calculations

Below we describe alternative way to specify correlation treatment in EOM-CC/CI calculations. These keywords will
be eventually phased out. By default, the level of correlation of the EOM part of the wave function (i.e., maximum
excitation level in the EOM operators R) is set to match CORRELATION, however, one can mix different correlation
levels for the reference and EOM states by using EOM_CORR. To request a CI calculation, set CORRELATION = CI
and select type of CI expansion by EOM_CORR. The table below shows default and allowed CORRELATION and
EOM_CORR combinations.
Default

Allowed

EOM_CORR

EOM_CORR

CI

none

CIS(D)
CCSD, OD

CIS(D)
CISD

CIS, CIS(D)
CISD
SDT, DT
N/A

CORRELATION

SD(fT)
SD(dT), SD(fT)
SD(dT), SD(fT), SD(sT)
SDT, DT

Target states
EE, SF
EE, SF, IP
EE, SF, DSF
EE, SF
EE, SF, IP, EA, DIP
EE, IP, EA
EE, SF, fake IP/EA
IP
EE, SF, IP, EA, DIP, DSF

CCMAN /
CCMAN2
y/n
y/n
y/n
y/n
y/y
n/y
y/n
y/n
y/n

Table 7.1: Default and allowed CORRELATION and EOM_CORR combinations as well as valid target state types. The
last column shows if a method is available in CCMAN or CCMAN2.

Table 7.7.13.1 shows the correct combinations of CORRELATION and EOM_CORR for standard EOM and CI models.
The most relevant EOM-CC input options follow.
EE_STATES
Sets the number of excited state roots to find. For closed-shell reference, defaults into
EE_SINGLETS. For open-shell references, specifies all low-lying states.
TYPE:
INTEGER/INTEGER ARRAY
DEFAULT:
0 Do not look for any excited states.
OPTIONS:
[i, j, k . . .] Find i excited states in the first irrep, j states in the second irrep etc.
RECOMMENDATION:
None

349

Chapter 7: Open-Shell and Excited-State Methods
Method
CIS

CORRELATION

EOM_CORR

Target states selection

CI

CIS

SF-CIS
CIS(D)

CI
CI

CIS
CIS(D)

SF-CIS(D)
CISD

CI
CI

CIS(D)
CISD

SF-CISD
IP-CISD
CISDT

CI
CI
CI

CISD
CISD
SDT

SF-CISDT
EOM-EE-CCSD

CI
CCSD

SDT or DT

EE_STATES
EE_SNGLETS, EE_TRIPLETS
SF_STATES
EE_STATES
EE_SNGLETS, EE_TRIPLETS
SF_STATES
EE_STATES
EE_SNGLETS, EE_TRIPLETS
SF_STATES
IP_STATES
EE_STATES
EE_SNGLETS, EE_TRIPLETS
SF_STATES

EOM-SF-CCSD
EOM-IP-CCSD
EOM-EA-CCSD
EOM-DIP-CCSD

CCSD
CCSD
CCSD
CCSD

EOM-2SF-CCSD
EOM-EE-(2,3)

CCSD
CCSD

SDT or DT
SDT

EOM-SF-(2,3)
EOM-IP-(2,3)
EOM-SF-CCSD(dT)
EOM-SF-CCSD(fT)
EOM-IP-CCSD(dT)
EOM-IP-CCSD(fT)
EOM-IP-CCSD(sT)

CCSD
CCSD
CCSD
CCSD
CCSD
CCSD
CCSD

SDT
SDT
SD(dT)
SD(fT)
SD(dT)
SD(fT)
SD(sT)

EE_STATES
EE_SNGLETS, EE_TRIPLETS
SF_STATES
IP_STATES
EA_STATES
DIP_STATES
DIP_SNGLETS, DIP_TRIPLETS
DSF_STATES
EE_STATES
EE_SNGLETS, EE_TRIPLETS
SF_STATES
IP_STATES
SF_STATES
SF_STATES
IP_STATES
IP_STATES
IP_STATES

Table 7.2: Commonly used EOM and CI models. ’SINGLETS’ and ’TRIPLETS’ are only available for closed-shell
references.

EE_SINGLETS
Sets the number of singlet excited state roots to find. Valid only for closed-shell references.
TYPE:
INTEGER/INTEGER ARRAY
DEFAULT:
0 Do not look for any excited states.
OPTIONS:
[i, j, k . . .] Find i excited states in the first irrep, j states in the second irrep etc.
RECOMMENDATION:
None

Chapter 7: Open-Shell and Excited-State Methods
EE_TRIPLETS
Sets the number of triplet excited state roots to find. Valid only for closed-shell references.
TYPE:
INTEGER/INTEGER ARRAY
DEFAULT:
0 Do not look for any excited states.
OPTIONS:
[i, j, k . . .] Find i excited states in the first irrep, j states in the second irrep etc.
RECOMMENDATION:
None
SF_STATES
Sets the number of spin-flip target states roots to find.
TYPE:
INTEGER/INTEGER ARRAY
DEFAULT:
0 Do not look for any excited states.
OPTIONS:
[i, j, k . . .] Find i SF states in the first irrep, j states in the second irrep etc.
RECOMMENDATION:
None
DSF_STATES
Sets the number of doubly spin-flipped target states roots to find.
TYPE:
INTEGER/INTEGER ARRAY
DEFAULT:
0 Do not look for any DSF states.
OPTIONS:
[i, j, k . . .] Find i doubly spin-flipped states in the first irrep, j states in the second irrep etc.
RECOMMENDATION:
None
IP_STATES
Sets the number of ionized target states roots to find. By default, β electron will be removed (see
EOM_IP_BETA).
TYPE:
INTEGER/INTEGER ARRAY
DEFAULT:
0 Do not look for any IP states.
OPTIONS:
[i, j, k . . .] Find i ionized states in the first irrep, j states in the second irrep etc.
RECOMMENDATION:
None

350

Chapter 7: Open-Shell and Excited-State Methods
EOM_IP_ALPHA
Sets the number of ionized target states derived by removing α electron (Ms = − 12 ).
TYPE:
INTEGER/INTEGER ARRAY
DEFAULT:
0 Do not look for any IP/α states.
OPTIONS:
[i, j, k . . .] Find i ionized states in the first irrep, j states in the second irrep etc.
RECOMMENDATION:
None
EOM_IP_BETA
Sets the number of ionized target states derived by removing β electron (Ms = 21 , default for
EOM-IP).
TYPE:
INTEGER/INTEGER ARRAY
DEFAULT:
0 Do not look for any IP/β states.
OPTIONS:
[i, j, k . . .] Find i ionized states in the first irrep, j states in the second irrep etc.
RECOMMENDATION:
None
EA_STATES
Sets the number of attached target states roots to find. By default, α electron will be attached
(see EOM_EA_ALPHA).
TYPE:
INTEGER/INTEGER ARRAY
DEFAULT:
0 Do not look for any EA states.
OPTIONS:
[i, j, k . . .] Find i EA states in the first irrep, j states in the second irrep etc.
RECOMMENDATION:
None
EOM_EA_ALPHA
Sets the number of attached target states derived by attaching α electron (Ms = 21 , default in EOMEA).
TYPE:
INTEGER/INTEGER ARRAY
DEFAULT:
0 Do not look for any EA states.
OPTIONS:
[i, j, k . . .] Find i EA states in the first irrep, j states in the second irrep etc.
RECOMMENDATION:
None

351

Chapter 7: Open-Shell and Excited-State Methods

352

EOM_EA_BETA
Sets the number of attached target states derived by attaching β electron (Ms =− 12 , EA-SF).
TYPE:
INTEGER/INTEGER ARRAY
DEFAULT:
0 Do not look for any EA states.
OPTIONS:
[i, j, k . . .] Find i EA states in the first irrep, j states in the second irrep etc.
RECOMMENDATION:
None
DIP_STATES
Sets the number of DIP roots to find. For closed-shell reference, defaults into DIP_SINGLETS.
For open-shell references, specifies all low-lying states.
TYPE:
INTEGER/INTEGER ARRAY
DEFAULT:
0 Do not look for any DIP states.
OPTIONS:
[i, j, k . . .] Find i DIP states in the first irrep, j states in the second irrep etc.
RECOMMENDATION:
None
DIP_SINGLETS
Sets the number of singlet DIP roots to find. Valid only for closed-shell references.
TYPE:
INTEGER/INTEGER ARRAY
DEFAULT:
0 Do not look for any singlet DIP states.
OPTIONS:
[i, j, k . . .] Find i DIP singlet states in the first irrep, j states in the second irrep etc.
RECOMMENDATION:
None
DIP_TRIPLETS
Sets the number of triplet DIP roots to find. Valid only for closed-shell references.
TYPE:
INTEGER/INTEGER ARRAY
DEFAULT:
0 Do not look for any DIP triplet states.
OPTIONS:
[i, j, k . . .] Find i DIP triplet states in the first irrep, j states in the second irrep etc.
RECOMMENDATION:
None

Note: It is a symmetry of a transition rather than that of a target state which is specified in excited state calculations.
The symmetry of the target state is a product of the symmetry of the reference state and the transition. For
closed-shell molecules, the former is fully symmetric and the symmetry of the target state is the same as that
of transition, however, for open-shell references this is not so.
Note: For the XX_STATES options, Q-C HEM will increase the number of roots if it suspects degeneracy, or change it
to a smaller value, if it cannot generate enough guess vectors to start the calculations.

Chapter 7: Open-Shell and Excited-State Methods

353

EOM_FAKE_IPEA
If TRUE, calculates fake EOM-IP or EOM-EA energies and properties using the diffuse orbital
trick. Default for EOM-EA and Dyson orbital calculations in CCMAN.
TYPE:
LOGICAL
DEFAULT:
FALSE (use proper EOM-IP code)
OPTIONS:
FALSE, TRUE
RECOMMENDATION:
None. This feature only works for CCMAN.

Note: When EOM_FAKE_IPEA is set to TRUE, it can change the convergence of Hartree-Fock iterations compared
to the same job without EOM_FAKE_IPEA, because a very diffuse basis function is added to a center of symmetry before the Hartree-Fock iterations start. For the same reason, BASIS2 keyword is incompatible with
EOM_FAKE_IPEA. In order to read Hartree-Fock guess from a previous job, you must specify EOM_FAKE_IPEA
(even if you do not request for any correlation or excited states) in that previous job. Currently, the second moments of electron density and Mulliken charges and spin densities are incorrect for the EOM-IP/EA-CCSD
target states.
EOM_USER_GUESS
Specifies if user-defined guess will be used in EOM calculations.
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
TRUE Solve for a state that has maximum overlap with a trans-n specified in $eom_user_guess.
RECOMMENDATION:
The orbitals are ordered by energy, as printed in the beginning of the CCMAN2 output. Not
available in CCMAN.
EOM_SHIFT
Specifies energy shift in EOM calculations.
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
n corresponds to n · 10−3 hartree shift (i.e., 11000 = 11 hartree); solve for eigenstates around this
value.
RECOMMENDATION:
Not available in CCMAN.

Chapter 7: Open-Shell and Excited-State Methods

354

EOM_NGUESS_DOUBLES
Specifies number of excited state guess vectors which are double excitations.
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
n Include n guess vectors that are double excitations
RECOMMENDATION:
This should be set to the expected number of doubly excited states, otherwise they may not be
found.
EOM_NGUESS_SINGLES
Specifies number of excited state guess vectors that are single excitations.
TYPE:
INTEGER
DEFAULT:
Equal to the number of excited states requested
OPTIONS:
n Include n guess vectors that are single excitations
RECOMMENDATION:
Should be greater or equal than the number of excited states requested, unless .
EOM_PRECONV_SINGLES
When not zero, singly excited vectors are converged prior to a full excited states calculation. Sets
the maximum number of iterations for pre-converging procedure.
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
0 do not pre-converge
1 pre-converge singles
RECOMMENDATION:
Sometimes helps with problematic convergence.

Note: In CCMAN, setting EOM_PRECONV_SINGLES = N would result in N Davidson iterations pre-converging singles.
EOM_PRECONV_DOUBLES
When not zero, doubly excited vectors are converged prior to a full excited states calculation.
Sets the maximum number of iterations for pre-converging procedure
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
0 Do not pre-converge
N Perform N Davidson iterations pre-converging doubles.
RECOMMENDATION:
Occasionally necessary to ensure a doubly excited state is found. Also used in DSF calculations
instead of EOM_PRECONV_SINGLES
Note: Not available in CCMAN2.

Chapter 7: Open-Shell and Excited-State Methods
EOM_PRECONV_SD
When not zero, EOM vectors are pre-converged prior to a full excited states calculation. Sets the
maximum number of iterations for pre-converging procedure.
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
0 do not pre-converge
N perform N Davidson iterations pre-converging singles and doubles.
RECOMMENDATION:
Occasionally necessary to ensure that all low-lying states are found. Also, very useful in
EOM(2,3) calculations.
None
Note: Not available in CCMAN2.
EOM_DAVIDSON_CONVERGENCE
Convergence criterion for the RMS residuals of excited state vectors.
TYPE:
INTEGER
DEFAULT:
5 Corresponding to 10−5
OPTIONS:
n Corresponding to 10−n convergence criterion
RECOMMENDATION:
Use the default. Normally this value be the same as EOM_DAVIDSON_THRESHOLD.
EOM_DAVIDSON_THRESHOLD
Specifies threshold for including a new expansion vector in the iterative Davidson diagonalization. Their norm must be above this threshold.
TYPE:
INTEGER
DEFAULT:
00103 Corresponding to 0.00001
OPTIONS:
abcde Integer code is mapped to abc × 10−(de+2) , i.e., 02505->2.5×10−6
RECOMMENDATION:
Use the default unless converge problems are encountered. Should normally be set to the same
values as EOM_DAVIDSON_CONVERGENCE, if convergence problems arise try setting to a value
slightly larger than EOM_DAVIDSON_CONVERGENCE.
EOM_DAVIDSON_MAXVECTORS
Specifies maximum number of vectors in the subspace for the Davidson diagonalization.
TYPE:
INTEGER
DEFAULT:
60
OPTIONS:
n Up to n vectors per root before the subspace is reset
RECOMMENDATION:
Larger values increase disk storage but accelerate and stabilize convergence.

355

Chapter 7: Open-Shell and Excited-State Methods
EOM_DAVIDSON_MAX_ITER
Maximum number of iteration allowed for Davidson diagonalization procedure.
TYPE:
INTEGER
DEFAULT:
30
OPTIONS:
n User-defined number of iterations
RECOMMENDATION:
Default is usually sufficient
EOM_IPEA_FILTER
If TRUE, filters the EOM-IP/EA amplitudes obtained using the diffuse orbital implementation
(see EOM_FAKE_IPEA). Helps with convergence.
TYPE:
LOGICAL
DEFAULT:
FALSE (EOM-IP or EOM-EA amplitudes will not be filtered)
OPTIONS:
FALSE, TRUE
RECOMMENDATION:
None

Note: Not available in CCMAN2.
CC_FNO_THRESH
Initialize the FNO truncation and sets the threshold to be used for both cutoffs (OCCT and
POVO).
TYPE:
INTEGER
DEFAULT:
None
OPTIONS:
range 0000-10000
abcd
Corresponding to ab.cd%
RECOMMENDATION:
None
CC_FNO_USEPOP
Selection of the truncation scheme.
TYPE:
INTEGER
DEFAULT:
1 OCCT
OPTIONS:
0 POVO
RECOMMENDATION:
None

356

Chapter 7: Open-Shell and Excited-State Methods
SCALE_NUCLEAR_CHARGE
Scales charge of each nuclei by a certain value. The nuclear repulsion energy is calculated for
the unscaled nuclear charges.
TYPE:
INTEGER
DEFAULT:
0 No scaling.
OPTIONS:
n A total positive charge of (1+n/100)e is added to the molecule.
RECOMMENDATION:
NONE
ADD_CHARGED_CAGE
Add a point charge cage of a given radius and total charge.
TYPE:
INTEGER
DEFAULT:
0 No cage.
OPTIONS:
0 No cage.
1 Dodecahedral cage.
2 Spherical cage.
RECOMMENDATION:
Spherical cage is expected to yield more accurate results, especially for small radii.
CAGE_RADIUS
Defines radius of the charged cage.
TYPE:
INTEGER
DEFAULT:
225
OPTIONS:
n radius is n/100 Å.
RECOMMENDATION:
None
CAGE_POINTS
Defines number of point charges for the spherical cage.
TYPE:
INTEGER
DEFAULT:
100
OPTIONS:
n Number of point charges to use.
RECOMMENDATION:
None

357

Chapter 7: Open-Shell and Excited-State Methods
CAGE_CHARGE
Defines the total charge of the cage.
TYPE:
INTEGER
DEFAULT:
400 Add a cage charged +4e.
OPTIONS:
n Total charge of the cage is n/100 a.u.
RECOMMENDATION:
None

358

Chapter 7: Open-Shell and Excited-State Methods

7.7.14

Examples

Example 7.31 EOM-EE-OD and EOM-EE-CCSD calculations of the singlet excited states of formaldehyde
$molecule
0 1
O
C 1 R1
H 2 R2
H 2 R2
R1
R2
A
$end

=
=
=

1
1

A
A

3

180.

1.4
1.0
120.

$rem
METHOD
BASIS
EE_STATES
$end

eom-od
6-31+g
[2,2,2,2]

@@@
$molecule
read
$end
$rem
METHOD
BASIS
EE_SINGLETS
EE_TRIPLETS
$end

eom-ccsd
6-31+g
[2,2,2,2]
[2,2,2,2]

Example 7.32 EOM-EE-CCSD calculations of the singlet excited states of PYP using Cholesky decomposition
$molecule
0 1
...too long to enter...
$end
$rem
METHOD
BASIS
PURECART
N_FROZEN_CORE
CC_T_CONV
CC_E_CONV
CHOLESKY_TOL
EE_SINGLETS
$end

eom-ccsd
aug-cc-pVDZ
1112
fc
4
6
2
using CD/1e-2 threshold
[2,2]

359

360

Chapter 7: Open-Shell and Excited-State Methods

Example 7.33 EOM-SF-CCSD calculations for methylene from high-spin 3 B2 reference
$molecule
0 3
C
H 1 rCH
H 1 rCH 2 aHCH
rCH
aHCH
$end

= 1.1167
= 102.07

$rem
METHOD
BASIS
SCF_GUESS
SF_STATES
$end

eom-ccsd
6-31G*
core
[2,0,0,2]

Two singlet A1 states and singlet and triplet B2 states

Example 7.34 EOM-SF-MP2 calculations for SiH2 from high-spin 3 B2 reference. Both energies and properties are
computed.
$molecule
0 3
Si
H 1 1.5145
H 1 1.5145 2 92.68
$end
$rem
BASIS
UNRESTRICTED
SCF_CONVERGENCE
METHOD
SF_STATES
CC_EOM_PROP_TE
$end

=
=
=
=
=
=

cc-pVDZ
true
8
eom-mp2
[1,1,0,0]
true
! Compute  of excited states

Example 7.35 EOM-IP-CCSD calculations for NO3 using closed-shell anion reference
$molecule
-1 1
N
O 1 r1
O 1 r2
O 1 r2
r1
r2
A2
$end

2 A2
2 A2

3 180.0

= 1.237
= 1.237
= 120.00

$rem
METHOD
BASIS
IP_STATES
$end

eom-ccsd
6-31G*
[1,1,2,1]

ground and excited

states of the radical

Chapter 7: Open-Shell and Excited-State Methods

Example 7.36 EOM-IP-CCSD calculation using FNO with OCCT=99%.
$molecule
0 1
O
H 1 1.0
H 1 1.0
$end

2

100.

$rem
METHOD
BASIS
IP_STATES
CC_FNO_THRESH
$end

eom-ccsd
6-311+G(2df,2pd)
[1,0,1,1]
9900
99% of the total natural population recovered

Example 7.37 EOM-IP-MP2 calculation of the three low lying ionized states of the phenolate anion
$molecule
0 1
C
-0.189057
H
-0.709319
C
1.194584
H
1.762373
C
1.848872
H
2.923593
C
1.103041
H
1.595604
C
-0.283047
H
-0.862269
C
-0.929565
O
-2.287040
H
-2.663814
$end
$rem
THRESH
CC_MEMORY
BASIS
METHOD
IP_STATES
$end

-1.215927
-2.157526
-1.155381
-2.070036
0.069673
0.111621
1.238842
2.196052
1.185547
2.095160
-0.042566
-0.159171
0.725029

-0.000922
-0.001587
-0.000067
-0.000230
0.000936
0.001593
0.001235
0.002078
0.000344
0.000376
-0.000765
-0.001759
0.001075

16
30000
6-31+g(d)
eom-mp2
[3]

Example 7.38 EOM-EE-MP2T calculation of the H2 excitation energies
$molecule
0 1
H
0.0000
H
0.0000
$end
$rem
THRESH
BASIS
METHOD
EE_STATES
$end

0.0000
0.0000

0.0000
0.7414

16
cc-pvdz
eom-mp2t
[3,0,0,0,0,0,0,0]

361

Chapter 7: Open-Shell and Excited-State Methods

Example 7.39 EOM-EA-CCSD calculation of CN using user-specified guess
$molecule
+1 1
C
N 1 1.1718
$end
$rem
METHOD
BASIS
EA_STATES
CC_EOM_PROP
EOM_USER_GUESS
$end

= eom-ccsd
6-311+g*
[1,1,1,1]
true
true
! attach to HOMO, HOMO+1, and HOMO+3

=
=
=
=

$eom_user_guess
1 2 4
$end

Example 7.40 DSF-CIDT calculation of methylene starting with quintet reference
$molecule
0 5
C
H 1 CH
H 1 CH 2 HCH
CH = 1.07
HCH = 111.0
$end
$rem
METHOD
BASIS
DSF_STATES
EOM_NGUESS_SINGLES
EOM_NGUESS_DOUBLES
$end

cisdt
6-31G
[0,2,2,0]
0
2

362

Chapter 7: Open-Shell and Excited-State Methods

363

Example 7.41 EOM-EA-CCSD job for cyano radical. We first do Hartree-Fock calculation for the cation in the basis
set with one extremely diffuse orbital (EOM_FAKE_IPEA) and use these orbitals in the second job. We need make
sure that the diffuse orbital is occupied using the OCCUPIED keyword. No SCF iterations are performed as the diffuse
electron and the molecular core are uncoupled. The attached states show up as “excited” states in which electron is
promoted from the diffuse orbital to the molecular ones.
$molecule
+1 1
C
N 1 bond
bond
$end

1.1718

$rem
METHOD
BASIS
PURECART
SCF_CONVERGENCE
EOM_FAKE_IPEA
$end

hf
6-311+G*
111
8
true

@@@
$molecule
0 2
C
N 1 bond
bond
$end

1.1718

$rem
BASIS
PURECART
SCF_GUESS
MAX_SCF_CYCLES
METHOD
CC_DOV_THRESH
EA_STATES
EOM_FAKE_IPEA
$end
$occupied
1 2 3 4 5 6 14
1 2 3 4 5 6
$end

6-311+G*
111
read
0
eom-ccsd
2501
use thresh for CC iters with convergence problems
[2,0,0,0]
true

Chapter 7: Open-Shell and Excited-State Methods

364

Example 7.42 DIP-EOM-CCSD calculation of methylene with charged cage stabilization.
$molecule
-2 1
C
0.000000
H -0.989216
H
0.989216
$end

0.000000
0.000000
0.000000

$rem
BASIS
SCF_ALGORITHM
SYMMETRY
METHOD
CC_SYMMETRY
DIP_SINGLETS
DIP_TRIPLETS
EOM_DAVIDSON_CONVERGENCE
CC_EOM_PROP
ADD_CHARGED_CAGE
CAGE_RADIUS
CAGE_CHARGE
CAGE_POINTS
CC_MEMORY
$end

=
=
=
=
=
=
=
=
=
=
=
=
=
=

0.106788
-0.320363
-0.320363

6-311g(d,p)
diis_gdm
false
eom-ccsd
false
[1] ! Compute one EOM-DIP singlet state
[1] ! Compute one EOM-DIP triplet state
5
true ! Compute excited state properties
2
! Install a charged sphere around the molecule
225 ! Radius = 2.25 A
500 ! Charge = +5 a.u.
100 ! Place 100 point charges
256 ! Use 256Mb of memory, increase for larger jobs

Example 7.43 EOM-EE-CCSD calculation of excited states in NO− using scaled nuclear charge stabilization method.
$molecule
-1 1
N -1.08735
O
1.08735
$end

0.0000
0.0000

$rem
INPUT_BOHR
BASIS
SYMMETRY
CC_SYMMETRY
METHOD
EE_SINGLETS
EE_TRIPLETS
CC_REF_PROP
CC_EOM_PROP
CC_MEMORY
SCALE_NUCLEAR_CHARGE
$end

=
=
=
=
=
=
=
=
=
=
=

0.0000
0.0000

true
6-31g
false
false
eom-ccsd
[2] ! Compute two EOM-EE singlet excited states
[2] ! Compute two EOM-EE triplet excited states
true ! Compute ground state properties
true ! Compute excited state properties
256 ! Use 256Mb of memory, increase for larger jobs
180 ! Adds +1.80e charge to the molecule

Chapter 7: Open-Shell and Excited-State Methods

365

Example 7.44 EOM-EE-CCSD calculation for phenol with user-specified guess requesting the EE transition from the
occupied orbital number 24 (3 A") to the virtual orbital number 2 (23 A’)
$molecule
0 1
C
0.935445
C
0.262495
C
-1.130915
C
-1.854154
C
-1.168805
C
0.220600
O
2.298632
H
2.681798
H
0.823779
H
-1.650336
H
-2.939976
H
-1.722580
H
0.768011
$end

-0.023376
1.197399
1.215736
0.026814
-1.188579
-1.220808
-0.108788
0.773704
2.130309
2.170478
0.044987
-2.123864
-2.158602

$rem
METHOD
BASIS
CC_MEMORY
MEM_STATIC
CC_T_CONV
CC_E_CONV
EE_STATES
EOM_DAVIDSON_CONVERGENCE
EOM_USER_GUESS
$end
$eom_user_guess
24
2
$end

0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000

EOM-CCSD
6-31+G(d,p)
3000
ccman2 memory
250
4
T-amplitudes convergence threshold
6
Energy convergence threshold
[0,1] Calculate 1 A" states
5
Convergence threshold for the Davidson procedure
true
Use user guess from $eom_user_guess section

366

Chapter 7: Open-Shell and Excited-State Methods

Example 7.45 Complex-scaled EOM-EE calculation for He. All roots of Ag symmetry are computed (full diagonalization)
$molecule
0 1
He 0
$end

0

0.0

$rem
COMPLEX_CCMAN
METHOD
BASIS
PURECART
EE_SINGLETS
EOM_DAVIDSON_CONV
EOM_DAVIDSON_THRESH
EOM_NGUESS_SINGLES
EOM_NGUESS_DOUBLES
CC_MEMORY
MEM_TOTAL
$end

1
engage complex_ccman
EOM-CCSD
gen
use general basis
1111
[2000,0,0,0,0,0,0,0] compute all Ag excitations
5
5
2000
Number of guess singles
2000
Number of guess doubles
2000
3000

$complex_ccman
CS_HF 1
CS_ALPHA 1000
CS_THETA 300
$end
$basis
He
0
S
4
1.000000
2.34000000E+02
2.58700000E-03
3.51600000E+01
1.95330000E-02
7.98900000E+00
9.09980000E-02
2.21200000E+00
2.72050000E-01
S
1
1.000000
6.66900000E-01
1.00000000E+00
S
1
1.000000
2.08900000E-01
1.00000000E+00
P
1
1.000000
3.04400000E+00
1.00000000E+00
P
1
1.000000
7.58000000E-01
1.00000000E+00
D
1
1.000000
1.96500000E+00
1.00000000E+00
S
1
1.000000
5.13800000E-02
1.00000000E+00
P
1
1.000000
1.99300000E-01
1.00000000E+00
D
1
1.000000
4.59200000E-01
1.00000000E+00
S
1
1.000000
2.44564000E-02
1.00000000E+00
S
1
1.000000
1.2282000E-02
1.00000000E+00
S
1
1.000000
6.1141000E-03
1.00000000E+00
P
1
1.0
8.130000e-02
1.0
P
1
1.0
4.065000e-02
1.0
P
1
1.0
2.032500e-02
1.0
D
1
1.0
2.34375e-01
1.0
D
1
1.0
1.17187e-01
1.0

Use complex HF
Set alpha equal 1
Set theta (angle) equals 0.3 (radian)

367

Chapter 7: Open-Shell and Excited-State Methods

Example 7.46 CAP-augmented EOM-EA-CCSD calculation for N−
2 . aug-cc-pVTZ basis augmented by the 3s3p3d
diffuse functions placed in the COM. Two EA states are computed for CAP strength η=0.002
$molecule
0 1
N
0.0
N
0.0
Gh 0.0
$end

0.0 -0.54875676501
0.0 0.54875676501
0.0 0.0

$rem
COMPLEX_CCMAN
METHOD
BASIS
EA_STATES
CC_MEMORY
MEM_TOTAL
CC_EOM_PROP
$end

1
engage complex_ccman
EOM-CCSD
gen
use general basis
[0,0,2,0,0,0,0,0] compute electron attachment energies
5000
ccman2 memory
2000
true
compute excited state properties

$complex_ccman
CS_HF
CAP_ETA
CAP_X
CAP_Y
CAP_Z
CAP_TYPE
$end

1
200
2760
2760
4880
1

$basis
N
0
aug-cc-pvtz
****
Gh
0
S
1
1.000000
2.88000000E-02
S
1
1.000000
1.44000000E-02
S
1
1.000000
0.72000000E-02
P
1
1.000000
2.45000000E-02
P
1
1.000000
1.22000000E-02
P
1
1.000000
0.61000000E-02
D
1
1.000000
0.755000000E-01
D
1
1.000000
0.377500000E-01
D
1
1.000000
0.188750000E-01
****
$end

Use
Set
Set
Set
Set
Use

complex HF
strength of CAP potential
length of the box along x
length of the box along y
length of the box along z
cuboid CAP

1.00000000E+00
1.00000000E+00
1.00000000E+00
1.00000000E+00
1.00000000E+00
1.00000000E+00
1.00000000E+00
1.00000000E+00
1.00000000E+00

0.002
dimension
dimension
dimension

368

Chapter 7: Open-Shell and Excited-State Methods

Example 7.47 CAP-EOM-EE calculation of water, with wave-function analysis of state and transition properties
$molecule
0 1
O
0.00000000
H
0.00000000
H
0.00000000
$end

0.00000000
-1.44761450
1.44761450

$rem
METHOD
BASIS
CC_MEMORY
MEM_TOTAL
SCF_CONVERGENCE
CC_CONVERGENCE
EOM_DAVIDSON_CONVERGENCE
CC_EOM_PROP
CC_FULLRESPONSE
CC_TRANS_PROP
COMPLEX_CCMAN
EE_STATES
INPUT_BOHR
! WFA KEYWORDS
STATE_ANALYSIS
MOLDEN_FORMAT
NTO_PAIRS
POP_MULLIKEN
$end
$complex_ccman
CS_HF
CAP_TYPE
CAP_ETA
CAP_X
CAP_Y
CAP_Z
$end

1
1
10000
2000
2500
2500

0.13594219
-1.07875060
-1.07875060

eom-ccsd
6-31G**
2000
2500
12
11
11
TRUE
FALSE
TRUE
1
[1,0,2,0]
TRUE
true
true
4
true

369

Chapter 7: Open-Shell and Excited-State Methods

Example 7.48 Formaldehyde, calculating EOM-IP-CCSD-S(D) and EOM-IP-MP2-S(D) energies of 4 valence ionized
states
$molecule
0 1
C
H 1 1.096135
H 1 1.096135
O 1 1.207459
$end
$rem
METHOD
BASIS
IP_STATES
$end

2
2

116.191164
121.904418

3

-180.000000 0

eom-ccsd-s(d)
6-31G*
[1,1,1,1]

@@@
$molecule
read
$end
$rem
METHOD
BASIS
IP_STATES
$end

eom-mp2-s(d)
6-31G*
[1,1,1,1]

Example 7.49 Formaldehyde, calculating EOM-EE-CCSD states with C-PCM method.
$molecule
0 1
O
C,1,R1
H,2,R2,1,A
H,2,R2,1,A,3,180.
R1 = 1.4
R2 = 1.0
A = 120.
$end
$rem
METHOD
BASIS
EE_STATES
SOLVENT_METHOD
$end
$pcm
theory
$end

eom-ccsd
cc-pvdz
[4]
pcm

cpcm

$solvent
dielectric
4.34
dielectric_infi 1.829
$end

Chapter 7: Open-Shell and Excited-State Methods

370

Example 7.50 NO−
2 , calculating EOM-IP-CCSD states with C-PCM method.
$molecule
-1 1
N1
O2 N1 RNO
O3 N1 RNO O2 AONO
RNO = 1.305
AONO = 106.7
$end
$rem
METHOD
BASIS
IP_STATES
SOLVENT_METHOD
$end
$pcm
theory
$end

eom-ccsd
cc-pvdz
[2]
pcm

cpcm

$solvent
dielectric
4.34
dielectric_infi 1.829
$end

7.7.15

Non-Hartree-Fock Orbitals in EOM Calculations

In cases of problematic open-shell references, e.g., strongly spin-contaminated doublet, triplet or quartet states, one
may choose to use DFT orbitals. This can be achieved by first doing DFT calculation and then reading the orbitals and
turning Hartree-Fock off (by setting SCF_GUESS = READ MAX_SCF_CYCLES = 0 in the CCMAN or CCMAN2 job).
In CCMAN, a more convenient way is just to specify EXCHANGE, e.g., if EXCHANGE = B3LYP, B3LYP orbitals will
be computed and used.
Note: Using non-HF exchange in CCMAN2 is not possible.

7.7.16

Analytic Gradients and Properties for the CCSD and EOM-XX-CCSD Methods

The coupled-cluster package in Q-C HEM can calculate properties of target EOM states including permanent dipoles,
static polarizabilities, hS 2 i and hR2 i values, nuclear gradients (and geometry optimizations). The target state of interest
is selected by CC_STATE_TO_OPT $rem, which specifies the symmetry and the number of the EOM state. In addition
to state properties, calculations of various interstate properties are available (transition dipoles, two-photon absorption
transition moments (and cross-sections), spin-orbit couplings).
Analytic gradients are available for the CCSD and all EOM-CCSD methods for both closed- and open-shell references
(UHF and RHF only), including frozen core/virtual functionality 72 (see also Section 6.13). These calculations should
be feasible whenever the corresponding single-point energy calculation is feasible.
Note: Gradients for ROHF and non-HF (e.g., B3LYP) orbitals are not yet available.
For the CCSD and EOM-CCSD wave functions, Q-C HEM currently can calculate permanent and transition dipole
moments, oscillator strengths, hR2 i (as well as XX, YY and ZZ components separately, which is useful for assigning
different Rydberg states, e.g., 3px vs. 3s, etc.), and the hS 2 i values. Interface of the CCSD and EOM-CCSD codes
with the NBO 5.0 package is also available. Furthermore, excited state analyses can be requested for EOM-CCSD

Chapter 7: Open-Shell and Excited-State Methods

371

excited states. For EOM-MP2, only state properties (dipole moments, hR2 i, hS 2 i are available). Similar functionality
is available for some EOM-OD and CI models (CCMAN only).
Analysis of the real- and complex-valued EOM-CC wave functions can also be performed; see Sections 7.7.24 and
11.2.6. NTO analysis for EOM-IP/EA/SF states is, obviously, only available for the transitions between the EOM
states, so CC_STATE_TO_OPT keyword needs to be used, as in calculations of transition properties.
Users must be aware of the point group symmetry of the system being studied and also the symmetry of the excited
(target) state of interest. It is possible to turn off the use of symmetry using the CC_SYMMETRY. If set to FALSE the
molecule will be treated as having C1 symmetry and all states will be of A symmetry.
Q-C HEM allows flexible control of interstate properties calculations, by using CC_TRANS_PROP rem or rem section $trans_prop: the user can request the transitions between all computed EOM target states and the reference
state (CC_TRANS_PROP = 1) or the calculations of all transition properties between all computed EOM target states
(CC_TRANS_PROP = 2). By default, the reference state is the CCSD reference. To compute transition properties relative
to a particular EOM state, use CC_STATE_TO_OPT.
By default, only one-electron properties are computed. To activate calculations of two-electron properties, such as
NACs, SOCs, 2PA, additional keywords should be activated, as described below.
The $trans_prop rem section allows the user to specify precisely which properties and for which pairs of states to
computed. When $trans_prop section is present in the input, it disables CC_TRANS_PROP rem.
$trans_prop
state_list
ee_singlets 1 1
ee_triplets 1 2
ref
end_list
state_pair_list
3 1
3 2
end_pairs
calc nac
state_list
ref
ee_singlets 0 0
end_list
calc dipole soc
calc opdm_norm

!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!

Start a list of states
state 1: EE singlet with irrep = 1 and istate = 1
state 2: EE triplet with irrep = 1 and istate = 2
state 3: Reference state (can be CC or MP2, but the latter NYI
in transition prop driver)
End list
Start to specify pairs of states,
transition from state 3 to state 1 (known bug here: CC state
needs to be 1st one)
transition from state 3 to state 2 (known bug here: CC state
needs to be 1st one)
End list of pairs
Compute NAC for all transition pairs listed before this keyword
Start another list of states (user is able to request multiple
state lists for multiple tasks)
reference state
zero means all requested irreps/istate in $rem

! Compute transition dipole and SOC
! Compute norm of transition OPDM

Notes about $trans_prop rem section:
1. calc computes properties for the first pair list (or state list) before it.
2. The pair list is optional: if there is no pair list, all possible combinations within the state list will be considered.
3. Options after calc include: nac, soc, dyson, 2pa, dipole, default, pcm, opdm_norm, wfa. Currently, only some of them are implemented.
4. $trans_prop control for CVS-EOM-CCSD properties is not yet implemented.
Note: $trans_prop section is a new feature and is still under development — use on your own risk. Eventually, this
section will replace other controls and will become a default.

Chapter 7: Open-Shell and Excited-State Methods
7.7.16.1

372

Transition moments and cross-sections for two-photon absorption within EOM-EE-CCSD

Calculation of transition moments and cross-sections for two-photon absorption for EOM-EE-CCSD wave functions
is available in Q-C HEM (CCMAN2 only). Both CCSD-EOM and EOM-EOM transitions can be computed. The
formalism is described in Ref. 93. This feature is available both for canonical and RI/CD implementations. Relevant
keywords are CC_EOM_2PA (turns on the calculation, controls NTO calculation), CC_STATE_TO_OPT (used for EOMEOM transitions); additional customization can be performed using the $2pa section. The quantity printed out is the
microscopic cross-section δ T P A (also known as rotationally averaged 2PA strength), as defined in Eq. (30) of Ref. 93.
The $2pa section is used to specify the range of frequency-pairs satisfying the resonance condition. If $2pa section
is absent in the input, the transition moments are computed for 2 degenerate photons with total energy matching the
excitation energy of each target EOM state (for CCSD-EOM) or each EOM-EOM energy difference (for EOM-EOM
transitions): 2hν = Eex
$2pa
N_2PA_POINTS 6
OMEGA_1 500000 10000

Non-degenerate resonant 2PA
Number of frequency pairs
Scans 500 cm$^{-1}$ to 550 cm$^{-1}$
in steps of 10 cm$^{-1}$

$end
N_2PA_POINTS is the number of frequency pairs across the spectrum. The first value associated with OMEGA_1 is
the frequency ×1000 in cm−1 at the start of the spectrum and the second value is the step size ×1000 in cm−1 . The
frequency of the second photon at each step is determined within the code as the excitation energy minus OMEGA_1.

To gain insight into computed cross sections for 2PA, one can perform NTO analysis of the response one-particle density
matrices 95 . To activate NTO analysis of the 2PA response one-particle transition density matrices, set STATE_ANALYSIS
= TRUE, MOLDEN_FORMAT = TRUE (to export the orbitals as M OL D EN files), NTO_PAIRS (specifies the number of
orbitals to print). The NTO analysis will be performed for the full 2PA response one-particle transition density matrices
as well as the normalized ωDMs (see Ref. 95 for more details).
7.7.16.2

Calculations of Spin-Orbit Couplings Using EOM-CC Wave Functions

Calculations of spin-orbit couplings (SOCs) for EOM-CC wave functions is available in CCMAN2. 32 We employ
a perturbative approach in which SOCs are computed as matrix elements of the respective part of the Breit-Pauli
Hamiltonian using zero-order non-relativistic wave functions. Both the full two-electron treatment and the meanfield approximation (a partial account of the two-electron contributions) are available for the EOM-EE/SF/IP/EA wave
functions, as well as between the CCSD reference and EOM-EE/SF. To enable SOC calculation, transition properties
between EOM states must be enabled via CC_TRANS_PROP, and SOC requested using CALC_SOC. By default, oneelectron and mean-field two-electron couplings will be computed. Full two-electron coupling calculation is activated
by setting CC_EOM_PROP_TE.
As with other EOM transition properties, the initial EOM state is set by CC_STATE_TO_OPT, and couplings are computed between that state and all other EOM states. In the absence of CC_STATE_TO_OPT, SOCs are computed between
the reference state and all EOM-EE or EOM-SF states.
Note: In a spin-restricted case, such as EOM-EE calculations using closed-shell reference state, SOCs between the
singlet and triplet EOM manifolds cannot be computed (only SOCs between the reference state and EOM
triplets can be calculated). To compute SOCs between EOM-EE singlets and EOM-EE triplets, run the same
job with UNRESTRICTED = TRUE, such that triplets and singlets appear in the same manifold.
7.7.16.3

Calculations of Non-Adiabatic Couplings Using EOM-CC Wave Functions

Calculations of non-adiabatic (derivative) couplings (NACs) for EOM-CC wave functions is available in CCMAN2.
We employ Szalay’s approach in which couplings are computed by a modified analytic gradient code, via “summed

Chapter 7: Open-Shell and Excited-State Methods

373

states”: 122



1
G(I+J) − GI − GJ ,
(7.52)
2
where, GI , GJ , and GIJ are analytic gradients for states I, J, and a fictitious summed state |ΨI+J i ≡ |ΨI i + |ΨJ i.
Currently, NACs for EE/IP/EA are available. 33
hxIJ ≡ hΨI |H x |ΨJ i =

Note: Note that the individual components of the NAC vector depend on the molecular orientation.

7.7.16.4

Calculations of Static Polarizabilities for CCSD and EOM-CCSD Wave Functions

Calculation of the static dipole polarizability for the CCSD and EOM-EE/SF wave function is available in CCMAN2.
CCSD polarizabilities are calculated as second derivatives of the CCSD energy. 94 Only the response of the cluster amplitudes is taken into the account; orbital relaxation is not included. Currently, this feature is available for the canonical
implementation only. Relevant keywords are CC_POL (turns on the calculation), EOM_POL (turns on the calculation
for EOM states, otherwise, only the CCSD polarizability will be computed), and CC_REF_PROP/CC_FULLRESPONSE
(both must be set to TRUE).
Note: EOM-CCSD polarizabilities are available for EE and SF wave functions only.

7.7.17

EOM-CC Optimization and Properties Job Control
CC_STATE_TO_OPT
Specifies which state to optimize (or from which state compute EOM-EOM inter-state properties).
TYPE:
INTEGER ARRAY
DEFAULT:
None
OPTIONS:
[i,j] optimize the jth state of the ith irrep.
RECOMMENDATION:
None

Note: The state number should be smaller or equal to the number of excited states calculated in the corresponding
irrep.

Note: If analytic gradients are not available, the finite difference calculations will be performed and the symmetry
will be turned off. In this case, CC_STATE_TO_OPT should be specified assuming C1 symmetry, i.e., as [1,N]
where N is the number of state to optimize (the states are numbered from 1).

Chapter 7: Open-Shell and Excited-State Methods
CC_EOM_PROP
Whether or not the non-relaxed (expectation value) one-particle EOM-CCSD target state properties will be calculated. The properties currently include permanent dipole moment, the second
moments hX 2 i, hY 2 i, and hZ 2 i of electron density, and the total hR2 i = hX 2 i + hY 2 i + hZ 2 i
(in atomic units). Incompatible with JOBTYPE=FORCE, OPT, FREQ.
TYPE:
LOGICAL
DEFAULT:
FALSE (no one-particle properties will be calculated)
OPTIONS:
FALSE, TRUE
RECOMMENDATION:
Additional equations (EOM-CCSD equations for the left eigenvectors) need to be solved for
properties, approximately doubling the cost of calculation for each irrep. The cost of the
one-particle properties calculation itself is low. The one-particle density of an EOM-CCSD
target state can be analyzed with NBO or LIBWFA packages by specifying the state with
CC_STATE_TO_OPT and requesting NBO = TRUE and CC_EOM_PROP = TRUE.
CC_TRANS_PROP
Whether or not the transition dipole moment (in atomic units) and oscillator strength for the
EOM-CCSD target states will be calculated. By default, the transition dipole moment is calculated between the CCSD reference and the EOM-CCSD target states. In order to calculate
transition dipole moment between a set of EOM-CCSD states and another EOM-CCSD state,
the CC_STATE_TO_OPT must be specified for this state.
TYPE:
INTEGER
DEFAULT:
0 (no transition properties will be calculated)
OPTIONS:
1 (calculate transition properties between all computed EOM state and the reference state)
2 (calculate transition properties between all pairs of EOM states)
RECOMMENDATION:
NONE
Additional equations (for the left EOM-CCSD eigenvectors plus lambda CCSD equations in case if transition properties between the CCSD reference and EOM-CCSD target states are requested) need to be
solved for transition properties, approximately doubling the computational cost. The cost of the transition
properties calculation itself is low.
Note: When $trans_prop section is present in the input, it disables CC_TRANS_PROP rem.

374

Chapter 7: Open-Shell and Excited-State Methods
CC_EOM_2PA
Whether or not the transition moments and cross-sections for two-photon absorption will be calculated. By default, the transition moments are calculated between the CCSD reference and the
EOM-CCSD target states. In order to calculate transition moments between a set of EOM-CCSD
states and another EOM-CCSD state, the CC_STATE_TO_OPT must be specified for this state. If
2PA NTO analysis is requested, the CC_EOM_2PA value is redundant as long as CC_EOM_2PA
> 0.
TYPE:
INTEGER
DEFAULT:
0 (do not compute 2PA transition moments)
OPTIONS:
1 Compute 2PA using the fastest algorithm (use σ̃-intermediates for canonical
and σ-intermediates for RI/CD response calculations).
2 Use σ-intermediates for 2PA response equation calculations.
3 Use σ̃-intermediates for 2PA response equation calculations.
RECOMMENDATION:
Additional response equations (6 for each target state) will be solved, which increases the cost
of calculations. The cost of 2PA moments is about 10 times that of energy calculation. Use the
default algorithm. Setting CC_EOM_2PA > 0 turns on CC_TRANS_PROP.
CALC_SOC
Whether or not the spin-orbit couplings between CC/EOM/ADC/CIS/TDDFT electronic states
will be calculated. In the CC/EOM-CC suite, by default the couplings are calculated between
the CCSD reference and the EOM-CCSD target states. In order to calculate couplings between
EOM states, CC_STATE_TO_OPT must specify the initial EOM state.
TYPE:
LOGICAL
DEFAULT:
FALSE (no spin-orbit couplings will be calculated)
OPTIONS:
FALSE, TRUE
RECOMMENDATION:
One-electron and mean-field two-electron SOCs will be computed by default. To enable full
two-electron SOCs, two-particle EOM properties must be turned on (see CC_EOM_PROP_TE).
CALC_NAC
Whether or not non-adiabatic couplings will be calculated for the EOM-CC, CIS, and TDDFT
wave functions.
TYPE:
INTEGER
DEFAULT:
0 (do not compute NAC)
OPTIONS:
1 NYI for EOM-CC
2 Compute NACs using Szalay’s approach (this what needs to be specified for EOM-CC).
RECOMMENDATION:
Additional response equations will be solved and gradients for all EOM states and for summed
states will be computed, which increases the cost of calculations. Request only when needed and
do not ask for too many EOM states.

375

Chapter 7: Open-Shell and Excited-State Methods
CC_POL
Whether or not the static polarizability for the CCSD wave function will be calculated.
TYPE:
LOGICAL
DEFAULT:
FALSE (CCSD static polarizability will not be calculated)
OPTIONS:
FALSE, TRUE
RECOMMENDATION:
Static polarizabilities are expensive since they require solving three additional response equations. Do no request this property unless you need it.
EOM_POL
Whether or not the static polarizability for the EOM-CCSD wave function will be calculated.
TYPE:
LOGICAL
DEFAULT:
FALSE (EOM polarizability will not be calculated)
OPTIONS:
FALSE, TRUE
RECOMMENDATION:
Static polarizabilities are expensive since they require solving three additional response equations. Do no request this property unless you need it.
EOM_REF_PROP_TE
Request for calculation of non-relaxed two-particle EOM-CC properties. The two-particle properties currently include hS 2 i. The one-particle properties also will be calculated, since the additional cost of the one-particle properties calculation is inferior compared to the cost of hS 2 i. The
variable CC_EOM_PROP must be also set to TRUE. Alternatively, CC_CALC_SSQ can be used to
request hS 2 i calculation.
TYPE:
LOGICAL
DEFAULT:
FALSE (no two-particle properties will be calculated)
OPTIONS:
FALSE, TRUE
RECOMMENDATION:
The two-particle properties are computationally expensive since they require calculation and use
of the two-particle density matrix (the cost is approximately the same as the cost of an analytic
gradient calculation). Do not request the two-particle properties unless you really need them.
CC_FULLRESPONSE
Fully relaxed properties (including orbital relaxation terms) will be computed. The variable
CC_EOM_PROP must be also set to TRUE.
TYPE:
LOGICAL
DEFAULT:
FALSE (no orbital response will be calculated)
OPTIONS:
FALSE, TRUE
RECOMMENDATION:
Not available for non-UHF/RHF references. Only available for EOM/CI methods for which
analytic gradients are available.

376

Chapter 7: Open-Shell and Excited-State Methods
CC_SYMMETRY
Controls the use of symmetry in coupled-cluster calculations
TYPE:
LOGICAL
DEFAULT:
TRUE
OPTIONS:
TRUE
Use the point group symmetry of the molecule
FALSE Do not use point group symmetry (all states will be of A symmetry).
RECOMMENDATION:
It is automatically turned off for any finite difference calculations, e.g. second derivatives.
STATE_ANALYSIS
Activates excited state analyses using LIBWFA.
TYPE:
LOGICAL
DEFAULT:
FALSE (no excited state analyses)
OPTIONS:
TRUE, FALSE
RECOMMENDATION:
Set to TRUE if excited state analysis is required, but also if plots of densities or orbitals are
needed. For details see section 11.2.6.

377

Chapter 7: Open-Shell and Excited-State Methods

378

Chapter 7: Open-Shell and Excited-State Methods
7.7.17.1

379

Examples

Example 7.51 Geometry optimization for the excited open-shell singlet state, 1 B2 , of methylene followed by the
calculations of the fully relaxed one-electron properties using EOM-EE-CCSD
$molecule
0 1
C
H 1 rCH
H 1 rCH 2 aHCH
rCH
aHCH
$end

= 1.083
= 145.

$rem
JOBTYPE
METHOD
BASIS
SCF_GUESS
SCF_CONVERGENCE
EE_SINGLETS
EOM_NGUESS_SINGLES
CC_STATE_TO_OPT
EOM_DAVIDSON_CONVERGENCE
$end

OPT
EOM-CCSD
cc-pVTZ
CORE
9
[0,0,0,1]
2
[4,1]
9
use tighter convergence for EOM amplitudes

@@@
$molecule
read
$end
$rem
METHOD
BASIS
SCF_GUESS
EE_SINGLETS
EOM_NGUESS_SINGLES
CC_EOM_PROP
CC_FULLRESPONSE
$end

EOM-CCSD
cc-pVTZ
READ
[0,0,0,1]
2
1 calculate properties for EOM states
1 use fully relaxed properties

Example 7.52 Property and transition property calculation on the lowest singlet state of CH2 using EOM-SF-CCSD
$molecule
0 3
C
H 1 rch
H 1 rch 2 ahch
rch = 1.1167
ahch = 102.07
$end
$rem
METHOD
BASIS
SCF_GUESS
SCF_CONVERGENCE
SF_STATES
CC_EOM_PROP
CC_TRANS_PROP
CC_STATE_TO_OPT
$end

eom-ccsd
cc-pvtz
core
9
[2,0,0,3]
Get three 1^B2 and two 1^A1 SF states
1
1
[4,1] First EOM state in the 4th irrep

Chapter 7: Open-Shell and Excited-State Methods

380

Example 7.53 Geometry optimization with tight convergence for the 2 A1 excited state of CH2 Cl, followed by calculation of non-relaxed and fully relaxed permanent dipole moment and hS 2 i.
$molecule
0 2
H
C
1 CH
CL 2 CCL
H
2 CH
CH
CCL
CCLH
DIH
$end

=
=
=
=

1
3

CCLH
CCLH

1

DIH

1.096247
2.158212
122.0
180.0

$rem
JOBTYPE
METHOD
BASIS
SCF_GUESS
EOM_DAVIDSON_CONVERGENCE
CC_T_CONV
EE_STATES
CC_STATE_TO_OPT
EOM_NGUESS_SINGLES
GEOM_OPT_TOL_GRADIENT
GEOM_OPT_TOL_DISPLACEMENT
GEOM_OPT_TOL_ENERGY
$end

OPT
EOM-CCSD
6-31G* Basis Set
SAD
9
EOM amplitude convergence
9
CCSD amplitudes convergence
[0,0,0,1]
[4,1]
2
2
2
2

@@@
$molecule
read
$end
$rem
METHOD
BASIS
SCF_GUESS
EE_STATES
EOM_NGUESS_SINGLES
CC_EOM_PROP
CC_EOM_PROP_TE
$end

EOM-CCSD
6-31G* Basis Set
READ
[0,0,0,1]
2
1
calculate one-electron properties
1
and two-electron properties (S^2)

@@@
$molecule
read
$end
$rem
METHOD
BASIS
SCF_GUESS
EE_STATES
EOM_NGUESS_SINGLES
CC_EOM_PROP
CC_EOM_PROP_TE
CC_FULLRESPONSE
$end

EOM-CCSD
6-31G* Basis Set
READ
[0,0,0,1]
2
1 calculate one-electron properties
1 and two-electron properties (S^2)CC_EXSTATES_PROP 1
1 same as above, but do fully relaxed properties

Chapter 7: Open-Shell and Excited-State Methods

381

Example 7.54 CCSD calculation on three A2 and one B2 state of formaldehyde. Transition properties will be calculated between the third A2 state and all other EOM states
$molecule
0 1
O
C 1 1.4
H 2 1.0
H 2 1.0
$end

1
1

120
120

$rem
BASIS
METHOD
EE_STATES
CC_STATE_TO_OPT
CC_TRANS_PROP
$end

3 180

6-31+G
EOM-CCSD
[0,3,0,1]
[2,3]
true

Example 7.55 EOM-IP-CCSD geometry optimization of X 2 B2 state of H2 O+ .
$molecule
0 1
H
0.774767
O
0.000000
H
-0.774767
$end
$rem
JOBTYPE
METHOD
BASIS
IP_STATES
CC_STATE_TO_OPT
$end

0.000000
0.000000
0.000000

opt
eom-ccsd
6-311G
[0,0,0,1]
[4,1]

0.458565
-0.114641
0.458565

382

Chapter 7: Open-Shell and Excited-State Methods

Example 7.56 CAP-EOM-EA-CCSD geometry optimization of the 2 B1 anionic resonance state of formaldehyde. The
applied basis is aug-cc-pVDZ augmented by 3s3p diffuse functions on heavy atoms.
$molecule
0 1
C
0.0000000000
O
0.0000000000
H
0.9478180646
H
-0.9478180646
$end

0.0000000000
0.0000000000
0.0000000000
0.0000000000

$rem
JOBTYPE
METHOD
BASIS
SCF_CONVERGENCE
CC_CONVERGENCE
EOM_DAVIDSON_CONVERGENCE
EA_STATES
CC_STATE_TO_OPT
XC_GRID
COMPLEX_CCMAN
$end

opt
eom-ccsd
gen
9
9
9
[0,0,0,2]
[4,1]
000250000974
1

$complex_ccman
CS_HF 1
CAP_TYPE 1
CAP_ETA 60
CAP_X 3850
CAP_Y 2950
CAP_Z 6100
$end
$basis
H
0
S
3

S
P
S
P

1.00
13.0100000
1.96200000
0.444600000
1 1.00
0.122000000
1 1.00
0.727000000
1 1.00
0.297400000E-01
1 1.00
0.141000000

0.196850000E-01
0.137977000
0.478148000
1.00000000
1.00000000
1.00000000
1.00000000

****
C
0
S
8

S

1.00
6665.00000
1000.00000
228.000000
64.7100000
21.0600000
7.49500000
2.79700000
0.521500000
8 1.00
6665.00000
1000.00000
228.000000
64.7100000
21.0600000
7.49500000
2.79700000
0.521500000

0.692000000E-03
0.532900000E-02
0.270770000E-01
0.101718000
0.274740000
0.448564000
0.285074000
0.152040000E-01
-0.146000000E-03
-0.115400000E-02
-0.572500000E-02
-0.233120000E-01
-0.639550000E-01
-0.149981000
-0.127262000
0.544529000

0.5721328608
-0.7102635035
1.1819748108
1.1819748108

Chapter 7: Open-Shell and Excited-State Methods

383

Example 7.57 Calculating resonant 2PA with degenerate photons.
$molecule
0 1
O
H 1 0.959
H 1 0.959 2 104.654
$end
$rem
METHOD
BASIS
EE_SINGLETS
CC_TRANS_PROP
CC_EOM_2PA
$end

eom-ccsd
aug-cc-pvtz
[1,0,0,0]
1A_1 state
1
Compute transition properties
1
Calculate 2PA cross-sections using the fastest algorithm

Example 7.58 Non-degenerate, resonant 2PA scan over a range of frequency pairs.
$molecule
0 1
O
H 1 0.959
H 1 0.959 2 104.654
$end
$rem
METHOD
BASIS
EE_SINGLETS
CC_TRANS_PROP
CC_EOM_2PA
$end

eom-ccsd
aug-cc-pvtz
[2,0,0,0] Two A_1 states
1
Calculate transition properties
1
Calculate 2PA cross-sections using the fastest algorithm

$2pa
n_2pa_points 11
omega_1 500000 5000
$end

Example 7.59 Resonant 2PA with degenerate photons between two excited states.
$molecule
0 1
O
H 1 0.959
H 1 0.959 2 104.654
$end
$rem
METHOD
BASIS
EE_SINGLETS
CC_STATE_TO_OPT
CC_TRANS_PROP
CC_EOM_2PA
$end

eom-ccsd
aug-cc-pvtz
[2,0,0,0] Two A_1 states
[1,1]
"Reference" state for transition properties is 1A_1 state
1
Compute transition properties
1
Calculate 2PA cross-sections using the fastest algorithm

384

Chapter 7: Open-Shell and Excited-State Methods

Example 7.60 Computation of spin-orbit couplings between closed-shell singlet and MS = 1 triplet state in NH using
EOM-SF-CCSD
$molecule
0 3
N
H N 1.0450
$end
$rem
METHOD
BASIS
SF_STATES
CC_TRANS_PROP
CALC_SOC
CC_STATE_TO_OPT
$end

=
=
=
=
=
=

eom-ccsd
6-31g
[1,2,0,0]
true
true
[1,1]

Example 7.61 Computation of non-adiabatic couplings between EOM-EE states within triplet (first job) and singlet
(second job) manifolds
$molecule
+1 1
H
He
$end

0.00000
0.00000

$rem
JOBTYPE
BASIS
METHOD
INPUT_BOHR
EE_TRIPLETS
cc_eom_prop
SYM_IGNORE
CALC_NAC
eom_davidson_convergence
scf_convergence
cc_convergence
$end

0.00000
0.00000

0.0
3.0

=
=
=
=
=
=
=
=
=
=
=

FORCE
cc-pVDZ
EOM-CCSD
true
[2]
true
true Do not reorient molecule and turn off symmetry
2
Invoke Szalay NAC
9
tight davidson convergence
9
Hartree-Fock convergence threshold 1e-9
9

=
=
=
=
=
=
=
=
=
=

FORCE
cc-pVDZ
EOM-CCSD
true
[2]
singlets
true Do not reorient molecule and turn off symmetry
2
Invoke Szalay NAC
9
tight davidson convergence
9
Hartree-Fock convergence threshold 1e-9
9

@@@
$molecule
read
$end
$rem
JOBTYPE
BASIS
METHOD
INPUT_BOHR
EE_STATES
SYM_IGNORE
CALC_NAC
eom_davidson_convergence
scf_convergence
cc_convergence
$end

385

Chapter 7: Open-Shell and Excited-State Methods

Example 7.62 Calculation of the static dipole polarizability of the CCSD wave function of Helium.
$molecule
0 1
He
$end
$rem
METHOD
BASIS
CC_REF_PROP
CC_POL
CC_DIIS_SIZE
CC_FULLRESPONSE
$end

7.7.18

ccsd
cc-pvdz
1
2
15
1

EOM(2,3) Methods for Higher-Accuracy and Problematic Situations (CCMAN only)

In the EOM-CC(2,3) approach, 51 the transformed Hamiltonian H̄ is diagonalized in the basis of the reference, singly,
doubly, and triply excited determinants, i.e., the excitation operator R is truncated at triple excitations. The excitation
operator T , however, is truncated at double excitation level, and its amplitudes are found from the CCSD equations,
just like for EOM-CCSD [or EOM-CC(2,2)] method.
The accuracy of the EOM-CC(2,3) method closely follows that of full EOM-CCSDT [which can be also called EOMCC(3,3)], whereas computational cost of the former model is less.
The inclusion of triple excitations is necessary for achieving chemical accuracy (1 kcal/mol) for ground state properties.
It is even more so for excited states. In particular, triple excitations are crucial for doubly excited states, 51 excited states
of some radicals and SF calculations (diradicals, triradicals, bond-breaking) when a reference open-shell state is heavily
spin-contaminated. Accuracy of EOM-CCSD and EOM-CC(2,3) is compared in Table 7.7.18.
System
Singly-excited electronic states
Doubly-excited electronic states
Severe spin-contamination of the reference
Breaking single bond (EOM-SF)
Breaking double bond (EOM-2SF)

EOM-CCSD
0.1–0.2 eV
≥ 1 eV
∼ 0.5 eV
0.1–0.2 eV
∼ 1 eV

EOM-CC(2,3)
0.01 eV
0.1–0.2 eV
≤ 0.1 eV
0.01 eV
0.1–0.2 eV

Table 7.3: Performance of the EOM-CCSD and EOM-CC(2,3) methods

The applicability of the EOM-EE/SF-CC(2,3) models to larger systems can be extended by using their active-space
variants, in which triple excitations are restricted to semi-internal ones.
Since the computational scaling of EOM-CC(2,3) method is O(N 8 ), these calculations can be performed only for
relatively small systems. Moderate size molecules (10 heavy atoms) can be tackled by either using the active space
implementation or tiny basis sets. To achieve high accuracy for these systems, energy additivity schemes can be
used. For example, one can extrapolate EOM-CCSDT/large basis set values by combining large basis set EOM-CCSD
calculations with small basis set EOM-CCSDT ones.
Running the full EOM-CC(2,3) calculations is straightforward, however, the calculations are expensive with the bottlenecks being storage of the data on a hard drive and the CPU time. Calculations with around 80 basis functions are
possible for a molecule consisting of four first row atoms (NO dimer). The number of basis functions can be larger for
smaller systems.

Chapter 7: Open-Shell and Excited-State Methods

386

Note: In EE calculations, one needs to always solve for at least one low-spin root in the first symmetry irrep in order
to obtain the correlated EOM energy of the reference. The triples correction to the total reference energy must
be used to evaluate EOM-(2,3) excitation energies.
Note: EOM-CC(2,3) works for EOM-EE, EOM-SF, and EOM-IP/EA. In EOM-IP, “triples” correspond to 3h2p excitations, and the computational scaling of EOM-IP-CC(2,3) is less.

7.7.19

Active-Space EOM-CC(2,3): Tricks of the Trade (CCMAN only)

Active space calculations are less demanding with respect to the size of a hard drive. The main bottlenecks here are
the memory usage and the CPU time. Both arise due to the increased number of orbital blocks in the active space
calculations. In the current implementation, each block can contain from 0 up to 16 orbitals of the same symmetry
irrep, occupancy, and spin-symmetry. For example, for a typical molecule of C2v symmetry, in a small/moderate basis
set (e.g., TMM in 6-31G*), the number of blocks for each index is:
occupied: (α + β) × (a1 + a2 + b1 + b2 ) = 2 × 4 = 8
virtuals: (α + β) × (2a1 + a2 + b1 + 2b2 ) = 2 × 6 = 12
(usually there are more than 16 a1 and b2 virtual orbitals).
In EOM-CCSD, the total number of blocks is O2 V 2 = 82 × 122 = 9216. In EOM-CC(2,3) the number of blocks in the
EOM part is O3 V 3 = 83 × 123 = 884736. In active space EOM-CC(2,3), additional fragmentation of blocks occurs
to distinguish between the restricted and active orbitals. For example, if the active space includes occupied and virtual
orbitals of all symmetry irreps (this will be a very large active space), the number of occupied and virtual blocks for
each index is 16 and 20, respectively, and the total number of blocks increases to 3.3×107 . Not all of the blocks contain
real information, some blocks are zero because of the spatial or spin-symmetry requirements. For the C2v symmetry
group, the number of non-zero blocks is about 10–12 times less than the total number of blocks, i.e., 3 × 106 . This
is the number of non-zero blocks in one vector. Davidson diagonalization procedure requires (2*MAX_VECTORS +
2*NROOTS) vectors, where MAX_VECTORS is the maximum number of vectors in the subspace, and NROOTS is
the number of the roots to solve for. Taking NROOTS = 2 and MAX_VECTORS = 20, we obtain 44 vectors with the
total number of non-zero blocks being 1.3 × 108 .
In CCMAN implementation, each block is a logical unit of information. Along with real data, which are kept on a hard
drive at all the times except of their direct usage, each non-zero block contains an auxiliary information about its size,
structure, relative position with respect to other blocks, location on a hard drive, and so on. The auxiliary information
about blocks is always kept in memory. Currently, the approximate size of this auxiliary information is about 400 bytes
per block. It means, that in order to keep information about one vector (3 × 106 blocks), 1.2 Gb of memory is required!
The information about 44 vectors amounts 53 Gb. Moreover, the huge number of blocks significantly slows down the
code.
To make the calculations of active space EOM-CC(2,3) feasible, we need to reduce the total number of blocks. One
way to do this is to reduce the symmetry of the molecule to lower or C1 symmetry group (of course, this will result in
more expensive calculation). For example, lowering the symmetry group from C2v to Cs would results in reducing the
total number of blocks in active space EOM-CC(2,3) calculations in about 26 = 64 times, and the number of non-zero
blocks in about 30 times (the relative portion of non-zero blocks in Cs symmetry group is smaller compared to that in
C2v ).
Alternatively, one may keep the MAX_VECTORS and NROOTS parameters of Davidson’s diagonalization procedure
as small as possible (this mainly concerns the MAX_VECTORS parameter). For example, specifying MAX_VECTORS
= 12 instead of 20 would require 30% less memory.
One more trick concerns specifying the active space. In a desperate situation of a severe lack of memory, should the two
previous options fail, one can try to modify (increase) the active space in such a way that the fragmentation of active
and restricted orbitals would be less. For example, if there is one restricted occupied b1 orbital and one active occupied

Chapter 7: Open-Shell and Excited-State Methods

387

B1 orbital, adding the restricted b1 to the active space will reduce the number of blocks, by the price of increasing the
number of FLOPS. In principle, adding extra orbital to the active space should increase the accuracy of calculations,
however, a special care should be taken about the (near) degenerate pairs of orbitals, which should be handled in the
same way, i.e., both active or both restricted.

7.7.20

Job Control for EOM-CC(2,3)

EOM-CC(2,3) is invoked by METHOD=EOM-CC(2,3). The following options are available:
EOM_PRECONV_SD
Solves the EOM-CCSD equations, prints energies, then uses EOM-CCSD vectors as initial vectors in EOM-CC(2,3). Very convenient for calculations using energy additivity schemes.
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
n Do n SD iterations
RECOMMENDATION:
Turning this option on is recommended
CC_REST_AMPL
Forces the integrals, T , and R amplitudes to be determined in the full space even though the
CC_REST_OCC and CC_REST_VIR keywords are used.
TYPE:
LOGICAL
DEFAULT:
TRUE
OPTIONS:
FALSE Do apply restrictions
TRUE
Do not apply restrictions
RECOMMENDATION:
None
CC_REST_TRIPLES
Restricts R3 amplitudes to the active space, i.e., one electron should be removed from the active
occupied orbital and one electron should be added to the active virtual orbital.
TYPE:
INTEGER
DEFAULT:
1
OPTIONS:
1 Applies the restrictions
RECOMMENDATION:
None

Chapter 7: Open-Shell and Excited-State Methods

388

CC_REST_OCC
Sets the number of restricted occupied orbitals including frozen occupied orbitals.
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
n Restrict n occupied orbitals.
RECOMMENDATION:
None
CC_REST_VIR
Sets the number of restricted virtual orbitals including frozen virtual orbitals.
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
n Restrict n virtual orbitals.
RECOMMENDATION:
None

To select the active space, orbitals can be reordered by specifying the new order in the $reorder_mosection. The section
consists of two rows of numbers (α and β sets), starting from 1, and ending with n, where n is the number of the last
orbital specified.
Example 7.63 Example $reorder_mo section with orbitals 16 and 17 swapped for both α and β electrons
$reorder_mo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 17 16
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 17 16
$end

Chapter 7: Open-Shell and Excited-State Methods
7.7.20.1

389

Examples

Example 7.64 EOM-SF(2,3) calculations of methylene.
$molecule
0 3
C
H 1 CH
H 1 CH 2 HCH
CH = 1.07
HCH = 111.0
$end
$rem
METHOD
BASIS
SF_STATES
N_FROZEN_CORE
N_FROZEN_VIRTUAL
EOM_PRECONV_SD
$end

eom-cc(2,3)
6-31G
[2,0,0,2]
1
1
20 Get EOM-CCSD energies first (max_iter=20).

Example 7.65 This is active-space EOM-SF(2,3) calculations for methane with an elongated CC bond. HF MOs
should be reordered as specified in the $reorder_mosection such that active space for triples consists of sigma and
sigma* orbitals.
$molecule
0 3
C
H 1 CH
H 1 CHX
H 1 CH
H 1 CH
CH
HCH
A120
CHX
$end

=
=
=
=

2
2
2

HCH
HCH
HCH

3
4

A120
A120

1.086
109.4712206
120.
1.8

$rem
METHOD
BASIS
SF_STATES
N_FROZEN_CORE
EOM_PRECONV_SD
CC_REST_TRIPLES
CC_REST_AMPL
CC_REST_OCC
CC_REST_VIR
PRINT_ORBITALS
$end
$reorder_mo
1 2 5 4 3
1 2 3 4 5
$end

eom-cc(2,3)
6-31G*
[1,0]
1
20
does eom-ccsd first, max_iter=20
1
triples are restricted to the active space only
0
ccsd and eom singles and doubles are full-space
4
specifies active space
17 specifies active space
10 (number of virtuals to print)

390

Chapter 7: Open-Shell and Excited-State Methods

Example 7.66 EOM-IP-CC(2,3) calculation of three lowest electronic states of water cation.
$molecule
0 1
H
0.774767
O
0.000000
H -0.774767
$end
$rem
METHOD
BASIS
IP_STATES
$end

7.7.21

0.000000
0.000000
0.000000

0.458565
-0.114641
0.458565

eom-cc(2,3)
6-311G
[1,0,1,1]

Non-Iterative Triples Corrections to EOM-CCSD and CCSD

The effect of triple excitations to EOM-CCSD energies can be included via perturbation theory in an economical N 7
computational scheme. Using EOM-CCSD wave functions as zero-order wave functions, the second order triples
correction to the µth EOM-EE or SF state is:
∆Eµ(2) = −

abc
abc
(µ)
(µ)σijk
1 X X σ̃ijk
abc
36
Dijk − ωµ
i,j,k a,b,c

(7.53)

where i, j and k denote occupied orbitals, and a, b and c are virtual orbital indices. ωµ is the EOM-CCSD excitation
energy of the µth state. The quantities σ̃ and σ are:
abc
σ̃ijk
(µ)
abc
σijk
(µ)

= hΦ0 |(L1µ + L2µ )(He(T1 +T2 ) )c |Φabc
ijk i
=

(T1 +T2 )
hΦabc
(R0µ
ijk |[He

(7.54)

+ R1µ + R2µ )]c |Φ0 i

abc
where, the L and R are left and right eigen-vectors for µth state. Two different choices of the denominator, Dijk
,
abc
define the (dT) and (fT) variants of the correction. In (fT), Dijk
is just Hartree-Fock orbital energy differences.
A more accurate (but not fully orbital invariant) (dT) correction employs the complete three body diagonal of H̄,
(T1 +T2 )
abc
hΦabc
)C |Φabc
ijk |(He
ijk i, Dijk as a denominator. For the reference (e.g., a ground-state CCSD wave function), the
(fT) and (dT) corrections are identical to the CCSD(2)T and CR-CCSD(T)L corrections of Piecuch and coworkers. 102

The EOM-SF-CCSD(dT) and EOM-SF-CCSD(fT) methods yield a systematic improvement over EOM-SF-CCSD
bringing the errors below 1 kcal/mol. For theoretical background and detailed benchmarks, see Ref. 80.
Similar corrections are available for EOM-IP-CCSD, 81 where triples correspond to 3h2p excitations and EOM-EACCSD, where triples correspond to 2h3p excitations.
Note: Due to the orbital non-invariance problem, using (dT) correction is discouraged.
Note: EOM-IP-CCSD(fT) correction is now available both in CCMAN and CCMAN2

7.7.21.1

Job Control for Non-Iterative Triples Corrections

Triples corrections are requested by using METHOD or EOM_CORR:

.

Chapter 7: Open-Shell and Excited-State Methods
METHOD
Specifies the calculation method.
TYPE:
STRING
DEFAULT:
No default value
OPTIONS:
EOM-CCSD(DT) EOM-CCSD(dT), available for EE, SF, and IP
EOM-CCSD(FT) EOM-CCSD(fT), available for EE, SF, IP, and EA
EOM-CCSD(ST) EOM-CCSD(sT), available for IP
RECOMMENDATION:
None
EOM_CORR
Specifies the correlation level.
TYPE:
STRING
DEFAULT:
None No correction will be computed
OPTIONS:
SD(DT) EOM-CCSD(dT), available for EE, SF, and IP
SD(FT) EOM-CCSD(fT), available for EE, SF, IP, and EA
SD(ST) EOM-CCSD(sT), available for IP
RECOMMENDATION:
None

Note: In CCMAN2, EOM-IP-CCSD(fT) can be computed with or without USE_LIBPT = TRUE.

391

Chapter 7: Open-Shell and Excited-State Methods
7.7.21.2

Examples

Example 7.67 EOM-EE-CCSD(fT) calculation of CH+
$molecule
1 1
C
H C CH
CH
$end

= 2.137130

$rem
INPUT_BOHR
METHOD
BASIS
EE_STATES
EOM_DAVIDSON_MAX_ITER
$end

true
eom-ccsd(ft)
general
[1,0,1,1]
60 increase number of Davidson iterations

$basis
H
0
S
3

S

1.00
19.24060000
2.899200000
0.6534000000
1 1.00
0.1776000000
1 1.00
0.0250000000
1 1.00
1.00000000

S
P

0.3282800000E-01
0.2312080000
0.8172380000
1.000000000
1.000000000
1.00000000

****
C
0
S
6

S

1.00
4232.610000
634.8820000
146.0970000
42.49740000
14.18920000
1.966600000
1 1.00
5.147700000
1 1.00
0.4962000000
1 1.00
0.1533000000
1 1.00
0.0150000000
4 1.00
18.15570000
3.986400000
1.142900000
0.3594000000
1 1.00
0.1146000000
1 1.00
0.0110000000
1 1.00
0.750000000

S
S
S
P

P
P
D
****
$end

0.2029000000E-02
0.1553500000E-01
0.7541100000E-01
0.2571210000
0.5965550000
0.2425170000
1.000000000
1.000000000
1.000000000
1.000000000
0.1853400000E-01
0.1154420000
0.3862060000
0.6400890000
1.000000000
1.000000000
1.00000000

392

393

Chapter 7: Open-Shell and Excited-State Methods

Example 7.68 EOM-SF-CCSD(dT) calculations of methylene
$molecule
0 3
C
H 1 CH
H 1 CH 2 HCH
CH = 1.07
HCH = 111.0
$end
$rem
METHOD
BASIS
SF_STATES
N_FROZEN_CORE
N_FROZEN_VIRTUAL
$end

eom-ccsd(dt)
6-31G
[2,0,0,2]
1
1

Example 7.69 EOM-IP-CCSD(dT) calculations of Mg
$molecule
0 1
Mg
0.000000
$end
$rem
JOBTYPE
METHOD
BASIS
IP_STATES
$end

7.7.22

0.000000

0.000000

sp
eom-ccsd(dt)
6-31g
[1,0,0,0,0,1,1,1]

Potential Energy Surface Crossing Minimization

Potential energy surface crossing optimization procedure finds energy minima on crossing seams. On the seam, the
potential surfaces are degenerated in the subspace perpendicular to the plane defined by two vectors: the gradient
difference
∂
g=
(E1 − E2 )
(7.55)
∂q
and the derivative coupling

h=

Ψ1

∂H
Ψ2
∂q


(7.56)

At this time Q-C HEM is unable to locate crossing minima for states which have non-zero derivative coupling. Fortunately, often this is not the case. Minima on the seams of conical intersections of states of different multiplicity can
be found as their derivative coupling is zero. Minima on the seams of intersections of states of different point group
symmetry can be located as well.
To run a PES crossing minimization, CCSD and EOM-CCSD methods must be employed for the ground and excited
state calculations respectively.
Note: MECP optimization is only available for methods with analytic gradients. Finite-difference evaluation of two
gradients is not possible.

Chapter 7: Open-Shell and Excited-State Methods
7.7.22.1

394

Job Control Options

XOPT_STATE_1, XOPT_STATE_2
Specify two electronic states the intersection of which will be searched.
TYPE:
[INTEGER, INTEGER, INTEGER]
DEFAULT:
No default value (the option must be specified to run this calculation)
OPTIONS:
[spin, irrep, state]
spin = 0
Addresses states with low spin,
see also EE_SINGLETS or IP_STATES,EA_STATES.
spin = 1
Addresses states with high spin,
see also EE_TRIPLETS.
irrep
Specifies the irreducible representation to which
the state belongs, for C2v point group symmetry
irrep = 1 for A1 , irrep = 2 for A2 ,
irrep = 3 for B1 , irrep = 4 for B2 .
state
Specifies the state number within the irreducible
representation, state = 1 means the lowest excited
state, state = 2 is the second excited state, etc..
0, 0, -1
Ground state.
RECOMMENDATION:
Only intersections of states with different spin or symmetry can be calculated at this time.

Note: The spin can only be specified when using closed-shell RHF references. In the case of open-shell references
all states are treated together, see also EE_STATES. E.g., in SF calculations use spin=0 regardless of what is the
actual multiplicity of the target state.
XOPT_SEAM_ONLY
Orders an intersection seam search only, no minimization is to perform.
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
TRUE
Find a point on the intersection seam and stop.
FALSE Perform a minimization of the intersection seam.
RECOMMENDATION:
In systems with a large number of degrees of freedom it might be useful to locate the seam first
setting this option to TRUE and use that geometry as a starting point for the minimization.

395

Chapter 7: Open-Shell and Excited-State Methods
7.7.22.2

Examples

Example 7.70 Minimize the intersection of Ã1 B2 and B̃1 A2 states of the N+
3 ion using EOM-CCSD method
$molecule
1 1
N1
N2 N1 rnn
N3 N2 rnn N1 annn
rnn=1.46
annn=70.0
$end
$rem
JOBTYPE
METHOD
BASIS
EE_SINGLETS
XOPT_STATE_1
XOPT_STATE_2
XOPT_SEAM_ONLY
GEOM_OPT_TOL_GRADIENT
$end
$opt
CONSTRAINT
stre 1 2
stre 2 3
ENDCONSTRAINT
$end

opt
eom-ccsd
6-31g
[0,2,0,2]
[0,4,1]
[0,2,2]
true
100

C2v point group symmetry
1B2 low spin state
2A2 low spin state
Find the seam only

Set constraints on the N-N bond lengths
1.46
1.46

@@@
$molecule
READ
$end
$rem
JOBTYPE
METHOD
BASIS
EE_SINGLETS
XOPT_STATE_1
XOPT_STATE_2
GEOM_OPT_TOL_GRADIENT
$end

opt
eom-ccsd
6-31g
[0,2,0,2]
[0,4,1]
[0,2,2]
30

Optimize the intersection seam

396

Chapter 7: Open-Shell and Excited-State Methods

Example 7.71 Minimize the intersection of Ã2 A1 and B̃2 B1 states of the NO2 molecule using EOM-IP-CCSD method
$molecule
-1 1
N1
O2 N1
O3 N1

rno
rno

O2

aono

rno = 1.3040
aono = 106.7
$end
$rem
JOBTYPE
UNRESTRICTED
METHOD
BASIS
IP_STATES
EOM_FAKE_IPEA
XOPT_STATE_1
XOPT_STATE_2
GEOM_OPT_TOL_GRADIENT
$END

7.7.23

opt
true
eom-ccsd
6-31g
[1,0,1,0]
1
[0,1,1]
[0,3,1]
30

Optimize the intersection seam

C2v point group symmetry
1A1 low spin state
1B1 low spin state
Tighten gradient tolerance

Dyson Orbitals for Ionized or Attached States within the EOM-CCSD Formalism

Dyson orbitals can be used to compute total photodetachment/photoionization cross-sections, as well as angular distribution of photoelectrons. A Dyson orbital is the overlap between the N-electron molecular wave function and the
N − 1/N + 1 electron wave function of the corresponding cation/anion:
Z
1
d
φ (1) =
ΨN (1, . . . , n) ΨN −1 (2, . . . , n) d2 · · · dn
(7.57)
N −1
Z
1
ΨN (2, . . . , n + 1), ΨN +1 (1, . . . , n + 1) d2 · · · d(n + 1)
(7.58)
φd (1) =
N +1
For the Hartree-Fock wave functions and within Koopmans’ approximation, these are just the canonical HF orbitals.
For correlated wave functions, Dyson orbitals are linear combinations of the reference molecular orbitals:
X
φd =
γp φp
(7.59)
p

γp = ΨN p+ ΨN −1
γp = Ψ

N

pΨ

N +1

(7.60)
(7.61)

The calculation of Dyson orbitals is straightforward within the EOM-IP/EA-CCSD methods, where cation/anion and
initial molecule states are defined with respect to the same MO basis. Since the left and right CC vectors are not the
same, one can define correspondingly two Dyson orbitals (left and right):
γpR = Φ0 eT1 +T2 LEE p+ RIP eT1 +T2 Φ0
γpL = Φ0 eT1 +T2 LIP p REE eT1 +T2 Φ0

(7.62)

The norm of these orbitals is proportional to the one-electron character of the transition.
Dyson orbitals also offer qualitative insight visualizing the difference between molecular and ionized/attached states.
In ionization/photodetachment processes, these orbitals can be also interpreted as the wave function of the leaving
electron. For additional details, see Refs. 97 and 98. Dyson orbitals can be used for computing total and differential
photoelectron cross-sections using a stand-alone ezDyson code 40 .

Chapter 7: Open-Shell and Excited-State Methods

397

Dyson orbitals can be computed both for valence states and core-level states (see Section 7.7.5 for calculations of
Dyson orbitals within FC-CVS-EOM framework).

7.7.23.1

Dyson Orbitals Job Control

The calculation of Dyson orbitals is implemented for the ground (reference) and excited states ionization/electron
attachment. To obtain the ground state Dyson orbitals one needs to run an EOM-IP/EA-CCSD calculation, request
transition properties calculation by setting CC_TRANS_PROP = TRUE and CC_DO_DYSON = TRUE. The Dyson orbitals
decomposition in the MO basis is printed in the output, for all transitions between the reference and all IP/EA states.
At the end of the file, also the coefficients of the Dyson orbitals in the AO basis are available.
Two implementations of Dyson orbitals are currently available: (i) the original implementation in CCMAN; and (ii)
new implementation in CCMAN2. The CCMAN implementation is using a diffuse orbital trick (i.e., EOM_FAKE_IPEA
will be automatically set to TRUE in these calculations). Note: this implementation has a bug affecting the values of
norms of Dyson orbitals (the shapes are correct); thus, using this code is strongly discouraged. The CCMAN2 implementation has all types of initial states available: Dyson orbitals from ground CC, excited EOM-EE, and spin-flip
EOM-SF states; it is fully compatible with all helper features for EOM calculations, like FNO, RI, Cholesky decomposition. The CCMAN2 implementation can use a user-specified EOM guess (using EOM_USER_GUESS keyword
and $eom_user_guess section), which is recommended for highly excited states (such as core-ionized states). In addition, CCMAN2 can calculate Dyson orbitals involving meta-stable states (see Section 7.7.6) and core-level states (see
Section 7.7.5).
For calculating Dyson orbitals between excited or spin-flip states from the reference configuration and IP/EA states,
same CC_TRANS_PROP = TRUE and CC_DO_DYSON = TRUE keywords have to be added to the combination of usual
EOM-IP/EA-CCSD and EOM-EE-CCSD or EOM-SF-CCSD calculations. (However, note the separate keyword
CC_DO_DYSON_EE = TRUE for CCMAN.) The IP_STATES keyword is used to specify the target ionized states. The
attached states are specified by EA_STATES. The EA-SF states are specified by EOM_EA_BETA. The excited (or spinflipped) states are specified by EE_STATES and SF_STATES. The Dyson orbital decomposition in MO and AO bases
is printed for each EE/SF-IP/EA pair of states first for reference, then for all excited states in the order: CC-IP/EA1,
CC-IP/EA2,. . ., EE/SF1 - IP/EA1, EE/SF1 - IP/EA2,. . ., EE/SF2 - IP/EA1, EE/SF2 - IP/EA2,. . ., and so on. CCMAN
implementation keeps reference transitions separate, in accordance with separating keywords.
CC_DO_DYSON
CCMAN2: starts all types of Dyson orbitals calculations. Desired type is determined by requesting corresponding EOM-XX transitions CCMAN: whether the reference-state Dyson orbitals
will be calculated for EOM-IP/EA-CCSD calculations.
TYPE:
LOGICAL
DEFAULT:
FALSE (the option must be specified to run this calculation)
OPTIONS:
TRUE/FALSE
RECOMMENDATION:
none

Chapter 7: Open-Shell and Excited-State Methods

398

CC_DO_DYSON_EE
Whether excited-state or spin-flip state Dyson orbitals will be calculated for EOM-IP/EA-CCSD
calculations with CCMAN.
TYPE:
LOGICAL
DEFAULT:
FALSE (the option must be specified to run this calculation)
OPTIONS:
TRUE/FALSE
RECOMMENDATION:
none

Dyson orbitals are most easily visualized by setting GUI = 2 and reading the resulting checkpoint file into IQ MOL. In
addition to the canonical orbitals, the Dyson orbitals will appear under the Surfaces item in the Model View. For stepby-step instructions, see ezDyson manual 40 . Alternatively Dyson orbitals can be plotted using IANLTY = 200 and the
$plots utility. Only the sizes of the box need to be specified, followed by a line of zeros:
$plots
comment
10
-2
10
-2
10
-2
0
0
$plots

2
2
2
0

0

All Dyson orbitals on the Cartesian grid will be written in the resulting plot.mo file (only CCMAN). For RHF(UHF)
lr
rl
rl
lr
lr
reference, the columns order in plot.mo is: φlr
1 α (φ1 β) φ1 α (φ1 β) φ2 α (φ2 β) . . .
In addition, setting the MAKE_CUBE_FILES keyword to TRUE will create cube files for Dyson orbitals which can
be viewed with VMD or other programs (see Section 11.5.4 for details). This option is available for CCMAN and
rl
lr rl
CCMAN2. The Dyson orbitals will be written to files mo.1.cube, mo.2.cube, . . . in the order φlr
1 φ1 φ2 φ2 . . ..
For meta-stable states, the real and imaginary parts of the Dyson orbitals are written to separate files in the order
rl
lr
rl
lr
rl
lr
rl
Re(φlr
1 ) Re(φ1 ) Re(φ2 ) Re(φ2 ) . . . Im(φ1 ) Im(φ1 ) Im(φ2 ) Im(φ2 ) . . .
Visualization via the M OL D EN format is currently not available.

Chapter 7: Open-Shell and Excited-State Methods
7.7.23.2

399

Examples

Example 7.72 Plotting grd-ex and ex-grd state Dyson orbitals for ionization of the oxygen molecule. The target states
of the cation are 2 Ag and 2 B2u . Works for CCMAN only.
$molecule
0 3
O
0.000
O
1.222
$end

0.000
0.000

$rem
BASIS
METHOD
IP_STATES
CC_TRANS_PROP
CC_DO_DYSON
IANLTY
$end

0.000
0.000

6-31G*
eom-ccsd
[1,0,0,0,0,0,1,0] Target EOM-IP states
true request transition OPDMs to be calculated
true calculate Dyson orbitals
200

$plots
plots excited states densities and trans densities
10
-2
2
10
-2
2
10
-2
2
0
0
0
0
$plots

400

Chapter 7: Open-Shell and Excited-State Methods

Example 7.73 Plotting ex-ex state Dyson orbitals between the 1st 2 A1 excited state of the HO radical and the the 1st
A1 and A2 excited states of HO− . Works for CCMAN only.
$molecule
-1 1
H
0.000
O
1.000
$end

0.000
0.000

$rem
METHOD
BASIS
IP_STATES
EE_STATES
CC_TRANS_PROP
CC_DO_DYSON_EE
IANLTY
$end

0.000
0.000

$plots
plot excited
10
-2
10
-2
10
-2
0
0
$plots

eom-ccsd
6-31G*
[1,0,0,0]
[1,1,0,0]
true
true
200

states of HO radical
excited states of HOcalculate transition properties
calculate Dyson orbitals for ionization from ex. states

states densities and trans densities
2
2
2
0
0

Example 7.74 Dyson orbitals for ionization of CO molecule; A1 and B1 ionized states requested.
$molecule
0 1
O
C O 1.131
$end
$rem
CORRELATION
BASIS
PURECART
IP_STATES
CCMAN2
CC_DO_DYSON
CC_TRANS_PROP
PRINT_GENERAL_BASIS
$end

CCSD
cc-pVDZ
111
[1,0,1,0]
true
true
true
true

5d, will be required for ezDyson
(A1,A2,B1,B2)

necessary for Dyson orbitals job
will be required for ezDyson

401

Chapter 7: Open-Shell and Excited-State Methods

Example 7.75 Dyson orbitals for ionization of H2 O; core (A1 ) state requested — ionization from O(1s).
$molecule
0 1
O
H1 O 0.955
H2 O 0.955
$end

H1

104.5

$rem
CORRELATION
BASIS
PURECART
IP_STATES
EOM_USER_GUESS
CCMAN2
CC_DO_DYSON
CC_TRANS_PROP
PRINT_GENERAL_BASIS
$end

CCSD
cc-pVTZ
111
[1,0,0,0]
1
true
true
true
true

5d, will be required for ezDyson
(A1,A2,B1,B2)
on, further defined in $eom_user_guess

necessary for Dyson orbitals job
will be required for ezDyson

$eom_user_guess
1
$end

Example 7.76 Dyson orbitals for ionization of NO molecule using EOM-EA and a closed-shell cation reference; A1
and B2 states requested.
$molecule
+1 1
N
0.00000
O
0.00000
$end

0.00000
0.00000

$rem
CORRELATION
BASIS
PURECART
EA_STATES
CCMAN2
CC_DO_DYSON
CC_TRANS_PROP
PRINT_GENERAL_BASIS
$end

0.00000
1.02286

CCSD
aug-cc-pVTZ
111
5d, will be required for ezDyson
[1,0,0,1] (A1,A2,B1,B2)
true
true
true
necessary for Dyson orbitals job
true
will be required for ezDyson

Chapter 7: Open-Shell and Excited-State Methods

Example 7.77 Dyson orbitals for detachment from the meta-stable 2 Πg state of N−
2.
$molecule
0 1
N
0.0
N
0.0
GH 0.0
$end

0.0
0.0
0.0

0.55
-0.55
0.0

$rem
METHOD
EA_STATES
CC_MEMORY
MEM_STATIC
BASIS
COMPLEX_CCMAN
CC_TRANS_PROP
CC_DO_DYSON
MAKE_CUBE_FILES
IANLTY
$end

EOM-CCSD
[0,0,2,0,0,0,0,0]
5000
1000
GEN
TRUE
TRUE
TRUE
TRUE
200

$complex_ccman
CS_HF
CAP_TYPE
CAP_X
CAP_Y
CAP_Z
CAP_ETA
$end

1
1
2760
2760
4880
400

$plots
plot Dyson orbitals
50 -10.0 10.0
50 -10.0 10.0
50 -10.0 10.0
0 0 0 0
$end
$basis
N
0
S
8
1.000000
1.14200000E+04
1.71200000E+03
3.89300000E+02
1.10000000E+02
3.55700000E+01
1.25400000E+01
4.64400000E+00
5.11800000E-01
S
8
1.000000
1.14200000E+04
1.71200000E+03
3.89300000E+02
1.10000000E+02
3.55700000E+01
1.25400000E+01
4.64400000E+00
5.11800000E-01
S
1
1.000000
1.29300000E+00
S
1
1.000000
1.78700000E-01
P
3
1.000000
2.66300000E+01
5.94800000E+00
1.74200000E+00

5.23000000E-04
4.04500000E-03
2.07750000E-02
8.07270000E-02
2.33074000E-01
4.33501000E-01
3.47472000E-01
-8.50800000E-03
-1.15000000E-04
-8.95000000E-04
-4.62400000E-03
-1.85280000E-02
-5.73390000E-02
-1.32076000E-01
-1.72510000E-01
5.99944000E-01
1.00000000E+00
1.00000000E+00
1.46700000E-02
9.17640000E-02
2.98683000E-01

402

403

Chapter 7: Open-Shell and Excited-State Methods

Example 7.78 Dyson orbitals for ionization of triplet O2 and O−
2 at slightly stretched (relative to the equilibrium O2
geometry); B3g states are requested.
$comment
EOM-IP-CCSD/6-311+G* and EOM-EA-CCSD/6-311+G* levels of theory,
UHF reference. Start from O2:
1) detach electron - ionization of neutral (alpha IP).
2) attach electron, use EOM-EA w.f. as initial state
- ionization of anion (beta EA).
$end
$molecule
0 3
O
0.00000
O
0.00000
$end

0.00000
0.00000

$rem
CORRELATION
BASIS
PURECART
EOM_IP_ALPHA
EOM_EA_BETA
CCMAN2
CC_DO_DYSON
CC_TRANS_PROP
PRINT_GENERAL_BASIS
$end

0.00000
1.30000

CCSD
6-311(3+)G*
2222
6d, will be required for ezDyson
[0,0,0,1,0,0,0,0] (Ag,B1g,B2g,B3g,Au,B1u,B2u,B3u)
[0,0,0,1,0,0,0,0] (Ag,B1g,B2g,B3g,Au,B1u,B2u,B3u)
true
true
true
necessary for Dyson orbitals job
true
will be required for ezDyson

Example 7.79 Dyson orbitals for ionization of formaldehyde from the first excited state AND from the ground state
$molecule
0 1
O
1.535338855
C
1.535331598
H
1.535342484
H
1.535342484
$end
$rem
CORRELATION
BASIS
PURECART
CCMAN2
EE_STATES
EOM_IP_ALPHA
EOM_IP_BETA
CC_TRANS_PROP
CC_DO_DYSON
PRINT_GENERAL_BASIS
$end

0.000000000
-0.000007025
0.937663512
-0.937656488

CCSD
6-31G*
2222
true
[1]
[1]
[1]
true
true
true

-0.438858006
0.767790994
1.362651452
1.362672535

6d, will be required for ezDyson
new Dyson code

necessary for Dyson orbitals job
will be required for ezDyson

Chapter 7: Open-Shell and Excited-State Methods

404

Example 7.80 Dyson orbitals for core ionization of Li atom use Li+ as a reference, get neutral atom via EOM-EA get
1st excitation for the cation via EOM-EE totally: core ionization AND 1st ionization of Li atom
$molecule
+1 1
Li
0.00000
$end

0.00000

$rem
CORRELATION
BASIS
PURECART
CCMAN2
EE_STATES
EA_STATES
EOM_NGUESS_SINGLES
CC_TRANS_PROP
CC_DO_DYSON
PRINT_GENERAL_BASIS
$end

0.00000

CCSD
6-311+G*
2222
6d, will be required for ezDyson
true
new Dyson code
[1,0,0,0,0,0,0,0]
[1,0,0,0,0,0,0,0]
5
to converge to the lowest EA state
true
necessary for Dyson orbitals job
true
true
will be required for ezDyson

Example 7.81 Dyson orbitals for ionization of CH2 from high-spin triplet reference and from the lowest SF state
$molecule
0 3
C
H 1 rCH
H 1 rCH 2 aHCH
rCH
aHCH
$end

= 1.1167
= 102.07

$rem
CORRELATION
BASIS
SCF_GUESS
CCMAN2
CC_SYMMETRY
SF_STATES
EOM_IP_ALPHA
EOM_EA_BETA
CC_TRANS_PROP
CC_DO_DYSON
GUI
$end

7.7.24

CCSD
6-31G*
core
true
false
[1]
[2]
[2]
true
true
2

new Dyson code

one should be careful to request
meaningful spin for EA/IP state(s)
necessary for Dyson orbitals job

Interpretation of EOM/CI Wave Functions and Orbital Numbering

Analysis of the leading wave function amplitudes is always necessary for determining the character of the state (e.g.,
HOMO → LUMO excitation, open-shell diradical, etc.). The CCMAN module print out leading EOM/CI amplitudes
using its internal orbital numbering scheme, which is printed in the beginning. The typical CCMAN EOM-CCSD
output looks like:
Root 1 Conv-d yes Tot Ene= -113.722767530 hartree (Ex Ene 7.9548 eV),
U1^2=0.858795, U2^2=0.141205 ||Res||=4.4E-07
Right U1:

405

Chapter 7: Open-Shell and Excited-State Methods
Value
0.5358
0.5358
-0.2278
-0.2278

i
7(
7(
7(
7(

B2
B2
B2
B2

)
)
)
)

B
A
B
A

->
->
->
->
->

a
17(
17(
18(
18(

B2
B2
B2
B2

)
)
)
)

B
A
B
A

This means that this state is derived by excitation from occupied orbital #7 (which has b2 symmetry) to virtual orbital
#17 (which is also of b2 symmetry). The two leading amplitudes correspond to β → β and α → α excitation (the spin
part is denoted by A or B). The orbital numbering for this job is defined by the following map:
The orbitals are ordered and numbered as follows:
Alpha orbitals:
Number Energy
Type
Symmetry ANLMAN number
0
-20.613
AOCC
A1
1A1
1
1
-11.367
AOCC
A1
2A1
2
2
-1.324
AOCC
A1
3A1
3
3
-0.944
AOCC
A1
4A1
4
4
-0.600
AOCC
A1
5A1
5
5
-0.720
AOCC
B1
1B1
6
6
-0.473
AOCC
B1
2B1
7
7
-0.473
AOCC
B2
1B2
8
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21

0.071
0.100
0.290
0.327
0.367
0.454
0.808
1.196
1.295
1.562
2.003
0.100
0.319
0.395
0.881
1.291
1.550
0.040
0.137
0.330
0.853
1.491

AVIRT
AVIRT
AVIRT
AVIRT
AVIRT
AVIRT
AVIRT
AVIRT
AVIRT
AVIRT
AVIRT
AVIRT
AVIRT
AVIRT
AVIRT
AVIRT
AVIRT
AVIRT
AVIRT
AVIRT
AVIRT
AVIRT

A1
A1
A1
A1
A1
A1
A1
A1
A1
A1
A1
B1
B1
B1
B1
B1
B1
B2
B2
B2
B2
B2

6A1
7A1
8A1
9A1
10A1
11A1
12A1
13A1
14A1
15A1
16A1
3B1
4B1
5B1
6B1
7B1
8B1
2B2
3B2
4B2
5B2
6B2

Total number:

9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30

The first column is CCMAN’s internal numbering (e.g., 7 and 17 from the example above). This is followed by the
orbital energy, orbital type (frozen, restricted, active, occupied, virtual), and orbital symmetry. Note that the orbitals
are blocked by symmetries and then ordered by energy within each symmetry block, (i.e., first all occupied a1 , then all
a2 , etc.), and numbered starting from 0. The occupied and virtual orbitals are numbered separately, and frozen orbitals
are excluded from CCMAN numbering. The two last columns give numbering in terms of the final ANLMAN printout
(starting from 1), e.g., our occupied orbital #7 will be numbered as 1B2 in the final printout. The last column gives
the absolute orbital number (all occupied and all virtuals together, starting from 1), which is often used by external
visualization routines.

406

Chapter 7: Open-Shell and Excited-State Methods

CCMAN2 numbers orbitals by their energy within each irrep keeping the same numbering for occupied and virtual
orbitals. This numbering is exactly the same as in the final printout of the SCF wave function analysis. Orbital energies
are printed next to the respective amplitudes. For example, a typical CCMAN2 EOM-CCSD output will look like that:
EOMEE-CCSD transition 2/A1
Total energy = -75.87450159 a.u. Excitation energy = 11.2971 eV.
R1^2 = 0.9396 R2^2 = 0.0604 Res^2 = 9.51e-08
Amplitude
0.6486
0.6486
-0.1268
-0.1268

Orbitals with energies
1 (B2) A
-0.5101
1 (B2) B
-0.5101
3 (A1) A
-0.5863
3 (A1) B
-0.5863

->
->
->
->

2 (B2)
0.1729
2 (B2)
0.1729
4 (A1)
0.0404
4 (A1)
0.0404

A
B
A
B

which means that for this state, the leading EOM amplitude corresponds to the transition from the first b2 orbital (orbital energy −0.5101) to the second b2 orbital (orbital energy 0.1729).
The most complete analysis of EOM-CC calculations is afforded by deploying a general wave-function analysis tool
contained in the libwa module and described in Section 11.2.6. The EOM-CC state analysis is activated by setting
STATE_ANALYSIS = TRUE. In addition, keywords controlling calculations of state and interstate properties should be
set up accordingly.
Note: Wave function analysis is only available for CCMAN2.
Example 7.82 Wave function analysis of the EOM-IP states (He+
3 ).
$molecule
0 1
He
He
1
He
2

R1
R1

1

A

R1 = 1.236447
A = 180.00
$end
$rem
METHOD
BASIS
IP_STATES
CC_EOM_PROP
CC_STATE_TO_OPT
CC_TRANS_PROP
STATE_ANALYSIS
MOLDEN_FORMAT
NTO_PAIRS
$end

7.8

=
=
=
=
=
=
=
=
=

EOM-CCSD
6-31G
[1,0,0,0,0,1,0,0]
true
Analyze state properties (state OPDM)
[1,1] Compute transition properties wrt 1st EOM state of 1st irrep
true
Analyze transitions (transition OPDM)
true
true
2

Correlated Excited State Methods: The ADC(n) Family

The ADC(n) family of correlated excited state methods is a series of size-extensive excited state methods based on
perturbation theory. Each order n of ADC presents the excited state equivalent to the well-known nth order Møller-

407

Chapter 7: Open-Shell and Excited-State Methods

Plesset perturbation theory for the ground state. Currently, the ADC variants ADC(0), ADC(1), ADC(2)-s, ADC(2)-x
and ADC(3) are implemented in Q-C HEM. 42,137 The “resolution-of-the-identity” approximation can be used with any
ADC variant. Additionally, there are spin-opposite scaling versions of both ADC(2) variants available. 57,137 Coreexcited states for the simulation of X-ray absorption spectra can be computed exploiting the core-valence separation
(CVS) approximation. Currently, the CVS-ADC(1), CVS-ADC(2)-s, CVS-ADC(2)-x and CVS-ADC(3) methods are
available. 130–132,137

7.8.1

The Algebraic Diagrammatic Construction (ADC) Scheme

The Algebraic Diagrammatic Construction (ADC) scheme of the polarization propagator is an excited state method
originating from Green’s function theory. It has first been derived employing the diagrammatic perturbation expansion
of the polarization propagator using the Møller-Plesset partition of the Hamiltonian. 111 An alternative derivation is
available in terms of the intermediate state representation (ISR), 112 which will be presented in the following.
As starting point for the derivation of ADC equations via ISR serves the exact N electron ground state ΨN
0 . From
ΨN
a
complete
set
of
correlated
excited
states
is
obtained
by
applying
physical
excitation
operators
Ĉ
.
J
0

with

Ψ̄N
= ĈJ ΨN
J
0

(7.63)

n o n
o
ĈJ = c†a ci ; c†a c†b ci cj , i < j, a < b; . . .

(7.64)

Yet, the resulting excited states do not form an orthonormal basis. To construct an orthonormal basis out of the |Ψ̄N
J i
the Gram-Schmidt orthogonalization scheme is employed successively on the excited states in the various excitation
classes starting from the exact ground state, the singly excited states, the doubly excited states etc.. This procedure
eventually yields the basis of intermediate states {|Ψ̃N
J i} in which the Hamiltonian of the system can be represented
forming the Hermitian ADC matrix
D
E
N
N
MIJ = Ψ̃N
Ĥ
−
E
Ψ̃
(7.65)
I
0
J
Here, the Hamiltonian of the system is shifted by the exact ground state energy E0N . The solution of the secular ISR
equation
MX = XΩ, with X† X = 1
(7.66)
yields the exact excitation energies Ωn as eigenvalues. From the eigenvectors the exact excited states in terms of the
intermediate states can be constructed as
E
X
ΨN
XnJ Ψ̃N
(7.67)
n =
J
J

This also allows for the calculation of dipole transition moments via
X † D
N
N
=
XnJ Ψ̃N
Tn = ΨN
n µ̂ Ψ0
J µ̂ Ψ0 ,

(7.68)

J

as well as excited state properties via
N
On = ΨN
n ô Ψn =

X

D
†
N
XnI
XnJ Ψ̃N
I ô ΨJ ,

(7.69)

I,J

where On is the property associated with operator ô.
Up to now, the exact N -electron ground state has been employed in the derivation of the ADC scheme, thereby resulting
in exact excitation energies and exact excited state wave functions. Since the exact ground state is usually not known,
a suitable approximation must be used in the derivation of the ISR equations. An obvious choice is the nth order
Møller-Plesset ground state yielding the nth order approximation of the ADC scheme. The appropriate ADC equations
have been derived in detail up to third order in Refs. 125–127. Due to the dependency on the Møller-Plesset ground
state the nth order ADC scheme should only be applied to molecular systems whose ground state is well described by
the respective MP(n) method.

Chapter 7: Open-Shell and Excited-State Methods

408

As in Møller-Plesset perturbation theory, the first ADC scheme which goes beyond the non-correlated wave function
methods in Section 7.2 is ADC(2). ADC(2) is available in a strict and an extended variant which are usually referred
to as ADC(2)-s and ADC(2)-x, respectively. The strict variant ADC(2)-s scales with the 5th power of the basis set.
The quality of ADC(2)-s excitation energies and corresponding excited states is comparable to the quality of those
obtained with CIS(D) (Section 7.6) or CC2. More precisely, excited states with mostly single excitation character are
well-described by ADC(2)-s, while excited states with double excitation character are usually found to be too high
in energy. The ADC(2)-x variant which scales as the sixth power of the basis set improves the treatment of doubly
excited states, but at the cost of introducing an imbalance between singly and doubly excited states. As result, the
excitation energies of doubly excited states are substantially decreased in ADC(2)-x relative to the states possessing
mostly single excitation character with the excitation energies of both types of states exhibiting relatively large errors.
Still, ADC(2)-x calculations can be used as a diagnostic tool for the importance doubly excited states in the low-energy
region of the spectrum by comparing to ADC(2)-s results. A significantly better description of both singly and doubly
excited states is provided by the third order ADC scheme ADC(3). The accuracy of excitation energies obtained with
ADC(3) is almost comparable to CC3, but at computational costs that scale with the sixth power of the basis set only. 42

7.8.2

Resolution of the Identity ADC Methods

Similar to MP2 and CIS(D), the ADC equations can be reformulated using the resolution-of-the-identity (RI) approximation. This significantly reduces the cost of the integral transformation and the storage requirements. Although it
does not change the overall computational scaling of O(N 5 ) for ADC(2)-s or O(N 6 ) for ADC(2)-x with the system
size, employing the RI approximation will result in computational speed-up of calculations of larger systems.
The RI approximation can be used with all available ADC methods. It is invoked as soon as an auxiliary basis set is
specified using AUX_BASIS.

7.8.3

Spin Opposite Scaling ADC(2) Models

The spin-opposite scaling (SOS) approach originates from MP2 where it was realized that the same spin contributions
can be completely neglected, if the opposite spin components are scaled appropriately. In a similar way it is possible
to simplify the second order ADC equations by neglecting the same spin contributions in the ADC matrix, while the
opposite-spin contributions are scaled with appropriate semi-empirical parameters. 47,57,135
Starting from the SOS-MP2 ground state the same scaling parameter cT = 1.3 is introduced into the ADC equations to
scale the t2 amplitudes. This alone, however, does not result in any computational savings or substantial improvements
of the ADC(2) results. In addition, the opposite spin components in the ph/2p2h and 2p2h/ph coupling blocks have to
be scaled using a second parameter cc to obtain a useful SOS-ADC(2)-s model. With this model the optimal value of
the parameter cc has been found to be 1.17 for the calculation of singlet excited states. 135
To extend the SOS approximation to the ADC(2)-x method yet another scaling parameter cx for the opposite spin
components of the off-diagonal elements in the 2p2h/2p2h block has to be introduced. Here, the optimal values of the
scaling parameters have been determined as cc = 1.0 and cx = 0.9 keeping cT unchanged. 57
The spin-opposite scaling models can be invoked by setting METHOD to either SOSADC(2) or SOSADC(2)-x. By default,
the scaling parameters are chosen as the optimal values reported above, i.e. cT = 1.3 and cc = 1.17 for ADC(2)-s and
cT = 1.3, cc = 1.0, and cx = 0.9 for ADC(2)-x. However, it is possible to adjust any of the three parameters by setting
ADC_C_T, ADC_C_C, or ADC_C_X, respectively.

7.8.4

Core-Excitation ADC Methods

Core-excited electronic states are located in the high energy X-ray region of the spectrum. Thus, to compute coreexcited states using standard diagonalization procedures, which usually solve for the energetically lowest-lying excited

409

Chapter 7: Open-Shell and Excited-State Methods

states first, requires the calculation of a multitude of excited states. This is computationally very expensive and only
feasible for calculations on very small molecules and small basis sets.
The core-valence separation (CVS) approximation solves the problem by neglecting the couplings between core and
valence excited states a priori. 3,27 Thereby, the ADC matrix acquires a certain block structure which allows to solve
only for core-excited states. The application of the CVS approximation is justified, since core and valence excited states
are energetically well separated and the coupling between both types of states is very small. To achieve the separation
of core and valence excited states the CVS approximation forces the following types of two-electron integrals to zero
hIp|qri = hpI|qri = hpq|Iri = hpq|rIi = 0
hIJ|pqi = hpq|IJi = 0

(7.70)

hIJ|Kpi = hIJ|pKi = hIp|JKi = hpI|JKi = 0,
where capital letters I, J, K refer to core orbitals while lower-case letters p, q, r denote non-core occupied or virtual
orbitals.
The core-valence approximation is currently available of ADC models up to third order (including the extended variant). 130–132 It can be invoked by setting METHOD to the respective ADC model prefixed by CVS. Besides the general
ADC related keywords, two additional keywords in the $rem block are necessary to control CVS-ADC calculations:
• ADC_CVS = TRUE switches on the CVS-ADC calculation
• CC_REST_OCC = n controls the number of core orbitals included in the excitation space. The integer n corresponds to the n energetically lowest core orbitals.
Example: cytosine with the molecular formula C4 H5 N3 O includes one oxygen atom. To calculate O 1s core-excited
states, CC_REST_OCC has to be set to 1, because the 1s orbital of oxygen is the energetically lowest. To obtain the N 1s
core excitations, the integer has to be set to 4, because the 1s orbital of the oxygen atom is included as well, since it
is energetically below the three 1s orbitals of the nitrogen atoms. Accordingly, to simulate the C 1s XAS spectrum of
cytosine, CC_REST_OCC must be set to 8.
To obtain the best agreement with experimental data, one should use the CVS-ADC(2)-x method in combination with
at least a diffuse triple-ζ basis set. 130–132

7.8.5

Spin-Flip ADC Methods

The spin-flip (SF) method 58–60,70 is used for molecular systems with few-reference wave functions like diradicals,
bond-breaking, rotations around single bonds, and conical intersections. Starting from a triplet (ms = 1) ground state
reference a spin-flip excitation operator {ĈJ } = {c†aβ ciα ; c†aβ c†bσ ciα cjσ , a < b, i < j} is introduced, which flipped
the spin of one electron while singlet and (ms = 0) triplet excited target states are yielded. The spin-flip method is
implemented for the ADC(2) (strict and extended) and the ADC(3) methods. 70 Note that high-spin (ms = 1) triplet
states can be calculated with the SF-ADC method as well using a closed-shell singlet reference state. The number of
spin-flip states that shall be calculated is controlled with the $rem variable SF_STATES.

7.8.6

Properties and Visualization

The calculation of excited states using the ADC MAN module yields by default the usual excitation energies and the
excitation amplitudes, as well as the transition dipole moments, oscillator strengths, and the norm of the doubles
part of the amplitudes (if applicable). In addition, the calculation of excited state properties, like dipole moments,
and transition properties between excited states can be requested by setting the $rem variables ADC_PROP_ES and
ADC_PROP_ES2ES, respectively. Resonant two-photon absorption cross-sections of the excited states can be computed
as well, using either sum-over-states expressions or the matrix inversion technique. The calculation via sum-over-state
expressions is automatically activated, if ADC_PROP_ES2ES is set. The accuracy of the results, however, strongly

410

Chapter 7: Open-Shell and Excited-State Methods

depends on the number of states which are included in the summation, i.e. the number of states computed. At least, 2030 excited states (per irreducible representation) are required to yield useful results for the two-photon absorption crosssections. Alternatively, the resonant two-photon absorption cross-sections can be calculated by setting ADC_PROP_TPA
to TRUE. In this case, the computation of a large number of excited states is avoided and there is no dependence on
the number of excited states. Instead, an additional linear matrix equation has to be solved for every excited state for
which the two-photon absorption cross-section is computed. Thus, the obtained resonant two-photon absorption crosssections are usually more reliable. The quantity printed out is the microscopic cross-section (also known as rotationally
averaged 2PA strength). Specifically, the value 30 × δTmP is printed out where δTmP is defined in Eq. (13) of Ref. 137.
Furthermore, the ADC MAN module allows for the detailed analysis of the excited states and export of various types of
excited state related orbitals and densities. This can be activated by setting the keyword STATE_ANALYSIS. Details on
the available analyses and export options can be found in section 11.2.6.

7.8.7

Excited States in Solution with ADC/SS-PCM

ADC MAN is interfaced to the versatile polarizable-continuum model (PCM) implemented in Q-C HEM (Section 12.2.2)
and may thus be employed for the calculation and analysis of excited-state wave functions and transitions in solution,
or more general in dielectric environments. The interface follows the state-specific approach, and supports a selfconsistent equilibration of the solvent field for long-lived excited states commonly referred to as equilibrium solvation,
as well as the calculation of perturbative corrections for vertical transitions, known as non-equilibrium solvation (see
also section 12.2.2.3). Combining both approaches, virtually all photochemically relevant processes can be modeled,
including ground- and excited-state absorption, fluorescence, phosphorescence, as well as photochemical reactivity.
Requiring only the electron-densities of ground- and excited states of the solute as well as and the dielectric constant
and refractive index of the solvent, ADC/SS-PCM is straightforward to set up and supports all orders and variants of
ADC for which densities are available via the ISR. This includes all levels of canonical ADC, SOS-ADC, SF-ADC for
electronically complicated situations, as well as CVS-ADC for the description of core-excited states with and without
the resolution of the identity and frozen-core approximation, restricted closed-shell references as well unrestricted openshell. The computed solvent-relaxed wave functions can be visualized and analyzed using the interface to LIBWFA.
Although we give a short introduction to the theory in this section, it is limited to a brief, qualitative overview with only
the most important equations, leaving out major aspects such as the polarization work. For a comprehensive, formal
introduction to the theory please be referred to Refs. 91 and 92 as well as Sections 12.2.2 and 12.2.2.3.
7.8.7.1

Modeling the Absorption Spectrum in Solution

(A) Theory
Let us begin with a brief review of the theoretical and technical aspects of the calculation of absorption spectra in
solution. For this purpose, one would typically employ the perturbative, state-specific approach in combination with an
ADC of second or third order. The first step is a self-consistent reaction field calculation [SCRF, Hartree-Fock with a
PCM, Eq. (7.71)].
E0 = h0|Ĥ vac + R̂(0)|0i
(7.71)
The PCM is formally represented by the reaction- or solvent-field operator R̂. In practice, R̂ is a set of point charges
placed on the molecular surface, which are optimized together with the orbitals during the SCF procedure. Note that
since R̂ accounts for the self-induced polarization of the solute, it depends on solute’s wave function (here the ground
state), which will in the following be indicated in the subscript R̂0 .
After the SCRF is converged, the final surface charges and the respective operator can, according to the Franck-Condon
principle, be separated into a “slow” solvent-nuclei related ( − n2 ) and “fast”, solvent-electron related (n2 ) component
(using Marcus partitioning, eq. (7.72)), and are stored on disk.
R̂(0) ≡ R̂0tot = R̂0f + R̂0s
qf =

n2 − 1 total
q
−1

qs =

 − n2 total
q
= q − qf
−1

(7.72)
(7.73)

411

Chapter 7: Open-Shell and Excited-State Methods

The “polarized” MOs resulting from the SCRF step are subjected to ordinary MP/ADC calculations, which yield the
correlation energy for the ground and excitation energies for the excited states, which are both added to the HF energy
to obtain the total MP ground and ADC excited-state energies. However, since the MOs contain the interaction with
the complete “frozen” solvent-polarization of the SCF ground-state density (R̂tot , i.e., both components), the resulting
excitation energies violate the Franck-Condon principle, which requires the solvents electronic degrees of freedom (fast
component of the polarization) to be relaxed. Furthermore, the solvent field has been obtained for the SCF density,
which more often than not provides a poor description of the electrostatic nature of the solute and may in turn lead to
systematic errors in the excitation energies.
To account for the relaxation of the solvent electrons and bring the excitation/excited-state energies in accordance with
Franck-Condon, we employ the perturbative ansatz shown in eq. (7.74), in which the fast component of the ground-state
solvent field R̂0f is replaced by the respective quantity computed for the excited-state density R̂if . In this framework, the
energy of an excited state i computed with the polarized MOs can be identified as the zeroth-order energy (eq. (7.76)),
while the first-order term becomes the difference between the interaction of the zeroth-order excited-state density with
the fast component of the ground- and excited-state solvent fields given in eq. (7.77).
EiNEq = hi|Ĥ vac + R̂0tot + λ(R̂if − R̂0f )|ii
NEq(0)

Ei

(0)
Ei
(1)
Ei

=
=
=

(0)

+ λEi

hi

(0)

|Ĥ

vac

hi

(0)

|R̂if

Ei

(1)

−

+

(7.74)

+ ...

(7.75)

R̂0tot |i(0) i

(7.76)

R̂0f |i(0) i

(7.77)

After the ADC iterations have converged, R̂if is computed from the respective excited-state density and R̂0f read from
disk to form the first-order correction. Adding this so-called ptSS-term to the zeroth order energy, one arrives at the
vertical energy in the non-equilibrium limit EiNEq . Since the ptSS-term accounts for the response of the electron density
of the implicit solvent molecules to excitation of the solute, its always negative and thus lowers the energy of the excited
state w.r.t. the ground state.
To eliminate problems resulting from the poor description of the ground-state solvent field at the SCF/HF level of
theory, we use an additive correction that is based on the MP2 ground-state density. In a nutshell, it replaces the
interaction between the potential of the difference density of the excited state V̂i(0) − V̂0MP with the SCF solvent field
Q̂0HF by the respective interaction with the MP solvent field Q̂0MP :
EPTD =

(V̂i(0) − V̂0MP ) · Q̂0MP

(7.78)

−

(V̂i(0) − V̂0MP ) · Q̂0HF

(7.79)

We will in the following refer to this as “perturbative, density-based” (PTD) correction and denote the respective
approach as ptSS(PTD). Accordingly, the non-PTD corrected results will be denoted ptSS(PTE).
In addition to the ptSS-PCM discussed above, a perturbative variant of the so-called linear response corrections (termed
ptLR presented in Ref. 91) is also available. In contrast to the ptSS approach, the ptLR corrections depend on the
transition rather than the state (difference) densities. Although the ptLR corrections are always computed and printed,
we discourage their use with the correlated ADC variants (2 and higher), for which the ptSS approach is better suited.
The ptSS and ptLR approaches are also available for TDDFT as described in Section 12.2.2.3.
A detailed, comprehensive introduction to the theory and implementation, as well as extensive benchmark data for the
non-equilibrium formalism in combination with all orders of ADC can be found in Ref. 91.
(B) Usage
The calculation of vertical transition energies with the ptSS-PCM approach is fairly straightforward. One just needs
to activate the PCM (set SOLVENT_METHOD in the $rem block to PCM), enable the non-equilibrium functionality
(set NONEQUILIBRIUM in the $pcm block to TRUE), and specify the solvent parameters, i.e., dielectric constant 
(DIELECTRIC) and squared refractive index n2 (DIELECTRIC_INFI) in the $solvent block.
Note: Symmetry will be disabled for all calculations with a PCM.
In the output of a ptSS-PCM calculation with any correlated ADC variant, multiple values are given for the total and
transition energies depending in the level of theory. For the correlated ADC variants these include:

Chapter 7: Open-Shell and Excited-State Methods

412

• Zeroth-order results, direct outcome of the ADC calculation with solvated orbitals, excited states are ordered
according to this value.
• First-order results, incl. only ptSS non-equilibrium corrections, termed ptSS(PTE).
• Corrected first-order results, incl. also the correlation correction, termed ptSS(PTD).
• Scaled and corrected first-order results, incl. an empirical scaling of the correlation correction, termed ptSS(PTD*).
We recommend to use the corrected first-order results since the PTD correction generally yields the most accurate results. The ptSS(PTD) approach is typically a very good approximation to the fully consistent, but more expensive PTED
scheme, in which the solvent field is made self-consistent with the MP2 density (see next section for the PTED scheme
and sample-jobs for a comparison of the two). Oscillator strengths are computed using the ptSS(PTD) energies. The
PTD* approach includes an empirical scaling of the PTD correction that was developed to improve the solvatochromicshifts of a series of nitro-aromatics with a cc-pVDZ basis. 91 However, we later found that for most other systems, the
scaling slightly worsens the agreement with the fully consistent PTED scheme. In addition to the compiled total and
transition energies, all contributing terms (ptSS, PTD etc.) are given separately. An even more verbose output detailing
all the integrals contributing to the 1st order corrections can be obtained by increasing PCM_PRINT to 1.
Note: The zeroth-order results are by no means identical to the gas-phase excitation energies, and in turn the ptSSterm is not the solvatochromic shift.
(C) Tips and Tricks
To model the absorption spectrum in polar solvents, it is advisable to use a structure optimized in the presence of a
PCM since the influence can be quite significant.
It should also be taken into account that a PCM, being a purely electrostatic model, lacks at the description of explicit
interactions like e.g. hydrogen bonds. If fairly strong h-bond donors/acceptors are present and a protic/Lewis-basic solvent is to be modeled, consider adding a few (one or maximum two per donor/acceptor site) explicit solvent molecules
(and optimize them together with the molecule in the presence of a PCM). A systematic investigation of this aspect for
two representative examples can be found in Ref. 91.
If large differences between the HF and MP description of the molecule exist (PTD terms > 0.2 eV), it is advisable
to employ the iterative ptSS(PTED) scheme described in the next section. Due to the inverse nature of the systematic
errors of ADC(2) and ADC(3), the best guess for the excitation energy in solution is usually the average of both values.
For the PCM, we recommend the formally exact and slightly more expensive integral-equation formalism (IEF)-PCM
variant (THEORY to IEFPCM in the $pcm block) in place of the approximate C-PCM, and otherwise default parameters.

7.8.7.2

Modeling Emission, Excited-State Absorption and Photochemical Reactivity

(A) Theory
To model emission/absorption of solvent-equilibrated excited states and/or to investigate their photochemical reactivity,
both components of the polarization have to be relaxed with respect to the desired state. This becomes evident considering that a full solvent-field equilibration (including the nuclear component) is essentially a geometry optimization
for the implicit solvent, and should thus be employed whenever the geometry of the solute is optimized for the desired
excited state. The Hamiltonian for a solvent-equilibrated state |ki simply reads
EkEqS = k Ĥ vac + R̂k k .

(7.80)

Since the interaction with solvent field of any state has to be introduced to the MOs in the SCF step of a calculation, a
solvent-field equilibration for excited states (and correlated ground states) is an iterative procedure requiring multiple
SCF+ADC calculations until convergence is achieved. This also means that a guess (typically from a ptSS calculation)
for the solvent field has to generated and used in the first SCF step. The MOs resulting from this first SCF are subjected
to an ADC calculation, providing a first excited-state density, for which a new solvent field is computed and employed
in the SCF step of the second iteration. This procedure is repeated until the solvent field (charges) and energies are

Chapter 7: Open-Shell and Excited-State Methods

413

converged and ultimately provides the total energy and wave function of the solvent-equilibrated excited state |ki, as
well as the out-of-equilibrium wave functions of other states. However, as the name already suggest, this state-specific
approach yields a meaningful energy only for the solvent-equilibrated reference state |ki. All other states have to be
treated in the non-equilibrium limit and subjected to the formalism presented in the previous section to be consistent
with the Franck-Condon principle. The respective generalization of the perturbative ansatz for the Hamiltonian for the
ith out-of-equilibrium state (e.g. the ground or any other excited state) in the field of the equilibrated state |ki reads as
NEq(k)

Ei

= i|Ĥ vac + R̂ktot + λ(R̂if − R̂kf )|i ,

(7.81)

which can be solved using the procedure introduced in the previous section.
While most of the technical aspects concerning the application of the model will be covered in the following, we highly
recommend to read at least the formalism and implementation section of Ref. 92 before using the model.
(B) Usage
The main switch for the state-specific equilibrium solvation (SS-PCM) is the variable EQSOLV in the $pcm block.
Setting it to TRUE will cause the SCF+ADC calculation to be carried out using the solvent field of the first excited
state (if that is found on disk), while any integer > 1 triggers the automatic solvent-field optimization and is interpreted
as the maximum number of steps. We recommend to use EQSOLV = 15. Note that to use the SS-PCM, the PCM
(SOLVENT_METHOD = PCM in $rem) and its non-equilibrium functionalities (NONEQUILIBRIUM = TRUE in $pcm )
have to be activated as well. Since the solvent-field iterations require an initial guess, a SS-PCM calculation is always
the second (or third...) step of a multi-step job.
Note: Any SS-PCM calculation requires a preceding converged ptSS-PCM calculation (i.e., EQSOLV = FALSE) for
the desired state to provide a guess for the initial solvent field, or it will crash during the SCF.
To create the input-file for a multi step job, add "@@@" at the end of the input for the first job and append the input
for the second job. See also section 3.6. Note that the solvent field used in the subsequent step is stored in the basis of
the molecular-surface elements and thus, the geometry of the molecule as well as parameters that affect construction
and discretization of the molecular cavity must not be changed between the jobs/steps. This, however, is not enforced.
The state for which the solvent field is to be optimized is specified using the variable EQSTATE in the $pcm block.
A value of 0 refers to the MP ground-state (for PTED calculations), 1 selects the energetically lowest excited state
(default), 2 the second lowest, and so on. The solvent field can be optimized for any singlet, triplet or spin-flip excited
state. However, only the desired class of states should be requested, i.e., either EE_SINGLETS or EE_TRIPLETS for
singlet references, and either EE_STATES or SF_STATES for triplet references. To compute, e.g., the phosphorescence
energy, only triplet states should be requested and EQSTATE would typically be set to 1.
Note: Computing multiple classes of excited states during the solvent-field iterations will confuse the state-ordering
logic of the program and yield the wrong results.
Convergence is controlled by EQS_CONV. Criteria are the SCF energy as well as the RMSD, MAD and largest single
difference of the surface-charges. The convergence should not need to be modified. It is per default derived from the
SCF convergence parameter (SCF_CONVERGENCE−4). The default value of 4 (since SCF_CONVERGENCE is 8 for
ADC calculations) corresponds to an maximum energy change of 2.72 meV and will yield converged total energies for
all states. Excitation energies and in particular the total energy of the reference state converge somewhat faster than the
SCF energy, and a value of 3 may save some time for the computation e.g. the emission energy of large solutes.
A typical ADC/SS-PCM calculation consists of three steps/subsequent jobs:

1. Generation of an initial guess for the solvent field using a non-equilibrium calculation (EQSTATE = FALSE). To
save time, this would typically be done with a smaller basis set and lower convergence criteria (e.g. ADC_DAVIDSON_CONVERG
4).
2. Converging the solvent field for the desired state. In this step it is advisable to compute as few states as possible
(maybe one more than EQSTATE to be aware of looming state crossings), and more importantly, only the desired
class of excited states.

Chapter 7: Open-Shell and Excited-State Methods

414

3. Computing all excited states of interest and their properties in the previously converged solvent field. If the
reference state of the solvent-field equilibration is a singlet state, this is straightforward and any number of
singlet/triplet states can be computed in this final step without further input. If singlet states are to be computed
in the solvent field of a triplet state, the additional variable EQS_REF in $pcm has to be set to tell the program
which state is to be treated as the reference state in this last step. For this purpose, EQS_REF is set to the desired
triplet state plus the number of converged singlet states. Hence, assuming 2 singlet states are to be calculated
along with the triplets in previously converged solvent field of T1 , and furthermore that both singlets converge,
EQS_REF needs to be set to 3 (this can not be realized using the variable EQSTATE, because we still want to use
the solvent field of the lowest triplet state stored in the previous step).
The self-consistent SS-PCM can also be used to compute a fully consistent solvent field for the MP2 ground state by
setting EQSTATE to 0. This is known as ptSS(PTED) approach and can improve vertical excitation energies if there are
large differences between the electrostatic properties of the SCF and MP ground states (large PTD corrections). In most
cases, however, the non-iterative PTD approach is a very good approximation to the PTED approach (see the sample
jobs below). The output of a PTED calculation is essentially identical to that of a ptSS calculation.
The program possesses limited intelligence in detecting the type of the calculation (PTED or EQS/SS-PCM) as well
as the target state of the solvent-field equilibration and will assemble and designate the ptSS corrections, total energies
and transition energies accordingly. This logic will be confused if multiple classes of states (e.g. singlet and triplets)
are computed simultaneously during the iterations, and/or if the ordering of the states changes. Nevertheless, in a final
job for a previously converged solvent field of a singlet reference singlet and triplet states can be computed together
yielding the correct output. However, if singlet states are computed in the converged solvent field of a triplet reference
the additional variable EQS_REF has to be set (and only then, see above).
In the “HF/MP2/MP3 Summary” section, zeroth (without ptSS) and first (with ptSS) order total energies of the respective ground state in the solvent field of the target state are given along with the ptSS term for a vertical transition from
the equilibrated state (emission). Note that to obtain the MP3 ground state energy during an ADC(3)/EQS calculation,
the ptSS term has to be added manually (it is printed in the MP2 Summary since we use the MP2 density for this
purpose).
In the “Excited-State Summary” section the reference state is distinguished from the out-of-equilibrium states. Respective total and transition energies are given along with the non-equilibrium corrections, transition moments and some
remarks. Note that for emission, in contrast to absorption, the ptSS term increases the transition energy as it lowers the
energy of the out-of-equilibrium ground state. The so-called "self-ptSS term" is a perturbative estimate of how much
the solvent field of this state is off from its equilibrium. Although the line in the output changes depending on the value
(from "not converged" to "reasonably converged" to "converged") it is not used in the actual convergence check. Note
that the self-ptSS term is computed with n2 (dielectric_infi) and not , as it probably should be. The self-ptSS term may
be used to judge how well a solvent field computed with a different methodology (basis, ADC order/variant) fits. In
such a case, values < 0.01 eV signal a reasonable agreement.
To calculate inter-state transition moments for excited-state absorption, the variable ADC_PROP_ES2ES has to be set to
TRUE. Unfortunately, transition energies have to be computed manually from the (first-order) total energies given in the
excited-state summary, since the transition energies given along with the state-to-state transition moments following
the excite-state summary are incorrect (missing non-equilibrium terms).
The progress of the solvent-field iteration and their convergence is reported following the “Excited-State Summary”.
(C) Tips and Tricks
To compute fluorescence and phosphorescence energies, solute geometry AND solvent field should both be optimized
for the suspected emitting state. Since hardly any programs can perform excited-state optimizations with the SS-PCM
solvent models, you will probably have to use gas-phase geometries. In our experience, the errors introduced by this
approximation are small to negligible (typically < 0.1 eV) in non-polar solvents, but can become significant in polar
solvents, in particular for polar charge-transfer states.
Concerning the predicted emission energies, we found that ADC(2)/SS-PCM typically over-stabilizes CT states, yielding emission energies that are too low. SOS-ADC(2) can improve this error, but does not eliminate it. In general, while

Chapter 7: Open-Shell and Excited-State Methods

415

emission energies are more accurate with (SOS-)ADC(2)/SS-PCM than with ADC(3)/SS-PCM, the latter affords better
relative state energies (see Ref. 92).
Keep in mind that the solute-solvent interaction of polar solvents with polar (e.g. charge-transfer) states can become
quite large (multiple eV), and may thus affect the ordering and nature of the excited states. This is quite typical for
charge-transfer states even in remotely polar solvents. If they are not the lowest state in the non-equilibrium calculation,
but say S2 , it is typically necessary to do one solvent-field iteration for the CT state (EQSOLV = 1 and EQSTATE = 2),
which brings the CT down to become S1 , and then continue the iterations with EQSOLV = 15 and EQSTATE = 1. It
is in general advisable to carefully check if the character and/or energetic ordering of the states changes during the
procedure, in particular for any equilibration of the solvent field for all but the lowest excited state (e.g. S2 or S3 ). But
even the solvent-field equilibration for a weakly polar S1 in a polar solvent can cause a more polar state with the same
dipole-vector to become the lowest state.
If the excited-states swap during the procedure, find out in which step the swap occurred and do only so many iterations
(i.e., set EQSOLV accordingly). In a following job, adjust the variable EQSTATE and continue the iterations. If states
start to mix when they get close, it might help to first set an artificially large dielectric constant to induce the change in
ordering, and then continue with the desired dielectric constant in a following job.
If performance is critical, the calculations may be accelerated by lowering the ADC convergence during the solvent-field
iterations (set ADC_DAVIDSON_CONVERGENCE = 5). The number of iterations may be reduced by first converging the
solvent field with a smaller basis/at a lower level of ADC followed by another job with the full basis/level of ADC.
However, in our experience ADC(2) and ADC(3) solvent fields for the same state differ quite significantly and the
approach probably does not help much. In contrast, the solvent field computed with a smaller basis (e.g. 3-21G, SVP)
is often a good approximation for that computed with a larger basis (e.g. def2-TZVP, see examples), such this may
actually help. It is in general advisable to compute just as many states as is necessary during the solvent-field iterations
and include higher lying excited states and triplets in the final job. In ADC(2) calculations for large systems, one should
always employ the resolution-of-the-identity approximation.
To save time in PTED jobs, it is suggested to disable the computation of excited states during the solvent-field iterations
of a PTED job by setting EE_SINGLETS (and/or EE_TRIPLETS/ EE_STATES) to 0 and compute the excited-states in a
final job once the reaction field is converged.
For more tips and examples see the sample jobs.

7.8.8

Frozen-Density Embedding: FDE-ADC methods

FDE-ADC 106 is a method to include interactions between an embedded species and its environment into an ADC(n)
calculation based on Frozen Density Embedding Theory (FDET). 133,134 FDE-ADC supports ADC and CVS-ADC
methods of orders 2s,2x and 3 and regular ADC job control keywords also apply.
The FDE-ADC method starts with generating an embedding potential using a MP(n) density for the embedded system
(A) and a DFT or HF density for the environment (B). A Hartree-Fock calculation is then carried out during which
the embedding potential is added to the Fock operator. The embedded Hartree-Fock orbitals act as an input for the
subsequent ADC calculation which yields the embedded properties (vertical excitation energies, oscillator strengths,
etc.). Further information on the FDE-ADC method and FDE-ADC job control are described in Section 12.7.1.

7.8.9

ADC Job Control

For an ADC calculation it is important to ensure that there are sufficient resources available for the necessary integral calculations and transformations. These resources are controlled using the $rem variables MEM_STATIC and
MEM_TOTAL. The memory used by ADC is currently 95% of the difference MEM_TOTAL - MEM_STATIC.
An ADC calculation is requested by setting the $rem variable METHOD to the respective ADC variant. Furthermore, the
number of excited states to be calculated has to be specified using one of the $rem variables EE_STATES, EE_SINGLETS,
or EE_TRIPLETS. The former variable should be used for open-shell or unrestricted closed-shell calculations, while

Chapter 7: Open-Shell and Excited-State Methods

416

the latter two variables are intended for restricted closed-shell calculations. Even though not recommended, it is
possible to use EE_STATES in a restricted calculation which translates into EE_SINGLETS, if neither EE_SINGLETS nor
EE_TRIPLETS is set. Similarly, the use EE_SINGLETS in an unrestricted calculation will translate into EE_STATES, if
the latter is not set as well.
All $rem variables to set the number of excited states accept either an integer number or a vector of integer numbers.
A single number specifies that the same number of excited states are calculated for every irreducible representation the
point group of the molecular system possesses (molecules without symmetry are treated as C1 symmetric). In contrast,
a vector of numbers determines the number of states for each irreducible representation explicitly. Thus, the length
of the vector always has to match the number of irreducible representations. Hereby, the excited states are labeled
according to the irreducible representation of the electronic transition which might be different from the irreducible
representation of the excited state wave function. Users can choose to calculate any molecule as C1 symmetric by
setting CC_SYMMETRY = FALSE.
METHOD
Controls the order in perturbation theory of ADC.
TYPE:
STRING
DEFAULT:
None
OPTIONS:
ADC(1)
Perform ADC(1) calculation.
ADC(2)
Perform ADC(2)-s calculation.
ADC(2)-x
Perform ADC(2)-x calculation.
ADC(3)
Perform ADC(3) calculation.
SOS-ADC(2)
Perform spin-opposite scaled ADC(2)-s calculation.
SOS-ADC(2)-x Perform spin-opposite scaled ADC(2)-x calculation.
CVS-ADC(1)
Perform ADC(1) calculation of core excitations.
CVS-ADC(2)
Perform ADC(2)-s calculation of core excitations.
CVS-ADC(2)-x Perform ADC(2)-x calculation of core excitations.
RECOMMENDATION:
None
EE_STATES
Controls the number of excited states to calculate.
TYPE:
INTEGER/ARRAY
DEFAULT:
0 Do not perform an ADC calculation
OPTIONS:
n>0
Number of states to calculate for each irrep or
[n1 , n2 , ...] Compute n1 states for the first irrep, n2 states for the second irrep, ...
RECOMMENDATION:
Use this variable to define the number of excited states in case of unrestricted or open-shell calculations. In restricted calculations it can also be used, if neither EE_SINGLETS nor EE_TRIPLETS
is given. Then, it has the same effect as setting EE_SINGLETS.

Chapter 7: Open-Shell and Excited-State Methods
EE_SINGLETS
Controls the number of singlet excited states to calculate.
TYPE:
INTEGER/ARRAY
DEFAULT:
0 Do not perform an ADC calculation of singlet excited states
OPTIONS:
n>0
Number of singlet states to calculate for each irrep or
[n1 , n2 , ...] Compute n1 states for the first irrep, n2 states for the second irrep, ...
RECOMMENDATION:
Use this variable to define the number of excited states in case of restricted calculations of singlet
states. In unrestricted calculations it can also be used, if EE_STATES not set. Then, it has the same
effect as setting EE_STATES.
EE_TRIPLETS
Controls the number of triplet excited states to calculate.
TYPE:
INTEGER/INTEGER ARRAY
DEFAULT:
0 Do not perform an ADC calculation of triplet excited states
OPTIONS:
n>0
Number of triplet states to calculate for each irrep or
[n1 , n2 , ...] Compute n1 states for the first irrep, n2 states for the second irrep, ...
RECOMMENDATION:
Use this variable to define the number of excited states in case of restricted calculations of triplet
states.
CC_SYMMETRY
Activates point-group symmetry in the ADC calculation.
TYPE:
LOGICAL
DEFAULT:
TRUE If the system possesses any point-group symmetry.
OPTIONS:
TRUE
Employ point-group symmetry
FALSE Do not use point-group symmetry
RECOMMENDATION:
None
ADC_PROP_ES
Controls the calculation of excited state properties (currently only dipole moments).
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
TRUE
Calculate excited state properties.
FALSE Do not compute state properties.
RECOMMENDATION:
Set to TRUE, if properties are required.

417

Chapter 7: Open-Shell and Excited-State Methods
ADC_PROP_ES2ES
Controls the calculation of transition properties between excited states (currently only transition
dipole moments and oscillator strengths), as well as the computation of two-photon absorption
cross-sections of excited states using the sum-over-states expression.
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
TRUE
Calculate state-to-state transition properties.
FALSE Do not compute transition properties between excited states.
RECOMMENDATION:
Set to TRUE, if state-to-state properties or sum-over-states two-photon absorption cross-sections
are required.
ADC_PROP_TPA
Controls the calculation of two-photon absorption cross-sections of excited states using matrix
inversion techniques.
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
TRUE
Calculate two-photon absorption cross-sections.
FALSE Do not compute two-photon absorption cross-sections.
RECOMMENDATION:
Set to TRUE, if to obtain two-photon absorption cross-sections.
STATE_ANALYSIS
Controls the analysis and export of excited state densities and orbitals (see 11.2.6 for details).
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
TRUE
Perform excited state analyses.
FALSE No excited state analyses or export will be performed.
RECOMMENDATION:
Set to TRUE, if detailed analysis of the excited states is required or if density or orbital plots are
needed.
ADC_C_T
Set the spin-opposite scaling parameter cT for an SOS-ADC(2) calculation. The parameter value
is obtained by multiplying the given integer by 10−3 .
TYPE:
INTEGER
DEFAULT:
1300 Optimized value cT = 1.3.
OPTIONS:
n Corresponding to n · 10−3
RECOMMENDATION:
Use the default.

418

Chapter 7: Open-Shell and Excited-State Methods
ADC_C_C
Set the spin-opposite scaling parameter cc for the ADC(2) calculation. The parameter value is
obtained by multiplying the given integer by 10−3 .
TYPE:
INTEGER
DEFAULT:
1170 Optimized value cc = 1.17 for ADC(2)-s or
1000 cc = 1.0 for ADC(2)-x
OPTIONS:
n Corresponding to n · 10−3
RECOMMENDATION:
Use the default.
ADC_C_X
Set the spin-opposite scaling parameter cx for the ADC(2)-x calculation. The parameter value is
obtained by multiplying the given integer by 10−3 .
TYPE:
INTEGER
DEFAULT:
1300 Optimized value cx = 0.9 for ADC(2)-x.
OPTIONS:
n Corresponding to n · 10−3
RECOMMENDATION:
Use the default.
ADC_NGUESS_SINGLES
Controls the number of excited state guess vectors which are single excitations. If the number
of requested excited states exceeds the total number of guess vectors (singles and doubles), this
parameter is automatically adjusted, so that the number of guess vectors matches the number of
requested excited states.
TYPE:
INTEGER
DEFAULT:
Equals to the number of excited states requested.
OPTIONS:
n User-defined integer.
RECOMMENDATION:

ADC_NGUESS_DOUBLES
Controls the number of excited state guess vectors which are double excitations.
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
n User-defined integer.
RECOMMENDATION:

419

Chapter 7: Open-Shell and Excited-State Methods
ADC_DO_DIIS
Activates the use of the DIIS algorithm for the calculation of ADC(2) excited states.
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
TRUE
Use DIIS algorithm.
FALSE Do diagonalization using Davidson algorithm.
RECOMMENDATION:
None.
ADC_DIIS_START
Controls the iteration step at which DIIS is turned on.
TYPE:
INTEGER
DEFAULT:
1
OPTIONS:
n User-defined integer.
RECOMMENDATION:
Set to a large number to switch off DIIS steps.
ADC_DIIS_SIZE
Controls the size of the DIIS subspace.
TYPE:
INTEGER
DEFAULT:
7
OPTIONS:
n User-defined integer
RECOMMENDATION:
None
ADC_DIIS_MAXITER
Controls the maximum number of DIIS iterations.
TYPE:
INTEGER
DEFAULT:
50
OPTIONS:
n User-defined integer.
RECOMMENDATION:
Increase in case of slow convergence.

420

Chapter 7: Open-Shell and Excited-State Methods
ADC_DIIS_ECONV
Controls the convergence criterion for the excited state energy during DIIS.
TYPE:
INTEGER
DEFAULT:
6 Corresponding to 10−6
OPTIONS:
n Corresponding to 10−n
RECOMMENDATION:
None
ADC_DIIS_RCONV
Convergence criterion for the residual vector norm of the excited state during DIIS.
TYPE:
INTEGER
DEFAULT:
6 Corresponding to 10−6
OPTIONS:
n Corresponding to 10−n
RECOMMENDATION:
None
ADC_DAVIDSON_MAXSUBSPACE
Controls the maximum subspace size for the Davidson procedure.
TYPE:
INTEGER
DEFAULT:
5× the number of excited states to be calculated.
OPTIONS:
n User-defined integer.
RECOMMENDATION:
Should be at least 2 − 4× the number of excited states to calculate. The larger the value the more
disk space is required.
ADC_DAVIDSON_MAXITER
Controls the maximum number of iterations of the Davidson procedure.
TYPE:
INTEGER
DEFAULT:
60
OPTIONS:
n Number of iterations
RECOMMENDATION:
Use the default unless convergence problems are encountered.

421

Chapter 7: Open-Shell and Excited-State Methods
ADC_DAVIDSON_CONV
Controls the convergence criterion of the Davidson procedure.
TYPE:
INTEGER
DEFAULT:
6 Corresponding to 10−6
OPTIONS:
n ≤ 12 Corresponding to 10−n .
RECOMMENDATION:
Use the default unless higher accuracy is required or convergence problems are encountered.
ADC_DAVIDSON_THRESH
Controls the threshold for the norm of expansion vectors to be added during the Davidson procedure.
TYPE:
INTEGER
DEFAULT:
Twice the value of ADC_DAVIDSON_CONV, but at maximum 10−14 .
OPTIONS:
n ≤ 14 Corresponding to 10−n
RECOMMENDATION:
Use the default unless convergence problems are encountered. The threshold value 10−n should
always be smaller than the convergence criterion ADC_DAVIDSON_CONV.
ADC_PRINT
Controls the amount of printing during an ADC calculation.
TYPE:
INTEGER
DEFAULT:
1 Basic status information and results are printed.
OPTIONS:
0 Quiet: almost only results are printed.
1 Normal: basic status information and results are printed.
2 Debug: more status information, extended information on timings.
...
RECOMMENDATION:
Use the default.
ADC_CVS
Activates the use of the CVS approximation for the calculation of CVS-ADC core-excited states.
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
TRUE
Activates the CVS approximation.
FALSE Do not compute core-excited states using the CVS approximation.
RECOMMENDATION:
Set to TRUE, if to obtain core-excited states for the simulation of X-ray absorption spectra. In
the case of TRUE, the $rem variable CC_REST_OCC has to be defined as well.

422

Chapter 7: Open-Shell and Excited-State Methods
CC_REST_OCC
Sets the number of restricted occupied orbitals including active core occupied orbitals.
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
n Restrict n energetically lowest occupied orbitals to correspond to the active core space.
RECOMMENDATION:
Example: cytosine with the molecular formula C4 H5 N3 O includes one oxygen atom. To calculate O 1s core-excited states, n has to be set to 1, because the 1s orbital of oxygen is the
energetically lowest. To obtain the N 1s core excitations, the integer n has to be set to 4, because
the 1s orbital of the oxygen atom is included as well, since it is energetically below the three 1s
orbitals of the nitrogen atoms. Accordingly, to simulate the C 1s spectrum of cytosine, n must
be set to 8.
SF_STATES
Controls the number of excited spin-flip states to calculate.
TYPE:
INTEGER
DEFAULT:
0 Do not perform a SF-ADC calculation
OPTIONS:
n>0
Number of states to calculate for each irrep or
[n1 , n2 , ...] Compute n1 states for the first irrep, n2 states for the second irrep, ...
RECOMMENDATION:
Use this variable to define the number of excited states in the case of a spin-flip calculation.
SF-ADC is available for ADC(2)-s, ADC(2)-x and ADC(3).

Keywords for SS-PCM control in $pcm:
EQSOLV
Main switch of the self-consistent SS-PCM procedure.
INPUT SECTION: $pcm
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
0
No self-consistent SS-PCM.
1
Single SS-PCM calculation (SCF+ADC) with the solvent field found on disk.
n >1 Do a maximum of n automatic solvent-field iterations.
RECOMMENDATION:
We recommend to use 15 steps max. Typical convergence is 3-5 steps. In difficult cases
6-12. If the solvent-field iteration do not converge in 15 steps, something is wrong.
Also make sure that a solvent field has been stored on disk by a previous job.

423

Chapter 7: Open-Shell and Excited-State Methods
EQSTATE
Specifies the state for which the solvent field is to be optimized.
INPUT SECTION: $pcm
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
0
MP2 ground state (for PTED approach)
1
energetically lowest excited state
2
2nd lowest excited state
...
RECOMMENDATION:
Given that only one class of excited states is calculated, the state will be selected according to its energetic position shown in the “Exited-State Summary” of the output file. A
maximum of 99 states is stored and can be selected.
EQS_CONV
Controls the convergence of the solvent-field iterations by setting the convergence criteria (a mixture of SCF energy and charge-vector). SCF energy criterion computes as
10−value eH
INPUT SECTION: $pcm
TYPE:
INTEGER
DEFAULT:
SCF_CONVERGENCE−4 = 4
OPTIONS:
3 May be sufficient for emission energies
4 Assured converged total energies (2.7 meV)
5 Really tight
RECOMMENDATION:
Use the default.
EQS_REF
Allows to specify which state is to be treated as the reference state in the ADC part of
the calculation. Does in contrast to EQSTATE not affect which solvent field is loaded in
the SCF step. Only has to be used when singlets are computed in the solvent field of a
triplet reference. Note that (converged) singlets states are always counted before triplets,
and thus to select T1 in a calculation with EE_SINGLETS = 2 this has to be set to 3.
INPUT SECTION: $pcm
TYPE:
INTEGER
DEFAULT:
Same as EQSTATE
OPTIONS:
1 First excited state
2 Second excited state
...
RECOMMENDATION:
Only needed when computing singlet states in the solvent field of a triplet reference.

424

Chapter 7: Open-Shell and Excited-State Methods

7.8.10

425

Examples

Example 7.83 Input for an ADC(2)-s calculation of singlet exited states of methane with D2 symmetry. In total six
excited states are requested corresponding to four (two) electronic transitions with irreducible representation B1 (B2 ).
$molecule
0 1
C
H
1 r0
H
1 r0
H
1 r0
H
1 r0

2
2
2

d0
d0
d0

3
4

d1
d1

r0 = 1.085
d0 = 109.4712206
d1 = 120.0
$end
$rem
METHOD
BASIS
MEM_TOTAL
MEM_STATIC
EE_SINGLETS
$end

adc(2)
6-31g(d,p)
4000
100
[0,4,2,0]

Example 7.84 Input for an unrestricted RI-ADC(2)-s calculation with C1 symmetry using DIIS. In addition, excited
state properties and state-to-state properties are computed.
$molecule
0 2
C
0.0
N
0.0
$end

0.0
0.0

$rem
METHOD
BASIS
AUX_BASIS
MEM_TOTAL
MEM_STATIC
CC_SYMMETRY
EE_STATES
ADC_DO_DIIS
ADC_PROP_ES
ADC_PROP_ES2ES
ADC_PROP_TPA
$end

-0.630969
0.540831

adc(2)
aug-cc-pVDZ
rimp2-aug-cc-pVDZ
4000
100
false
6
true
true
true
true

426

Chapter 7: Open-Shell and Excited-State Methods

Example 7.85 Input for a restricted CVS-ADC(2)-x calculation with C1 symmetry using 4 parallel CPU cores. In this
case, the 10 lowest nitrogen K-shell singlet excitations are computed.
$molecule
0 1
C
-5.17920
C
-3.85603
N
-2.74877
C
-5.23385
C
-2.78766
N
-4.08565
N
-3.73433
O
-1.81716
H
-4.50497
H
-2.79158
H
-4.10443
H
-6.08637
H
-6.17341
$end

2.21618
2.79078
2.08372
0.85443
0.70838
0.13372
4.14564
-0.02560
4.74117
4.50980
-0.88340
2.82445
0.29221

$rem
METHOD
EE_SINGLETS
ADC_DAVIDSON_MAXSUBSPACE
MEM_TOTAL
MEM_STATIC
THREADS
CC_SYMMETRY
BASIS
ADC_DAVIDSON_THRESH
SYMMETRY
ADC_DAVIDSON_MAXITER
ADC_CVS
CC_REST_OCC
$end

0.01098
0.05749
0.05569
-0.04040
0.01226
-0.03930
0.16144
0.00909
-0.12037
0.04490
-0.07575
0.02474
-0.07941

cvs-adc(2)-x
10
60
10000
1000
4
false
6-31G*
8
false
900
true
4

Example 7.86 Input for a restricted SF-ADC(2)-s calculation of the first three spin-flip target states of cyclobutadiene
without point group symmetry.
$molecule
0 3
C
0.000000
C
1.439000
C
1.439000
C
0.000000
H
-0.758726
H
2.197726
H
2.197726
H
-0.758726
$end
$rem
METHOD
MEM_TOTAL
MEM_STATIC
CC_SYMMETRY
BASIS
SF_STATES
$end

0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000

0.000000
0.000000
1.439000
1.439000
-0.758726
-0.758726
2.197726
2.197726

adc(2)
15000
1000
false
3-21G
3

The next example provides input for a restricted ADC(2)-x calculation of water with Cs symmetry. Four singlet A00

Chapter 7: Open-Shell and Excited-State Methods

427

excited states and two triplet A0 excited states are requested. For the first two states (1 1A00 and 1 3A0 ) the transition
densities as well as the attachment and detachment densities are exported into cube files.
Example 7.87 Restricted ADC(2)-x calculation of water with Cs symmetry.
$molecule
0 1
O
0.000
H
0.000
H
0.896
$end

0.000
0.000
0.000

$rem
METHOD
BASIS
THREADS
MEM_TOTAL
MEM_STATIC
EE_SINGLETS
EE_TRIPLETS
ADC_PROP_ES
MAKE_CUBE_FILES
$end

0.000
0.950
-0.317

adc(2)-x
6-31g(d,p)
2
3000
100
[0,4]
[2,0]
true
true

$plots
Plot transition and a/d densities
40 -3.0 3.0
40 -3.0 3.0
40 -3.0 3.0
0 0 2 2
1 2
1 2
$end

The next sample provides input for a ADC(2)-s/ptSS-PCM calculation of the five lowest singlet-excited states of N, N dimethylnitroaniline in diethyl ether. The PCM settings are all default values except THEORY, which is set to IEFPCM

428

Chapter 7: Open-Shell and Excited-State Methods
instead of the default CPCM.
Example 7.88 DC(2)-s/ptSS-PCM calculation of N, N -dimethylnitroaniline in diethyl ether.
$molecule
0 1
C
-4.263068
C
-5.030982
C
-4.428196
C
-3.009941
C
-2.243560
C
-2.871710
H
-4.740854
H
-2.502361
H
-1.166655
H
-6.104933
N
-5.178541
C
-6.632186
H
-6.998203
H
-7.038179
H
-7.001616
C
-4.531894
H
-3.912683
H
-5.298508
H
-3.902757
N
-2.070815
O
-0.842606
O
-2.648404
$end

2.512843
1.361365
0.076338
0.019036
1.171441
2.416638
3.480454
-0.932570
1.104642
1.461766
-1.053870
-0.969550
-0.462970
-1.975370
-0.431420
-2.358860
-2.476270
-3.126680
-2.507480
3.621238
3.510489
4.710370

$rem
THREADS
METHOD
BASIS
MEM_TOTAL
MEM_STATIC
ADC_PROP_ES
ADC_PRINT
EE_SINGLETS
ADC_DAVIDSON_MAXITER
PCM_PRINT
SOLVENT_METHOD
$end
$pcm
nonequilibrium
theory
Solver
vdwScale
$end
$solvent
dielectric
dielectric_infi
$end

4
adc(2)
3-21G
32000
2000
true
1
5
100
1
pcm

0.025391
0.007383
-0.021323
-0.030206
-0.011984
0.015684
0.047090
-0.052168
-0.020011
0.015870
-0.039597
-0.034925
0.860349
-0.051945
-0.910237
-0.066222
-0.957890
-0.075403
0.813678
0.033076
0.025476
0.054545

!increase print level
!invokes PCM solvent model

true
IEFPCM !default is CPCM, IEFPCM is more accurate
Inversion
1.20

4.34 !epsilon of Et2O
1.829 !n_square of Et2O

429

Chapter 7: Open-Shell and Excited-State Methods

The next job requires a rather complicated compound input file. The sample job computes ADC/SS-PCM EqS solventfield equilibration for the first excited singlet state of peroxinitrite in water, which can be used to compute the fluorescence energy. After generating a starting point in the first job (using a smaller basis and lower ADC convergence
criteria), the solvent-field iterations are carried out until convergence in the second job. In the third job, ADC(2) excited states are computed in the converged solvent field that was left on disk by the second Job. In the fourth job, we
additionally compute ADC(3) excited states. This mixed approach should in general be used with great caution. If the
self-ptSS term of the reference state becomes too large (>0.01 eV) like it is the case here, the fully consistent approach
should be used, meaning that the solvent reaction field should also be computed at the ADC(3) level. PCM settings are
all default values except THEORY, which is set to IEFPCM instead of the default CPCM.
Example 7.89 ADC/SS-PCM EQS solvent-field equilibration for the first excited singlet state of peroxinitrite in water.
$comment
ADC(2)/ptSS-PCM to generate starting point for the EqS
Step in the next Job
$end
$rem
THREADS
METHOD
BASIS
MEM_TOTAL
MEM_STATIC
EE_SINGLETS
ADC_PROP_ES
ADC_DAVIDSON_CONV
SOLVENT_METHOD
$end

2
adc(2)
3-21G !using a small basis to speed up this step
6000
1000
1
true
4
pcm

$pcm
nonequilibrium true
$end
$solvent !Water
dielectric
dielectric_infi
$end

78.4
1.76

$molecule
-1 1
N
-0.068642000000
O
0.349666000000
O
-0.948593000000
O
0.659040000000
$end
@@@

-0.600693000000
0.711166000000
0.200668000000
-0.386002000000

-0.723424000000
1.187490000000
-0.956940000000
0.402650000000

Chapter 7: Open-Shell and Excited-State Methods
$comment
ADC(2)/ptSS-PCM(EqS) solvent-field equilibration
for the first excited state
$end
$rem
THREADS
METHOD
BASIS
MEM_TOTAL
MEM_STATIC
EE_SINGLETS
ADC_PROP_ES
SOLVENT_METHOD
$end

2
adc(2)
6-31G*
6000
1000
2 !compute 2 singlets during the equilibration
true
pcm !activate PCM

$pcm
eqsolv
15
!maximum 15 steps, converges after 5
eqstate
1
!Equilibrate 1st excited state
eqs_conv
4
!Default convergence
theory
iefpcm
nonequilibrium true
$end
$solvent
dielectric
dielectric_infi
$end
$molecule
read
$end
@@@

78.4
1.76

430

Chapter 7: Open-Shell and Excited-State Methods

$comment
Compute ADC(2) excited states in the converged solvent field
$end
$rem
THREADS
METHOD
BASIS
MEM_TOTAL
MEM_STATIC
EE_SINGLETS
ADC_PROP_ES
ADC_PROP_ES2ES
SOLVENT_METHOD
$end

2
adc(2)
6-31G*
6000
1000
6 !compute 6 singlets
true
true !compute ES 2 ES transition moments for ESA
pcm

$pcm
eqsolv
true !only one calculation with converged field
eqstate 1
!Equilibrate 1st excited state
theory
iefpcm
nonequilibrium true
$end
$solvent
dielectric
78.4
dielectric_infi 1.76
$end
$molecule
read
$end
@@@
$comment
We can also compute ADC(3) excited states in the
converged ADC(2) solvent field and use the selfptSS term as diagnostic.
$end
$rem
THREADS
METHOD
BASIS
MEM_TOTAL
MEM_STATIC
EE_SINGLETS
EE_TRIPLETS
ADC_PROP_ES
SOLVENT_METHOD
$end

2
adc(3)
6-31G*
6000
1000
3 !compute 3 singlets
1
!and 1 triplet
true
pcm

431

Chapter 7: Open-Shell and Excited-State Methods

432

$pcm
eqsolv
true !only one calculation with converged field
eqstate 1
!Equilibrate 1st excited state
theory
iefpcm
nonequilibrium true
$end
$solvent
dielectric
78.4
dielectric_infi 1.76
$end
$molecule
read
$end
The next sample job provides the input for a RI-ADC(2)/ptSS-PCM(PTED) calculation for the five lowest excited
states of peroxinitrite in water. After generating a starting point in the first job, which also provides the ptSS(PTE) and
ptSS(PTD) results for comparison, the solvent-field is equilibrated for the MP density in the second job. During the
iterations, the calculation of excited states is disabled to speed up the calculation. In the third job, five excited states are
computed at the RI-ADC(2)/ptSS(PTED) level of theory. Although the PTD corrections for this molecule are unusually
large, a comparison of the PTE, PTD and PTD* results from the first job with the PTED results from the third job will
reveal a reasonable agreement between the fully consistent PTED and the perturbative PTD approaches. In the fourth
job, excited states are calculated with a larger basis set. The self-ptSS term of the MP ground state will be quite small,

433

Chapter 7: Open-Shell and Excited-State Methods
showing that the solvent-field computed with the smaller SVP basis is a good approximation.

Example 7.90 RI-ADC(2)/ptSS-PCM(PTED) calculation for the five lowest excited states of peroxinitrite in water.
$comment
RI-ADC(2)/ptSS-PCM to generate starting point for
the PTED iterations in the next Job and provide
PTE and PTD energies for comparing with PTED
$end
$rem
THREADS
METHOD
BASIS
AUX_BASIS
MEM_TOTAL
MEM_STATIC
EE_SINGLETS
ADC_PROP_ES
SOLVENT_METHOD
$end

2
adc(2)
def2-SVP
rimp2-VDZ
6000
1000
5
true
pcm

$pcm
nonequilibrium true
$end
$solvent !Water
dielectric
78.4
dielectric_infi 1.76
$end
$molecule
-1 1
N
-0.068642000000
O
0.349666000000
O
-0.948593000000
O
0.659040000000
$end

-0.600693000000
0.711166000000
0.200668000000
-0.386002000000

-0.723424000000
1.187490000000
-0.956940000000
0.402650000000

@@@

$comment
RI-ADC(2)/ptSS-PTED solvent-field equilibration for
the MP ground state. No excited states are computed
$end
$rem
THREADS
METHOD
BASIS
AUX_BASIS
MEM_TOTAL
MEM_STATIC
EE_SINGLETS
ADC_PROP_ES
SOLVENT_METHOD
$end
$pcm

2
adc(2)
def2-SVP
rimp2-VDZ
6000
1000
0 !dont compute ES
true
pcm !activate PCM

Chapter 7: Open-Shell and Excited-State Methods
eqsolv
15
!maximum 15 steps
eqstate
0
!Equilibrate MP ground state
eqs_conv
5
!higher convergence
theory
iefpcm
nonequilibrium true
$end
$solvent
dielectric
78.4
dielectric_infi 1.76
$end
$molecule
read
$end
@@@
$comment
Compute ADC(2)/ptSS-PTED excited states in the
converged solvent field
$end
$rem
THREADS
METHOD
BASIS
AUX_BASIS
MEM_TOTAL
MEM_STATIC
EE_SINGLETS
ADC_PROP_ES
SOLVENT_METHOD
$end

2
adc(2)
def2-SVP
rimp2-VDZ
6000
1000
5 !compute 5 singlets
true
pcm

$pcm
eqsolv
true !only one calculation with converged field
eqstate 0
!Equilibrate MP ground state
theory
iefpcm
nonequilibrium true
$end
$solvent
dielectric 78.4
dielectric_infi 1.76
$end
$molecule
read
$end
@@@

434

Chapter 7: Open-Shell and Excited-State Methods

435

$comment
We can also compute the ES in the converged field
with a larger basis and without RI in the stored
solvent-field.
$end
$rem
THREADS
METHOD
BASIS
MEM_TOTAL
MEM_STATIC
EE_SINGLETS
ADC_PROP_ES
SOLVENT_METHOD
$end

2
adc(2)
def2-TZVP
6000
1000
3 !compute 3 singlets
true
pcm

$pcm
eqsolv true !only one calculation with converged field
eqstate 0 !Equilibrate MP ground state
theory iefpcm
nonequilibrium true
$end
$solvent
dielectric 78.4
dielectric_infi 1.76
$end
$molecule
read
$end

7.9

Restricted Active Space Spin-Flip (RAS-SF) and Configuration Interaction (RAS-CI)

The restricted active space spin-flip (RAS-SF) is a special form of configuration interaction that is capable of describing
the ground and low-lying excited states with moderate computational cost in a single-reference formulation, 5,17,21,144
including strongly correlated systems. The RAS-SF approach is essentially a much lower computational cost alternative
to Complete Active Space SCF (CASSCF) methods. RAS-SF typically works by performing a full CI calculation within
an active space that is defined by the half-occupied orbitals of a restricted open shell HF (ROHF) reference determinant.
In this way the difficulties of state-specific orbital optimization in CASSCF are bypassed. Single excitations into (hole)
and out of (particle) the active space provide state-specific relaxation instead. Unlike most CI-based methods, RAS-SF
is size-consistent, as well as variational, and, the increase in computational cost with system size is modest for a fixed
number of spin flips. Beware, however, for the increase in cost as a function of the number of spin-flips is exponential!
RAS-SF has been shown to be capable of tackling multiple low-lying electronic states in polyradicals and reliably
predicting ground state multiplicities. 4,5,16,21,143,144
RAS-SF can also be viewed as one particular case of a more general RAS-CI family of methods. For instance, instead
of defining the active space via spin-flipping as above, initial orbitals of other types can be read in, and electronic
excitations calculated this way may be viewed as a RAS-EE-CI method (though size-consistency will generally be

436

Chapter 7: Open-Shell and Excited-State Methods

lost). Similar to EOM-CC approaches (see Section 7.7), other target RAS-CI wave functions can be constructed starting
from any electronic configuration as the reference and using a general excitation-type operator. For instance, one can
construct an ionizing variant that removes an arbitrary number of particles that is RAS-nIP-CI. An electron-attaching
variant is RAS-nEA-CI. 17
Q-C HEM features two versions of RAS-CI code with different, complementary, functionality. One code (invoked
by specifying CORRRELATION = RASCI) has been written by David Casanova; 17 below we will refer to this code as
RASCI1. The second implementation (invoked by specifying CORRRELATION = RASCI2) is primarily due to Paul
Zimmerman; 144 we will refer to it as RASCI2 below.
The RASCI1 code uses an integral-driven implementation (exact integrals) and spin-adaptation of the CI configurations
which results in a smaller diagonalization dimension. The current Q-C HEM implementation of RASCI1 only allows for
the calculation of systems with an even number of electrons, with the multiplicity of each state being printed alongside
the state energy. Shared memory parallel execution decreases compute time as all the underlying integrals routines are
parallelized.
The RASCI2 code includes the ability to simulate closed and open shell systems (i.e., even and odd numbers of electrons), fast integral evaluation using the resolution of the identity (RI) approximation, shared memory parallel operation,
and analysis of the hS 2 i values and natural orbitals. The natural orbitals are stored in the QCSCRATCH directory in a
folder called “NOs” in M OL D EN-readable format. Shared memory parallel is invoked as described in Section 2.8. A
RASCI2 input requires the specification of an auxiliary basis set analogous to RI-MP2 computations (see Section 6.6.1).
Otherwise, the active space as well as hole and particle excitations are specified in the same way as in RASCI1.
Note: Because RASCI2 uses the RI approximation, the total energies computed with the two codes will be slightly
different; however, the energy differences between different states should closely match each other.

7.9.1

The Restricted Active Space (RAS) Scheme

In the RAS formalism, we divide the orbital space into three subspaces called RAS1, RAS2 and RAS3 (Fig. 7.2). The
RAS-CI states are defined by the number of orbitals and the restrictions in each subspace.
...
max p electrons (particles)

RAS3

RAS2




active space

RAS1





min N − h electrons (holes)

...

Figure 7.2: Orbital subspaces in RAS-CI employing a ROHF triplet reference.
The single reference RAS-CI electronic wave functions are obtained by applying a spin-flipping or excitation operator
R̂ on the reference determinant φ(0) .
ΨRAS = R̂ φ(0)
(7.82)
The R̂ operator must obey the restrictions imposed in the subspaces RAS1, RAS2 and RAS3, and can be decomposed
as:
R̂ = r̂RAS2 + r̂h + r̂p + r̂hp + r̂2h + r̂2p + ...
(7.83)

437

Chapter 7: Open-Shell and Excited-State Methods

where r̂RAS2 contains all possible electronic promotions within the RAS2 space, that is, a reduced full CI, and the rest
of the terms generate configurations with different number of holes (h super-index) in RAS1 and electrons in RAS3 (p
super-index). The current implementation truncates this series up to the inclusion of hole and particle contributions,
i.e. the first three terms on the right hand side of Eq. (7.83).

7.9.2

Second-Order Perturbative Corrections to RAS-CI

In general, the RAS-CI family of methods within the hole and particle approximation is unable to capture the necessary
amounts of dynamic correlation for the computation of relative energies with chemical accuracy. The missed correlation
can be added on top of the RAS-CI wave function using multi-reference perturbation theory (MRPT). 18 The second
order energy correction, i.e. RASCI(2), can be expressed as:
E (2) = −

X |hk|Ĥ|0i|2
Ek − E0 + 

(7.84)

k

where 0 indicates the zero-order space and {|ki} is the complementary set of determinants. There is no natural choice
for the {Ek } excited energies in MRPT, and two different models are available within the RASCI(2) approach, that
is the Davidson-Kapuy and Epstein-Nesbet partitionings. As it is common practice in many second-order MRPT
corrections, the denominator energy differences in Eq. (7.84) can be level shifted with a parameter .

7.9.3

Short-Range Density Functional Correlation within RAS-CI

Alternatively, effective dynamic correlation can be introduced into the RAS-CI wave function by means of short-range
density functional correlation energy. The idea relies on the different ability of wave function methods and DFT to
treat non-dynamic and dynamic correlations. Concretely, the RAS-CI-srDFT (or RAS-srDFT) method 19 is based on
the range separation of the electron-electron Coulomb operator (V̂ee ) through the error function to describe long-range
interactions,

lr,µ
V̂ee
=

X erf(µrij )
i 0 Compute n RAS-CI states
RECOMMENDATION:
None. Only works with RASCI.
RAS_ELEC
Sets the number of electrons in RAS2 (active electrons).
TYPE:
INTEGER
DEFAULT:
None
OPTIONS:
n User-defined integer, n > 0
RECOMMENDATION:
None. Only works with RASCI.
RAS_ACT
Sets the number of orbitals in RAS2 (active orbitals).
TYPE:
INTEGER
DEFAULT:
None
OPTIONS:
n User-defined integer, n > 0
RECOMMENDATION:
None. Only works with RASCI.

Chapter 7: Open-Shell and Excited-State Methods
RAS_OCC
Sets the number of orbitals in RAS1
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
n User-defined integer, n > 0
RECOMMENDATION:
These are the initial doubly occupied orbitals (RAS1) before including hole type of excitations.
The RAS1 space starts from the lowest orbital up to RAS_OCC, i.e. no frozen orbitals option
available yet. Only works with RASCI.
RAS_DO_HOLE
Controls the presence of hole excitations in the RAS-CI wave function.
TYPE:
LOGICAL
DEFAULT:
TRUE
OPTIONS:
TRUE
Include hole configurations (RAS1 to RAS2 excitations)
FALSE Do not include hole configurations
RECOMMENDATION:
None. Only works with RASCI.
RAS_DO_PART
Controls the presence of particle excitations in the RAS-CI wave function.
TYPE:
LOGICAL
DEFAULT:
TRUE
OPTIONS:
TRUE
Include particle configurations (RAS2 to RAS3 excitations)
FALSE Do not include particle configurations
RECOMMENDATION:
None. Only works with RASCI.
RAS_AMPL_PRINT
Defines the absolute threshold (×102 ) for the CI amplitudes to be printed.
TYPE:
INTEGER
DEFAULT:
10 0.1 minimum absolute amplitude
OPTIONS:
n User-defined integer, n ≥ 0
RECOMMENDATION:
None. Only works with RASCI.

439

Chapter 7: Open-Shell and Excited-State Methods
RAS_ACT_ORB
Sets the user-selected active orbitals (RAS2 orbitals).
TYPE:
INTEGER ARRAY
DEFAULT:
From RAS_OCC+1 to RAS_OCC+RAS_ACT
OPTIONS:
[i, j, k...] The number of orbitals must be equal to the RAS_ACT variable
RECOMMENDATION:
None. Only works with RASCI.
RAS_NATORB
Controls the computation of the natural orbital occupancies.
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
TRUE
Compute natural orbital occupancies for all states
FALSE Do not compute natural orbital occupancies
RECOMMENDATION:
None. Only works with RASCI.
RAS_NATORB_STATE
Allows to save the natural orbitals of a RAS-CI computed state.
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
i Saves the natural orbitals for the i-th state
RECOMMENDATION:
None. Only works with RASCI.
RAS_GUESS_CS
Controls the number of closed shell guess configurations in RAS-CI.
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
n Imposes to start with n closed shell guesses
RECOMMENDATION:
Only relevant for the computation of singlet states. Only works with RASCI.

440

Chapter 7: Open-Shell and Excited-State Methods
RAS_SPIN_MULT
Specifies the spin multiplicity of the roots to be computed
TYPE:
INTEGER
DEFAULT:
1 Singlet states
OPTIONS:
0
Compute any spin multiplicity
2n + 1 User-defined integer, n ≥ 0
RECOMMENDATION:
Only for RASCI, which at present only allows for the computation of systems with an even
number of electrons. Thus, RAS_SPIN_MULT only can take odd values.
RAS_PT2
Perform second-order perturbative correction to RAS-CI energy
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
TRUE
Compute RASCI(2) energy corrections
FALSE Do not compute RASCI(2) energy corrections
RECOMMENDATION:
None. Only works with RASCI.
RAS_PT2_PARTITION
Specifies the partitioning scheme in RASCI(2)
TYPE:
INTEGER
DEFAULT:
1 Davidson-Kapuy (DK) partitioning
OPTIONS:
2 Epstein-Nesbet (EN) partitioning
0 Do both DK and EN partitionings
RECOMMENDATION:
Only for RASCI if RAS_PT2 is set to true.
RAS_PT2_VSHIFT
Defines the energy level shift (×103 au) in RASCI(2)
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
n User-defined integer
RECOMMENDATION:
Only for RASCI if RAS_PT2 is set to true.

441

Chapter 7: Open-Shell and Excited-State Methods
RAS_SRDFT
Perform short-range density functional RAS-CI calculation
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
TRUE
Compute RASCI-srDFT states and energies
FALSE Do not perform a RASCI-srDFT calculation
RECOMMENDATION:
None. Only works with RASCI. RAS_SRDFT_EXC and RAS_SRDFT_COR need to be set.
RAS_SRDFT_EXC
Define short-range exchange functional
TYPE:
STRING
DEFAULT:
No default
OPTIONS:
NAME Use RAS_SRDFT_EXC = NAME, where NAME is
one of the short-range exchange functionals listed in Section 5.3.2
RECOMMENDATION:
None.
RAS_SRDFT_COR
Define short-range correlation functional
TYPE:
STRING
DEFAULT:
No default
OPTIONS:
NAME Use RAS_SRDFT_COR = NAME, where NAME is
one of the short-range correlation functionals listed in Section 5.3.3
RECOMMENDATION:
None
RAS_OMEGA
Sets the Coulomb range-separation parameter within the RAS-CI-srDFT method.
TYPE:
INTEGER
DEFAULT:
400 (ω = 0.4 bohr−1 )
OPTIONS:
n Corresponding to ω = n/1000, in units of bohr−1
RECOMMENDATION:
None. Range-separation parameter is typical indicated by ω or µ

442

Chapter 7: Open-Shell and Excited-State Methods

443

RAS_SRDFT_DAMP
Sets damping factor (α < 1) in the RAS-CI-srDFT method.
TYPE:
INTEGER
DEFAULT:
5000 (α = 0.5)
OPTIONS:
n Corresponding to α = n/10000
RECOMMENDATION:
Modify in case of convergence issues along the RAS-CI-srDFT iterations
RAS_NFRAG
If n > 0 activates the excitation analysis in RASCI
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
n Number of fragments to be considered
RECOMMENDATION:
Only for RASCI. The printed information level is controlled by RAS_PRINT
RAS_NFRAG_ATOMS
Sets the number of atoms in each fragment.
TYPE:
INTEGER ARRAY
DEFAULT:
None
OPTIONS:
[i, j, k...] The sum of the numbers must be equal to the total number of atoms in the systems
RECOMMENDATION:
None. Only works with RASCI.
RAS_FRAG_SETS
Sets the number of atoms in each fragment.
TYPE:
INTEGER ARRAY
DEFAULT:
[NOcc,NAct,NVir] Number of orbitals within RAS1, RAS2 and RAS3 spaces
OPTIONS:
[i, j, k...] Defines sets of canonical MOs to be localized into n fragments
RECOMMENDATION:
Setting within RAS1, RAS2 and RAS3 spaces alleviates the computational cost of the localization procedure. It might also result in improved fragment orbitals. Only works with RASCI.

7.9.6

Job Control Options for RASCI2

At present the RASCI1 and RASCI2 implementations employ different keywords (which will be reconciled in a future
version). This subsection applies to RASCI2 (even and odd electron systems, determinant-driven algorithm using the
resolution of the identity approximation).

Chapter 7: Open-Shell and Excited-State Methods

444

The use of the RAS-CI2 methodology is controlled by setting the CORRELATION = RASCI2 and EXCHANGE = HF.
The minimum input also requires specifying the desired (non-zero) value for RAS_N_ROOTS, and the number of active
occupied and virtual orbital comprising the “active” RAS2 space. RASCI2 calculations also require specification of an
auxiliary basis via AUX_BASIS.
RAS_N_ROOTS
Sets the number of RAS-CI roots to be computed.
TYPE:
INTEGER
DEFAULT:
None
OPTIONS:
n n > 0 Compute n RAS-CI states
RECOMMENDATION:
None. Only works with RASCI2
RAS_ACT_OCC
Sets the number of occupied orbitals to enter the RAS active space.
TYPE:
Integer
DEFAULT:
None
OPTIONS:
n user defined integer
RECOMMENDATION:
None. Only works with RASCI2
RAS_ACT_VIR
Sets the number of virtual orbitals to enter the RAS active space.
TYPE:
Integer
DEFAULT:
None
OPTIONS:
n user defined integer
RECOMMENDATION:
None. Only works with RASCI2.
RAS_ACT_DIFF
Sets the number of alpha vs. beta electrons
TYPE:
Integer
DEFAULT:
None
OPTIONS:
n user defined integer
RECOMMENDATION:
Set to 1 for an odd number of electrons or a cation, -1 for an anion. Only works with RASCI2.

Other $rem variables that can be used to control the evaluation of RASCI2 calculations are SET_ITER for the maximum
number of Davidson iterations, and N_FROZEN_CORE and N_FROZEN_VIRTUAL to exclude core and/or virtual orbitals
from the RASCI2 calculation.

Chapter 7: Open-Shell and Excited-State Methods

7.9.7

445

Examples

Example 7.91 Input for a RAS-2SF-CI calculation of three states of the DDMX tetraradical using RASCI1. The
active space (RAS2) contains 4 electrons in the 4 singly occupied orbitals in the ROHF quintet reference. Natural
orbital occupancies are requested.
$molecule
0 5
C
0.000000
C
-1.222482
C
-2.390248
H
-2.344570
H
-3.363161
C
-1.215393
H
-2.150471
C
0.000000
C
1.215393
H
2.150471
C
1.222482
C
2.390248
H
2.344570
H
3.363161
$end
$rem
EXCHANGE
CORRELATION
BASIS
UNRESTRICTED
MEM_TOTAL
MEM_STATIC
RAS_ROOTS
RAS_ACT
RAS_ELEC
RAS_OCC
RAS_SPIN_MULT
RAS_NATORB
$end

0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000

1.092150
0.303960
1.015958
2.095067
0.537932
-1.155471
-1.702536
-1.769131
-1.155471
-1.702536
0.303960
1.015958
2.095067
0.537932

hf
rasci
6-31g
false
4000
100
3
4
4
25
0
true

Example 7.92 Input for a RAS-2IP-CI calculation of triplet states of F2 molecule using the dianion closed shell F2−
2
as the reference determinant. RASCI1 code is used
$molecule
-2 1
F
F 1 1.4136
$end
$rem
EXCHANGE
CORRELATION
BASIS
MEM_TOTAL
MEM_STATIC
RAS_ROOTS
RAS_ACT
RAS_ELEC
RAS_OCC
RAS_SPIN_MULT
$end

hf
rasci
cc-pVTZ
4000
100
2
6
10
4
3

446

Chapter 7: Open-Shell and Excited-State Methods

Example 7.93 Input for a FCI/6-31G calculation of water molecule expanding the RAS2 space to the entire molecular
orbital set. RASCI code is used.
$molecule
0 1
O
0.000
H -0.762
H
0.762
$end

0.000
0.000
0.000

$rem
EXCHANGE
CORRELATION
BASIS
MEM_TOTAL
MEM_STATIC
RAS_ROOTS
RAS_ACT
RAS_ELEC
RAS_OCC
RAS_SPIN_MULT
RAS_DO_HOLE
RAS_DO_PART
$end

0.120
-0.479
-0.479

hf
rasci
6-31G
4000
100
1
13
10
0
1
false
false

Example 7.94 Methylene single spin-flip calculation using RASCI2
$molecule
0 3
C
0.0000000
H
-0.8611113
H
0.8611113
$end
$rem
EXCHANGE
CORRELATION
BASIS
AUX_BASIS
UNRESTRICTED
RAS_ACT_OCC
RAS_ACT_VIR
RAS_ACT_DIFF
RAS_N_ROOTS
SET_ITER
$end

0.0000000
0.0000000
0.0000000

0.0000000
0.6986839
0.6986839

HF
RASCI2
cc-pVDZ
rimp2-cc-pVDZ
false
1
! # alpha electrons
1
! # virtuals in active space
0
! # set to 1 for odd # of e-s
4
25
! number of iterations in RASCI

447

Chapter 7: Open-Shell and Excited-State Methods

Example 7.95 Two methylene separated by 10 Å; double spin-flip calculation using RASCI2. Note that the hS 2 i
values for this case will not be uniquely defined at the triply-degenerate ground state.
$molecule
0 5
C
0.0000000
H
-0.8611113
H
0.8611113
C
0.0000000
H
-0.8611113
H
0.8611113
$end
$rem
EXCHANGE
CORRELATION
BASIS
AUX_BASIS
RAS_ACT_OCC
RAS_ACT_VIR
RAS_ACT_DIFF
UNRESTRICTED
RAS_N_ROOTS
SET_ITER
$end

0.0000000
0.0000000
0.0000000
10.0000000
10.0000000
10.0000000

0.0000000
0.6986839
0.6986839
0.0000000
0.6986839
0.6986839

HF
RASCI2
cc-pVDZ
rimp2-cc-pVDZ
2
! # alpha electrons
2
! # virtuals in active space
0
! # set to 1 for odd # of e-s
false
8
25

Example 7.96 RASCI2 calculation of the nitrogen cation using double spin-flip.
$molecule
1 6
N
N
1 4.5
$end
$rem
EXCHANGE
CORRELATION
BASIS
AUX_BASIS
RAS_ACT_OCC
RAS_ACT_VIR
RAS_ACT_DIFF
UNRESTRICTED
N_FROZEN_CORE
N_FROZEN_VIRTUAL
RAS_N_ROOTS
SET_ITER
$end

7.10

HF
RASCI2
6-31G*
rimp2-VDZ
3
! # alpha electrons
3
! # virtuals in active space
1
! # for odd # e-s, cation
false
2
2
8
25

Core Ionization Energies and Core-Excited States

In experiments using high-energy radiation (such as X-ray spectroscopy, EXAFS, NEXAFS, XAS) core electrons can
be ionized or excited to low-lying virtual orbitals. There are several ways to compute ionization or excitation energies
of core electrons in Q-C HEM. Standard approaches for excited and ionized states need to be modified to tackle corelevel states, because these states have very high energies and are embedded in the ionization continuum (i.e., they are
Feschbach resonances 110 ).

Chapter 7: Open-Shell and Excited-State Methods

448

The most robust and accurate strategy is to invoke many-body methods, such as EOM or ADC, together with the corevalence separation (CVS) approximation 27 . In this approach, the excitations involving core electrons are decoupled
from the rest of the configurational space. This allows one to reduce computational costs and decouple the highly
excited core states from the continuum. These methods are described in Sections 7.7.5 and 7.8.4.
Within EOM-CC formalism, one can also use an approximate EOM-EE/IP methods in which the target states are
described by single excitations and double excitations are treated perturbatively; these methods are described in Section 7.7.10. While being moderately useful, these methods are less accurate than the CVS-EOM variants 110 .
In addition, one can use the ∆E approach, which amounts to a simple energy difference calculation in which core
ionization is computed from energy differences computed for the neutral and core-ionized state. It is illustrated by
example 97 below.
Example 7.97 Q-C HEM input for calculating chemical shift for 1s-level of methane (CH4 ). The first job is just an SCF
calculation to obtain the orbitals and CCSD energy of the neutral. The second job solves the HF and CCSD equations
for the core-ionized state.
$molecule
0,1
C
0.000000
H
0.631339
H
-0.631339
H
-0.631339
H
0.631339
$end
$rem
EXCHANGE
CORRELATION
BASIS
MAX_CIS_CYCLES
$end

0.000000
0.631339
-0.631339
0.631339
-0.631339

=
=
=
=

@@@
$molecule
+1,2
C
0.000000
H
0.631339
H
-0.631339
H
-0.631339
H
0.631339
$end
$rem
UNRESTRICTED
EXCHANGE
BASIS
MAX_CIS_CYCLES
SCF_GUESS
CORRELATION
MOM_START
$end

HF
CCSD
6-31G*
100

0.000000
0.631339
-0.631339
0.631339
-0.631339

=
=
=
=
=
=
=

0.000000
0.631339
0.631339
-0.631339
-0.631339

0.000000
0.631339
0.631339
-0.631339
-0.631339

TRUE
HF
6-31G*
100
read Read MOs from previous job and use occupied as specified below
CCSD
1 Do not reorder orbitals in SCF procedure!

$occupied
1 2 3 4 5
2 3 4 5
$end

In this job, we first compute the HF and CCSD energies of neutral CH4 : ESCF = −40.1949062375 and ECCSD =
−40.35748087 (HF orbital energy of the neutral gives the Koopmans IE, which is 11.210 hartree = 305.03 eV). In
the second job, we do the same for core-ionized CH4 . To obtain the desired SCF solution, MOM_START option and

Chapter 7: Open-Shell and Excited-State Methods

449

$occupied keyword are used. The resulting energies are ESCF = −29.4656758483 (hS 2 i = 0.7730) and ECCSD =
−29.64793957. Thus, ∆ECCSD = (40.357481 − 29.647940) = 10.709 hartree = 291.42 eV.
This approach can be further extended to obtain multiple excited states involving core electrons by performing CIS,
TDDFT, or EOM-EE calculations.
Note: This approach often leads to convergence problems in correlated calculations.
Finally, one can also use the following trick illustrated by example 98.
Example 7.98 Q-C HEM input for calculating chemical shift for 1s-level of methane (CH4 ) using EOM-IP. Here we
solve SCF as usual, then reorder the MOs such that the core orbital becomes the “HOMO”, then solve the CCSD and
EOM-IP equations with all valence orbitals frozen and the core orbital being active.
$molecule
0,1
C
0.000000
H
0.631339
H
-0.631339
H
-0.631339
H
0.631339
$end
$rem
EXCHANGE
BASIS
MAX_CIS_CYCLES
CORRELATION
CCMAN2
N_FROZEN_CORE
IP_STATES
$end

0.000000
0.631339
-0.631339
0.631339
-0.631339

=
=
=
=
=
=
=

0.000000
0.631339
0.631339
-0.631339
-0.631339

HF
6-31G*
100
CCSD
false
4 Freeze all valence orbitals
[1,0,0,0] Find one EOM_IP state

$reorder_mo
5 2 3 4 1
5 2 3 4 1
$end

Here we use EOM-IP to compute core-ionized states. Since core states are very high in energy, we use “frozen core”
trick to eliminate valence ionized states from the calculation. That is, we reorder MOs such that our core is the
last occupied orbital and then freeze all the rest. The so computed EOM-IP energy is 245.57 eV. From the EOM-IP
amplitude, we note that this state of a Koopmans character (dominated by single core ionization); thus, canonical HF
MOs provide good representation of the correlated Dyson orbital. The same strategy can be used to compute coreexcited states.
Note: The accuracy of this approach is rather poor and is similar to the Koopmans approximation.

7.10.1

Calculations of States Involving Core Excitation/Ionization with (TD)DFT

TDDFT is not suited to describe the extended X-ray absorption fine structure (EXAFS) region, wherein the core electron is ejected and scattered by the neighboring atoms. Core-excitation energies computed with TDDFT with standard
hybrid functionals are many electron volts too low compared with experiment. Exchange-correlation functionals specifically designed to treat core excitations are available in Q-C HEM. These short-range corrected (SRC) functionals are a
modification of the more familiar long-range corrected functionals (discussed in Section 5.6). However, in SRC-DFT
the short-range component of the Coulomb operator is predominantly Hartree-Fock exchange, while the mid to longrange component is primarily treated with standard DFT exchange. These functionals can be invoked by using the
SRC_DFT rem. In addition, a number of parameters (OMEGA, OMEGA2, HF_LR, HF_SR) that control the shape of the
short and long-range Hartree-Fock components need to be specified. Full details of these functionals and appropriate
values for the parameters can be found in Refs. 10,13. An example of how to use these functionals is given below. For

Chapter 7: Open-Shell and Excited-State Methods

450

the K-shell of heavy elements (2nd row of the periodic table) relativistic effects become increasing important and a
further correction for these effects is required. Also calculations for L-shell excitations are complicated by core-hole
spin orbit coupling.

7.11

Real-Time SCF Methods (RT-TDDFT, RT-HF, OSCF2)

Linear response calculations are the most efficient way to predict resonant electronic response frequencies and intensities. Q-C HEM can also propagate mean-field theories (HF, DFT) in real-time with and without dissipation and finitetemperature effects. These methods time-evolve the Liouville-von Neumann equation of motion for a one-particle
density operator, ρ̂(t):


(7.90)
(i/~)ρ̂(t) = F̂ {ρ}, ρ̂(t)
These real-time methods are useful for simulating attosecond to picosecond timescales, density response to multiple
or strong applied fields, and timescales of energy relaxation. Symmetry, non-singlet densities, Meta-GGAs and 5thrung functionals are not supported for real-time propagation, and nuclei are fixed. The CPU cost of a single time step
is comparable to that of a single SCF cycle, if the modified-midpoint unitary transformation (MMUT) propagation
algorithm 73 is employed, and no more than a few times the cost of a single SCF cycle if predictor/corrector algorithms
(which facilitate stable time propagation using much larger time steps) are used. 141 Memory costs are similar to those
of a ground-state SCF calculation, which for large systems, or those with a large density of states, represents a dramatic
reduction in the memory requirement relative to linear-reponse TDDFT. The number of required Fock-build steps can
be estimated from the default electronic time-step, which is 0.02 atomic units (4.8 × 10−4 fs). The real-time code
exploits real-time shared-memory parallelism, and the use of 8 or more cores (–nt 8) is suggested. The input file of
a basic real-time propagation is relatively simple, but sophisticated jobs require additional input files, and generate
additional output.
Example 7.99 Q-C HEM Basic RT-TDDFT job for BH3.
$molecule
0 1
B
0.000000
H
1.115609
H
-0.332368
H
-0.782205
$end
$rem
SCF_GUESS
GEN_SCFMAN
RTTDDFT
BASIS
THRESH
SCF_CONVERGENCE
EXCHANGE
SYMMETRY
SYM_IGNORE
SCF_ALGORITHM
UNRESTRICTED
MAX_SCF_CYCLES
$end

0.000000
-0.322048
1.137184
-0.814474

0.000000
0.260309
0.111382
-0.375294

core
true
1
3-21g
10
9
b3lyp
false
TRUE
DIIS
FALSE
200

Before executing this job, a directory (/logs) must be made in the output directory of the job to collect most of the
results of the propagation. By default, a real-time job begins from the ground state SCF density, applies a weak,
brief (0.07 atomic time units) oscillating electric field to the y-axis of a molecule which excites a superposition of
all y-polarized excited states, and outputs the resulting time-dependent dipole moment, energy and other quantities to
the logs directory. The parameters of the time-dependent run are printed when the propagation begins. During the
propagation the state of the molecule and propagation is summarized in the Q-C HEM output file, including estimates

Chapter 7: Open-Shell and Excited-State Methods

451

of the elapsed and remaining simulation wall-time. Propagation stops when MaxIter time steps are exceeded, or will
stop prematurely if the density matrix becomes nonphysical. The logs directory will be populated by several text, white
space delimited tables by default. The ./logs/Pol.csv file is a table consisting of time (in a.u.), µx , µy , µz , total energy,
trρ̂, the gap, the electronic energy, and two unused columns. The ./logs/Fourier_x.csv file contains Fourier transform
F[µx (t)], in the format of its total value in energy units (eV) followed by its negative imaginary part and then its real
part. Analogous files ./logs/Fourier_y.csv and ./logs/Fourier_z.csv are also created.
By adjusting the options of the propagation with a file TDSCF.prm, a much larger amount of output can be generated,
including the electron and hole densities at all times as sequential M OL D EN files (*.mol) viewable with the free package
G ABEDIT (gabedit.sourceforge.net). The *.mol files generated by the real-time code have fractional orbital
occupation numbers, and do not render properly in viewers other than G ABEDIT to our knowledge. So long as the
applied field is weak and has short duration the positions at which peaks appear in the Fourier are the same as a
linear-response TDDFT-RPA job (note: not the same as a LR-TDDFT-TDA job), as shown in the sample. In an energyconserving job, the width of the peak is inversely proportional to the duration of the signal sample used to construct the
Fourier transform. If the applied field pulse is long (> 1 a.u.), or has strong intensity (> 0.01 a.u.) non-linear response
can be studied. Non-linear effects in real-time SCF are an area of active investigation and development.
The absorption cross-section σii (ω) in the i direction, i ∈ {x, y, z}, can be obtained from the imaginary part of the
frequency-dependent polarizability αii (ω):




4πω
σii (ω) =
= αii (ω) ,
(7.91)
c
and the rotationally-averaged absorption spectrum (within the dipole approximation) is
A(ω) =

1
3



σxx (ω) + σyy (ω) + σzz (ω) .

(7.92)

The frequency-dependent polarizability αij (ω) is obtained from the Fourier transform of µi (t), for a perturbing field
in the j direction: 141


F µi (t)
.
αij (ω) = 
(7.93)
F Ej (t)
Thus to compute the spectrum in Eq. (7.92), three RT-TDDFT simulations are required, perturbing separately in the x,
y, and z directions.
Because of the large number of floating point arguments used to control a real-time job, a separate input file TDSCF.prm
in the same directory as the Q-C HEM input file is used for parameters. The file is two columns of plain text. The first
column is a string naming the parameter (which must match the case and spelling printed in the output exactly to be
interpreted correctly), and the second column is a floating point number. The number must only be followed by a new
line. Several inputs are interpreted as true = 1 or false = 0. The most useful parameters are discussed in this section.

Chapter 7: Open-Shell and Excited-State Methods
The code will signal if this file is not found and default values will be supplied instead.
Example 7.100 Q-C HEM Typical TDSCF.prm for a real-time B3LYP calculation.
dt 0.02
Stabilize 0
TCLOn 0
MaxIter 80000
ApplyImpulse 1
ApplyCw 0
FieldFreq 0.7
Tau 0.07
FieldAmplitude 0.001
ExDir 1.0
EyDir 1.0
EzDir 1.0
Print 0
StatusEvery 10
SaveDipoles 1
DipolesEvery 2
SavePopulations 0
SaveFockEnergies 0
WriteDensities 0
SaveEvery 500
FourierEvery 5000
MMUT 1
LFLPPC 0

Parameter String
dt
Stabilize
MaxIter
ApplyImpulse
ApplyCw
FieldFreq
Tau
FieldAmplitude
ExDir
EyDir
EzDir
Print
StatusEvery
SaveDipoles
DipolesEvery
SavePopulations
SaveFockEnergies
WriteDensities
SaveEvery
FourierEvery
FourierZoom
MMUT
LFLPPC

Explanation
timestep (atomic units) > 0.02 may be unstable.
> 0 Forces positive occupation numbers.
Maximum Timesteps
0 = No applied gaussian impulse
0 = No Continuous Wave, 1 = Cosine applied field.
Frequency of applied field (atomic units)
Time-variance of Gaussian Impulse (atomic units)
Max. amplitude of field (atomic units)
Field Polarization Vector (x-component)
Field Polarization Vector (y-component)
Field Polarization Vector (z-component)
Value > 0 makes print more debug output.
Iterations between status in output file
0 = No Pol.csv generated.
Iterations between samples of Dipole
1 = Saves diagonal of the density
1 = Saves diagonal of the Fock matrix
> 0 generates .mol files readable with G ABEDIT
Iterations between writing of all files in /logs
Iterations between Fourier Transform
A zoom parameter which controls resolution of FT.
Modified midpoint unitary transform propagator calculation 73 (default)
Predictor/corrector propagator 141
Table 7.4: TDSCF.prm Parameters

Any undocumented options not discussed above are not officially supported at this time.

452

453

Chapter 7: Open-Shell and Excited-State Methods

7.12

Visualization of Excited States

As methods for ab initio calculations of excited states are becoming increasingly more routine, questions arise concerning how best to extract chemical meaning from such calculations. There are several approaches for analyzing
molecular excited states; they are based on reduced one-particle density matrices (OPDMs). The two objects exploited
in this analysis are: (i) the difference between the ground- and excited-state OPDMs and (ii) the transition OPDM
connecting the ground and excited state. In the case of CIS and TDDFT/TDA wave functions, both quantities are identical and can be directly mapped into the CIS amplitudes; however, for correlated wave functions the two objects are
not the same. The most basic analysis includes calculation of attachment/detachment density 44 and natural transition
orbitals. 82 These quantities allow one to arrive to a most compact description of an excited state. More detailed analysis allows one to derive additional insight about the nature of the excited state. Detailed description and illustrative
examples can be found in Refs. 104,105.
This section describes the theoretical background behind attachment/detachment analysis and natural transition orbitals,
while details of the input for creating data suitable for plotting these quantities is described separately in Chapter 11,
which also describes additional excited-state analysis tools. For historical reasons, there are duplicate implementations
of some features. For example, CIS and TDDFT wave functions can be analyzed using an original built-in code and by
using a more recent module, LIBWFA.

7.12.1

Attachment/Detachment Density Analysis

Consider the one-particle density matrices of the initial and final states of interest, P1 and P2 respectively. Assuming
that each state is represented in a finite basis of spin-orbitals, such as the molecular orbital basis, and each state is at
the same geometry. Subtracting these matrices yields the difference density
∆ = P1 − P2

(7.94)

Now, the eigenvectors of the one-particle density matrix P describing a single state are termed the natural orbitals, and
provide the best orbital description that is possible for the state, in that a CI expansion using the natural orbitals as the
single-particle basis is the most compact. The basis of the attachment/detachment analysis is to consider what could
be termed natural orbitals of the electronic transition and their occupation numbers (associated eigenvalues). These are
defined as the eigenvectors U defined by
U† ∆U = δ
(7.95)
The sum of the occupation numbers δp of these orbitals is then
tr(∆) =

N
X

δp = n

(7.96)

p=1

where n is the net gain or loss of electrons in the transition. The net gain in an electronic transition which does not
involve ionization or electron attachment will obviously be zero.
The detachment density
D = UdU†

(7.97)

is defined as the sum of all natural orbitals of the difference density with negative occupation numbers, weighted by
the absolute value of their occupations where d is a diagonal matrix with elements
dp = − min(δp , 0)

(7.98)

The detachment density corresponds to the electron density associated with single particle levels vacated in an electronic
transition or hole density.
The attachment density
A = UaU†

(7.99)

Chapter 7: Open-Shell and Excited-State Methods

454

is defined as the sum of all natural orbitals of the difference density with positive occupation numbers where a is a
diagonal matrix with elements
ap = max(δp , 0)
(7.100)
The attachment density corresponds to the electron density associated with the single particle levels occupied in the
transition or particle density. The difference between the attachment and detachment densities yields the original
difference density matrix
∆=A−D
(7.101)

7.12.2

Natural Transition Orbitals

In certain situations, even the attachment/detachment densities may be difficult to analyze. An important class of examples are systems with multiple chromophores, which may support exciton states consisting of linear combinations
of localized excitations. For such states, both the attachment and the detachment density are highly delocalized and
occupy basically the same region of space. 68 Lack of phase information makes the attachment/detachment densities difficult to analyze, while strong mixing of the canonical MOs means that excitonic states are also difficult to characterize
in terms of MOs.
Analysis of these and other excited states is greatly simplified by constructing Natural Transition Orbitals (NTOs)
for the excited states. (The basic idea behind NTOs is rather old 78 and has been rediscovered several times; 82,88
these orbitals were later shown to be equivalent to CIS natural orbitals. 121 ) Let T denote the transition density matrix
from an excited-state calculation. The dimension of this matrix is O × V , where O and V denote the number of
occupied and virtual MOs, respectively. The NTOs are defined by transformations U and V obtained by singular value
decomposition (SVD) of the matrix T, i.e., 88
UTV† = Λ
(7.102)
The matrices U and V are unitary and Λ is diagonal, with the latter containing at most O non-zero elements. The
matrix U is a unitary transformation from the canonical occupied MOs to a set of NTOs that together represent the
“hole” orbital that is left by the excited electron, while V transforms the canonical virtual MOs into a set of NTOs
representing the excited electron. (Equivalently, the “holes” are the eigenvectors of the O × O matrix TT† and
the particles are eigenvectors of the V × V matrix T† T. 82 ) These “hole” and “particle” NTOs come in pairs, and
their relative importance in describing the excitation is governed by the diagonal elements of Λ, which are excitation
amplitudes in the NTO basis. By virtue of the SVD in Eq. (7.102), any excited state may be represented using at most
O excitation amplitudes and corresponding hole/particle NTO pairs. (The‘ discussion here assumes that V ≥ O, which
is typically the case except possibly in minimal basis sets. Although it is possible to use the transpose of Eq. (7.102) to
obtain NTOs when V < O, this has not been implemented in Q-C HEM due to its limited domain of applicability.)
The SVD generalizes the concept of matrix diagonalization to the case of rectangular matrices, and therefore reduces
as much as possible the number of non-zero outer products needed for an exact representation of T. In this sense,
the NTOs represent the best possible particle/hole picture of an excited state. The detachment density is recovered as
the sum of the squares of the “hole” NTOs, while the attachment density is precisely the sum of the squares of the
“particle” NTOs. Unlike the attachment/detachment densities, however, NTOs preserve phase information, which can
be very helpful in characterizing the diabatic character (e.g., ππ ∗ or nπ ∗ ) of excited states in complex systems. Even
when there is more than one significant NTO amplitude, as in systems of electronically-coupled chromophores, 68 the
NTOs still represent a significant compression of information, as compared to the canonical MO basis.
NTOs are available within Q-C HEM for CIS, RPA, TDDFT, ADC, and EOM-CC methods. For the correlated wave
functions (EOM-CC and ADC), they can be computed using LIBWFA module. The simplest way to visualize the NTOs
is to generate them in a format suitable for viewing with the freely-available M OL D EN or M AC M OL P LT programs, as
described in Chapter 11.

455

Chapter 7: Open-Shell and Excited-State Methods

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Chapter 8

Basis Sets
8.1

Introduction

A basis set is a set of functions combined linearly to model molecular orbitals. Basis functions can be considered as
representing the atomic orbitals of the atoms and are introduced in quantum chemical calculations because the equations
defining the molecular orbitals are otherwise very difficult to solve.
Many standard basis sets have been carefully optimized and tested over the years. In principle, a user would employ the
largest basis set available in order to model molecular orbitals as accurately as possible. In practice, the computational
cost grows rapidly with the size of the basis set so a compromise must be sought between accuracy and cost. If this
is systematically pursued, it leads to a “theoretical model chemistry”, 9 that is, a well-defined energy procedure (e.g.,
Hartree-Fock) in combination with a well-defined basis set.
Basis sets have been constructed from Slater, Gaussian, plane wave and delta functions. Slater functions were initially employed because they are considered “natural” and have the correct behavior at the origin and in the asymptotic
regions. However, the two-electron repulsion integrals (ERIs) encountered when using Slater basis functions are expensive and difficult to evaluate. Delta functions are used in several quantum chemistry programs. However, while
codes incorporating delta functions are simple, thousands of functions are required to achieve accurate results, even for
small molecules. Plane waves are widely used and highly efficient for calculations on periodic systems, but are not so
convenient or natural for molecular calculations.
The most important basis sets are contracted sets of atom-centered Gaussian functions. The number of basis functions
used depends on the number of core and valence atomic orbitals, and whether the atom is light (H or He) or heavy
(everything else). Contracted basis sets have been shown to be computationally efficient and to have the ability to yield
chemical accuracy (see Appendix B). The Q-C HEM program has been optimized to exploit basis sets of the contracted
Gaussian function type and has a large number of built-in standard basis sets (developed by Dunning and Pople, among
others) which the user can access quickly and easily.
The selection of a basis set for quantum chemical calculations is very important. It is sometimes possible to use small
basis sets to obtain good chemical accuracy, but calculations can often be significantly improved by the addition of
diffuse and polarization functions. Consult the literature and review articles 5,7,9–11 to aid your selection and see the
section “Further Reading” at the end of this Chapter.

8.2

Built-In Basis Sets

Q-C HEM is equipped with many standard basis sets, 1 and allows the user to specify the required basis set by its standard
symbolic representation. The available built-in basis sets include the following types:

Chapter 8: Basis Sets

462

• Pople basis sets
• Dunning basis sets
• Correlation consistent Dunning basis sets
• Ahlrichs basis sets
• Jensen polarization consistent basis sets
• Karlsruhe "def2" basis sets
• The universal Gaussian basis set (UGBS)
In addition, Q-C HEM supports the following features:
• Extra diffuse functions available for high quality excited state calculations.
• Standard polarization functions.
• Basis sets are requested by symbolic representation.
• s, p, sp, d, f and g angular momentum types of basis functions.
• Maximum number of shells per atom is 100.
• Pure and Cartesian basis functions.
• Mixed basis sets (see section 8.5).
• Basis set superposition error (BSSE) corrections.
The following $rem keyword controls the basis set:
BASIS
Sets the basis set to be used
TYPE:
STRING
DEFAULT:
No default basis set
OPTIONS:
General, Gen User-defined. See section below
Symbol
Use standard basis sets as in the table below
Mixed
Use a combination of different basis sets
RECOMMENDATION:
Consult literature and reviews to aid your selection.

8.3

Basis Set Symbolic Representation

Examples are given in the tables below and follow the standard format generally adopted for specifying basis sets. The
single exception applies to additional diffuse functions. These are best inserted in a similar manner to the polarization
functions; in parentheses with the light atom designation following heavy atom designation. (i.e., heavy, light). Use a
period (.) as a place-holder (see examples).

8.3.1

Customization

Q-C HEM offers a number of standard and special customization features. One of the most important is that of supplying
additional diffuse functions. Diffuse functions are often important for studying anions and excited states of molecules,
and for the latter several sets of additional diffuse functions may be required. These extra diffuse functions can be

463

Chapter 8: Basis Sets

STO−j(k+, l+)G(m, n)
j −21(k+, l+)G(m, n)
j − 31(k+, l+)G(m, n)
j − 311(k+, l+)G(m, n)

j
2,3,6
3
4,6
6

k

l

a

b

a

b

a

b

a

b

m
d
2d
3d
df ,2df ,3df

n
p
2p
3p
pd,2pd,3pd

Table 8.1: Summary of Pople type basis sets available in Q-C HEM. m and nrefer to the polarization functions on heavy
and light atoms respectively. a k is the number of sets of diffuse functions on heavy b l is the number of sets of diffuse
functions on light atoms.
Symbolic Name
STO-2G
STO-3G
STO-6G
3-21G
4-31G
6-31G
6-311G
G3LARGE
G3MP2LARGE

Atoms Supported
H, He, Li→Ne, Na→Ar, K, Ca, Sr
H, He, Li→Ne, Na→Ar, K→Kr, Rb→Sb
H, He, Li→Ne, Na→Ar, K→Kr
H, He, Li→Ne, Na→Ar, K→Kr, Rb→Xe, Cs
H, He, Li→Ne, P→Cl
H, He, Li→Ne, Na→Ar, K→Zn
H, He, Li→Ne, Na→Ar, Ga→Kr
H, He, Li→Ne, Na→Ar, K→Kr
H, He, Li→Ne, Na→Ar, Ga→Kr

Table 8.2: Atoms supported for Pople basis sets available in Q-C HEM (see the Table below for specific examples).
Symbolic Name
3-21G
3-21+G
3-21G*
6-31G
6-31+G
6-31G*
6-31G(d,p)
6-31G(.,+)G
6-31+G*
6-311G
6-311+G
6-311G*
6-311G(d,p)
G3LARGE
G3MP2LARGE

Atoms Supported
H, He, Li → Ne, Na → Ar, K →Kr, Rb → sXe,
Cs
H, He, Na → Cl, Na → Ar, K, Ca, Ga → Kr
Na → Ar
H, He, Li → Ne, Na → Ar, K → Zn, Ga → Kr
H, He, Li → Ne, Na → Ar, Ga → Kr
H, He, Li → Ne, Na → Ar, K → Zn, Ga → Kr
H, He, Li → Ne, Na → Ar, K → Zn, Ga → Kr
H, He, Li → Ne, Na → Ar, Ga → Kr
H, He, Li → Ne, Na → Ar, Ga → Kr
H, He, Li → Ne, Na → Ar, Ga → Kr
H, He, Li → Ne, Na → Ar
H, He, Li → Ne, Na → Ar, Ga → Kr
H, He, Li → Ne, Na → Ar, Ga → Kr
H, He, Li → Ne, Na → Ar, K → Kr
H, He, Li → Ne, Na → Ar, Ga → Kr

Table 8.3: Examples of extended Pople basis sets.

k
l
m
n

SV(k+, l+)(md, np), DZ(k+, l+)(md, np), TZ(k+, l+)(md, np)
# sets of heavy atom diffuse functions
# sets of light atom diffuse functions
# sets of d functions on heavy atoms
# sets of p functions on light atoms

Table 8.4: Summary of Dunning-type basis sets available in Q-C HEM.

464

Chapter 8: Basis Sets

Symbolic Name
SV
DZ
TZ

Atoms Supported
H, Li → Ne
H, Li → Ne, Al → Cl
H, Li → Ne

Table 8.5: Atoms supported for old Dunning basis sets available in Q-C HEM.

Symbolic Name
SV
SV*
SV(d,p)
DZ
DZ+
DZ++
DZ*
DZ**
DZ(d,p)
TZ
TZ+
TZ++
TZ*
TZ**
TZ(d,p)

Atoms Supported
H, Li → Ne
H, B → Ne
H, B → Ne
H, Li → Ne, Al→Cl
H, B → Ne
H, B → Ne
H, Li → Ne
H, Li → Ne
H, Li → Ne
H, Li→Ne
H, Li→Ne
H, Li→Ne
H, Li→Ne
H, Li→Ne
H, Li→Ne

Table 8.6: Examples of extended Dunning basis sets.

Symbolic Name
cc-pVDZ
cc-pVTZ
cc-pVQZ
cc-pV5Z
cc-pCVDZ
cc-pCVTZ
cc-pCVQZ
cc-pCV5Z
aug-cc-pVDZ
aug-cc-pVTZ
aug-cc-pVQZ
aug-cc-pV5Z
aug-cc-pCVDZ
aug-cc-pCVTZ
aug-cc-pCVQZ
aug-cc-pCV5Z

Atoms Supported
H → Ar, Ca, Ga → Kr
H → Ar, Ca, Ga → Kr
H → Ar, Ca, Ga → Kr
H → Ar, Ca, Ga → Kr
H → Ar (H and He use cc-pVDZ)
H → Ar (H and He use cc-pVTZ)
H → Ar (H and He use cc-pVQZ)
H, He, B → Ar (H and He use cc-pV5Z)
H → Ar, Ga → Kr
H → Ar, Ga → Kr
H → Ar, Ga → Kr
H, He, B → Ne, Al → Ar, Ga → Kr
H → Ar (H and He use aug-cc-pVDZ)
H → Ar (H and He use aug-cc-pVTZ)
H → Ar (H and He use aug-cc-pVQZ)
He, He, B → Ne, Al → Ar (H and He use aug-cc-pV5Z)

Table 8.7: Atoms supported Dunning correlation-consistent basis sets available in Q-C HEM.

465

Chapter 8: Basis Sets
Symbolic Name
TZV
VDZ
VTZ

Atoms Supported
Li → Kr
H → Kr
H → Kr

Table 8.8: Atoms supported for Ahlrichs basis sets available in Q-C HEM.
Symbolic Name
pcseg-0, pcseg-1, pcseg-2, pcseg-3, pcseg-4
(pc-0, pc-1, pc-2, pc-3, pc-4)
pcJ-0, pcJ-1, pcJ-2, pcJ-3, pcJ-4
pcS-0, pcS-1, pcS-2, pcS-3, pcS-4

Atoms Supported
H → Kr
H → Kr
H → Ar, except Li, Be, Na, Mg
H → Ar

Table 8.9: Atoms supported for Jensen polarization consistent basis sets available in Q-C HEM. The pcseg-n sets should
be preferred in stead of pc-n, as they are more efficient in Q-C HEM.
generated from the standard diffuse functions by applying a scaling factor to the exponent of the original diffuse
function. This yields a geometric series of exponents for the diffuse functions which includes the original standard
functions along with more diffuse functions.
When using very large basis sets, especially those that include many diffuse functions, or if the system being studied
is very large, linear dependence in the basis set may arise. This results in an over-complete description of the space
spanned by the basis functions, and can cause a loss of uniqueness in the molecular orbital coefficients. Consequently,
the SCF may be slow to converge or behave erratically. Q-C HEM will automatically check for linear dependence in the
basis set, and will project out the near-degeneracies, if they exist. This will result in there being slightly fewer molecular
orbitals than there are basis functions. Q-C HEM checks for linear-dependence by considering the eigenvalues of the
overlap matrix. Very small eigenvalues are an indication that the basis set is close to being linearly dependent. The size
at which the eigenvalues are considered to be too small is governed by the $rem variable BASIS_LIN_DEP_THRESH.
By default this is set to 6, corresponding to a threshold of 10−6 . This has been found to give reliable results, however,
if you have a poorly behaved SCF, and you suspect there maybe linear dependence in you basis, the threshold should
be increased.
Symbolic Name
def-mSVP
def2-SV(P), def2-SVP, def2-SVPD
def2-TZVP, def2-TZVPP, def2-TZVPD, def2-TZVPPD
def2-QZVP, def2-QZVPP, def2-QZVPD, def2-QZVPPD
UGBS

Atoms Supported
H-Kr (Na-Kr are identical to def2-SV(P))
He-Kr, Rb-Rn (with def2-ECP)
He-Kr, Rb-Rn (with def2-ECP)
He-Kr, Rb-Rn (with def2-ECP)
H-Rn

Table 8.10: Atoms supported for Karlsruhe “def2" basis sets and the universal Gaussian basis set (UGBS) available in
Q-C HEM.

Chapter 8: Basis Sets

466

PRINT_GENERAL_BASIS
Controls print out of built in basis sets in input format
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
TRUE
Print out standard basis set information
FALSE Do not print out standard basis set information
RECOMMENDATION:
Useful for modification of standard basis sets.
BASIS_LIN_DEP_THRESH
Sets the threshold for determining linear dependence in the basis set
TYPE:
INTEGER
DEFAULT:
6 Corresponding to a threshold of 10−6
OPTIONS:
n Sets the threshold to 10−n
RECOMMENDATION:
Set to 5 or smaller if you have a poorly behaved SCF and you suspect linear dependence in you
basis set. Lower values (larger thresholds) may affect the accuracy of the calculation.

8.4
8.4.1

User-Defined Basis Sets ($basis)
Introduction

Users may, on occasion, prefer to use non-standard basis, and it is possible to declare user-defined basis sets in QC HEM input (see Chapter 3 on Q-C HEM inputs). The format for inserting a non-standard user-defined basis set is both
logical and flexible, and is described in detail in the job control section below.
Note that the SAD guess is not currently supported with non-standard or user-defined basis sets. The simplest alternative
is to specify the GWH or CORE options for SCF_GUESS, but these are relatively ineffective other than for small basis
sets. The recommended alternative is to employ basis set projection by specifying a standard basis set for the BASIS2
keyword. See the section in Chapter 4 on initial guesses for more information.

8.4.2

Job Control

In order to use a user-defined basis set the BASIS $rem must be set to GENERAL or GEN.
When using a non-standard basis set which incorporates d or higher angular momentum basis functions, the $rem
variable PURECART needs to be initiated. This $rem variable indicates to the Q-C HEM program how to handle the
angular form of the basis functions. As indicated above, each integer represents an angular momentum type which can
be defined as either pure (1) or Cartesian (2). For example, 111 would specify all g, f and d basis functions as being in
the pure form. 121 would indicate g- and d- functions are pure and f -functions Cartesian.

467

Chapter 8: Basis Sets
PURECART
INTEGER
TYPE:
Controls the use of pure (spherical harmonic) or Cartesian angular forms
DEFAULT:
2111 Cartesian h-functions and pure g, f, d functions
OPTIONS:
hgf d Use 1 for pure and 2 for Cartesian.
RECOMMENDATION:
This is pre-defined for all standard basis sets

In standard basis sets all functions are pure, except for the d functions in n-21G–type bases (e.g., 3-21G) and n-31G
bases (e.g., 6-31G, 6-31G*,6-31+G*, . . .). In particular, the 6-311G series uses pure functions for both d and f .

8.4.3

Format for User-Defined Basis Sets

The format for the user-defined basis section is as follows:
$basis
X
L
α1
α2
..
.
αK

0
K
C1Lmin
C2Lmin
..
.
Lmin
CK

scale
C1Lmin +1
C2Lmin +1
..
.
Lmin +1
CK

...
...
..
.
...

C1Lmax
C2Lmax
..
.
Lmax
CK

****
$end

where
X
L
K
scale
αi
CiL

Atomic symbol of the atom (atomic number not accepted)
Angular momentum symbol (S, P, SP, D, F, G)
Degree of contraction of the shell (integer)
Scaling to be applied to exponents (default is 1.00)
Gaussian primitive exponent (positive real number)
Contraction coefficient for each angular momentum (non-zero real numbers).

Atoms are terminated with **** and the complete basis set is terminated with the $end keyword terminator. No blank
lines can be incorporated within the general basis set input. Note that more than one contraction coefficient per line is
one required for compound shells like SP. As with all Q-C HEM input deck information, all input is case-insensitive.

468

Chapter 8: Basis Sets

8.4.4

Example

Example 8.1 Example of adding a user-defined non-standard basis set. Note that since d, f and g functions are
incorporated, the $rem variable PURECART must be set. Note the use of BASIS2 for the initial guess.
$molecule
0 1
O
H O oh
H O oh

2

hoh

oh =
1.2
hoh = 110.0
$end
$rem
EXCHANGE
BASIS
BASIS2
PURECART
$end
$basis
H 0
S 2 1.00
1.30976
0.233136
****
O 0
S 2 1.00
49.9810
8.89659
SP 2 1.00
1.94524
0.49336
D 1 1.00
0.39000
F 1 1.00
4.10000
G 1 1.00
3.35000
****
$end

8.5

hf
gen
sto-3g
112

user-defined general basis
sto-3g orbitals as initial guess
Cartesian d functions, pure f and g

0.430129
0.678914

0.430129
0.678914
0.049472
0.963782

0.511541
0.612820

1.000000
1.000000
1.000000

Mixed Basis Sets

In addition to defining a custom basis set, it is also possible to specify different standard basis sets for different atoms.
For example, in a large alkene molecule the hydrogen atoms could be modeled by the STO-3G basis, while the carbon
atoms have the larger 6-31G(d) basis. This can be specified within the $basis block using the more familiar basis set
labels.
Note: (1) It is not possible to augment a standard basis set in this way; the whole basis needs to be inserted as for
a user-defined basis (angular momentum, exponents, contraction coefficients) and additional functions added.
Standard basis set exponents and coefficients can be easily obtained by setting the PRINT_GENERAL_BASIS
$rem variable to TRUE.
(2) The PURECART flag must be set for all general basis input containing d angular momentum or higher
functions, regardless of whether standard basis sets are entered in this non-standard manner.

469

Chapter 8: Basis Sets

The user can also specify different basis sets for atoms of the same type, but in different parts of the molecule. This
allows a larger basis set to be used for the active region of a system, and a smaller basis set to be used in the less
important regions. To enable this the BASIS keyword must be set to MIXED and a $basis section included in the input
deck that gives a complete specification of the basis sets to be used. The format is exactly the same as for the userdefined basis, except that the atom number (as ordered in the $molecule section) must be specified in the field after
the atomic symbol. A basis set must be specified for every atom in the input, even if the same basis set is to be used
for all atoms of a particular element. Custom basis sets can be entered, and the shorthand labeling of basis sets is also
supported.
The use of different basis sets for a particular element means the global potential energy surface is no longer unique.
The user should exercise caution when using this feature of mixed basis sets, especially during geometry optimizations
and transition state searches.
Example 8.2 Example of adding a user defined non-standard basis set. The user is able to specify different standard
basis sets for different atoms.
$molecule
0 1
O
H O oh
H O oh

2

hoh

oh =
1.2
hoh = 110.0
$end
$rem
EXCHANGE
BASIS
PURECART
BASIS2
$end
$basis
H 0
6-31G
****
O 0
6-311G(d)
****
$end

hf
General
2
sto-3g

user-defined general basis
Cartesian D functions
use STO-3G as initial guess

470

Chapter 8: Basis Sets

Example 8.3 Example of using a mixed basis set for methanol. The user is able to specify different standard basis sets
for some atoms and supply user-defined exponents and contraction coefficients for others. This might be particularly
useful in cases where the user has constructed exponents and contraction coefficients for atoms not defined in a standard
basis set so that only the non-defined atoms need have the exponents and contraction coefficients entered. Note that a
basis set has to be specified for every atom in the molecule, even if the same basis is to be used on an atom type.
$molecule
0 1
C
0.0000000
H
0.9153226
H
0.0000000
H
-0.9153226
O
0.0000000
H
0.0000000
$end
$rem
EXCHANGE
BASIS
$end

0.0148306
0.5361067
-1.0112551
0.5361067
-0.0695490
0.8662925

hf
mixed

$basis
C 1
3-21G
****
O 2
S 3 1.00
3.22037000E+02
4.84308000E+01
1.04206000E+01
SP 2 1.00
7.40294000E+00
1.57620000E+00
SP 1 1.00
3.73684000E-01
SP 1 1.00
8.45000000E-02
****
H 3
6-31(+,+)G(d,p)
****
H 4
sto-3g
****
H 5
sto-3g
****
H 6
sto-3g
****
$end

8.6

0.7155831
1.0707116
1.1374379
1.0707116
-0.6801243
-1.0101622

user-defined mixed basis

5.92394000E-02
3.51500000E-01
7.07658000E-01
-4.04453000E-01
1.22156000E+00

2.44586000E-01
8.53955000E-01

1.00000000E+00

1.00000000E+00

1.00000000E+00

1.00000000E+00

Dual Basis Sets

There are several types of calculation that can be performed within Q-C HEM using two atomic orbital basis sets instead
of just one as we have been assuming in this chapter so far. Such calculations are said to involve dual basis sets. Typically iterations are performed in a smaller, primary, basis, which is specified by the $rem keyword BASIS2. Examples
of calculations that can be performed using dual basis sets include:

471

Chapter 8: Basis Sets
• An improved initial guess for an SCF calculation in the large basis. See Section 4.4.5.

• Dual basis self-consistent field calculations (Hartree-Fock and density functional theory). See discussion in
Section 4.7.
• Density functional perturbative corrections by “triple jumping”. See Section 4.8.
• Dual basis MP2 calculations. See discussion in Section 6.6.1.
BASIS2
Defines the (small) second basis set.
TYPE:
STRING
DEFAULT:
No default for the second basis set.
OPTIONS:
Symbol
Use standard basis sets as for BASIS.
BASIS2_GEN
General BASIS2
BASIS2_MIXED Mixed BASIS2
RECOMMENDATION:
BASIS2 should be smaller than BASIS. There is little advantage to using a basis larger than a
minimal basis when BASIS2 is used for initial guess purposes. Larger, standardized BASIS2
options are available for dual-basis calculations as discussed in Section 4.7 and summarized in
Table 4.7.3.

In addition to built-in basis sets for BASIS2, it is also possible to enter user-defined second basis sets using an additional
$basis2 input section, whose syntax generally follows the $basis input section documented above in Section 8.4.

8.7

Auxiliary Basis Sets for RI (Density Fitting)

While atomic orbital standard basis sets are used to expand one-electron functions such as molecular orbitals, auxiliary
basis sets are also used in many Q-C HEM jobs to efficiently approximate products of one-electron functions, such as
arise in electron correlation methods.
For a molecule of fixed size, increasing the number of basis functions per atom, n, leads to O(n4 ) growth in the number
of significant four-center two-electron integrals, since the number of non-negligible product charge distributions, |µνi,
grows as O(n2 ). As a result, the use of large (high-quality) basis expansions is computationally costly. Perhaps the
most practical way around this “basis set quality” bottleneck is the use of auxiliary basis expansions. 6,8,12 The ability
to use auxiliary basis sets to accelerate a variety of electron correlation methods, including both energies and analytical
gradients, is a major feature of Q-C HEM.
The auxiliary basis {|Ki} is used to approximate products of Gaussian basis functions:
X
K
|µνi ≈ |f
µνi =
|KiCµν

(8.1)

K

Auxiliary basis expansions were introduced long ago, and are now widely recognized as an effective and powerful
approach, which is sometimes synonymously called resolution of the identity (RI) or density fitting (DF). When using
auxiliary basis expansions, the rate of growth of computational cost of large-scale electronic structure calculations with
n is reduced to approximately n3 .
If n is fixed and molecule size increases, auxiliary basis expansions reduce the pre-factor associated with the computation, while not altering the scaling. The important point is that the pre-factor can be reduced by 5 or 10 times or more.
Such large speedups are possible because the number of auxiliary functions required to obtain reasonable accuracy, X,
has been shown to be only about 3 or 4 times larger than N .

472

Chapter 8: Basis Sets

The auxiliary basis expansion coefficients, C, are determined by minimizing the deviation between the fitted distribution and the actual distribution, hµν − µ
fν|µν − µ
fνi, which leads to the following set of linear equations:
X
L
hK |L iCµν
= hK |µν i
(8.2)
L

Evidently solution of the fit equations requires only two- and three-center integrals, and as a result the (four-center)
two-electron integrals can be approximated as the following optimal expression for a given choice of auxiliary basis
set:
X
L
K
f =
hµν|λσi ≈ hf
µν|λσi
Cµν
hL|KiCλσ
(8.3)
K,L

In the limit where the auxiliary basis is complete (i.e. all products of AOs are included), the fitting procedure described
above will be exact. However, the auxiliary basis is invariably incomplete (as mentioned above, X ≈ 3N ) because this
is essential for obtaining increased computational efficiency.
More details on Q-C HEM’s use of RI methods is given in Section 6.6 on RI-MP2 and related methods, Section 6.15
on pairing methods, Section 6.8.5 on coupled cluster methods, Section 4.6.6 on DFT methods, and Section 7.9 on
restricted active space methods. In the remainder of this section we focus on documenting the input associated with the
auxiliary basis itself.
Q-C HEM contains a variety of built-in auxiliary basis sets, that can be specified by the $rem keyword AUX_BASIS.
AUX_BASIS
Sets the auxiliary basis set to be used
TYPE:
STRING
DEFAULT:
No default auxiliary basis set
OPTIONS:
General, Gen User-defined. As for BASIS
Symbol
Use standard auxiliary basis sets as in the table below
Mixed
Use a combination of different basis sets
RECOMMENDATION:
Consult literature and EMSL Basis Set Exchange to aid your selection.

Symbolic Name
RIMP2-VDZ
RIMP2-TZVPP
RIMP2-cc-pVDZ
RIMP2-cc-pVTZ
RIMP2-cc-pVQZ
RIMP2-aug-cc-pVDZ
RIMP2-aug-cc-pVTZ
RIMP2-aug-cc-pVQZ

Atoms Supported
H, He, Li → Ne, Na → Ar, K → Br
H, He, Li → Ne, Na → Ar, Ga → Kr
H, He, Li → Ne, Na → Ar, Ga → Kr
H, He, Li → Ne, Na → Ar, Ga → Kr
H, He, Li → Ne, Na → Ar, Ga → Kr
H, He, B → Ne, Al → Ar, Ga → Kr
H, He, B → Ne, Al → Ar, Ga → Kr
H, He, B → Ne, Al → Ar, Ga → Kr

Table 8.11: Built-in auxiliary basis sets available in Q-C HEM for electron correlation.
In addition to built-in auxiliary basis sets, it is also possible to enter user-defined auxiliary basis sets using an $aux_basis
input section, whose syntax generally follows the $basis input section documented above in Section 8.4.

473

Chapter 8: Basis Sets

8.8

Ghost Atoms and Basis Set Superposition Error

When calculating intermolecular interaction energies, a naïve calculation of the energy difference
∆EAB = EAB − EA − EB

(8.4)

usually results in severe overestimation of the interaction energy, even if all three energies in Eq. (8.4) are computed
at a good level of theory. This phenomenon, known as basis set superposition error (BSSE), is an artifact of an
unbalanced approximation, namely, that the dimer energy EAB is computed in a more flexible basis set as compared
to the two monomer energies. Although BSSE disappears in the complete basis-set limit, it does so extremely slowly:
in (H2 O)6 , for example, an MP2/aug-cc-pVQZ calculation of the interaction energy is still a bit more than 1 kcal/mol
away from the MP2 complete-basis limit. 13 Short of computing all energies in very large basis sets and extrapolating
to the complete-basis limit, the conventional solution to the BSSE problem is the counterpoise correction, originally
proposed by Boys and Bernardi. 4 Here, one corrects for BSSE by computing the monomer energies EA and EB in the
dimer basis set, with the idea being that this results in a more balanced treatment of ∆EAB .
In truth the average of the counterpoise-corrected and uncorrected results is often a better approximation than either of
them individually, but in any case one needs the counterpoise-corrected result. This requires basis functions to be placed
at arbitrary points in space, not just those defined by the nuclear centers; these are usually termed “floating centers”
or “ghost atoms”. Ghost atoms have zero nuclear charge but can support a user-defined basis set. Their positions are
specified in the $molecule section alongside all the other atoms (atomic symbol: Gh), and their intended basis functions
are specified in one of two ways:
1. Via a user-defined $basis section, using BASIS = MIXED.
2. Placing “@” next to an atomic symbol in the $molecule section designates it as a ghost atom supporting the same
basis functions as the corresponding atom, so that a $basis section is not required.
Examples of either procedure appear below.
The calculation of ∆EAB in Eq. (8.4) requires three separate electronic structure calculations but this process can be
performed automatically using the Q-C HEM’s machinery based on absolutely-localized molecular orbitals (ALMOs).
This machinery is much more versatile and is described in detail later so we will not discuss the automatic procedure

474

Chapter 8: Basis Sets
here; see Section 13.4.3 for that.

Example 8.4 A calculation on a water monomer in the presence of the full dimer basis set. The energy will be slightly
lower than that without the ghost atom functions due to the greater flexibility of the basis set.
$molecule
0 1
O
1.68668
H
1.09686
H
1.09686
Gh -1.45451
Gh -2.02544
Gh -2.02544
$end
$rem
METHOD
BASIS
$end

-0.00318
0.01288
0.01288
0.01190
-0.04298
-0.04298

0.000000
-0.741096
0.741096
0.000000
-0.754494
0.754494

mp2
mixed

$basis
O 1
6-31G*
****
H 2
6-31G*
****
H 3
6-31G*
****
O 4
6-31G*
****
H 5
6-31G*
****
H 6
6-31G*
****
$end

Example 8.5 A calculation on ammonia in the presence of the basis set of ammonia borane.
$molecule
0 1
N
0.0000
H
0.9507
H
-0.4752
H
-0.4755
@B
0.0000
@H
0.5859
@H
0.5857
@H
-1.1716
$end
$rem
METHOD
BASIS
PURECART
$end

0.0000
0.0001
-0.8234
0.8233
0.0000
1.0146
-1.0147
0.0001

0.7288
1.0947
1.0947
1.0947
-0.9379
-1.2474
-1.2474
-1.2474

B3LYP
6-31G(d,p)
1112

Chapter 8: Basis Sets

475

References and Further Reading
[1] Basis sets were obtained from the Extensible Computational Chemistry Environment Basis Set Database, Version 1.0, as developed and distributed by the Molecular Science Computing Facility, Environmental and Molecular Sciences Laboratory which is part of the Pacific Northwest Laboratory, P.O. Box 999, Richland, Washington 99352, USA, and funded by the U.S. Department of Energy. The Pacific Northwest Laboratory is a multiprogram laboratory operated by Battelle Memorial Institute for the U.S. Department of Energy under contract
DE-AC06-76RLO 1830. Contact David Feller, Karen Schuchardt or Don Jones for further information.
[2] Ground-State Methods (Chapters 4 and 6).
[3] Effective Core Potentials (Chapter 9).
[4] S. F. Boys and F. Bernardi. Mol. Phys., 19:553, 1970. DOI: 10.1080/00268977000101561.
[5] E. R. Davidson and D. Feller. Chem. Rev., 86:681, 1986. DOI: 10.1021/cr00074a002.
[6] B. I. Dunlap. Phys. Chem. Chem. Phys., 2:2113, 2000. DOI: 10.1039/b000027m.
[7] D. Feller and E. R. Davidson. In K. B. Lipkowitz and D. B. Boyd, editors, Reviews in Computational Chemistry,
volume 1, page 1. Wiley-VCH, New York, 1990. DOI: 10.1002/9780470125786.ch1.
[8] M. Feyereisen, G. Fitzgerald, and A. Komornicki. Chem. Phys. Lett., 208:359, 1993. DOI: 10.1016/00092614(93)87156-W.
[9] W. J. Hehre, L. Radom, P. v. R. Schleyer, and J. A. Pople. Ab Initio Molecular Orbital Theory. Wiley, New York,
1986.
[10] S. Huzinaga. Comp. Phys. Rep., 2:281, 1985. DOI: 10.1016/0167-7977(85)90003-6.
[11] F. Jensen. Introduction to Computational Chemistry. Wiley, New York, 1994.
[12] Y. Jung, A. Sodt, P. M. W. Gill, and M. Head-Gordon. Proc. Natl. Acad. Sci. USA, 102:6692, 2005. DOI:
0.1073/pnas.0408475102.
[13] R. M. Richard, K. U. Lao, and J. M. Herbert. J. Phys. Chem. Lett., 4:2674, 2013. DOI: 10.1021/jz401368u.

Chapter 9

Effective Core Potentials
9.1

Introduction

The application of quantum chemical methods to elements in the lower half of the Periodic Table is more difficult than
for the lighter atoms. There are two key reasons for this:
• the number of electrons in heavy atoms is large
• relativistic effects in heavy atoms are often non-negligible
Both of these problems stem from the presence of large numbers of core electrons and, given that such electrons do
not play a significant direct role in chemical behavior, it is natural to ask whether it is possible to model their effects
in some simpler way. Such enquiries led to the invention of Effective Core Potentials (ECPs) or pseudopotentials. For
reviews of relativistic effects in chemistry, see for example Refs. 4,7,9,13,17,32.
If we seek to replace the core electrons around a given nucleus by a pseudopotential, while affecting the chemistry as
little as possible, the pseudopotential should have the same effect on nearby valence electrons as the core electrons.
The most obvious effect is the simple electrostatic repulsion between the core and valence regions but the requirement
that valence orbitals must be orthogonal to core orbitals introduces additional subtler effects that cannot be neglected.
One of the key issues in the development of ECPs is the definition of the “core”. So-called “large-core” ECPs include
all shells except the outermost one, but “small-core” ECPs include all except the outermost two shells. Although the
small-core ECPs are more expensive to use (because more electrons are treated explicitly), it is often found that their
enhanced accuracy justifies their use.
When an ECP is constructed, it is usually based either on non-relativistic, or quasi-relativistic all-electron calculations. As one might expect, the quasi-relativistic ECPs tend to yield better results than their non-relativistic brethren,
especially for atoms beyond the 3d block
Q-C HEM’s ECP package is integrated with its electron correlation and DFT packages. Of course, no correlation or
exchange-correlation energy due to the core electrons is included when using an ECP in a DFT or correlated method,
respectively.
The most widely used ECPs today are of the form first proposed by Kahn et al. in the 1970s. 22 These model the effects
of the core by a one-electron operator U (r) whose matrix elements are simply added to the one-electron Hamiltonian
matrix. The ECP operator is given by
U (r) = UL (r) +

L−1
X

+l
X

`=0 m=−l

|Y`m i Ul (r) hY`m |

(9.1)

477

Chapter 9: Effective Core Potentials
where the radial potentials have the form
U` (r) =

K
X̀

D`k rn`k e−η`k r

2

(9.2)

k=1

P
and m |Y`m i hY`m | is the spherical harmonic projector of angular momentum `. In practice, n`k = −2, −1 or 0 and
L rarely exceeds 5. In addition, UL (r) contains a Coulombic term Nc /r, where Nc is the number of core electrons.

9.2

ECP Fitting

The ECP matrix elements are arguably the most difficult one-electron integrals in existence. Indeed, using current
methods, the time taken to compute the ECP integrals can exceed the time taken to compute the far more numerous
electron repulsion integrals. Q-C HEM 5.0 implements a state-of-the-art ECP implementation 28 based on efficient
recursion relations and upper bounds. This method relies on a restricted radial potential U` (r), where the radial power
is only ever zero, i.e. n = 0. Whilst true for some ECPs, such as the Stuttgart-Bonn sets, many other ECPs have
radial potentials containing n = −2 and n = −1 terms. To overcome this challenge, we fit these ECP radial potentials
using only n = 0 terms. Each n = −2 and n = −1 term is expanded as a sum of three n = 0 terms, each
with independent contraction coefficient D`k and Gaussian exponent η`k . The Gaussian exponents are given by a
predetermined recipe and the contraction coefficients are computed in a least squares fitting procedure. The errors
introduced by the ECP fitting are insignificant and of the same order as those introduced by numerical integration
present in other ECP methods. For the built-in ECPs, fitted variants of each are now provided in the $QCAUX directory
e.g. fit-LANL2DZ. For user-defined ECPs with n = −2 or n = −1 terms, Q-C HEM will perform a fit at run time with
the additional rem keyword “ECP_FIT = TRUE".

9.3
9.3.1

Built-In ECPs
Overview

Q-C HEM is equipped with several standard ECP sets which are specified using the ECP keyword within the $rem block.
The built-in ECPs, which are described in some detail at the end of this Chapter, fall into four families:
• The Hay-Wadt (or Los Alamos) sets (fit-HWMB and fit-LANL2DZ)
• The Stevens-Basch-Krauss-Jansien-Cundari set (fit-SBKJC)
• The Christiansen-Ross-Ermler-Nash-Bursten sets (fit-CRENBS and fit-CRENBL)
• The Stuttgart-Bonn sets (SRLC and SRSC)
Besides the ones above, a common “def2-ECP" needs to be used with Karlsruhe basis sets for elements Rb-Rn (see
section 8.3).
References and information about the definition and characteristics of most of these sets can be found at the EMSL site
of the Pacific Northwest National Laboratory: 1
http://www.emsl.pnl.gov/forms/basisform.html
Each of the built-in ECPs comes with a matching orbital basis set for the valence electrons. In general, it is advisable
to use these together and, if you select a basis set other than the matching one, Q-C HEM will print a warning message
in the output file. If you omit the BASIS $rem keyword entirely, Q-C HEM will automatically provide the matching one.
The following $rem variable controls which ECP is used:

Chapter 9: Effective Core Potentials

478

ECP
Defines the effective core potential and associated basis set to be used
TYPE:
STRING
DEFAULT:
No ECP
OPTIONS:
General, Gen User defined. ($ecp keyword required)
Symbol
Use standard ECPs discussed above.
RECOMMENDATION:
ECPs are recommended for first row transition metals and heavier elements. Consul the reviews
for more details.

9.3.2

Combining ECPs

If you wish, you can use different ECP sets for different elements in the system. This is especially useful if you would
like to use a particular ECP but find that it is not available for all of the elements in your molecule. To combine different
ECP sets, you set the ECP and BASIS keywords to “GEN” or (equivalently) “GENERAL”, and then add a $ecp block and
a $basis block to your input file. In each of these blocks, you must name the ECP and the orbital basis set that you wish
to use, separating each element by “****”. There is also a built-in combination that can be invoked specifying ECP =
fit-LACVP. It assigns automatically 6-31G* or other suitable type basis sets for atoms H–Ar, while uses fit-LANL2DZ
for heavier atoms.

9.3.3

Examples

Example 9.1 Computing the HF/fit-LANL2DZ energy of AgCl at a bond length of 2.4 Å.
$molecule
0 1
Ag
Cl Ag
r
$end

=

r

$rem
METHOD
ECP
BASIS
$end

2.4

hf
Hartree-Fock calculation
fit-lanl2dz
Using the Hay-Wadt ECP
lanl2dz
And the matching basis set

479

Chapter 9: Effective Core Potentials

Example 9.2 Computing the single point energy of HI with B3LYP/def2-SV(P) (using def2-ECP for I).
$molecule
0 1
H
0.0
I
0.0
$end

0.0
0.0

$rem
METHOD
BASIS
ECP
SYMMETRY
SYM_IGNORE
THRESH
SCF_CONVERGENCE
$end

0.0
1.5

b3lyp
def2-sv(p)
def2-ecp
false
true
14
8

Example 9.3 Optimization of the structure of Se8 using HF/fit-LANL2DZ, followed by a single-point energy calculation at the MP2/fit-LANL2DZ level.
$molecule
0 1
x1
x2
x1
Se1 x1
Se2 x1
Se3 x1
Se4 x1
Se5 x2
Se6 x2
Se7 x2
Se8 x2

xx
sx
sx
sx
sx
sx
sx
sx
sx

x2
x2
x2
x2
x1
x1
x1
x1

90.
90.
90.
90.
90.
90.
90.
90.

Se1
Se2
Se3
Se1
Se5
Se6
Se7

90.
90.
90.
45.
90.
90.
90.

xx = 1.2
sx = 2.8
$end
$rem
JOBTYPE
METHOD
ECP
$end

opt
hf
fit-lanl2dz

@@@
$molecule
read
$end
$rem
METHOD
ECP
SCF_GUESS
$end

mp2
MP2 correlation energy
fit-lanl2dz
Hay-Wadt ECP and basis
read
Read in the MOs

Chapter 9: Effective Core Potentials

480

Example 9.4 Computing the HF geometry of CdBr2 using the Stuttgart relativistic ECPs. The small-core ECP and
basis are employed on the Cd atom and the large-core ECP and basis on the Br atoms.
$molecule
0 1
Cd
Br1 Cd
Br2 Cd

r
r

Br1

180.0

r = 2.4
$end
$rem
JOBTYPE
METHOD
ECP
BASIS
PURECART
$end

opt
hf
gen
gen
1

Geometry optimization
Hartree-Fock theory
Combine ECPs
Combine basis sets
Use pure d functions

$ecp
Cd
srsc
****
Br
srlc
****
$end
$basis
Cd
srsc
****
Br
srlc
****
$end

9.4

User-Defined ECPs

Many users will find that the library of built-in ECPs is adequate for their needs. However, if you need to use an ECP
that is not built into Q-C HEM, you can enter it in much the same way as you can enter a user-defined orbital basis set;
see Chapter 8.

9.4.1

Job Control for User-Defined ECPs

To apply a user-defined ECP, you must set the ECP and BASIS keywords in $rem to GEN. You then add a $ecp block
that defines your ECP, element by element, and a $basis block that defines your orbital basis set, separating elements
by asterisks.
The syntax within the $basis block is described in Chapter 8. The syntax for each record within the $ecp block is as
follows:.
$ecp
For each atom that will bear an ECP
Chemical symbol for the atom

Chapter 9: Effective Core Potentials
ECP name ; the L value for the ECP ; number of core electrons removed
For each ECP component (in the order unprojected, P̂0 , P̂1 , , P̂L−1
The component name
The number of Gaussians in the component
For each Gaussian in the component
The power of r ; the exponent ; the contraction coefficient
A sequence of four asterisks (i.e., ****)
$end
Note: (1) All of the information in the $ecp block is case-insensitive.
(2) The power of r (which includes the Jacobian r2 factor) must be 0, 1 or 2.
(3) If an r0 or r1 term is included you must include the rem keyword “ECP_FIT = TRUE".

481

482

Chapter 9: Effective Core Potentials

9.4.2

Example

Example 9.5 Optimizing the HF geometry of AlH3 using a user-defined ECP and basis set on Al and the 3-21G basis
on H.
$molecule
0 1
Al
H1 Al
H2 Al
H3 Al

r
r
r

H1
H1

120.0
120.0

H2

180.0

r = 1.6
$end
$rem
JOBTYPE
opt
METHOD
hf
ECP
gen
BASIS
gen
ECP_FIT = TRUE
$end

Geometry optimization
Hartree-Fock theory
User-defined ECP
User-defined basis

$ecp
Al
Stevens_ECP 2 10
d potential
1
1
1.95559 -3.03055
s-d potential
2
0
7.78858
6.04650
2
1.99025 18.87509
p-d potential
2
0
2.83146
3.29465
2
1.38479
6.87029
****
$end
$basis
Al
SP 3 1.00
0.90110
0.44950
0.14050
SP 1 1.00
0.04874
****
H
3-21G
****
$end

-0.30377
0.13382
0.76037

-0.07929
0.16540
0.53015

0.32232

0.47724

483

Chapter 9: Effective Core Potentials

9.5

ECPs and Electron Correlation

The ECP package is integrated with the electron correlation package and it is therefore possible to apply any of QC HEM’s post-Hartree-Fock methods to systems in which some of the atoms may bear pseudopotentials. Of course, the
correlation energy contribution arising from core electrons that have been replaced by an ECP is not included. In this
sense, correlation energies with ECPs are comparable to correlation energies from frozen-core calculations. However,
the use of ECPs effectively removes both core electrons and the corresponding virtual (unoccupied) orbitals.
Any of the local, gradient-corrected and hybrid functionals discussed in Chapter 5 may be used and you may also perform ECP calculations with user-defined hybrid functionals. In a DFT calculation with ECPs, the exchange-correlation
energy is obtained entirely from the non-core electrons. This will be satisfactory if there are no chemically important
cores/valence effects but may introduce significant errors if not, particularly if you are using a “large-core” ECP.
Example 9.6 Optimization of the structure of Se8 using HF/fit-LANL2DZ, followed by a single-point energy calculation at the MP2/fit-LANL2DZ level.
$molecule
0 1
x1
x2
x1
Se1 x1
Se2 x1
Se3 x1
Se4 x1
Se5 x2
Se6 x2
Se7 x2
Se8 x2

xx
sx
sx
sx
sx
sx
sx
sx
sx

x2
x2
x2
x2
x1
x1
x1
x1

90.
90.
90.
90.
90.
90.
90.
90.

Se1
Se2
Se3
Se1
Se5
Se6
Se7

90.
90.
90.
45.
90.
90.
90.

xx = 1.2
sx = 2.8
$end
$rem
JOBTYPE
METHOD
ECP
$end

opt
hf
fit-lanl2dz

@@@
$molecule
read
$end
$rem
JOBTYPE
METHOD
ECP
SCF_GUESS
$end

9.6

sp
Single-point energy
mp2
MP2 correlation energy
fit-lanl2dz
Hay-Wadt ECP and basis
read
Read in the MOs

Forces and Vibrational Frequencies with ECPs

It is important to be able to optimize geometries using pseudopotentials and for this purpose Q-C HEM contains analytical first derivatives of the nuclear potential energy term for ECPs.
The ECP package is also integrated with the vibrational analysis package and it is therefore possible to compute the

Chapter 9: Effective Core Potentials

484

vibrational frequencies (and hence the infrared and Raman spectra) of systems in which some of the atoms may bear
ECPs.
Q-C HEM cannot calculate analytic second derivatives of the nuclear potential-energy term when ECPs are used, and
must therefore resort to finite difference methods. However, for HF and DFT calculations, it can compute analytic
second derivatives for all other terms in the Hamiltonian. The program takes full advantage of this by only computing
the potential-energy derivatives numerically, and adding these to the analytically calculated second derivatives of the
remaining energy terms.
There is a significant speed advantage associated with this approach as, at each finite-difference step, only the potentialenergy term needs to be calculated. This term requires only three-center integrals, which are far fewer in number and
much cheaper to evaluate than the four-center, two-electron integrals associated with the electron-electron interaction
terms. Readers are referred to Table 10.1 for a full list of the analytic derivative capabilities of Q-C HEM.
Example 9.7 Structure and vibrational frequencies of TeO2 using Hartree-Fock theory and the Stuttgart relativistic
large-core ECPs. Note that the vibrational frequency job reads both the optimized structure and the molecular orbitals
from the geometry optimization job that precedes it. Note also that only the second derivatives of the potential energy
term will be calculated by finite difference, all other terms will be calculated analytically.
$molecule
0 1
Te
O1 Te
O2 Te

r
r

O1

a

r = 1.8
a = 108
$end
$rem
JOBTYPE
METHOD
ECP
$end

opt
hf
srlc

@@@
$molecule
read
$end
$rem
JOBTYPE
METHOD
ECP
SCF_GUESS
$end

9.7

freq
hf
srlc
read

A Brief Guide to Q-C HEM’s Built-In ECPs

The remainder of this Chapter consists of a brief reference guide to Q-C HEM’s built-in ECPs. The ECPs vary in their
complexity and their accuracy and the purpose of the guide is to enable the user quickly and easily to decide which
ECP to use in a planned calculation.
The following information is provided for each ECP:
• The elements for which the ECP is available in Q-C HEM. This is shown on a schematic Periodic Table by
shading all the elements that are not supported.

485

Chapter 9: Effective Core Potentials
• The literature reference for each element for which the ECP is available in Q-C HEM.

• The matching orbital basis set that Q-C HEM will use for light (i.e.. non-ECP atoms). For example, if the user
requests SRSC ECPs—which are defined only for atoms beyond argon—Q-C HEM will use the 6-311G* basis
set for all atoms up to Ar.
• The core electrons that are replaced by the ECP. For example, in the fit-LANL2DZ ECP for the Fe atom, the core
is [Ne], indicating that the 1s, 2s and 2p electrons are removed.
• The maximum spherical harmonic projection operator that is used for each element. This often, but not always,
corresponds to the maximum orbital angular momentum of the core electrons that have been replaced by the
ECP. For example, in the fit-LANL2DZ ECP for the Fe atom, the maximum projector is of P -type.
• The number of valence basis functions of each angular momentum type that are present in the matching orbital
basis set. For example, in the matching basis for the fit-LANL2DZ ECP for the Fe atom, there the three s shells,
three p shells and two d shells. This basis is therefore almost of triple-split valence quality.

9.7.1

The fit-HWMB ECP at a Glance

a

a

b

c

d
×
×
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××
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××
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×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
××××××××××××××××××××××××××××××××××××××××××××××××××××××××

fit-HWMB is not available for shaded elements
(a)
(b)
(c)
(d)

No ECP; Pople STO-3G basis used
Wadt & Hay (Ref. 39)
Hay & Wadt (Ref. 20)
Hay & Wadt (Ref. 19)

486

Chapter 9: Effective Core Potentials
Element
H–He
Li–Ne
Na–Ar
K–Ca
Sc–Cu
Zn
Ga–Kr
Rb–Sr
Y–Ag
Cd
In–Xe
Cs–Ba
La
Hf–Au
Hg
Tl–Bi

9.7.2

Core
none
none
[Ne]
[Ne]
[Ne]
[Ar]
[Ar]+3d
[Ar]+3d
[Ar]+3d
[Kr]
[Kr]+4d
[Kr]+4d
[Kr]+4d
[Kr]+4d+4f
[Xe]+4f
[Xe]+4f+5d

Max Projector
none
none
P
P
P
D
D
D
D
D
D
D
D
F
F
F

Valence
(1s)
(2s,1p)
(1s,1p)
(2s,1p)
(2s,1p,1d)
(1s,1p,1d)
(1s,1p)
(2s,1p)
(2s,1p,1d)
(1s,1p,1d)
(1s,1p)
(2s,1p)
(2s,1p,1d)
(2s,1p,1d)
(1s,1p,1d)
(1s,1p)

The fit-LANL2DZ ECP at a Glance

a

a

b

c

d
×
×
×
××
××
××
××
××
××
××
××
××
××
×
×
××
××
××
××
××
××
××
××
××
××
××
×

×
×
××
××
××
××
××
××
××
××
××
××
×
×
×
××
××
××
××
××
××
××
××
××
××
×
××

×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
e
f
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
××××××××××××××××××××××××××××××××××××
×
××××××××

fit-LANL2DZ is not available for shaded elements
(a)
(b)
(c)
(d)
(e)

No ECP; Pople 6-31G basis used
Wadt & Hay (Ref. 39)
Hay & Wadt (Ref. 20)
Hay & Wadt (Ref. 19)
Hay (Ref. 18)

Note that Q-C HEM 4.2.2 and later versions also support LANL2DZ-SV basis, which employs SV basis functions
(instead of 6-31G) on H, Li-He elements (like some other quantum chemistry packages).

487

Chapter 9: Effective Core Potentials
Element
H–He
Li–Ne
Na–Ar
K–Ca
Sc–Cu
Zn
Ga–Kr
Rb–Sr
Y–Ag
Cd
In–Xe
Cs–Ba
La
Hf–Au
Hg
Tl
Pb–Bi
U–Pu

9.7.3

Core
none
none
[Ne]
[Ne]
[Ne]
[Ar]
[Ar]+3d
[Ar]+3d
[Ar]+3d
[Kr]
[Kr]+4d
[Kr]+4d
[Kr]+4d
[Kr]+4d+4f
[Xe]+4f
[Xe]+4f+5d
[Xe]+4f+5d
[Xe]+4f+5d

Max Projector
none
none
P
P
P
D
D
D
D
D
D
D
D
F
F
F
F
F

Valence
(2s)
(3s,2p)
(2s,2p)
(3s,3p)
(3s,3p,2d)
(2s,2p,2d)
(2s,2p)
(3s,3p)
(3s,3p,2d)
(2s,2p,2d)
(2s,2p)
(3s,3p)
(3s,3p,2d)
(3s,3p,2d)
(2s,2p,2d)
(2s,2p,2d)
(2s,2p)
(3s,3p,2d,2f)

The fit-SBKJC ECP at a Glance
a

a
b

b

c
×
×
×
××
××
××
××
××
××
××
××
××
××
×
×
××
××
××
××
××
××
××
××
××
××
××
×

d
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
××××××××××××××××××××××××××××××××××××××××××××××××××××××××
×

fit-SBKJC is not available for shaded elements
(a)
(b)
(c)
(d)

No ECP; Pople 3-21G basis used
Stevens, Basch, & M. Krauss (Ref. 36)
Stevens, Krauss, Basch, & Jasien (Ref. 37)
Cundari & Stevens (Ref. 8)

488

Chapter 9: Effective Core Potentials
Element
H–He
Li–Ne
Na–Ar
K–Ca
Sc–Ga
Ge–Kr
Rb–Sr
Y–In
Sn–Xe
Cs–Ba
La
Ce–Lu
Hf–Tl
Pb–Rn

9.7.4

Core
none
[He]
[Ne]
[Ar]
[Ne]
[Ar]+3d
[Kr]
[Ar]+3d
[Kr]+4d
[Xe]
[Kr]+4d
[Kr]+4d
[Kr]+4d+4f
[Xe]+4f+5d

Max Projector
none
S
P
P
P
D
D
D
D
D
F
D
F
F

Valence
(2s)
(2s,2p)
(2s,2p)
(2s,2p)
(4s,4p,3d)
(2s,2p)
(2s,2p)
(4s,4p,3d)
(2s,2p)
(2s,2p)
(4s,4p,3d)
(4s,4p,1d,1f)
(4s,4p,3d)
(2s,2p)

The fit-CRENBS ECP at a Glance

a

a
b
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
××××××××××××
×

c
d

×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
××××××××××××××××××××××××××××××××××××××××××××××××××××××××
×

fit-CRENBS is not available for shaded elements
(a)
(b)
(c)
(d)

No ECP; Pople STO-3G basis used
Hurley, Pacios, Christiansen, Ross, & Ermler (Ref. 21)
LaJohn, Christiansen, Ross, Atashroo & Ermler (Ref. 26)
Ross, Powers, Atashroo, Ermler, LaJohn & Christiansen (Ref. 33)

489

Chapter 9: Effective Core Potentials
Element
H–He
Li–Ne
Na–Ar
K–Ca
Sc–Zn
Ga–Kr
Y–Cd
In–Xe
La
Hf–Hg
Tl–Rn

9.7.5

Core
none
none
none
none
[Ar]
[Ar]+3d
[Kr]
[Kr]+4d
[Xe]
[Xe]+4f
[Xe]+4f+5d

Max Projector
none
none
none
none
P
D
D
D
D
F
F

Valence
(1s)
(2s,1p)
(3s,2p)
(4s,3p)
(1s,0p,1d)
(1s,1p)
(1s,1p,1d)
(1s,1p)
(1s,1p,1d)
(1s,1p,1d)
(1s,1p)

The fit-CRENBL ECP at a Glance
a

a
b

b
c
d
e

g
f
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)

h

No ECP; Pople 6-311G* basis used
Pacios & Christiansen (Ref. 31)
Hurley, Pacios, Christiansen, Ross, & Ermler (Ref. 21)
LaJohn, Christiansen, Ross, Atashroo, & Ermler (Ref. 26)
Ross, Powers, Atashroo, Ermler, LaJohn, & Christiansen (Ref. 33)
Ermler, Ross, & Christiansen (Ref. 12)
Ross, Gayen, & Ermler (Ref. 34)
Nash, Bursten, & Ermler (Ref. 29)

490

Chapter 9: Effective Core Potentials
Element
H–He
Li–Ne
Na–Mg
Al–Ar
K–Ca
Sc–Zn
Ga–Kr
Rb–Sr
Y–Cd
In–Xe
Cs–La
Ce–Lu
Hf–Hg
Tl–Rn
Fr–Ra
Ac–Pu
Am–Lr

9.7.6

Core
none
[He]
[He]
[Ne]
[Ne]
[Ne]
[Ar]
[Ar]+3d
[Ar]+3d
[Kr]
[Kr]+4d
[Xe]
[Kr]+4d+4f
[Xe]+4f
[Xe]+4f+5d
[Xe]+4f+5d
[Xe]+4f+5d

Max Projector
none
S
S
P
P
P
P
D
D
D
D
D
F
F
F
F
F

Valence
(3s)
(4s,4p)
(6s,4p)
(4s,4p)
(5s,4p)
(7s,6p,6d)
(3s,3p,4d)
(5s,5p)
(5s,5p,4d)
(3s,3p,4d)
(5s,5p,4d)
(6s,6p,6d,6f)
(5s,5p,4d)
(3s,3p,4d)
(5s,5p,4d)
(5s,5p,4d,4f)
(0s,2p,6d,5f)

The SRLC ECP at a Glance
a

a

b
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
××
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
f
c ×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
g
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
h
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
××××××××××××××××××××××××××××××××××××××××××××

d

e

i

×
××
××
××
×
×
××
××
××
××
××
××
××
×

×
×
××
××
××
××
××
××
××
××
××
××
××
××
××
××
××
××
××
××
××
××
××
××
××
××
××
××
××
××
××
××
××
××
××
××
××
××
××
××
××
××
××
××
××
××
××
××
××
××
××
××
××
××
××
××
××
××
××
×
×
××
×
××
××
××
××
××
××
××
××
××
××
××
××
××
××
××
××
××
××
××
××
××
××
××
××
××
××
××
××
××
××
××
××
××
××
××
××
××
××
××
××
××
××
××
××
××
××
××
××
××
××
××
××
××
××
××
××
××
××
××
×

j

SRLC is not available for shaded elements
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)

No ECP; Pople 6-31G basis used
Fuentealba, Preuss, Stoll, & von Szentpály (Ref. 14)
Fuentealba, von Szentpály, Preuss, & Stoll (Ref. 16)
Bergner, Dolg, Küchle, Stoll, & Preuss (Ref. 6
Nicklass, Dolg, Stoll, & Preuss, (Ref. 30)
Schautz, Flad, & Dolg (Ref. 35)
Fuentealba, Stoll, von Szentpály, Schwerdtfeger, & Preuss (Ref. 15)
von Szentpály, Fuentealba, Preuss, & Stoll (Ref. 38)
Küchle, Dolg, Stoll, & Preuss (Ref. 24)

491

Chapter 9: Effective Core Potentials
Element
H–He
Li–Be
B–N
O–F
Ne
Na–P
S–Cl
Ar
K–Ca
Zn
Ga–As
Se–Br
Kr
Rb–Sr
In–Sb
Te–I
Xe
Cs–Ba
Hg–Bi
Po–At
Rn
Ac–Lr

9.7.7

Core
none
[He]
[He]
[He]
[He]
[Ne]
[Ne]
[Ne]
[Ar]
[Ar]+3d
[Ar]+3d
[Ar]+3d
[Ar]+3d
[Kr]
[Kr]+4d
[Kr]+4d
[Kr]+4d
[Xe]
[Xe]+4f+5d
[Xe]+4f+5d
[Xe]+4f+5d
[Xe]+4f+5d

Max Projector
none
P
D
D
D
D
D
F
D
D
F
F
G
D
F
F
G
D
G
G
G
G

Valence
(2s)
(2s,2p)
(2s,2p)
(2s,3p)
(4s,4p,3d,1f)
(2s,2p)
(2s,3p)
(4s,4p,3d,1f)
(2s,2p)
(3s,2p)
(2s,2p)
(2s,3p)
(4s,4p,3d,1f)
(2s,2p)
(2s,2p)
(2s,3p)
(4s,4p,3d,1f)
(2s,2p)
(2s,2p,1d)
(2s,3p,1d)
(2s,2p,1d)
(5s,5p,4d,3f,2g)

The SRSC ECP at a Glance

a

a
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
××××××××××××××××××××××××

d
b

c

×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
××××××××××××

e

f
g

SRSC is not available for shaded elements
(a)
(b)
(c)
(d)
(e)
(f)
(g)

No ECP; Pople 6-311G* basis used
Leininger, Nicklass, Küchle, Stoll, Dolg, & Bergner (Ref. 27)
Kaupp, Schleyer, Stoll, & Preuss (Ref. 23)
Dolg, Wedig, Stoll, & Preuss (Ref. 11)
Andrae, Häußermann, Dolg, Stoll, & Preuss (Ref. 5)
Dolg, Stoll, & Preuss (Ref. 10)
Küchle, Dolg, Stoll, & Preuss (Ref. 25)

×
×
×
×
××
××
×
××
×
××
××

492

Chapter 9: Effective Core Potentials
Element
H–Ar
Li–Ne
Na–Ar
K
Ca
Sc–Zn
Rb
Sr
Y–Cd
Cs
Ba
Ce–Yb
Hf–Pt
Au
Hg
Ac–Lr

9.7.8

Core
none
none
none
[Ne]
[Ne]
[Ne]
[Ar]+3d
[Ar]+3d
[Ar]+3d
[Kr]+4d
[Kr]+4d
[Ar]+3d
[Kr]+4d+4f
[Kr]+4d+4f
[Kr]+4d+4f
[Kr]+4d+4f

Max Projector
none
none
none
F
F
D
F
F
F
F
F
G
G
F
G
G

Valence
(3s)
(4s,3p,1d)
(6s,5p,1d)
(5s,4p)
(4s,4p,2d)
(6s,5p,3d)
(5s,4p)
(4s,4p,2d)
(6s,5p,3d)
(5s,4p)
(3s,3p,2d,1f)
(5s,5p,4d,3f)
(6s,5p,3d)
(7s,3p,4d)
(6s,6p,4d)
(8s,7p,6d,4f)

The Karlsruhe “def2” ECP at a Glance

For elements Rb–Rn (not including the lanthanides), all the Karlsruhe “def2" basis sets are paired with a common set
of ECPs. 40 It is briefly summarized in the table below (the number of valence basis functions depend on the basis set
in use so it is not presented):
Element
H–Kr
Rb–Xe
Cs–La
Hf–Rn

Core
none
[Ar]+3d
[Kr]+4d
[Kr]+4d+4f

Max Projector
none
D
D
D

493

Chapter 9: Effective Core Potentials

References and Further Reading
[1] Basis sets were obtained from the Extensible Computational Chemistry Environment Basis Set Database, Version 1.0, as developed and distributed by the Molecular Science Computing Facility, Environmental and Molecular Sciences Laboratory which is part of the Pacific Northwest Laboratory, P.O. Box 999, Richland, Washington 99352, USA, and funded by the U.S. Department of Energy. The Pacific Northwest Laboratory is a multiprogram laboratory operated by Battelle Memorial Institute for the U.S. Department of Energy under contract
DE-AC06-76RLO 1830. Contact David Feller, Karen Schuchardt or Don Jones for further information.
[2] Ground-State Methods (Chapters 4 and 6).
[3] Basis Sets (Chapter 8).
[4] J. Almlöf and O. Gropen. In K. B. Lipkowitz and D. B. Boyd, editors, Reviews in Computational Chemistry,
volume 8, page 203. Wiley-VCH, New York, 1996. DOI: 10.1002/9780470125854.ch4.
[5] D. Andrae, U. Häußermann, M. Dolg, H. Stoll, and H. Preuß.
10.1007/BF01114537.

Theor. Chem. Acc., 77:123, 1990.

DOI:

[6] A. Bergner, M. Dolg, W. Küchle, H. Stoll, and H. Preuß.
10.1080/00268979300103121.

Mol. Phys., 80:1431, 1993.

DOI:

[7] P. A. Christiansen, W. C. Ermler, and K. S. Pitzer. Annu. Rev. Phys. Chem., 36:407, 1985. DOI: 10.1146/annurev.pc.36.100185.002203.
[8] T. R. Cundari and W. J. Stevens. J. Chem. Phys., 98:5555, 1993. DOI: 10.1063/1.464902.
[9] T. R. Cundari, M. T. Benson, M. L. Lutz, and S. O. Sommerer. In K. B. Lipkowitz and D. B. Boyd,
editors, Reviews in Computational Chemistry, volume 8, page 145. Wiley-VCH, New York, 1996. DOI:
10.1002/9780470125854.ch3.
[10] M. Dolg and H. Preuss. J. Chem. Phys., 90:1730, 1989. DOI: 10.1063/1.456066.
[11] M. Dolg, U. Wedig, H. Stoll, and H. Preuss. J. Chem. Phys., 86:866, 1987. DOI: 10.1063/1.452288.
[12] W. C. Ermler, R. B. Ross, and P. A. Christiansen.
10.1002/qua.560400611.

Int. J. Quantum Chem., 40:829, 1991.

DOI:

[13] G. Frenking, I. Antes, M. Boehme, S. Dapprich, A. W. Ehlers, V. Jonas, A. Neuhaus, M. Otto, R. Stegmann,
A. Veldkamp, and S. F. Vyboishchikov. In K. B. Lipkowitz and D. B. Boyd, editors, Reviews in Computational
Chemistry, volume 8, page 63. Wiley-VCH, New York, 1996. DOI: 10.1002/9780470125854.ch2.
[14] P. Fuentealba, H. Preuss, H. Stoll, and L. von Szentpály. Chem. Phys. Lett., 89:418, 1982. DOI: 10.1016/00092614(82)80012-2.
[15] P. Fuentealba, H. Stoll, L. von Szentpály, P. Schwerdtfeger, and H. Preuss. J. Phys. B, 16:L323, 1983.
[16] P. Fuentealba, L. von Szentpály, H. Preuss, and H. Stoll. J. Phys. B, 18:1287, 1985. DOI: 10.1088/00223700/18/7/010.
[17] M. S. Gordon and T. R. Cundari. Coord. Chem. Rev., 147:87, 1996. DOI: 10.1016/0010-8545(95)01133-1.
[18] P. J. Hay. J. Chem. Phys., 79:5469, 1983. DOI: 10.1063/1.445665.
[19] P. J. Hay and W. R. Wadt. J. Chem. Phys., 82:270, 1985. DOI: 10.1063/1.448975.
[20] P. J. Hay and W. R. Wadt. J. Chem. Phys., 82:299, 1985. DOI: 10.1063/1.448975.
[21] M. M. Hurley, L. F. Pacios, and P. A. Christiansen. J. Chem. Phys., 84:6840, 1986. DOI: 10.1063/1.450689.
[22] L. R. Kahn and W. A. Goddard III. J. Chem. Phys., 56:2685, 1972. DOI: 10.1063/1.1677597.

494

Chapter 9: Effective Core Potentials
[23] M. Kaupp, P. v. R. Schleyer, H. Stoll, and H. Preuss.
doi.org/10.1063/1.459993.

J. Chem. Phys., 94:1360, 1991.

DOI:

[24] W. Küchle, M. Dolg, H. Stoll, and H. Preuss. Mol. Phys., 74:1245, 1991. DOI: 10.1080/00268979100102941.
[25] W. Küchle, M. Dolg, H. Stoll, and H. Preuss. J. Chem. Phys., 100:7535, 1994. DOI: 10.1063/1.466847.
[26] L. A. LaJohn, P. A. Christiansen, R. B. Ross, T. Atashroo, and W. C. Ermler. J. Chem. Phys., 87:2812, 1987.
DOI: 10.1063/1.453069.
[27] T. Leininger, A. Nicklass, W. Küchle, H. Stoll, M. Dolg, and A. Bergner. Chem. Phys. Lett., 255:274, 1996. DOI:
10.1016/0009-2614(96)00382-X.
[28] S. C. McKenzie, E. Epifanovsky, G. M. J. Barca A. T. B. Gilbert, and P. M. W. Gill. J. Phys. Chem. A, 122:3066,
2018. DOI: 10.1021/acs.jpca.7b12679.
[29] C. S. Nash and B. E. Bursten. J. Chem. Phys., 106:5133, 1997. DOI: 10.1063/1.473992.
[30] A. Nicklass, M. Dolg, H. Stoll, and H. Preuss. J. Chem. Phys., 102:8942, 1995. DOI: 10.1063/1.468948.
[31] L. F. Pacios and P. A. Christiansen. J. Chem. Phys., 82:2664. DOI: 10.1063/1.448263.
[32] P. Pyykko. Chem. Rev., 88:563, 1988. DOI: 10.1021/cr00085a006.
[33] R. B. Ross, J. M. Powers, T. Atashroo, W. C. Ermler, L. A. LaJohn, and P. A. Christiansen. J. Chem. Phys., 93:
6654, 1990. DOI: 10.1063/1.458934.
[34] R. B. Ross, S. Gayen, and W. C. Ermler. J. Chem. Phys., 100:8145, 1994. DOI: 10.1063/1.466809.
[35] F. Schautz, H.-J. Flad, and M. Dolg. Theor. Chem. Acc., 99:231, 1998. DOI: 10.1007/s002140050331.
[36] W. J. Stevens, H. Basch, and M. Krauss. J. Chem. Phys., 81:6026, 1984. DOI: 10.1063/1.447604.
[37] W. J. Stevens, M. Krauss, H. Basch, and P. G. Jasien. Can. J. Chem., 70:612, 1992. DOI: 10.1139/v92-085.
[38] L. von Szentpály, P. Fuentealba, H. Preuss, and H. Stoll. Chem. Phys. Lett., 93:555, 1982. DOI: 10.1016/00092614(82)83728-7.
[39] W. R. Wadt and P. J. Hay. J. Chem. Phys., 82:284, 1985. DOI: 10.1063/1.448800.
[40] F. Weigend and R. Ahlrichs. Phys. Chem. Chem. Phys., 7:3297, 2005. DOI: 10.1039/b508541a.

Chapter 10

Exploring Potential Energy Surfaces:
Searches for Critical Points and Molecular
Dynamics
10.1

Equilibrium Geometries and Transition-State Structures

10.1.1

Overview

Molecular potential energy surfaces rely on the Born-Oppenheimer separation of nuclear and electronic motion. Minima on such energy surfaces correspond to the classical picture of equilibrium geometries, and transition state structures
correspond to first-order saddle points. Both equilibrium and transition-state structures are stationary points for which
the energy gradient vanishes. Characterization of such critical points requires consideration of the eigenvalues of the
Hessian (second derivative matrix): minimum-energy, equilibrium geometries possess Hessians whose eigenvalues are
all positive, whereas transition-state structures are defined by a Hessian with precisely one negative eigenvalue. (The
latter is therefore a local maximum along the reaction path between minimum-energy reactant and product structures,
but a minimum in all directions perpendicular to this reaction path.
The quality of a geometry optimization algorithm is of major importance; even the fastest integral code in the world
will be useless if combined with an inefficient optimization algorithm that requires excessive numbers of steps to
converge. Q-C HEM incorporates a geometry optimization package (O PTIMIZE—see Appendix A) developed by the
late Jon Baker over more than ten years.
The key to optimizing a molecular geometry successfully is to proceed from the starting geometry to the final geometry
in as few steps as possible. Four factors influence the path and number of steps:
• starting geometry
• optimization algorithm
• quality of the Hessian (and gradient)
• coordinate system
Q-C HEM controls the last three of these, but the starting geometry is solely determined by the user, and the closer
it is to the converged geometry, the fewer optimization steps will be required. Decisions regarding the optimization
algorithm and the coordinate system are generally made by the O PTIMIZE package (i.e., internally, within Q-C HEM)
to maximize the rate of convergence. Although users may override these choices in many cases, this is not generally
recommended.

496

Chapter 10: Exploring Potential Energy Surfaces: Critical Points and Molecular Dynamics
Level of Theory
(Algorithm)
DFT
HF
ROHF
MP2
(V)OD
(V)QCCD
CIS (except RO)
CFMM

Analytical
Gradients
3
3
3
3
3
3
3
3

Maximum Angular
Momentum Type
h
h
h
h
h
h
h
h

Analytical
Hessian
3
3
7
7
7
7
3
7

Maximum Angular
Momentum Type
f
f

f

Table 10.1: Gradients and Hessians currently available for geometry optimizations with maximum angular momentum
types for analytical derivative calculations (for higher angular momentum, derivatives are computed numerically).
Analytical Hessians are not yet available for meta-GGA functionals such as BMK and the M05 and M06 series.
Another consideration when trying to minimize the optimization time concerns the quality of the gradient and Hessian.
A higher-quality Hessian (i.e., analytical versus approximate) will in many cases lead to faster convergence, in the
sense of requiring fewer optimization steps. However, the construction of an analytical Hessian requires significant
computational effort and may outweigh the advantage of fewer optimization cycles. Currently available analytical
gradients and Hessians are summarized in Table 10.1.
Features of Q-C HEM’s geometry and transition-state optimization capabilities include:
• Cartesian, Z-matrix or internal coordinate systems
• Eigenvector Following (EF) or GDIIS algorithms
• Constrained optimizations
• Equilibrium structure searches
• Transition structure searches
• Hessian-free characterization of stationary points
• Initial Hessian and Hessian update options
• Reaction pathways using intrinsic reaction coordinates (IRC)
• Optimization of minimum-energy crossing points (MECPs) along conical seams

10.1.2

Job Control

Obviously a level of theory, basis set, and starting molecular geometry must be specified to begin a geometry optimization or transition-structure search. These aspects are described elsewhere in this manual, and this section describes
job-control variables specific to optimizations.

Chapter 10: Exploring Potential Energy Surfaces: Critical Points and Molecular Dynamics
JOBTYPE
Specifies the calculation.
TYPE:
STRING
DEFAULT:
Default is single-point, which should be changed to one of the following options.
OPTIONS:
OPT
Equilibrium structure optimization.
TS
Transition structure optimization.
RPATH Intrinsic reaction path following.
RECOMMENDATION:
Application-dependent.
GEOM_OPT_HESSIAN
Determines the initial Hessian status.
TYPE:
STRING
DEFAULT:
DIAGONAL
OPTIONS:
DIAGONAL Set up diagonal Hessian.
READ
Have exact or initial Hessian. Use as is if Cartesian, or transform
if internals.
RECOMMENDATION:
An accurate initial Hessian will improve the performance of the optimizer, but is expensive to
compute.
GEOM_OPT_COORDS
Controls the type of optimization coordinates.
TYPE:
INTEGER
DEFAULT:
−1
OPTIONS:
0
Optimize in Cartesian coordinates.
1
Generate and optimize in internal coordinates, if this fails abort.
−1 Generate and optimize in internal coordinates, if this fails at any stage of the
optimization, switch to Cartesian and continue.
2
Optimize in Z-matrix coordinates, if this fails abort.
−2 Optimize in Z-matrix coordinates, if this fails during any stage of the
optimization switch to Cartesians and continue.
RECOMMENDATION:
Use the default; delocalized internals are more efficient.

497

498

Chapter 10: Exploring Potential Energy Surfaces: Critical Points and Molecular Dynamics
GEOM_OPT_TOL_GRADIENT
Convergence on maximum gradient component.
TYPE:
INTEGER
DEFAULT:
300 ≡ 300 × 10−6 tolerance on maximum gradient component.
OPTIONS:
n Integer value (tolerance = n × 10−6 ).
RECOMMENDATION:
Use the default.
To converge GEOM_OPT_TOL_GRADIENT and one
GEOM_OPT_TOL_DISPLACEMENT and GEOM_OPT_TOL_ENERGY must be satisfied.
GEOM_OPT_TOL_DISPLACEMENT
Convergence on maximum atomic displacement.
TYPE:
INTEGER
DEFAULT:
1200 ≡ 1200 × 10−6 tolerance on maximum atomic displacement.
OPTIONS:
n Integer value (tolerance = n × 10−6 ).
RECOMMENDATION:
Use the default.
To converge GEOM_OPT_TOL_GRADIENT and one
GEOM_OPT_TOL_DISPLACEMENT and GEOM_OPT_TOL_ENERGY must be satisfied.
GEOM_OPT_TOL_ENERGY
Convergence on energy change of successive optimization cycles.
TYPE:
INTEGER
DEFAULT:
100 ≡ 100 × 10−8 tolerance on maximum (absolute) energy change.
OPTIONS:
n Integer value (tolerance = value n × 10−8 ).
RECOMMENDATION:
Use the default.
To converge GEOM_OPT_TOL_GRADIENT and one
GEOM_OPT_TOL_DISPLACEMENT and GEOM_OPT_TOL_ENERGY must be satisfied.

of

of

of

GEOM_OPT_MAX_CYCLES
Maximum number of optimization cycles.
TYPE:
INTEGER
DEFAULT:
50
OPTIONS:
n User defined positive integer.
RECOMMENDATION:
The default should be sufficient for most cases. Increase if the initial guess geometry is poor, or
for systems with shallow potential wells.

Chapter 10: Exploring Potential Energy Surfaces: Critical Points and Molecular Dynamics
GEOM_OPT_PRINT
Controls the amount of O PTIMIZE print output.
TYPE:
INTEGER
DEFAULT:
3 Error messages, summary, warning, standard information and gradient print out.
OPTIONS:
0 Error messages only.
1 Level 0 plus summary and warning print out.
2 Level 1 plus standard information.
3 Level 2 plus gradient print out.
4 Level 3 plus Hessian print out.
5 Level 4 plus iterative print out.
6 Level 5 plus internal generation print out.
7 Debug print out.
RECOMMENDATION:
Use the default.
GEOM_OPT_SYMFLAG
Controls the use of symmetry in O PTIMIZE.
TYPE:
LOGICAL
DEFAULT:
TRUE
OPTIONS:
TRUE
Make use of point group symmetry.
FALSE Do not make use of point group symmetry.
RECOMMENDATION:
Use the default.
GEOM_OPT_MODE
Determines Hessian mode followed during a transition state search.
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
0 Mode following off.
n Maximize along mode n.
RECOMMENDATION:
Use the default, for geometry optimizations.

499

Chapter 10: Exploring Potential Energy Surfaces: Critical Points and Molecular Dynamics
GEOM_OPT_MAX_DIIS
Controls maximum size of subspace for GDIIS.
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
0
Do not use GDIIS.
-1 Default size = min(NDEG, NATOMS, 4) NDEG = number of molecular
degrees of freedom.
n Size specified by user.
RECOMMENDATION:
Use the default or do not set n too large.
GEOM_OPT_DMAX
Maximum allowed step size. Value supplied is multiplied by 10−3 .
TYPE:
INTEGER
DEFAULT:
300 = 0.3
OPTIONS:
n User-defined cutoff.
RECOMMENDATION:
Use the default.
GEOM_OPT_UPDATE
Controls the Hessian update algorithm.
TYPE:
INTEGER
DEFAULT:
-1
OPTIONS:
-1 Use the default update algorithm.
0 Do not update the Hessian (not recommended).
1 Murtagh-Sargent update.
2 Powell update.
3 Powell/Murtagh-Sargent update (TS default).
4 BFGS update (OPT default).
5 BFGS with safeguards to ensure retention of positive definiteness
(GDISS default).
RECOMMENDATION:
Use the default.

500

Chapter 10: Exploring Potential Energy Surfaces: Critical Points and Molecular Dynamics
GEOM_OPT_LINEAR_ANGLE
Threshold for near linear bond angles (degrees).
TYPE:
INTEGER
DEFAULT:
165 degrees.
OPTIONS:
n User-defined level.
RECOMMENDATION:
Use the default.
FDIFF_STEPSIZE
Displacement used for calculating derivatives by finite difference.
TYPE:
INTEGER
DEFAULT:
100 Corresponding to 0.001 Å. For calculating second derivatives.
OPTIONS:
n Use a step size of n × 10−5 .
RECOMMENDATION:
Use the default except in cases where the potential surface is very flat, in which case a larger
value should be used. See FDIFF_STEPSIZE_QFF for third and fourth derivatives.

501

Chapter 10: Exploring Potential Energy Surfaces: Critical Points and Molecular Dynamics

502

Example 10.1 As outlined, the rate of convergence of the iterative optimization process is dependent on a number
of factors, one of which is the use of an initial analytic Hessian. This is easily achieved by instructing Q-C HEM to
calculate an analytic Hessian and proceed then to determine the required critical point
$molecule
0 1
O
H 1 oh
H 1 oh 2 hoh
oh = 1.1
hoh = 104
$end
$rem
JOBTYPE
METHOD
BASIS
$end

freq
Calculate an analytic Hessian
hf
6-31g(d)

$comment
Now proceed with the Optimization making sure to read in the analytic
Hessian (use other available information too).
$end
@@@
$molecule
read
$end
$rem
JOBTYPE
METHOD
BASIS
SCF_GUESS
GEOM_OPT_HESSIAN
$end

10.1.3

opt
hf
6-31g(d)
read
read
Have the initial Hessian

Hessian-Free Characterization of Stationary Points

Q-C HEM allows the user to characterize the stationary point found by a geometry optimization or transition state
search without performing a full analytical Hessian calculation, which is sometimes unavailable or computationally
unaffordable. This is achieved via a finite difference Davidson procedure developed by Sharada et al. 52 For a geometry
optimization, it solves for the lowest eigenvalue of the Hessian (λ1 ) and checks if λ1 > 0 (a negative λ1 indicates a
saddle point); for a TS search, it solves for the lowest two eigenvalues, and λ1 < 0 and λ2 > 0 indicate a transition
state. The lowest eigenvectors of the updated P-RFO (approximate) Hessian at convergence are used as the initial guess
for the Davidson solver.
The cost of this Hessian-free characterization method depends on the rate of convergence of the Davidson solver.
For example, to characterize an energy minimum, it requires 2 × Niter total energy + gradient calculations, where
Niter is the number of iterations that the Davidson algorithm needs to converge, and “2" is for forward and backward
displacements on each iteration. According to Ref. 52, this method can be much more efficient than exact Hessian
calculation for substantially large systems.
Note: At the moment, this method does not support QM/MM or systems with fixed atoms.

Chapter 10: Exploring Potential Energy Surfaces: Critical Points and Molecular Dynamics

503

GEOM_OPT_CHARAC
Use the finite difference Davidson method to characterize the resulting energy minimum/transition state.
TYPE:
BOOLEAN
DEFAULT:
FALSE
OPTIONS:
FALSE do not characterize the resulting stationary point.
TRUE
perform a characterization of the stationary point.
RECOMMENDATION:
Set it to TRUE when the character of a stationary point needs to be verified, especially for a
transition structure.
GEOM_OPT_CHARAC_CONV
Overide the built-in convergence criterion for the Davidson solver.
TYPE:
INTEGER
DEFAULT:
0 (use the built-in default value 10−5 )
OPTIONS:
n Set the convergence criterion to 10−n .
RECOMMENDATION:
Use the default. If it fails to converge, consider loosening the criterion with caution.

Example 10.2 Geometry optimization of a triflate anion that converges to an eclipsed conformation, which is a first
order saddle point. This is verified via the finite difference Davidson method by setting GEOM_OPT_CHARAC to TRUE.
$molecule
-1 1
C 0.00000
F -1.09414
S 0.00000
O 1.25831
O -1.25831
O 0.00000
F 1.09414
F 0.00000
$end

-0.00078
-0.63166
0.00008
-0.72597
-0.72597
1.45286
-0.63166
1.26313

0.98436
1.47859
-0.94745
-1.28972
-1.28972
-1.28958
1.47859
1.47663

$rem
JOBTYPE
METHOD
GEOM_OPT_DMAX
BASIS
SCF_CONVERGENCE
THRESH
SYMMETRY
SYM_IGNORE
GEOM_OPT_TOL_DISPLACEMENT
GEOM_OPT_TOL_ENERGY
GEOM_OPT_TOL_GRADIENT
GEOM_OPT_CHARAC
$end

opt
BP86
50
6-311+G*
8
14
FALSE
TRUE
10
10
10
TRUE

Chapter 10: Exploring Potential Energy Surfaces: Critical Points and Molecular Dynamics

504

Example 10.3 TS search for alanine dipeptide rearrangement reaction beginning with a guess structure converges
correctly. The resulting TS structure is verified using the finite difference Davidson method.
$molecule
0
1
C
C
N
C
C
N
C
O
C
O
H
H
H
H
H
H
H
H
H
H
H
H
$end

3.21659
2.16708
1.21359
0.11616
-1.19613
-2.18193
-3.43891
2.19596
0.11486
-1.29658
3.25195
3.06369
4.20892
1.24786
0.25990
-2.02230
-3.60706
-4.29549
-3.36801
-0.68664
0.01029
1.06461

$rem
JOBTYPE
EXCHANGE
BASIS
SCF_MAX_CYCLES
SYMMETRY
SYM_IGNORE
$end

-1.41022
-0.35258
-0.16703
0.82394
0.03585
-0.02502
-0.74663
0.25708
1.96253
-0.59392
-2.14283
-1.95423
-0.93714
-0.78278
1.31404
0.38818
-1.48647
-0.06423
-1.25875
2.66864
1.65112
2.50818

freq
B3LYP
6-31G
250
false
true

@@@
$rem
JOBTYPE
SCF_GUESS
GEOM_OPT_DMAX
GEOM_OPT_MAX_CYCLES
EXCHANGE
BASIS
MAX_SCF_CYCLES
GEOM_OPT_HESSIAN
SYMMETRY
SYM_IGNORE
GEOM_OPT_CHARAC
$end
$molecule
read
$end

ts
read
100
1500
B3LYP
6-31G
250
read
false
true
true

-0.26053
-0.59607
0.41640
0.50964
0.74226
-0.18081
0.01614
-1.63440
-0.53088
1.85462
-1.08721
0.67666
-0.22851
1.21013
1.47973
-1.10143
-0.76756
0.04327
0.98106
-0.27269
-1.56461
-0.45885

Chapter 10: Exploring Potential Energy Surfaces: Critical Points and Molecular Dynamics

10.2

505

Improved Algorithms for Transition-Structure Optimization

Transition-structure searches tend to be more difficult (meaning, more likely to be unsuccessful) as compared to
minimum-energy (equilibrium) geometry optimizations. Odds of success can be enhanced via an initial guess structure
that is determined in an automated way, rather than simply “guessed” by the user. Several such automated algorithms
are available in Q-C HEM, and are described in this section.

10.2.1

Freezing String Method

Perhaps the most significant difficulty in locating transition states is to obtain a good initial guess of the geometry to
feed into a surface-walking algorithm. This difficulty becomes especially relevant for large systems, for which the
dimensionality of the search space is large. Interpolation algorithms are promising for locating good guesses of the
minimum-energy pathway connecting reactant and product states as well as approximate saddle-point geometries. For
example, the nudged elastic band method 17,36 and the string method 10 start from a certain initial reaction pathway
connecting the reactant and the product state, and then optimize in discretized path space towards the minimum-energy
pathway. The highest-energy point on the approximate minimum-energy pathway becomes a good initial guess for the
saddle-point configuration that can subsequently be used with any local surface-walking algorithm.
Inevitably, the performance of any interpolation method heavily relies on the choice of the initial reaction pathway, and
a poorly-chosen initial pathway can cause slow convergence, or possibly convergence to an incorrect pathway. The
growing-string method 42 and freezing-string method 6,51 offer solutions to this problem, in which two string fragments
(one representing the reactant state and the other representing the product state) are “grown” (i.e., increasingly-finely
defined) until the two fragments join. The freezing-string method offers a choice between Cartesian interpolation
and linear synchronous transit (LST) interpolation. It also allows the user to choose between conjugate gradient and
quasi-Newton optimization techniques.
Freezing-string calculations are requested by setting JOBTYPE = FSM in the $rem section. Additional job-control
keywords are described below, along with examples. Consult Refs. 6 and 51 for a guide to a typical use of this method.
FSM_NNODE
Specifies the number of nodes along the string
TYPE:
INTEGER
DEFAULT:
Undefined
OPTIONS:
N number of nodes in FSM calculation
RECOMMENDATION:
N = 15. Use 10 to 20 nodes for a typical calculation. Reaction paths that connect multiple
elementary steps should be separated into individual elementary steps, and one FSM job run for
each pair of intermediates. Use a higher number when the FSM is followed by an approximateHessian based transition state search (Section 10.2.2).
FSM_NGRAD
Specifies the number of perpendicular gradient steps used to optimize each node
TYPE:
INTEGER
DEFAULT:
Undefined
OPTIONS:
N Number of perpendicular gradients per node
RECOMMENDATION:
Anything between 2 and 6 should work, where increasing the number is only needed for difficult
reaction paths.

Chapter 10: Exploring Potential Energy Surfaces: Critical Points and Molecular Dynamics

506

FSM_MODE
Specifies the method of interpolation
TYPE:
INTEGER
DEFAULT:
2
OPTIONS:
1 Cartesian
2 LST
RECOMMENDATION:
In most cases, LST is superior to Cartesian interpolation.
FSM_OPT_MODE
Specifies the method of optimization
TYPE:
INTEGER
DEFAULT:
Undefined
OPTIONS:
1 Conjugate gradients
2 Quasi-Newton method with BFGS Hessian update
RECOMMENDATION:
The quasi-Newton method is more efficient when the number of nodes is high.

An example input appears below. Note that the $molecule section includes geometries for two optimized intermediates,
separated by ****. The order of the atoms is important, as Q-C HEM assumes that the nth atom in the reactant moves
toward the nth atom in the product. The FSM string is printed out in the file stringfile.txt, which contains Cartesian
coordinates of the structures that connect reactant to product. Each node along the path is labeled in this file, and its
energy is provided. The geometries and energies are also printed at the end of the Q-C HEM output file, where they are
labeled:
---------------------------------------STRING
---------------------------------------Finally, if MOLDEN_FORMAT is set to TRUE, then geometries along the string are printed in a M OL D EN-readable
format at the end of the Q-C HEM output file. The highest-energy node can be taken from this file and used to run
a transition structure search as described in section 10.1. If the string returns a pathway that is unreasonable, check

Chapter 10: Exploring Potential Energy Surfaces: Critical Points and Molecular Dynamics

507

whether the atoms in the two input geometries are in the correct order.
Example 10.4 Example of the freezing-string method.
$molecule
0 1
Si
1.028032
H
0.923921
H
1.294874
H
-1.713989
H
-1.532839
****
Si
0.000228
H
0.644754
H
1.047648
H
-0.837028
H
-0.855603
$end
$rem
JOBTYPE
FSM_NGRAD
FSM_NNODE
FSM_MODE
FSM_OPT_MODE
METHOD
BASIS
$end

10.2.2

-0.131573
-1.301934
0.900609
0.300876
0.232021

-0.779689
0.201724
0.318888
-0.226231
0.485307

-0.000484
-1.336958
1.052717
0.205648
0.079077

-0.000023
-0.064865
0.062991
-1.211126
1.213023

fsm
3
12
2
2
b3lyp
6-31G

Hessian-Free Transition-State Search

Once a guess structure to the transition state is obtained, standard eigenvector-following methods such as Baker’s partitioned rational-function optimization (P-RFO) algorithm 3 can be employed to refine the guess to the exact transition
state. The reliability of P-RFO depends on the quality of the Hessian input, which enables the method to distinguish
between the reaction coordinate (characterized by a negative eigenvalue) and the remaining degrees of freedom. In
routine calculations therefore, an exact Hessian is determined via frequency calculation prior to the P-RFO search.
Since the cost of evaluating an exact Hessian typically scales one power of system size higher than the energy or the
gradient, this step becomes impractical for systems containing large number of atoms.
The exact Hessian calculation can be avoided by constructing an approximate Hessian based on the output of FSM. 52
The tangent direction at the transition state guess on the FSM string is a good approximation to the Hessian eigenvector corresponding to the reaction coordinate. The tangent is therefore used to calculate the correct eigenvalue and
corresponding eigenvector by variationally minimizing the Rayleigh-Ritz ratio. 28 The reaction coordinate information
is then incorporated into a guess matrix which, in turn, is obtained by transforming a diagonal matrix in delocalized
internal coordinates 4,12 to Cartesian coordinates. The resulting approximate Hessian, by design, has a single negative
eigenvalue corresponding to the reaction coordinate. This matrix is then used in place of the exact Hessian as input to
the P-RFO method.
An example of this one-shot, Hessian-free approach that combines the FSM and P-RFO methods in order to determine

Chapter 10: Exploring Potential Energy Surfaces: Critical Points and Molecular Dynamics

508

the exact transition state from reactant and product structures is shown below:
Example 10.5
$molecule
0 1
Si
1.028032
H
0.923921
H
1.294874
H
-1.713989
H
-1.532839
****
Si
0.000228
H
0.644754
H
1.047648
H
-0.837028
H
-0.855603
$end
$rem
JOBTYPE
FSM_NGRAD
FSM_NNODE
FSM_MODE
FSM_OPT_MODE
METHOD
BASIS
SYMMETRY
SYM_IGNORE
$end

-0.131573
-1.301934
0.900609
0.300876
0.232021

-0.779689
0.201724
0.318888
-0.226231
0.485307

-0.000484
-1.336958
1.052717
0.205648
0.079077

-0.000023
-0.064865
0.062991
-1.211126
1.213023

fsm
3
18
2
2
b3lyp
6-31g
false
true

@@@
$rem
JOBTYPE
SCF_GUESS
GEOM_OPT_HESSIAN
MAX_SCF_CYCLES
GEOM_OPT_DMAX
GEOM_OPT_MAX_CYCLES
METHOD
BASIS
SYMMETRY
SYM_IGNORE
$end

ts
read
read
250
50
100
b3lyp
6-31g
false
true

$molecule
read
$end

10.2.3

Improved Dimer Method

Once a good approximation to the minimum energy pathway is obtained, e.g., with the help of an interpolation algorithm such as the growing string method, local surface walking algorithms can be used to determine the exact location
of the saddle point. Baker’s P-RFO method, 3 using either an approximate or an exact Hessian, has proven to be a very
powerful for this purpose, but does require calculation of a full Hessian matrix.
The dimer method, 16 on the other hand, is a mode-following algorithm that requires only the curvature along one direction in configuration space, rather than the full Hessian, which can be accomplished using only gradient evaluations.
This method is thus especially attractive for large systems where a full Hessian calculation might be prohibitively ex-

Chapter 10: Exploring Potential Energy Surfaces: Critical Points and Molecular Dynamics

509

pensive, or for saddle-point searches where the initial guess is such that the eigenvector of corresponding to the smallest
Hessian eigenvalue does not correspond to the desired reaction coordinate. An improved version of the original dimer
method 21,22 has been implemented in Q-C HEM, which significantly reduces the influence of numerical noise and thus
significantly reduces the cost of the algorithm.

10.3

Constrained Optimization

Constrained optimization refers to the optimization of molecular structures (transition state or minimum-energy) in
which certain parameters such as bond lengths, bond angles or dihedral angles are fixed. Jon Baker’s O PTIMIZE
package, implemented in Q-C HEM, makes it possible to handle constraints directly in delocalized internal coordinates
using the method of Lagrange multipliers (see Appendix A). Features of constrained optimization in Q-C HEM are:
• Starting geometries need not satisfy the requested constraints.
• Constrained optimization is performed in delocalized internal coordinates, which is typically the most efficient
coordinate system for optimization of large molecules.
• Q-C HEM’s free-format $opt section allows the user to apply constraints with ease.
Constraints are imposed via the $opt input section, whose format is shown below, and the various parts of this input
section are described below.
Note: As with the rest of the Q-C HEM input file, the $opt section is case-insensitive, but there should be no blank
space at the beginning of a line.
$opt
CONSTRAINT
stre atom1 atom2
...
bend atom1 atom2
...
outp atom1 atom2
...
tors atom1 atom2
...
linc atom1 atom2
...
linp atom1 atom2
...
ENDCONSTRAINT

value
atom3

value

atom3

atom4

value

atom3

atom4

value

atom3

atom4

value

atom3

atom4

value

FIXED
atom
coordinate_reference
...
ENDFIXED
DUMMY
idum
type
...
ENDDUMMY

list_length

CONNECT
atom
list_length

list

defining_list

Chapter 10: Exploring Potential Energy Surfaces: Critical Points and Molecular Dynamics

510

...
ENDCONNECT
$end

10.3.1

Geometry Optimization with General Constraints

CONSTRAINT and ENDCONSTRAINT define the beginning and end, respectively, of the constraint section of $opt
within which users may specify up to six different types of constraints:

interatomic distances
Values in Ångstroms; value > 0:
stre
atom1
atom2
value
angles
Values in degrees, 0 ≤ value ≤ 180; atom2 is the middle atom of the bend:
bend
atom1
atom2
atom3
value
out-of-plane-bends
Values in degrees, −180 ≤ value ≤ 180 atom2; angle between atom4 and the atom1–atom2–atom3 plane:
outp
atom1
atom2
atom3
atom4
value
dihedral angles
Values in degrees, −180 ≤ value ≤ 180; angle the plane atom1–atom2–atom3 makes with the plane atom2–atom3–
atom4:
tors
atom1
atom2
atom3
atom4
value
coplanar bends
Values in degrees, −180 ≤ value ≤ 180; bending of atom1–atom2–atom3 in the plane atom2–atom3–atom4:
linc
atom1
atom2
atom3
atom4
value
perpendicular bends
Values in degrees, −180 ≤ value ≤ 180; bending of atom1–atom2–atom3 perpendicular to the plane atom2–atom3–
atom4:
linp
atom1
atom2
atom3
atom4
value

10.3.2

Frozen Atoms

Absolute atom positions can be frozen with the FIXED section. The section starts with the FIXED keyword as the first
line and ends with the ENDFIXED keyword on the last. The format to fix a coordinate or coordinates of an atom is:
atom

coordinate_reference

coordinate_reference can be any combination of up to three characters X, Y and Z to specify the coordinate(s) to be
fixed: X, Y , Z, XY, XZ, YZ, XYZ. The fixing characters must be next to each other. e.g.,
FIXED
2 XY
ENDFIXED
means the x-coordinate and y-coordinate of atom 2 are fixed, whereas
FIXED
2 X Y
ENDFIXED

Chapter 10: Exploring Potential Energy Surfaces: Critical Points and Molecular Dynamics

511

will yield erroneous results.
Note: When the FIXED section is specified within $opt, the optimization will proceed in Cartesian coordinates.

10.3.3

Dummy Atoms

DUMMY defines the beginning of the dummy atom section and ENDDUMMY its conclusion. Dummy atoms are used to

help define constraints during constrained optimizations in Cartesian coordinates. They cannot be used with delocalized
internals.
All dummy atoms are defined with reference to a list of real atoms, that is, dummy atom coordinates are generated from
the coordinates of the real atoms from the dummy atoms defining list (see below). There are three types of dummy
atom:
1. Positioned at the arithmetic mean of up to seven real atoms in the defining list.
2. Positioned a unit distance along the normal to a plane defined by three atoms, centered on the middle atom of the
three.
3. Positioned a unit distance along the bisector of a given angle.
The format for declaring dummy atoms is:
DUMMY
idum
type
ENDDUMMY
idum
type
list_length
defining_list

list_length

defining_list

Center number of defining atom (must be one greater than the total number of real atoms
for the first dummy atom, two greater for second etc.).
Type of dummy atom (either 1, 2 or 3; see above).
Number of atoms in the defining list.
List of up to seven atoms defining the position of the dummy atom.

Once defined, dummy atoms can be used to define standard internal (distance, angle) constraints as per the constraints
section, above.
Note: The use of dummy atoms of type 1 has never progressed beyond the experimental stage.

10.3.4

Dummy Atom Placement in Dihedral Constraints

Bond and dihedral angles cannot be constrained in Cartesian optimizations to exactly 0◦ or ±180◦ . This is because the
corresponding constraint normals are zero vectors. Also, dihedral constraints near these two limiting values (within,
say 20◦ ) tend to oscillate and are difficult to converge.
These difficulties can be overcome by defining dummy atoms and redefining the constraints with respect to the dummy
atoms. For example, a dihedral constraint of 180◦ can be redefined to two constraints of 90◦ with respect to a suitably positioned dummy atom. The same thing can be done with a 180◦ bond angle (long a familiar use in Z-matrix
construction).
Typical usage is as shown in Table 10.2. Note that the order of atoms is important to obtain the correct signature on the
dihedral angles. For a 0◦ dihedral constraint, atoms J and K should be switched in the definition of the second torsion
constraint in Cartesian coordinates.
Note: In almost all cases the above discussion is somewhat academic, as internal coordinates are now best imposed
using delocalized internal coordinates and there is no restriction on the constraint values.

Chapter 10: Exploring Potential Energy Surfaces: Critical Points and Molecular Dynamics
Internal Coordinates
$opt
CONSTRAINT
tors I J K L 180.0
ENDCONSTRAINT
$end

512

Cartesian Coordinates
$opt
DUMMY
M2IJK
ENDDUMMY
CONSTRAINT
tors I J K M 90
tors M J K L 90
ENDCONSTRAINT
$end

Table 10.2: Comparison of dihedral angle constraint method for adopted coordinates.

10.3.5

Additional Atom Connectivity

Normally delocalized internal coordinates are generated automatically from the input Cartesian coordinates. This
is accomplished by first determining the atomic connectivity list (i.e., which atoms are formally bonded) and then
constructing a set of individual primitive internal coordinates comprising all bond stretches, all planar bends and all
proper torsions that can be generated based on the atomic connectivity. The delocalized internal are in turn constructed
from this set of primitives.
The atomic connectivity depends simply on distance and there are default bond lengths between all pairs of atoms in
the code. In order for delocalized internals to be generated successfully, all atoms in the molecule must be formally
bonded so as to form a closed system. In molecular complexes with long, weak bonds or in certain transition states
where parts of the molecule are rearranging or dissociating, distances between atoms may be too great for the atoms to
be regarded as formally bonded, and the standard atomic connectivity will separate the system into two or more distinct
parts. In this event, the generation of delocalized internal coordinates will fail. Additional atomic connectivity can be
included for the system to overcome this difficulty.
CONNECT defines the beginning of the additional connectivity section and ENDCONNECT the end. The format of the
CONNECT section is:

CONNECT
atom
list_length
ENDCONNECT

list

Chapter 10: Exploring Potential Energy Surfaces: Critical Points and Molecular Dynamics
atom
list_length
list

513

Atom for which additional connectivity is being defined.
Number of atoms in the list of bonded atoms.
List of up to 8 atoms considered as being bonded to the given atom.

Example 10.6 Methanol geometry optimization with constraints.
$comment
Methanol geom opt with constraints in bond length and bond angles.
$end
$molecule
0 1
C
0.14192
O
0.14192
H
1.18699
H -0.34843
H -0.34843
H -0.77395
$end

0.33268
-1.08832
0.65619
0.74268
0.74268
-1.38590

$rem
GEOM_OPT_PRINT
JOBTYPE
METHOD
BASIS
$end

0.00000
0.00000
0.00000
0.88786
-0.88786
0.00000

6
opt
hf
3-21g

$opt
CONSTRAINT
stre 1 6 1.8
bend 2 1 4 110.0
bend 2 1 5 110.0
ENDCONSTRAINT
$end

10.3.6

Application of External Forces

In 2009, three methods for optimizing the geometry of a molecule under a constant external force were introduced,
which were called Force-Modified Potential Energy Surface (FMPES), 39 External Force is Explicitly Included (EFEI), 48
and Enforced Geometry Optimization (EGO). 58 These methods are closely related, and the interested reader is referred
to Ref. 53 for a detailed discussion of the similarities and differences between them. For simplicity, we will stick to
the term EFEI in this Section. An EFEI calculation is a geometry optimization in which a constant that is equal to the
external force is added to the nuclear gradient of two atoms specified by the user. The external force is applied along
the vector connecting the two atoms, thus driving them apart. The geometry optimization converges when the restoring
force of the molecule is equal to the external force. The EFEI method can also be used in AIMD simulations (see
Section 10.7), in which case the force is added in every time step. The basic syntax for specifying EFEI calculations is
as follows.
$efei
atom1 atom2 f orce1
atom3 atom4 f orce2
...
$end

Here, atom1 and atom2 are the indices of the atoms to which a force is applied. force1 is the sum of the force values that
acts on atom1 and atom2 in nanoNewtons (nN). If this value is positive, a mechanical force of magnitude force1/2 acts
on each of these atoms, thus driving them apart. If it is negative, an attractive force acts between the atoms. Optionally,

Chapter 10: Exploring Potential Energy Surfaces: Critical Points and Molecular Dynamics

514

additional pairs of atoms that are subject to a force can be specified by adding lines in the $efei section.
Example 10.7 EFEI calculation of hydrogen peroxide with a constant stretching force of 2.5 nN acting on each oxygen
atom
$molecule
0 1
O
O
H
H
$end

-0.7059250062
0.7059250062
1.0662092915
-1.0662092915

$rem
JOBTYPE
EXCHANGE
BASIS
$end

-0.1776745132
0.1776745132
-0.5838921799
0.5838921799

-0.0698000882
-0.0698000882
0.4181150580
0.4181150580

opt
b3lyp
6-31G*

$efei
1 2 5
$end

10.4

Potential Energy Scans

It is often useful to scan the potential energy surface (PES), optimizing all other degrees of freedom for each particular
value of the scanned variable(s). Such a “relaxed” scan may provide a rough estimate of a pathway between reactant
and product—assuming the coordinate(s) for the scan has been chosen wisely—and is often used in development
of classical force fields to optimize dihedral angle parameters. Ramachandran plots, for example, are key tools for
studying conformational changes of peptides and proteins, and are essentially two-dimensional torsional scans.
In certain cases, relaxed scans might encounter some difficulties on optimizations. A “frozen” scan can be easier to
perform because of no geometry optimizations although it provides less information of real dynamics.
Q-C HEM supports one- and two-dimensional PES scans, by setting JOBTYPE equal to PES_SCAN in the $rem section.
In addition, a $scan input section with the following format should be specified, in the format below but with no more
than two bond-length, bond-angle, or torsional variables specified.
$scan
stre atom1
...
bend atom1
...
tors atom1
...
$end

atom2

value1 value2 incr

atom2

atom3

value1 value2 incr

atom2

atom3

atom4

value1 value2 incr

The first example below demonstrates how to scan the torsional potential of butane, which is a sequence of constrained

Chapter 10: Exploring Potential Energy Surfaces: Critical Points and Molecular Dynamics

515

optimizations with the C1–C2–C3–C4 dihedral angle fixed at −180◦ , −165◦ , −150◦ , . . ., 165◦ , 180◦ .
Example 10.8 One-dimensional torsional scan of butane
$molecule
0 1
C
C
C
C
H
H
H
H
H
H
H
H
H
H
$end
$rem
JOBTYPE
METHOD
BASIS
$end

1.934574
0.556601
-0.556627
-1.934557
2.720125
2.061880
2.062283
0.464285
0.464481
-0.464539
-0.464346
-2.062154
-2.720189
-2.061778

-0.128781
0.526657
-0.526735
0.128837
0.655980
-0.759501
-0.759765
1.168064
1.167909
-1.167976
-1.168166
0.759848
-0.655832
0.759577

-0.000151
0.000200
0.000173
-0.000138
-0.000236
-0.905731
0.905211
-0.903444
0.903924
0.903964
-0.903402
0.905185
-0.000229
-0.905748

pes_scan
hf
sto-3g

$scan
tors 1 2 3 4 -180 180 15
$end

The next example is a two-dimension potential scan. The first dimension is a scan of the C1–C2–C3–C4 dihedral angle
from −180◦ to 180◦ degree in 30◦ intervals; the second dimension is a scan of the C2–C3 bond length from 1.5 Å to

Chapter 10: Exploring Potential Energy Surfaces: Critical Points and Molecular Dynamics

516

1.6 Å in 0.05 Å increments.
Example 10.9 Two-dimensional torsional scan of butane
$molecule
0 1
C
C
C
C
H
H
H
H
H
H
H
H
H
H
$end
$rem
JOBTYPE
METHOD
BASIS
$end

1.934574
0.556601
-0.556627
-1.934557
2.720125
2.061880
2.062283
0.464285
0.464481
-0.464539
-0.464346
-2.062154
-2.720189
-2.061778

-0.128781
0.526657
-0.526735
0.128837
0.655980
-0.759501
-0.759765
1.168064
1.167909
-1.167976
-1.168166
0.759848
-0.655832
0.759577

-0.000151
0.000200
0.000173
-0.000138
-0.000236
-0.905731
0.905211
-0.903444
0.903924
0.903964
-0.903402
0.905185
-0.000229
-0.905748

pes_scan
hf
sto-3g

$scan
tors 1 2 3 4 -180 180 30
stre 2 3 1.5 1.6 0.05
$end

To perform a frozen PES scan, set FROZEN_SCAN to be TRUE and use input geometry in Z-matrix format. The example

Chapter 10: Exploring Potential Energy Surfaces: Critical Points and Molecular Dynamics

517

demonstrates a frozen PES of the C1–C2 bond stretching from 1.0 Åto 2.0 Åfor methanol.
Example 10.10 One-dimensional frozen PES scan of methanol
$molecule
0 1
C
O C RCO
H1 C RCH1
X C 1.0a
H2 C RCH2
H3 C RCH2
H4 O ROH

O
O
X
X
C

H1CO
XCO
H2CX
H2CX
HOC

H1
H1
H1
H1

180.0
90.0
-90.0
180.0

RCO = 1.421
RCH1 = 1.094
RCH2 = 1.094
ROH = 0.963
H1CO = 107.2
XCO = 129.9
H2CX = 54.25
HOC = 108.0
$end
$rem
jobtype
frozen_scan
exchange
correlatoin
basis
$end

pes_scan
true
s
vwn
3-21g

$scan
stre 1 2 1.0 2.0 0.5
$end

Q-C HEM also supports one-dimensional restrained PES scan for transition state search of typical SN 2 reactions. The
geometry restrains are
2
k (R12 ± R34 − R) ,
(10.1)
which is a harmonic potential applied to bias geometry optimization. R12 and R34 are two bond lengths in the reaction
coordinate. R constrains the range of R12 ± R34 , and k is a force constant. To perform a restrained PES scan, the
following format should be specified.
$scan
r12mr34
r12pr34
$end

atom1
atom1

atom2
atom2

atom3
atom3

atom4
atom4

Rmin
Rmin

Rmax
Rmax

incr
incr

force_constant
force_constant

Chapter 10: Exploring Potential Energy Surfaces: Critical Points and Molecular Dynamics

518

Example 10.11 One-dimensional restrained PES scan of chloromethane SN2 reaction
$molecule
-1 1
C
0.418808
Cl -0.775224
H
1.408172
H
0.147593
H
0.413296
Cl
1.947359
$end

-1.240869
-1.495584
-1.490565
-1.907736
-0.199000
1.619163

0.249048
1.586668
0.631227
-0.568952
-0.092071
-1.747832

$rem
jobtype pes_scan
exchange b3lyp
basis 6-31G*
$end
$scan
r12mr34 1 2 1 6 -2.0 2.0 0.2 1000.0
$end

10.5

Intrinsic Reaction Coordinate

The concept of a reaction path is chemically intuitive (a pathway from reactants to products) yet somewhat theoretically
ambiguous because most mathematical definitions depend upon the chosen coordinate system. Stationary points on a
potential energy surface are independent of this choice, but the path connecting them is not, and there exist various
mathematical definitions of a “reaction path”. Q-C HEM uses the intrinsic reaction coordinate (IRC) definition, as
originally defined by Fukui, 13 which has come to be a fairly standard choice in quantum chemistry. The IRC is
essentially sequence of small, steepest-descent paths going downhill from the transition state.
The reaction path is most unlikely to be a straight line and so by taking a finite step length along the direction of the
gradient you will leave the “true” reaction path. A series of small steepest descent steps will zig-zag along the actual
reaction path (a behavior known as “stitching”). Ishida et al. 25 developed a predictor-corrector algorithm, involving a
second gradient calculation after the initial steepest-descent step, followed by a line search along the gradient bisector
to get back on the path, and this algorithm was subsequently improved by Schmidt et al.. 49 This is the method that
Q-C HEM adopts. It cannot be used for the first downhill step from the transition state, since the gradient is zero, so
instead a step is taken along the Hessian mode whose frequency is imaginary.
The reaction path can be defined and followed in Z-matrix coordinates, Cartesian coordinates or mass-weighted Cartesian coordinates. The latter represents the “true” IRC as defined by Fukui. 13 If the rationale for following the reaction
path is simply to determine which local minima are connected by a given transition state, which, is arguably the major
use of IRC algorithms, then the choice of coordinates is irrelevant. In order to use the IRC code, the transition state
geometry and the exact Hessian must be available. These must be computed via two prior calculations, with JOBTYPE
= TS (transition structure search) and JOBTYPE = FREQ (Hessian calculation), respectively. Job control variables and
examples appear below.
An IRC calculation is invoked by setting JOBTYPE = RPATH in the $rem section, and additional $rem variables are
described below. IRC calculations may benefit from the methods discussed in Section 10.2 for obtaining good initial
guesses for transition-state structures.

Chapter 10: Exploring Potential Energy Surfaces: Critical Points and Molecular Dynamics
RPATH_COORDS
Determines which coordinate system to use in the IRC search.
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
0 Use mass-weighted coordinates.
1 Use Cartesian coordinates.
2 Use Z-matrix coordinates.
RECOMMENDATION:
Use the default.
RPATH_DIRECTION
Determines the direction of the eigenmode to follow. This will not usually be known prior to the
Hessian diagonalization.
TYPE:
INTEGER
DEFAULT:
1
OPTIONS:
1 Descend in the positive direction of the eigen mode.
-1 Descend in the negative direction of the eigen mode.
RECOMMENDATION:
It is usually not possible to determine in which direction to go a priori, and therefore both
directions will need to be considered.
RPATH_MAX_CYCLES
Specifies the maximum number of points to find on the reaction path.
TYPE:
INTEGER
DEFAULT:
20
OPTIONS:
n User-defined number of cycles.
RECOMMENDATION:
Use more points if the minimum is desired, but not reached using the default.
RPATH_MAX_STEPSIZE
Specifies the maximum step size to be taken (in 0.001 a.u.).
TYPE:
INTEGER
DEFAULT:
150 corresponding to a step size of 0.15 a.u..
OPTIONS:
n Step size = n/1000 a.u.
RECOMMENDATION:
None.

519

Chapter 10: Exploring Potential Energy Surfaces: Critical Points and Molecular Dynamics
RPATH_TOL_DISPLACEMENT
Specifies the convergence threshold for the step. If a step size is chosen by the algorithm that is
smaller than this, the path is deemed to have reached the minimum.
TYPE:
INTEGER
DEFAULT:
5000 Corresponding to 0.005 a.u.
OPTIONS:
n User-defined. Tolerance = n/1000000 a.u.
RECOMMENDATION:
Use the default. Note that this option only controls the threshold for ending the RPATH job
and does nothing to the intermediate steps of the calculation. A smaller value will provide
reaction paths that end closer to the true minimum. Use of smaller values without adjusting
RPATH_MAX_STEPSIZE, however, can lead to oscillations about the minimum.
RPATH_PRINT
Specifies the print output level.
TYPE:
INTEGER
DEFAULT:
2
OPTIONS:
n
RECOMMENDATION:
Use the default, as little additional information is printed at higher levels. Most of the output
arises from the multiple single point calculations that are performed along the reaction pathway.

Example 10.12
$molecule
0 1
C
H 1 1.20191
N 1 1.22178
$end
$rem
JOBTYPE
BASIS
METHOD
$end

2

72.76337

freq
sto-3g
hf

@@@
$molecule
read
$end
$rem
JOBTYPE
BASIS
METHOD
SCF_GUESS
RPATH_MAX_CYCLES
$end

rpath
sto-3g
hf
read
30

520

Chapter 10: Exploring Potential Energy Surfaces: Critical Points and Molecular Dynamics

521

10.6

Nonadiabatic Couplings and Optimization of Minimum-Energy Crossing Points

10.6.1

Nonadiabatic Couplings

Conical intersections are degeneracies between Born-Oppenheimer potential energy surfaces that facilitate non-adiabatic
transitions between excited states, i.e., internal conversion and intersystem crossing processes, both of which represent
a breakdown of the Born-Oppenheimer approximation. 20,35 Although simultaneous intersections between more than
two electronic states are possible, 35 consider for convenience the two-state case, and let


HJJ (R) HJK (R)
H=
.
(10.2)
∗
HKJ
(R) HKK (R)
denote the matrix representation of the vibronic (vibrational + electronic) Hamiltonian [Eq. (4.2)] in a basis of two
electronic states, J and K. (Electronic degrees of freedom have been integrated out of this expression, and R represents
the remaining, nuclear coordinates.) By definition, the Born-Oppenheimer states are the ones that diagonalize H at a
particular molecular geometry R, and thus two conditions must be satisfied in order to obtain degeneracy in the BornOppenheimer representation: HJJ = HKK and HJK = 0. As such, degeneracies between two Born-Oppenheimer
potential energy surfaces exist in subspaces of dimension Nint − 2, where Nint = 3Natoms − 6 is the number of
internal (vibrational) degrees of freedom (assuming the molecule is non-linear). This (Nint − 2)-dimensional subspace
is known as the seam space because the two states are degenerate everywhere within this space. In the remaining two
degrees of freedom, known as the branching space, the degeneracy between Born-Oppenheimer surfaces is lifted by an
infinitesimal displacement, which in a three-dimensional plot resembles a double cone about the point of intersection,
hence the name conical intersection.
The branching space is defined by the span of a pair of vectors gJK and hJK . The former is simply the difference in
the gradient vectors of the two states in question,
gJK =

∂EJ
∂EK
−
,
∂R
∂R

(10.3)

and is readily evaluated at any level of theory for which analytic energy gradients are available (or less-readily, via
finite difference, if they are not!). The definition of the non-adiabatic coupling vector hJK , on the other hand, is more
involved and not directly amenable to finite-difference calculations:
D


E
hJK = ΨJ ∂ Ĥ/∂R ΨK .
(10.4)
This is closely related to the derivative coupling vector


dJK = ΨJ ∂/∂R ΨK =

hJK
.
EJ − EK

(10.5)

The latter expression for dJK demonstrates that the coupling between states becomes large in regions of the potential
surface where the two states are nearly degenerate. The relative orientation and magnitudes of the vectors gJK and
hJK determined the topography around the intersection, i.e., whether the intersection is “peaked” or “sloped”; see
Ref. 20 for a pedagogical overview.
Algorithms to compute the non-adiabatic couplings dJK are not widely available in quantum chemistry codes, but
thanks to the efforts of the Herbert and Subotnik groups, they are available in Q-C HEM when the wave functions ΨJ
and ΨK , and corresponding electronic energies EJ and EK , are computed at the CIS or TDDFT level, 11,41,60,61 or at the
corresponding spin-flip (SF) levels of theory (SF-CIS or SF-TDDFT). The spin-flip implementation 60 is particularly
significant, because only that approach—and not traditional spin-conserving CIS or TDDFT—affords correct topology
around conical intersections that involve the ground state. To understand why, suppose that J in Eq. (10.2) represents
the ground state; call it J = 0 for definiteness. In linear response theory (TDDFT) or in CIS (by virtue of Brillouin’s
theorem), the coupling matrix elements between the reference (ground) state and all of the excited states vanish identically, hence H0K (R) ≡ 0. This means that there is only one condition to satisfy in order to obtain degeneracy, hence

Chapter 10: Exploring Potential Energy Surfaces: Critical Points and Molecular Dynamics

522

the branching space is one- rather than two-dimensional, for any conical intersection that involves the ground state. 31
(For intersections between two excited states, the topology should be correct.) In the spin-flip approach, however, the
reference state has a different spin multiplicity than the target states; if the latter have spin quantum number S, then the
reference state has spin S + 1. This has the effect that the ground state of interest (spin S) is treated as an excitation,
and thus on a more equal footing with excited states of the same spin, and it rigorously fixes the topology problem
around conical intersections. 60
Nonadiabatic (derivative) couplings are available for both CIS and TDDFT. The CIS non-adiabatic couplings can
be obtained from direct differentiation of the wave functions with respect to nuclear positions. 11,60 For TDDFT, the
same procedure can be carried out to calculate the approximate non-adiabatic couplings, in what has been termed
the “pseudo-wave function” approach. 40,60 Formally more rigorous TDDFT non-adiabatic couplings derived from
quadratic response theory are also available, although they are subject to certain undesirable, accidental singularities if for the two states J and K in Eq. (10.4), the energy difference |EJ − EK | is quasi-degenerate with the excitation
energy ωI = EI − E0 for some third state, I. 41,61 As such, the pseudo-wave function method is the recommended
approach for computing non-adiabatic couplings with TDDFT, although in the spin flip case the pseudo-wave function
approach is rigorously equivalent to the pseudo-wave function approach, and is free of singularities. 61
Finally, we note that there is some evidence that SF-TDDFT calculations are most accurate when used with functionals
containing ∼50% Hartree-Fock exchange, 24,50 and many studies with this method (see Ref. 20 for a survey) have used
the BH&HLYP functional, in which LYP correlation is combined with Becke’s “half and half” (BH&H) exchange
functional, consisting of 50% Hartree-Fock exchange and 50% Becke88 exchange (EXCHANGE = BHHLYP in QC HEM.)

10.6.2

Job Control and Examples

In order to perform non-adiabatic coupling calculations, the $derivative_coupling section must be given:
$derivative_coupling
one line comment
i, j, k, ...
$end

Nonadiabatic couplings will then be computed between all pairs of the states i, j, k, . . .; use “0” to request the HF or
DFT reference state, “1” for the first excited state, etc.
CALC_NAC
Determines whether we are calculating non-adiabatic couplings.
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
TRUE
Calculate non-adiabatic couplings.
FALSE Do not calculate non-adiabatic couplings.
RECOMMENDATION:
None.

Chapter 10: Exploring Potential Energy Surfaces: Critical Points and Molecular Dynamics

523

CIS_DER_NUMSTATE
Determines among how many states we calculate non-adiabatic couplings. These states must be
specified in the $derivative_coupling section.
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
0 Do not calculate non-adiabatic couplings.
n Calculate n(n − 1)/2 pairs of non-adiabatic couplings.
RECOMMENDATION:
None.
SET_QUADRATIC
Determines whether to include full quadratic response contributions for TDDFT.
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
TRUE
Include full quadratic response contributions for TDDFT.
FALSE Use pseudo-wave function approach.
RECOMMENDATION:
The pseudo-wave function approach is usually accurate enough and is free of accidental singularities. Consult Refs. 61 and 41 for additional guidance.

Example 10.13 Nonadiabatic couplings among the lowest five singlet states of ethylene, computed at the TD-B3LYP
level using the pseudo-wave function approach.
$molecule
0 1
C
1.85082356
H
2.38603593
H
0.78082359
C
2.52815456
H
1.99294220
H
3.59815453
$end
$rem
CIS_N_ROOTS
CIS_TRIPLETS
SET_ITER
CIS_DER_NUMSTATE
CALC_NAC
EXCHANGE
BASIS
$end

-1.78953123
-2.71605577
-1.78977646
-0.61573833
0.31078621
-0.61549310

4
false
50
5
true
b3lyp
6-31G*

$derivative_coupling
0 is the reference state
0 1 2 3 4
$end

0.00000000
0.00000000
0.00000000
0.00000000
0.00000000
0.00000000

Chapter 10: Exploring Potential Energy Surfaces: Critical Points and Molecular Dynamics

524

Example 10.14 Nonadiabatic couplings between S0 and S1 states of ethylene using BH&HLYP and spin-flip TDDFT.
$molecule
0 3
C
1.85082356
H
2.38603593
H
0.78082359
C
2.52815456
H
1.99294220
H
3.59815453
$end
$rem
SPIN_FLIP
UNRESTRICTED
CIS_N_ROOTS
CIS_TRIPLETS
SET_ITER
CIS_DER_NUMSTATE
CALC_NAC
EXCHANGE
BASIS
$end

-1.78953123
-2.71605577
-1.78977646
-0.61573833
0.31078621
-0.61549310

0.00000000
0.00000000
0.00000000
0.00000000
0.00000000
0.00000000

true
true
4
false
50
2
true
bhhlyp
6-31G*

$derivative_coupling
comment
1 3
$end

Example 10.15 Nonadiabatic couplings between S1 and S2 states of ethylene computed via quadratic response theory
at the TD-B3LYP level.
$molecule
0 1
C
1.85082356
H
2.38603593
H
0.78082359
C
2.52815456
H
1.99294220
H
3.59815453
$end
$rem
CIS_N_ROOTS
CIS_TRIPLETS
RPA
SET_ITER
CIS_DER_NUMSTATE
CALC_NAC
EXCHANGE
BASIS
SET_QUADRATIC
$end
$derivative_coupling
comment
1 2
$end

-1.78953123
-2.71605577
-1.78977646
-0.61573833
0.31078621
-0.61549310

0.00000000
0.00000000
0.00000000
0.00000000
0.00000000
0.00000000

4
false
true
50
2
true
b3lyp
6-31G*
true #include full quadratic response

Chapter 10: Exploring Potential Energy Surfaces: Critical Points and Molecular Dynamics

10.6.3

525

Minimum-Energy Crossing Points

The seam space of a conical intersection is really a (hyper)surface of dimension Nint − 2, and while the two electronic
states in question are degenerate at every point within this space, the electronic energy varies from one point to the next.
To provide a simple picture of photochemical reaction pathways, it is often convenient to locate the minimum-energy
crossing point (MECP) within this (Nint −2)-dimensional seam. Two separate minimum-energy pathway searches, one
on the excited state starting from the ground-state geometry and terminating at the MECP, and the other on the ground
state starting from the MECP and terminating at the ground-state geometry, then affords a photochemical mechanism.
(See Ref. 59 for a simple example.) In some sense, then, the MECP is to photochemistry what the transition state is
to reactions that occur on a single Born-Oppenheimer potential energy surface. One should be wary of pushing this
analogy too far, because whereas a transition state reasonably be considered to be a bottleneck point on the reaction
pathway, the path through a conical intersection may be downhill and perhaps therefore more likely to proceed from
one surface to the other at a point “near" the intersection, and in addition there can be multiple conical intersections
between the same pair of states so more than one photochemical mechanism may be at play. Such complexity could
be explored, albeit at significantly increased cost, using non-adiabatic “surface hopping" ab initio molecular dynamics,
as described in Section 10.7.6. Here we describe the computationally-simpler procedure of locating an MECP along a
conical seam.
Recall that the branching space around a conical intersection between electronic states J and K is spanned by two
vectors, gJK [Eq. (10.3)] and hJK [Eq. (10.4)]. While the former is readily available in analytic form for any electronic
structure method that has analytic excited-state gradients, the non-adiabatic coupling vector hJK is not available for
most methods. For this reason, several algorithms have been developed to optimize MECPs without the need to evaluate
hJK , and three such algorithms are available in Q-C HEM.
Martínez and coworkers 32 developed a penalty-constrained MECP optimization algorithm that consists of minimizing
the objective function

2 !


EI (R) − EJ (R)
1
Fσ (R) = 2 EI (R) + EJ (R) + σ
,
(10.6)
EI (R) − EJ (R) + α
where α is a fixed parameter to avoid singularities and σ is a Lagrange multiplier for a penalty function meant to drive
the energy gap to zero. Minimization of Fσ is performed iteratively for increasingly large values σ.
A second MECP optimization algorithm is a simplification of the penalty-constrained approach that we call the “direct”
method. Here, the gradient of the objective function is
G = PGmean + 2(EK − EJ )Gdiff ,

(10.7)

Gmean = 21 (GJ + GK )

(10.8)

where
is the mean energy gradient, with Gi = ∂Ei /∂R being the nuclear gradient for state i, and
Gdiff =

GK − GJ
||GK − GJ ||

(10.9)

is the normalized difference gradient. Finally,
P = 1 − Gdiff G>
diff

(10.10)

projects the gradient difference direction out of the mean energy gradient in Eq. (10.7). The algorithm then consists in
minimizing along the gradient G, with for the iterative cycle over a Lagrange multiplier, which can sometimes be slow
to converge.
The third and final MECP optimization algorithm that is available in Q-C HEM is the branching-plane updating method
developed by Morokuma and coworkers 33 and implemented in Q-C HEM by Zhang and Herbert. 59 This algorithm uses
a gradient that is similar to that in Eq. (10.7) but projects out not just Gdiff in Eq. (10.10) but also a second vector that
is orthogonal to it, representing an iteratively-updated approximation to the branching space.

Chapter 10: Exploring Potential Energy Surfaces: Critical Points and Molecular Dynamics

526

None of these three methods requires evaluation of non-adiabatic couplings, and all three can be used to optimize
MECPs at the CIS, SF-CIS, TDDFT, SF-TDDFT, and SOS-CIS(D0) levels. The direct algorithm can also be used for
EOM-XX-CCSD methods (XX = EE, IP, or EA). It should be noted that since EOM-XX-CCSD is a linear response
method, it suffers from the same topology problem around conical intersections involving the ground state that was
described in regards to TDDFT in Section 10.6.1. With spin-flip approaches, correct topology is obtained. 60
Analytic derivative couplings are available for (SF-)CIS and (SF-)TDDFT, so for these methods one can alternatively
employ an optimization algorithm that makes use of both gJK and hJK . Such an algorithm, due to Schlegel and
coworkers, 5 is available in Q-C HEM and consists of optimization along the gradient in Eq. (10.7) but with a projector
>
P = 1 − Gdiff G>
diff − yy

where
y=

(1 − xx> )hJK
,
||(1 − xx> )hJK ||

(10.11)

(10.12)

in place of the projector in Eq. (10.10). Equation (10.11) has the effect of projecting the span of gJK and hJK (i.e., the
branching space) out of state-averaged gradient in Eq. (10.7). The tends to reduce the number of iterations necessary
to converge the MECP, and since calculation of the (optional) hJK vector represents only a slight amount of overhead
on top of the (required) gJK vector, this last algorithm tends to yield significant speed-ups relative to the other three. 60
As such, it is the recommended choice for (SF-)CIS and (SF-)TDDFT.
It should be noted that while the spin-flip methods cure the topology problem around conical intersections that involve
the ground state, this method tends to exacerbate spin contamination relative to the corresponding spin-conserving
approaches. 62 While spin contamination is certainly present in traditional, spin-conserving CIS and TDDFT, it presents
the following unique challenge in spin-flip methods. Suppose, for definiteness, that one is interested in singlet excited
states. Then the reference state for the spin-flip methods should be the high-spin triplet. A spin-flipping excitation
will then generate S0 , S1 , S2 , . . . but will also generate the MS = 0 component of the triplet reference state, which
therefore appears in what is ostensibly the singlet manifold. Q-C HEM attempts to identify this automatically, based on a
threshold for hŜ 2 i, but severe spin contamination can sometimes defeat this algorithm, 59 hampering Q-C HEM’s ability
to distinguish singlets from triplets (in this particular example). An alternative might be the state-tracking procedure
that is described in Section 10.6.5.

10.6.4

Job Control and Examples

For MECP optimization, set MECP_OPT = TRUE in the $rem section, and note that the $derivative_coupling input
section discussed in Section 10.6.2 is not necessary in this case.
MECP_OPT
Determines whether we are doing MECP optimizations.
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
TRUE
Do MECP optimization.
FALSE Do not do MECP optimization.
RECOMMENDATION:
None.

Chapter 10: Exploring Potential Energy Surfaces: Critical Points and Molecular Dynamics
MECP_METHODS
Determines which method to be used.
TYPE:
STRING
DEFAULT:
BRANCHING_PLANE
OPTIONS:
BRANCHING_PLANE
Use the branching-plane updating method.
MECP_DIRECT
Use the direct method.
PENALTY_FUNCTION Use the penalty-constrained method.
RECOMMENDATION:
The direct method is stable for small molecules or molecules with high symmetry. The
branching-plane updating method is more efficient for larger molecules but does not work
if the two states have different symmetries. If using the branching-plane updating method,
GEOM_OPT_COORDS must be set to 0 in the $rem section, as this algorithm is available in
Cartesian coordinates only. The penalty-constrained method converges slowly and is suggested
only if other methods fail.
MECP_STATE1
Sets the first Born-Oppenheimer state for MECP optimization.
TYPE:
INTEGER/INTEGER ARRAY
DEFAULT:
None
OPTIONS:
[i,j] Find the jth excited state with the total spin i; j = 0 means the SCF ground state.
RECOMMENDATION:
i is ignored for restricted calculations; for unrestricted calculations, i can only be 0 or 1.
MECP_STATE2
Sets the second Born-Oppenheimer state for MECP optimization.
TYPE:
INTEGER/INTEGER ARRAY
DEFAULT:
None
OPTIONS:
[i,j] Find the jth excited state with the total spin i; j = 0 means the SCF ground state.
RECOMMENDATION:
i is ignored for restricted calculations; for unrestricted calculations, i can only be 0 or 1.
CIS_S2_THRESH
Determines whether a state is a singlet or triplet in unrestricted calculations.
TYPE:
INTEGER
DEFAULT:
120
OPTIONS:
n Sets the hŜ 2 i threshold to n/100
RECOMMENDATION:
For the default case, states with hŜ 2 i > 1.2 are treated as triplet states and other states are treated
as singlets.

527

Chapter 10: Exploring Potential Energy Surfaces: Critical Points and Molecular Dynamics

528

MECP_PROJ_HESS
Determines whether to project out the coupling vector from the Hessian when using branching
plane updating method.
TYPE:
LOGICAL
DEFAULT:
TRUE
OPTIONS:
TRUE
FALSE
RECOMMENDATION:
Use the default.

Example 10.16 MECP optimization of an intersection between the S2 and S3 states of NO−
2 , using the direct method
at the SOS-CIS(D0) level.
$molecule
-1 1
N1
O2 N1
O3 N1

RNO
RNO

O2

AONO

RNO = 1.50
AONO = 100
$end
$rem
JOBTYPE
METHOD
BASIS
AUX_BASIS
PURECART
CIS_N_ROOTS
CIS_TRIPLETS
CIS_SINGLETS
MEM_STATIC
MEM_TOTAL
MECP_OPT
MECP_STATE1
MECP_STATE2
MECP_METHODS
$end

=
=
=
=
=
=
=
=
=
=
=
=
=
=

opt
soscis(d0)
aug-cc-pVDZ
rimp2-aug-cc-pVDZ
1111
4
false
true
900
1950
true
[0,2]
[0,3]
mecp_direct

Chapter 10: Exploring Potential Energy Surfaces: Critical Points and Molecular Dynamics

529

Example 10.17 Optimization of the ethylidene MECP between S0 and S1 in C2 H2 , at the SF-TDDFT level using the
branching-plane updating method.
$molecule
0 3
C
0.044626
C
0.008905
H
0.928425
H -0.831032
H -0.009238
H
0.068314
$end

-0.2419240
0.6727548
-0.1459163
-0.1926895
0.9611331
-1.2533580

$rem
JOBTYPE
MECP_OPT
MECP_METHODS
MECP_PROJ_HESS
GEOM_OPT_COORDS
METHOD
SPIN_FLIP
UNRESTRICTED
BASIS
CIS_N_ROOTS
MECP_STATE1
MECP_STATE2
CIS_S2_THRESH
$end

0.357157
1.460500
-0.272095
-0.288529
2.479936
0.778847

opt
true
branching_plane
true ! project out y vector from the hessian
0 ! currently only works for Cartesian coordinate
bhhlyp
true
true
6-31G(d,p)
4
[0,1]
[0,2]
120

Example 10.18 Optimization of the twisted-pyramidalized ethylene MECP between S0 and S1 in C2 H2 using SFTDDFT.
$molecule
0 3
C -0.015889
C
0.012427
H
0.857876
H -0.936470
H
0.764557
H
0.740773
$end
$rem
JOBTYPE
MECP_OPT
MECP_METHODS
METHOD
SPIN_FLIP
UNRESTRICTED
BASIS
CIS_N_ROOTS
MECP_STATE1
MECP_STATE2
CIS_S2_THRESH
$end

0.073532
-0.002468
0.147014
-0.011696
0.663381
-0.869764

-0.059559
1.315694
-0.710529
-0.626761
1.762573
1.328583

opt
true
penalty_function
bhhlyp
true
true
6-31G(d,p)
4
[0,1]
[0,2]
120

Chapter 10: Exploring Potential Energy Surfaces: Critical Points and Molecular Dynamics

530

Example 10.19 Optimization of the B̃ 1 A2 and Ã1 B2 states of N+
3 using the direct method at the EOM-EE-CCSD
level.
$molecule
1 1
N1
N2 N1 rNN
N3 N2 rNN N1 aNNN
rNN = 1.54
aNNN = 50.0
$end
$rem
JOBTYPE
MECP_OPT
MECP_METHODS
METHOD
BASIS
EE_SINGLETS
XOPT_STATE_1
XOPT_STATE_2
CCMAN2
GEOM_OPT_TOL_GRADIENT
$end

opt
true
mecp_direct
eom-ccsd
6-31g
[0,2,0,2]
[0,2,2]
[0,4,1]
false
30

Example 10.20 Optimization of the ethylidene MECP between S0 and S1 , using BH&HLYP spin-flip TDDFT with
analytic derivative couplings.
$molecule
0 3
C
0.044626
C
0.008905
H
0.928425
H -0.831032
H -0.009238
H
0.068314
$end

-0.241924
0.672754
-0.145916
-0.192689
0.961133
-1.253358

$rem
JOBTYPE
MECP_OPT
MECP_METHODS
MECP_PROJ_HESS
GEOM_OPT_COORDS
MECP_STATE1
MECP_STATE2
UNRESTRICTED
SPIN_FLIP
CIS_N_ROOTS
CALC_NAC
CIS_DER_NUMSTATE
SET_ITER
EXCHANGE
BASIS
SYMMETRY_IGNORE
$end

0.357157
1.460500
-0.272095
-0.288529
2.479936
0.778847

opt
true
branching_plane
true
0
[0,1]
[0,2]
true
true
4
true
2
50
bhhlyp
6-31G(d,p)
true

Chapter 10: Exploring Potential Energy Surfaces: Critical Points and Molecular Dynamics

10.6.5

531

State-Tracking Algorithm

For optimizing excited-state geometries and other applications, it can be important to find and follow electronically
excited states of a particular character as the geometry changes. Various state-tracking procedures have been proposed
for such cases. 15,62 An excited-state, state-tracking algorithm available in Q-C HEM is based on the overlap of the
attachment/detachment densities at successive steps (Section 7.12.1). 8 Using the densities avoids any issues that may
be introduced by sign changes in the orbitals or configuration-interaction coefficients.
Two parameters are used to influence the choice of the electronic surface. One (γE ) controls the energy window for
states included in the search, and the other (γS ) controls how well the states must overlap in order to be considered
of the same character. These can be set by the user or generated automatically based on the magnitude of the nuclear
displacement. The energy window is defined relative to the estimated energy for the current step (i.e., Eest ± γE ),
which in turn is based on the energy, gradient and nuclear displacement of previous steps. This estimated energy is
specific to the type of calculation (e.g., geometry optimization).
The similarity metric for the overlap is defined as
S =1−

1
2



||∆A|| + ||∆D||

(10.13)

where ∆A = At+1 − At is the difference in attachment density matrices (Eq. (7.99)) and ∆D = Dt+1 − Dt is the
difference in detachment density matrices (Eq. (7.97)), at successive steps. Equation (10.13) uses the matrix spectral
norm,

1/2
||M|| = λmax M† M
(10.14)
where λmax is the largest eigenvalue of M.
The selected state always satisfies one of the following
1. It is the only state in the window defined by γE .
2. It is the state with the largest overlap, provided at least one state has S ≥ γS .
3. It is the nearest state energetically if all states in the window have S < γS , or if there are no states in the energy
window.
State-following can currently be used with CIS or TDDFT excited states and is initiated with the $rem variable
STATE_FOLLOW. It can be used with geometry optimization, ab initio molecular dynamics, 8 or with the freezing/
growing-string method. The desired state is specified using SET_STATE_DERIV for optimization or dynamics, or using SET_STATE_REACTANT and SET_STATE_PRODUCT for the freezing- or growing-string methods. The results for
geometry optimizations can be affected by the step size (GEOM_OPT_DMAX), and using a step size smaller than the
default value can provide better results. Also, it is often challenging to converge the strings in freezing/growing-string
calculations.
STATE_FOLLOW
Turns on state following.
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
FALSE Do not use state-following.
TRUE
Use state-following.
RECOMMENDATION:
None.

Chapter 10: Exploring Potential Energy Surfaces: Critical Points and Molecular Dynamics

532

FOLLOW_ENERGY
Adjusts the energy window for near states
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
0 Use dynamic thresholds, based on energy difference between steps.
n Search over selected state Eest ± n × 10−6 Eh .
RECOMMENDATION:
Use a wider energy window to follow a state diabatically, smaller window to remain on the
adiabatic state most of the time.
FOLLOW_OVERLAP
Adjusts the threshold for states of similar character.
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
0 Use dynamic thresholds, based on energy difference between steps.
n Percentage overlap for previous step and current step.
RECOMMENDATION:
Use a higher value to require states have higher degree of similarity to be considered the same
(more often selected based on energy).

10.7 Ab Initio Molecular Dynamics
Q-C HEM can propagate classical molecular dynamics trajectories on the Born-Oppenheimer potential energy surface
generated by a particular theoretical model chemistry (e.g., B3LYP/6-31G* or MP2/aug-cc-pVTZ). This procedure, in
which the forces on the nuclei are evaluated on-the-fly, is known variously as “direct dynamics”, “ab initio molecular
dynamics” (AIMD), or “Born-Oppenheimer molecular dynamics” (BOMD). In its most straightforward form, a BOMD
calculation consists of an energy + gradient calculation at each molecular dynamics time step, and thus each time step is
comparable in cost to one geometry optimization step. A BOMD calculation may be requested using any SCF energy +
gradient method available in Q-C HEM, including excited-state dynamics in cases where excited-state analytic gradients
are available. As usual, Q-C HEM will automatically evaluate derivatives by finite-difference if the analytic versions are
not available for the requested method, but in AIMD applications this is very likely to be prohibitively expensive.
While the number of time steps required in most AIMD trajectories dictates that economical (typically SCF-based)
underlying electronic structure methods are required, any method with available analytic gradients can reasonably be
used for BOMD, including (within Q-C HEM) HF, DFT, MP2, RI-MP2, CCSD, and CCSD(T). The RI-MP2 method,
especially when combined with Fock matrix and Z-vector extrapolation (as described below) is particularly effective
as an alternative to DFT-based dynamics.

10.7.1

Overview and Basic Job Control

Initial Cartesian coordinates and velocities must be specified for the nuclei. Coordinates are specified in the $molecule
section as usual, while velocities can be specified using a $velocity section with the form:
$velocity
vx,1 vy,1 vz,1
vx,2 vy,2 vz,2

Chapter 10: Exploring Potential Energy Surfaces: Critical Points and Molecular Dynamics

533

vx,N vy,N vz,N
$end

Here vx,i , vy,i , and vz,i are the x, y, and z Cartesian velocities of the ith nucleus, specified in atomic units (bohrs per
atomic unit of time, where 1 a.u. of time is approximately 0.0242 fs). The $velocity section thus has the same form as
the $molecule section, but without atomic symbols and without the line specifying charge and multiplicity. The atoms
must be ordered in the same manner in both the $velocity and $molecule sections.
As an alternative to a $velocity section, initial nuclear velocities can be sampled from certain distributions (e.g.,
Maxwell-Boltzmann), using the AIMD_INIT_VELOC variable described below. AIMD_INIT_VELOC can also be set
to QUASICLASSICAL, which triggers the use of quasi-classical trajectory molecular dynamics (see Section 10.7.5).
Although the Q-C HEM output file dutifully records the progress of any ab initio molecular dynamics job, the most
useful information is printed not to the main output file but rather to a directory called “AIMD” that is a subdirectory
of the job’s scratch directory. (All ab initio molecular dynamics jobs should therefore use the –save option described
in Section 2.7.) The AIMD directory consists of a set of files that record, in ASCII format, one line of information
at each time step. Each file contains a few comment lines (indicated by “#”) that describe its contents and which we
summarize in the list below.
• Cost: Records the number of SCF cycles, the total CPU time, and the total memory use at each dynamics step.
• EComponents: Records various components of the total energy (all in hartree).
• Energy: Records the total energy and fluctuations therein.
• MulMoments: If multipole moments are requested, they are printed here.
• NucCarts: Records the nuclear Cartesian coordinates x1 , y1 , z1 , x2 , y2 , z2 , . . . , xN , yN , zN at each time step, in
either bohrs or Ångstroms.
• NucForces: Records the Cartesian forces on the nuclei at each time step (same order as the coordinates, but given
in atomic units).
• NucVeloc: Records the Cartesian velocities of the nuclei at each time step (same order as the coordinates, but
given in atomic units).
• TandV: Records the kinetic and potential energy, as well as fluctuations in each.
• View.xyz: Cartesian-formatted version of NucCarts for viewing trajectories in an external visualization program.
For ELMD jobs, there are other output files as well:
• ChangeInF: Records the matrix norm and largest magnitude element of ∆F = F(t + δt) − F(t) in the basis of
Cholesky-orthogonalized AOs. The files ChangeInP, ChangeInL, and ChangeInZ provide analogous information
for the density matrix P and the Cholesky orthogonalization matrices L and Z defined in Ref. 18.
• DeltaNorm: Records the norm and largest magnitude element of the curvy-steps rotation angle matrix ∆ defined
in Ref. 18. Matrix elements of ∆ are the dynamical variables representing the electronic degrees of freedom.
The output file DeltaDotNorm provides the same information for the electronic velocity matrix d∆/dt.
• ElecGradNorm: Records the norm and largest magnitude element of the electronic gradient matrix FP − PF in
the Cholesky basis.
• dTfict: Records the instantaneous time derivative of the fictitious kinetic energy at each time step, in atomic units.
Ab initio molecular dynamics jobs are requested by specifying JOBTYPE = AIMD. Initial velocities must be specified
either using a $velocity section or via the AIMD_INIT_VELOC keyword described below. In addition, the following
$rem variables must be specified for any ab initio molecular dynamics job:

Chapter 10: Exploring Potential Energy Surfaces: Critical Points and Molecular Dynamics

534

AIMD_METHOD
Selects an ab initio molecular dynamics algorithm.
TYPE:
STRING
DEFAULT:
BOMD
OPTIONS:
BOMD
Born-Oppenheimer molecular dynamics.
CURVY Curvy-steps Extended Lagrangian molecular dynamics.
RECOMMENDATION:
BOMD yields exact classical molecular dynamics, provided that the energy is tolerably conserved. ELMD is an approximation to exact classical dynamics whose validity should be tested
for the properties of interest.
TIME_STEP
Specifies the molecular dynamics time step, in atomic units (1 a.u. = 0.0242 fs).
TYPE:
INTEGER
DEFAULT:
None.
OPTIONS:
User-specified.
RECOMMENDATION:
Smaller time steps lead to better energy conservation; too large a time step may cause the job to
fail entirely. Make the time step as large as possible, consistent with tolerable energy conservation.
AIMD_TIME_STEP_CONVERSION
Modifies the molecular dynamics time step to increase granularity.
TYPE:
INTEGER
DEFAULT:
1
OPTIONS:
n The molecular dynamics time step is TIME_STEP/n a.u.
RECOMMENDATION:
None
AIMD_STEPS
Specifies the requested number of molecular dynamics steps.
TYPE:
INTEGER
DEFAULT:
None.
OPTIONS:
User-specified.
RECOMMENDATION:
None.

Ab initio molecular dynamics calculations can be quite expensive, and thus Q-C HEM includes several algorithms
designed to accelerate such calculations. At the self-consistent field (Hartree-Fock and DFT) level, BOMD calculations
can be greatly accelerated by using information from previous time steps to construct a good initial guess for the new

Chapter 10: Exploring Potential Energy Surfaces: Critical Points and Molecular Dynamics

535

molecular orbitals or Fock matrix, thus hastening SCF convergence. A Fock matrix extrapolation procedure, 19 based
on a suggestion by Pulay and Fogarasi, 46 is available for this purpose.
The Fock matrix elements Fµν in the atomic orbital basis are oscillatory functions of the time t, and Q-C HEM’s
extrapolation procedure fits these oscillations to a power series in t:
Fµν (t) =

N
X

cn t n

(10.15)

n=0

The N + 1 extrapolation coefficients cn are determined by a fit to a set of M Fock matrices retained from previous time
steps. Fock matrix extrapolation can significantly reduce the number of SCF iterations required at each time step, but
for low-order extrapolations, or if SCF_CONVERGENCE is set too small, a systematic drift in the total energy may be
observed. Benchmark calculations testing the limits of energy conservation can be found in Ref. 19, and demonstrate
that numerically exact classical dynamics (without energy drift) can be obtained at significantly reduced cost.
Fock matrix extrapolation is requested by specifying values for N and M , as in the form of the following two $rem
variables:
FOCK_EXTRAP_ORDER
Specifies the polynomial order N for Fock matrix extrapolation.
TYPE:
INTEGER
DEFAULT:
0 Do not perform Fock matrix extrapolation.
OPTIONS:
N Extrapolate using an N th-order polynomial (N > 0).
RECOMMENDATION:
None
FOCK_EXTRAP_POINTS
Specifies the number M of old Fock matrices that are retained for use in extrapolation.
TYPE:
INTEGER
DEFAULT:
0 Do not perform Fock matrix extrapolation.
OPTIONS:
M Save M Fock matrices for use in extrapolation (M > N )
RECOMMENDATION:
Higher-order extrapolations with more saved Fock matrices are faster and conserve energy better
than low-order extrapolations, up to a point. In many cases, the scheme (N = 6, M = 12), in
conjunction with SCF_CONVERGENCE = 6, is found to provide about a 50% savings in computational cost while still conserving energy.

When nuclear forces are computed using underlying electronic structure methods with non-optimized orbitals (such as
MP2), a set of response equations must be solved. 2 While these equations are linear, their dimensionality necessitates
an iterative solution, 27,43 which, in practice, looks much like the SCF equations. Extrapolation is again useful here, 54
and the syntax for Z-vector (response) extrapolation is similar to Fock extrapolation.

Chapter 10: Exploring Potential Energy Surfaces: Critical Points and Molecular Dynamics

536

Z_EXTRAP_ORDER
Specifies the polynomial order N for Z-vector extrapolation.
TYPE:
INTEGER
DEFAULT:
0 Do not perform Z-vector extrapolation.
OPTIONS:
N Extrapolate using an N th-order polynomial (N > 0).
RECOMMENDATION:
None
Z_EXTRAP_POINTS
Specifies the number M of old Z-vectors that are retained for use in extrapolation.
TYPE:
INTEGER
DEFAULT:
0 Do not perform response equation extrapolation.
OPTIONS:
M Save M previous Z-vectors for use in extrapolation (M > N )
RECOMMENDATION:
Using the default Z-vector convergence settings, a (M, N ) = (4, 2) extrapolation was shown to
provide the greatest speedup. At this setting, a 2–3-fold reduction in iterations was demonstrated.

Assuming decent conservation, a BOMD calculation represents exact classical dynamics on the Born-Oppenheimer
potential energy surface. In contrast, so-called extended Lagrangian molecular dynamics (ELMD) methods make an
approximation to exact classical dynamics in order to expedite the calculations. ELMD methods—of which the most
famous is Car–Parrinello molecular dynamics—introduce a fictitious dynamics for the electronic (orbital) degrees of
freedom, which are then propagated alongside the nuclear degrees of freedom, rather than optimized at each time step
as they are in a BOMD calculation. The fictitious electronic dynamics is controlled by a fictitious mass parameter µ,
and the value of µ controls both the accuracy and the efficiency of the method. In the limit of small µ the nuclei and
the orbitals propagate adiabatically, and ELMD mimics true classical dynamics. Larger values of µ slow down the
electronic dynamics, allowing for larger time steps (and more computationally efficient dynamics), at the expense of
an ever-greater approximation to true classical dynamics.
Q-C HEM’s ELMD algorithm is based upon propagating the density matrix, expressed in a basis of atom-centered Gaussian orbitals, along shortest-distance paths (geodesics) of the manifold of allowed density matrices P. Idempotency of
P is maintained at every time step, by construction, and thus our algorithm requires neither density matrix purification,
nor iterative solution for Lagrange multipliers (to enforce orthogonality of the molecular orbitals). We call this procedure “curvy steps” ELMD, 18 and in a sense it is a time-dependent implementation of the GDM algorithm (Section 4.5)
for converging SCF single-point calculations.
The extent to which ELMD constitutes a significant approximation to BOMD continues to be debated. When assessing
the accuracy of ELMD, the primary consideration is whether there exists a separation of time scales between nuclear
oscillations, whose time scale τnuc is set by the period of the fastest vibrational frequency, and electronic oscillations,
whose time scale τelec may be estimated according to 18

τelec ≥

µ
εLUMO − εHOMO

1/2
(10.16)

A conservative estimate, suggested in Ref. 18, is that essentially exact classical dynamics is attained when τnuc >
10 τelec . In practice, we recommend careful benchmarking to insure that ELMD faithfully reproduces the BOMD
observables of interest.
Due to the existence of a fast time scale τelec , ELMD requires smaller time steps than BOMD. When BOMD is

Chapter 10: Exploring Potential Energy Surfaces: Critical Points and Molecular Dynamics

537

combined with Fock matrix extrapolation to accelerate convergence, it is no longer clear that ELMD methods are
substantially more efficient, at least in Gaussian basis sets. 19,46
The following $rem variables are required for ELMD jobs:
AIMD_FICT_MASS
Specifies the value of the fictitious electronic mass µ, in atomic units, where µ has dimensions
of (energy)×(time)2 .
TYPE:
INTEGER
DEFAULT:
None
OPTIONS:
User-specified
RECOMMENDATION:
Values in the range of 50–200 a.u. have been employed in test calculations; consult Ref. 18 for
examples and discussion.

10.7.2

Additional Job Control and Examples
AIMD_INIT_VELOC
Specifies the method for selecting initial nuclear velocities.
TYPE:
STRING
DEFAULT:
None
OPTIONS:
THERMAL
Random sampling of nuclear velocities from a Maxwell-Boltzmann
distribution. The user must specify the temperature in Kelvin via
the $rem variable AIMD_TEMP.
ZPE
Choose velocities in order to put zero-point vibrational energy into
each normal mode, with random signs. This option requires that a
frequency job to be run beforehand.
QUASICLASSICAL Puts vibrational energy into each normal mode. In contrast to the
ZPE option, here the vibrational energies are sampled from a
Boltzmann distribution at the desired simulation temperature. This
also triggers several other options, as described below.
RECOMMENDATION:
This variable need only be specified in the event that velocities are not specified explicitly in a
$velocity section.

AIMD_MOMENTS
Requests that multipole moments be output at each time step.
TYPE:
INTEGER
DEFAULT:
0 Do not output multipole moments.
OPTIONS:
n Output the first n multipole moments.
RECOMMENDATION:
None

Chapter 10: Exploring Potential Energy Surfaces: Critical Points and Molecular Dynamics
AIMD_TEMP
Specifies a temperature (in Kelvin) for Maxwell-Boltzmann velocity sampling.
TYPE:
INTEGER
DEFAULT:
None
OPTIONS:
User-specified number of Kelvin.
RECOMMENDATION:
This variable is only useful in conjunction with AIMD_INIT_VELOC = THERMAL. Note that the
simulations are run at constant energy, rather than constant temperature, so the mean nuclear
kinetic energy will fluctuate in the course of the simulation.
DEUTERATE
Requests that all hydrogen atoms be replaces with deuterium.
TYPE:
LOGICAL
DEFAULT:
FALSE Do not replace hydrogens.
OPTIONS:
TRUE Replace hydrogens with deuterium.
RECOMMENDATION:
Replacing hydrogen atoms reduces the fastest vibrational frequencies by a factor of 1.4, which
allow for a larger fictitious mass and time step in ELMD calculations. There is no reason to
replace hydrogens in BOMD calculations.

538

Chapter 10: Exploring Potential Energy Surfaces: Critical Points and Molecular Dynamics

539

Example 10.21 Simulating thermal fluctuations of the water dimer at 298 K.
$molecule
0 1
O
1.386977
H
1.748442
H
1.741280
O -1.511955
H -0.558095
H -1.910308
$end

0.011218
0.720970
-0.793653
-0.009629
0.008225
0.077777

$rem
JOBTYPE
AIMD_METHOD
METHOD
BASIS
TIME_STEP
AIMD_STEPS
AIMD_INIT_VELOC
AIMD_TEMP
FOCK_EXTRAP_ORDER
FOCK_EXTRAP_POINTS
$end

0.109098
-0.431026
-0.281811
-0.120521
0.047352
0.749067

aimd
bomd
b3lyp
6-31g*
20
1000
thermal
298
6
12

(20 a.u. = 0.48 fs)

request Fock matrix extrapolation

Example 10.22 Propagating F− (H2 O)4 on its first excited-state potential energy surface, calculated at the CIS level.
$molecule
-1
1
O -1.969902
H -2.155172
H -1.018352
O -1.974264
H -2.153919
H -1.023012
O -1.962151
H -2.143937
H -1.010860
O -1.957618
H -2.145835
H -1.005985
F
1.431477
$end

-1.946636
-1.153127
-1.980061
0.720358
1.222737
0.684200
1.947857
1.154349
1.980414
-0.718815
-1.221322
-0.682951
0.000499

$rem
JOBTYPE
AIMD_METHOD
METHOD
BASIS
ECP
PURECART
CIS_N_ROOTS
CIS_TRIPLETS
CIS_STATE_DERIV
AIMD_INIT_VELOC
AIMD_TEMP
TIME_STEP
AIMD_STEPS
$end

0.714962
1.216596
0.682456
1.942703
1.148346
1.980531
-0.723321
-1.226245
-0.682958
-1.950659
-1.158379
-1.978284
0.010220

aimd
bomd
hf
6-31+G*
SRLC
1111
3
false
1
thermal
150
25
827

propagate on first excited state

(500 fs)

Chapter 10: Exploring Potential Energy Surfaces: Critical Points and Molecular Dynamics

540

Example 10.23 Simulating vibrations of the NaCl molecule using ELMD.
$molecule
0 1
Na
0.000000
Cl
0.000000
$end
$rem
JOBTYPE
METHOD
ECP
$end

0.000000
0.000000

-1.742298
0.761479

freq
b3lyp
sbkjc

@@@
$molecule
read
$end
$rem
JOBTYPE
METHOD
ECP
TIME_STEP
AIMD_STEPS
AIMD_METHOD
AIMD_FICT_MASS
AIMD_INIT_VELOC
$end

10.7.3

aimd
b3lyp
sbkjc
14
500
curvy
360
zpe

Thermostats: Sampling the NVT Ensemble

Implicit in the discussion above was an assumption of conservation of energy, which implies dynamics run in the
microcanonical (N V E) ensemble. Alternatively, the AIMD code in Q-C HEM can sample the canonical (N V T )
ensemble with the aid of thermostats. These mimic the thermal effects of a surrounding temperature bath, and the time
average of a trajectory (or trajectories) then affords thermodynamic averages at a chosen temperature. This option is
appropriate in particular when multiple minima are thermally accessible. All sampled information is once again saved
in the AIMD/ subdirectory of the $QCSCRATCH directory for the job. Thermodynamic averages and error analysis
may be performed externally, using these data. Two commonly used thermostat options, both of which yield proper
canonical distributions of the classical molecular motion, are implemented in Q-C HEM and are described in more detail
below. Constant-pressure barostats (for N P T simulations) are not yet implemented.
As with any canonical sampling, the trajectory evolves at the mercy of barrier heights. Short trajectories will sample
only within the local minimum of the initial conditions, which may be desired for sampling the properties of a given
isomer, for example. Due to the energy fluctuations induced by the thermostat, the trajectory is neither guaranteed to
stay within this potential energy well nor guaranteed to overcome barriers to neighboring minima, except in the infinitesampling limit for the latter case, which is likely never reached in practice. Importantly, the user should note that the
introduction of a thermostat destroys the validity of any real-time trajectory information; thermostatted trajectories
should not be used to assess real-time dynamical observables, but only to compute thermodynamic averages.

10.7.3.1

Langevin Thermostat

A stochastic, white-noise Langevin thermostat (AIMD_THERMOSTAT = LANGEVIN) combines random “kicks” to the
nuclear momenta with a dissipative, friction term. The balance of these two contributions mimics the exchange of

Chapter 10: Exploring Potential Energy Surfaces: Critical Points and Molecular Dynamics

541

energy with a surrounding heat bath. The resulting trajectory, in the long-time sampling limit, generates the correct
canonical distribution. The implementation in Q-C HEM follows the velocity Verlet formulation of Bussi and Parrinello, 7 which remains a valid propagator for all time steps and thermostat parameters. The thermostat is coupled
to each degree of freedom in the simulated system. The MD integration time step (TIME_STEP) should be chosen in
the same manner as in an NVE trajectory. The only user-controllable parameter for this thermostat, therefore, is the
timescale over which the implied bath influences the trajectory. The AIMD_LANGEVIN_TIMESCALE keyword determines this parameter, in units of femtoseconds. For users who are more accustomed to thinking in terms of friction
strength, this parameter is proportional to the inverse friction. A small value of the timescale parameter yields a “tight”
thermostat, which strongly maintains the system at the chosen temperature but does not typically allow for rapid configurational flexibility. (Qualitatively, one may think of such simulations as sampling in molasses. This analogy, however,
only applies to the thermodynamic sampling properties and does not suggest any electronic role of the solvent!) These
small values are generally more appropriate for small systems, where the few degrees of freedom do not rapidly exchange energy and behave may behave in a non-ergodic fashion. Alternatively, large values of the time-scale parameter
allow for more flexible configurational sampling, with the tradeoff of more (short-term) deviation from the desired
average temperature. These larger values are more appropriate for larger systems since the inherent, microcanonical
exchange of energy within the large number of degrees of freedom already tends toward canonical properties. (Think of
this regime as sampling in a light, organic solvent.) Importantly, thermodynamic averages in the infinite-sampling limit
are completely independent of this time-scale parameter. Instead, the time scale merely controls the efficiency with
which the ensemble is explored. If maximum efficiency is desired, the user may externally compute lifetimes from the
time correlation function of the desired observable and minimize the lifetime as a function of this timescale parameter.
At the end of the trajectory, the average computed temperature is compared to the requested target temperature for
validation purposes.
Example 10.24 Canonical (N V T ) sampling using AIMD with the Langevin thermostat
$comment
Short example of using the Langevin thermostat
for canonical (NVT) sampling
$end
$molecule
0 1
H
O 1 1.0
H 2 1.0
$end

1

104.5

$rem
JOBTYPE
EXCHANGE
BASIS
AIMD_TIME_STEP
AIMD_STEPS
AIMD_THERMOSTAT
AIMD_INIT_VELOC
AIMD_TEMP
AIMD_LANGEVIN_TIMESCALE
$end

10.7.3.2

aimd
hf
sto-3g
20
!in au
100
langevin
thermal
298
!in K - initial conditions AND thermostat
100
!in fs

Nosé-Hoover Thermostat

An alternative thermostat approach is also available, namely, the Nosé-Hoover thermostat 34 (also known as a NoséHoover “chain”), which mimics the role of a surrounding thermal bath by performing a microcanonical (N V E) trajectory in an extended phase space. By allowing energy to be exchanged with a chain of fictitious particles that are coupled
to the target system, N V T sampling is properly obtained for those degrees of freedom that represent the real system.
(Only the target system properties are saved in $QCSCRATCH/AIMD for subsequent analysis and visualization, not

Chapter 10: Exploring Potential Energy Surfaces: Critical Points and Molecular Dynamics

542

the fictitious Nosé-Hoover degrees of freedom.) The implementation in Q-C HEM follows that of Martyna, 34 which
augments the original extended-Lagrangian approach of Nosé 37,38 and Hoover, 23 using a chain of auxiliary degrees of
freedom to restore ergodicity in stiff systems and thus afford the correct N V T ensemble. Unlike the Langevin thermostat, the collection of system and auxiliary chain particles can be propagated in a time-reversible fashion with no need
for stochastic perturbations.
Rather than directly setting the masses and force constants of the auxiliary chain particles, the Q-C HEM implementation
focuses instead, on the time scale of the thermostat, as was the case for the Langevin thermostat described above. The
time-scale parameter is controlled by the keyword NOSE_HOOVER_TIMESCALE, given in units of femtoseconds. The
only other user-controllable parameter for this function is the length of the Nosé-Hoover chain, which is typically
chosen to be 3–6 fictitious particles. Importantly, the version in Q-C HEM is currently implemented as a single chain
that is coupled to the system, as a whole. Comprehensive thermostatting in which every single degree of freedom
is coupled to its own thermostat, which is sometimes used for particularly stiff systems, is not implemented and for
such cases the Langevin thermostat is recommended instead. For large and/or fluxional systems, the single-chain
Nosé-Hoover approach is appropriate.
Example 10.25 Canonical (N V T ) sampling using AIMD with the Nosé-Hoover chain thermostat
$molecule
0 1
H
O 1 1.0
H 2 1.0 1 104.5
$end
$rem
JOBTYPE
EXCHANGE
BASIS
AIMD_TIME_STEP
AIMD_STEPS
AIMD_THERMOSTAT
AIMD_INIT_VELOC
AIMD_TEMP
NOSE_HOOVER_LENGTH
NOSE_HOOVER_TIMESCALE
$end

aimd
hf
sto-3g
20
!in au
100
nose_hoover
thermal
298
!in K - initial conditions AND thermostat
3
!chain length
100
!in fs

AIMD_THERMOSTAT
Applies thermostatting to AIMD trajectories.
TYPE:
INTEGER
DEFAULT:
none
OPTIONS:
LANGEVIN
Stochastic, white-noise Langevin thermostat
NOSE_HOOVER Time-reversible, Nosé-Hoovery chain thermostat
RECOMMENDATION:
Use either thermostat for sampling the canonical (NVT) ensemble.

Chapter 10: Exploring Potential Energy Surfaces: Critical Points and Molecular Dynamics

543

AIMD_LANGEVIN_TIMESCALE
Sets the timescale (strength) of the Langevin thermostat
TYPE:
INTEGER
DEFAULT:
none
OPTIONS:
n Thermostat timescale,asn n fs
RECOMMENDATION:
Smaller values (roughly 100) equate to tighter thermostats but may inhibit rapid sampling. Larger
values (≥ 1000) allow for more rapid sampling but may take longer to reach thermal equilibrium.
NOSE_HOOVER_LENGTH
Sets the chain length for the Nosé-Hoover thermostat
TYPE:
INTEGER
DEFAULT:
none
OPTIONS:
n Chain length of n auxiliary variables
RECOMMENDATION:
Typically 3-6
NOSE_HOOVER_TIMESCALE
Sets the timescale (strength) of the Nosé-Hoover thermostat
TYPE:
INTEGER
DEFAULT:
none
OPTIONS:
n Thermostat timescale, as n fs
RECOMMENDATION:
Smaller values (roughly 100) equate to tighter thermostats but may inhibit rapid sampling. Larger
values (≥ 1000) allow for more rapid sampling but may take longer to reach thermal equilibrium.

10.7.4

Vibrational Spectra

The inherent nuclear motion of molecules is experimentally observed by the molecules’ response to impinging radiation. This response is typically calculated within the mechanical and electrical harmonic approximations (second
derivative calculations) at critical-point structures. Spectra, including anharmonic effects, can also be obtained from
dynamics simulations. These spectra are generated from dynamical response functions, which involve the Fourier
transform of auto-correlation functions. Q-C HEM can provide both the vibrational spectral density from the velocity
auto-correlation function
Z ∞
D(ω) ∝
dt e−iωt h~v (0) · ~v (t)i
(10.17)
−∞

and infrared absorption intensity from the dipole auto-correlation function
Z ∞
ω
dt e−iωt h~
µ(0) · µ
~ (t)i
I(ω) ∝
2π −∞

(10.18)

These two features are activated by the AIMD_NUCL_VACF_POINTS and AIMD_NUCL_DACF_POINTS keywords, respectively, where values indicate the number of data points to include in the correlation function. Furthermore, the

Chapter 10: Exploring Potential Energy Surfaces: Critical Points and Molecular Dynamics

544

AIMD_NUCL_SAMPLE_RATE keyword controls the frequency at which these properties are sampled (entered as num-

ber of time steps). These spectra—generated at constant energy—should be averaged over a suitable distribution of
initial conditions. The averaging indicated in the expressions above, for example, should be performed over a Boltzmann distribution of initial conditions.
Note that dipole auto-correlation functions can exhibit contaminating information if the molecule is allowed to rotate/translate. While the initial conditions in Q-C HEM remove translation and rotation, numerical noise in the forces
and propagation can lead to translation and rotation over time. The trans/rot correction in Q-C HEM is activated by the
PROJ_TRANSROT keyword.
AIMD_NUCL_VACF_POINTS
Number of time points to use in the velocity auto-correlation function for an AIMD trajectory
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
0
Do not compute velocity auto-correlation function.
1 ≤ n ≤ AIMD_STEPS Compute velocity auto-correlation function for last n
time steps of the trajectory.
RECOMMENDATION:
If the VACF is desired, set equal to AIMD_STEPS.
AIMD_NUCL_DACF_POINTS
Number of time points to use in the dipole auto-correlation function for an AIMD trajectory
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
0
Do not compute dipole auto-correlation function.
1 ≤ n ≤ AIMD_STEPS Compute dipole auto-correlation function for last n
timesteps of the trajectory.
RECOMMENDATION:
If the DACF is desired, set equal to AIMD_STEPS.
AIMD_NUCL_SAMPLE_RATE
The rate at which sampling is performed for the velocity and/or dipole auto-correlation function(s). Specified as a multiple of steps; i.e., sampling every step is 1.
TYPE:
INTEGER
DEFAULT:
None.
OPTIONS:
1 ≤ n ≤ AIMD_STEPS Update the velocity/dipole auto-correlation function
every n steps.
RECOMMENDATION:
Since the velocity and dipole moment are routinely calculated for ab initio methods, this variable
should almost always be set to 1 when the VACF/DACF are desired.

Chapter 10: Exploring Potential Energy Surfaces: Critical Points and Molecular Dynamics

545

PROJ_TRANSROT
Removes translational and rotational drift during AIMD trajectories.
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
FALSE Do not apply translation/rotation corrections.
TRUE
Apply translation/rotation corrections.
RECOMMENDATION:
When computing spectra (see AIMD_NUCL_DACF_POINTS, for example), this option can be
used to remove artificial, contaminating peaks stemming from translational and/or rotational
motion. Recommend setting to TRUE for all dynamics-based spectral simulations.

10.7.5

Quasi-Classical Molecular Dynamics

So-called “quasi-classical” trajectories 26,44,45 (QCT) put vibrational energy into each mode in the initial velocity setup
step, which can improve on the results of purely classical simulations, for example in the calculation of photoelectron 29 or infrared spectra. 47 Improvements include better agreement of spectral linewidths with experiment at lower
temperatures and better agreement of vibrational frequencies with anharmonic calculations.
The improvements at low temperatures can be understood by recalling that even at low temperature there is nuclear motion due to zero-point motion. This is included in the quasi-classical initial velocities, thus leading to finite peak widths
even at low temperatures. In contrast to that the classical simulations yield zero peak width in the low temperature
limit, because the thermal kinetic energy goes to zero as temperature decreases. Likewise, even at room temperature
the quantum vibrational energy for high-frequency modes is often significantly larger than the classical kinetic energy.
QCT-MD therefore typically samples regions of the potential energy surface that are higher in energy and thus more
anharmonic than the low-energy regions accessible to classical simulations. These two effects can lead to improved
peak widths as well as a more realistic sampling of the anharmonic parts of the potential energy surface. However,
the QCT-MD method also has important limitations which are described below and that the user has to monitor for
carefully.
In our QCT-MD implementation the initial vibrational quantum numbers are generated as random numbers sampled
from a vibrational Boltzmann distribution at the desired simulation temperature. In order to enable reproducibility of
the results, each trajectory (and thus its set of vibrational quantum numbers) is denoted by a unique number using
the AIMD_QCT_WHICH_TRAJECTORY variable. In order to loop over different initial conditions, run trajectories with
different choices for AIMD_QCT_WHICH_TRAJECTORY. It is also possible to assign initial velocities corresponding to
an average over a certain number of trajectories by choosing a negative value. Further technical details of our QCT-MD
implementation are described in detail in Appendix A of Ref. 29.
AIMD_QCT_WHICH_TRAJECTORY
Picks a set of vibrational quantum numbers from a random distribution.
TYPE:
INTEGER
DEFAULT:
1
OPTIONS:
n
Picks the nth set of random initial velocities.
−n Uses an average over n random initial velocities.
RECOMMENDATION:
Pick a positive number if you want the initial velocities to correspond to a particular set of
vibrational occupation numbers and choose a different number for each of your trajectories. If
initial velocities are desired that corresponds to an average over n trajectories, pick a negative
number.

Chapter 10: Exploring Potential Energy Surfaces: Critical Points and Molecular Dynamics

546

Below is a simple example input for running a QCT-MD simulation of the vibrational spectrum of water. Most input
variables are the same as for classical MD as described above. The use of quasi-classical initial conditions is triggered
by setting the AIMD_INIT_VELOC variable to QUASICLASSICAL.
Example 10.26 Simulating the IR spectrum of water using QCT-MD.
$comment
Don’t forget to run a frequency calculation prior to this job.
$end
$molecule
0 1
O
0.000000
H
-1.475015
H
1.475015
$end

0.000000
0.000000
0.000000

$rem
JOBTYPE
INPUT_BOHR
METHOD
BASIS
SCF_CONVERGENCE
TIME_STEP
AIMD_STEPS
AIMD_TEMP
AIMD_PRINT
FOCK_EXTRAP_ORDER
FOCK_EXTRAP_POINTS
! IR SPECTRAL SAMPLING
AIMD_MOMENTS
AIMD_NUCL_SAMPLE_RATE
AIMD_NUCL_VACF_POINTS
! QCT-SPECIFIC SETTINGS
AIMD_INIT_VELOC
AIMD_QCT_WHICH_TRAJECTORY

0.520401
-0.557186
-0.557186

aimd
true
hf
3-21g
6
20 ! (in atomic units)
12500
6 ps total simulation time
12
2
6
! Use a 6th-order extrapolation
12 ! of the previous 12 Fock matrices
1
5
1000
quasiclassical
1
! Loop over several values to get
! the correct Boltzmann distribution.

$end

Other types of spectra can be calculated by calculating spectral properties along the trajectories. For example, we
observed that photoelectron spectra can be approximated quite well by calculating vertical detachment energies (VDEs)
along the trajectories and generating the spectrum as a histogram of the VDEs. 29 We have included several simple
scripts in the $QC/aimdman/tools subdirectory that we hope the user will find helpful and that may serve as the basis
for developing more sophisticated tools. For example, we include scripts that allow to perform calculations along a
trajectory (md_calculate_along_trajectory) or to calculate vertical detachment energies along a trajectory
(calculate_rel_energies).
Another application of the QCT code is to generate random geometries sampled from the vibrational wave function via a
Monte Carlo algorithm. This is triggered by setting both the AIMD_QCT_INITPOS and AIMD_QCT_WHICH_TRAJECTORY
variables to negative numbers, say −m and −n, and setting AIMD_STEPS to zero. This will generate m random geometries sampled from the vibrational wave function corresponding to an average over n trajectories at the user-specified
simulation temperature.

Chapter 10: Exploring Potential Energy Surfaces: Critical Points and Molecular Dynamics

547

AIMD_QCT_INITPOS
Chooses the initial geometry in a QCT-MD simulation.
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
0
Use the equilibrium geometry.
n
Picks a random geometry according to the harmonic vibrational wave function.
−n Generates n random geometries sampled from
the harmonic vibrational wave function.
RECOMMENDATION:
None.

For systems that are described well within the harmonic oscillator model and for properties that rely mainly on the
ground-state dynamics, this simple MC approach may yield qualitatively correct spectra. In fact, one may argue that it
is preferable over QCT-MD for describing vibrational effects at very low temperatures, since the geometries are sampled
from a true quantum distribution (as opposed to classical and quasi-classical MD). We have included another script in
the $QC/aimdman/tools directory to help with the calculation of vibrationally averaged properties (monte_geom).
Example 10.27 MC sampling of the vibrational wave function for HCl.
$comment
Generates 1000 random geometries for HCl based on the harmonic vibrational
wave function at 1 Kelvin. The wave function is averaged over 1000
sets of random vibrational quantum numbers (\ie{}, the ground state in
this case due to the low temperature).
$end
$molecule
0 1
H
0.000000
Cl
0.000000
$end

0.000000
0.000000

-1.216166
0.071539

$rem
JOBTYPE
aimd
METHOD
B3LYP
BASIS
6-311++G**
SCF_CONVERGENCE
1
SKIP_SCFMAN
1
MAX_SCF_CYCLES
0
XC_GRID
1
TIME_STEP
20
(in atomic units)
AIMD_STEPS
0
AIMD_INIT_VELOC
quasiclassical
AIMD_QCT_VIBSEED
1
AIMD_QCT_VELSEED
2
AIMD_TEMP
1
(in Kelvin)
! set aimd_qct_which_trajectory to the desired
! trajectory number
AIMD_QCT_WHICH_TRAJECTORY -1000
AIMD_QCT_INITPOS
-1000
$end

It is also possible make some modes inactive, i.e., to put vibrational energy into a subset of modes (all other are set
to zero). The list of active modes can be specified using the $qct_active_modes section. Furthermore, the vibrational
quantum numbers for each mode can be specified explicitly using the $qct_vib_distribution input section. It is also

Chapter 10: Exploring Potential Energy Surfaces: Critical Points and Molecular Dynamics

548

possible to set the phases using $qct_vib_phase (allowed values are 1 and −1). Below is a simple sample input:
Example 10.28 User control over the QCT variables.
$comment
Makes the 1st vibrational mode QCT-active; all other ones receive zero
kinetic energy. We choose the vibrational ground state and a positive
phase for the velocity.
$end
...
$qct_active_modes
1
$end
$qct_vib_distribution
0
$end
$qct_vib_phase
1
$end
...

Finally we turn to a brief description of the limitations of QCT-MD. Perhaps the most severe limitation stems from
the so-called “kinetic energy spilling” problem, 9 which means that there can be an artificial transfer of kinetic energy
between modes. This can happen because the initial velocities are chosen according to quantum energy levels, which
are usually much higher than those of the corresponding classical systems. Furthermore, the classical equations of
motion also allow for the transfer of non-integer multiples of the zero-point energy between the modes, which leads
to different selection rules for the transfer of kinetic energy. Typically, energy spills from high-energy into low-energy
modes, leading to spurious “hot” dynamics. A second problem is that QCT-MD is actually based on classical Newtonian
dynamics, which means that the probability distribution at low temperatures can be qualitatively wrong compared to
the true quantum distribution. 29
Q-C HEM implements a routine to monitor the kinetic energy within each normal mode along the trajectory and that
is automatically switched on for quasi-classical simulations. It is thus possible to monitor for trajectories in which the
kinetic energy in one or more modes becomes significantly larger than the initial energy. Such trajectories should be
discarded. (Alternatively, see Ref. 9 for a different approach to the zero-point leakage problem.) Furthermore, this
monitoring routine prints the squares of the (harmonic) vibrational wave function along the trajectory. This makes it
possible to weight low-temperature results with the harmonic quantum distribution to alleviate the failure of classical
dynamics for low temperatures.

10.7.6

Fewest-Switches Surface Hopping

As discussed in Section 10.6, optimization of minimum-energy crossing points (MECPs) along conical seams, followed
by optimization of minimum-energy pathways that connect these MECPs to other points of interest on ground- and
excited-state potential energy surfaces, affords an appealing one-dimensional picture of photochemical reactivity that
is analogous to the “reactant → transition state → product” picture of ground-state chemistry. Just as the ground-state
reaction is not obligated to proceed exactly through the transition-state geometry, however, an excited-state reaction
need not proceed precisely through the MECP and the particulars of nuclear kinetic energy can lead to deviations. This
is arguably more of an issue for excited-state reactions, where the existence of multiple conical intersections can easily
lead to multiple potential reaction mechanisms. AIMD potentially offers a way to sample over the available mechanisms
in order to deduce which ones are important in an automated way, but must be extended in the photochemical case to
reactions that involve more than one Born-Oppenheimer potential energy surface.

Chapter 10: Exploring Potential Energy Surfaces: Critical Points and Molecular Dynamics

549

The most widely-used trajectory-based method for non-adiabatic simulations is Tully’s “fewest-switches” surfacehopping (FSSH) algorithm. 14,57 In this approach, classical trajectories are propagated on a single potential energy
surface, but can undergo “hops” to a different potential surface in regions of near-degeneracy between surfaces. The
probability of these stochastic hops is governed by the magnitude of the non-adiabatic coupling [Eq. (10.4)]. Considering the ensemble average of a swarm of trajectories then provides information about, e.g., branching ratios for
photochemical reactions.
The FSSH algorithm, based on the AIMD code, is available in Q-C HEM for any electronic structure method where
analytic derivative couplings are available, which at present means CIS, TDDFT, and their spin-flip analogues (see
Section 10.6.1). The nuclear dynamics component of the simulation is specified just as in an AIMD calculation. Artificial decoherence can be added to the calculation at additional cost according to the augmented FSSH (AFSSH)
method, 30,55,56 which enforces stochastic wave function collapse at a rate proportional to the difference in forces between the trajectory on the active surface and position moments propagated the other surfaces. At every time step, the
component of the wave function on each active surface is printed to the output file. These amplitudes, as well as the
position and momentum moments (if AFSSH is requested), is also printed to a text file called SurfaceHopper located
in the $QC/AIMD sub-directory of the job’s scratch directory.
In order to request a FSSH calculation, only a few additional $rem variables must be added to those necessary for
an excited-state AIMD simulation. At present, FSSH calculations can only be performed using Born-Oppenheimer
molecular dynamics (BOMD) method. Furthermore, the optimized velocity Verlet (OVV) integration method is not
supported for FSSH calculations.
FSSH_LOWESTSURFACE
Specifies the lowest-energy state considered in a surface hopping calculation.
TYPE:
INTEGER
DEFAULT:
None
OPTIONS:
n Only states n and above are considered in a FSSH calculation.
RECOMMENDATION:
None
FSSH_NSURFACES
Specifies the number of states considered in a surface hopping calculation.
TYPE:
INTEGER
DEFAULT:
None
OPTIONS:
n n states are considered in the surface hopping calculation.
RECOMMENDATION:
Any states which may come close in energy to the active surface should be included in the surface
hopping calculation.

550

Chapter 10: Exploring Potential Energy Surfaces: Critical Points and Molecular Dynamics
FSSH_INITIALSURFACE
Specifies the initial state in a surface hopping calculation.
TYPE:
INTEGER
DEFAULT:
None
OPTIONS:
n An integer between FSSH_LOWESTSURFACE
FSSH_NSURFACES −1.
RECOMMENDATION:
None

and

FSSH_LOWESTSURFACE

AFSSH
Adds decoherence approximation to surface hopping calculation.
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
0 Traditional surface hopping, no decoherence.
1 Use augmented fewest-switches surface hopping (AFSSH).
RECOMMENDATION:
AFSSH will increase the cost of the calculation, but may improve accuracy for some systems.
See Refs. 30,55,56 for more detail.
AIMD_SHORT_TIME_STEP
Specifies a shorter electronic time step for FSSH calculations.
TYPE:
INTEGER
DEFAULT:
TIME_STEP

OPTIONS:
n Specify an electronic time step duration of n/AIMD_TIME_STEP_CONVERSION
a.u. If n is less than the nuclear time step variable TIME_STEP, the
electronic wave function will be integrated multiple times per nuclear time step,
using a linear interpolation of nuclear quantities such as the energy gradient and
derivative coupling. Note that n must divide TIME_STEP evenly.
RECOMMENDATION:
Make AIMD_SHORT_TIME_STEP as large as possible while keeping the trace of the density matrix close to unity during long simulations. Note that while specifying an appropriate duration
for the electronic time step is essential for maintaining accurate wave function time evolution,
the electronic-only time steps employ linear interpolation to estimate important quantities. Consequently, a short electronic time step is not a substitute for a reasonable nuclear time step.

+

Chapter 10: Exploring Potential Energy Surfaces: Critical Points and Molecular Dynamics
FSSH_CONTINUE
Restart a FSSH calculation from a previous run, using the file 396.0. When this is enabled,
the initial conditions of the surface hopping calculation will be set, including the correct wave
function amplitudes, initial surface, and position/momentum moments (if AFSSH) from the final
step of some prior calculation.
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
0 Start fresh calculation.
1 Restart from previous run.
RECOMMENDATION:
None

551

Chapter 10: Exploring Potential Energy Surfaces: Critical Points and Molecular Dynamics

552

Example 10.29 FSSH simulation. Note that analytic derivative couplings must be requested via CALC_NAC, but it is
unnecessary to include a $derivative_coupling section. The same is true for SET_STATE_DERIV, which will be set to
the initial active surface automatically. Finally, one must be careful to choose a small enough time step for systems that
have energetic access to a region of large derivative coupling, hence the choice for AIMD_TIME_STEP_CONVERSION
and TIME_STEP.
$molecule
0 1
C
-1.620294
C
-0.399206
C
-0.105193
H
-0.789110
C
1.069016
H
1.292495
C
1.956240
H
2.859680
C
1.666259
H
2.348104
C
0.495542
H
0.253287
O
-1.931045
H
-2.269528
$end

0.348677
-0.437493
-1.296810
-1.374693
-2.045054
-2.701157
-1.940324
-2.517019
-1.084065
-1.005765
-0.335701
0.325843
1.124872
0.227813

$rem
JOBTYPE
EXCHANGE
BASIS
CIS_N_ROOTS
SYMMETRY
SYM_IGNORE
CIS_SINGLETS
CIS_TRIPLETS
PROJ_TRANSROT
FSSH_LOWESTSURFACE
FSSH_NSURFACES
FSSH_INITIALSURFACE
AFSSH
CALC_NAC
AIMD_STEPS
TIME_STEP
AIMD_SHORT_TIME_STEP
AIMD_TIME_STEP_CONVERSION
AIMD_PRINT
AIMD_INIT_VELOC
AIMD_TEMP
AIMD_THERMOSTAT
AIMD_INTEGRATION
FOCK_EXTRAP_ORDER
FOCK_EXTRAP_POINTS
$end

-0.008838
-0.012535
-1.081340
-1.905080
-1.072304
-1.889686
0.002842
0.008420
1.071007
1.894140
1.065497
1.871866
0.911738
-0.865645

aimd
hf
3-21g
2
off
true
false
true
true
1
2 ! hop between T1 and T2
1 ! start on T1
0 ! no decoherence
true
500
14
2
1 ! Do not alter time_step duration
1
thermal
300 # K
4 # Langevin
vverlet
6
12

Chapter 10: Exploring Potential Energy Surfaces: Critical Points and Molecular Dynamics

553

10.8 Ab initio Path Integrals
10.8.1

Theory

Even in cases where the Born-Oppenheimer separation is valid, solving the electronic Schrödinger equation may only
be half the battle. The remainder involves the solution of the nuclear Schrödinger equation for its resulting eigenvalues
and eigenfunctions. This half is typically treated by the harmonic approximation at critical points, but anharmonicity,
tunneling, and low-frequency (“floppy”) motions can lead to extremely delocalized nuclear distributions, particularly
for protons and for non-covalent interactions.
While the Born-Oppenheimer separation allows for a local solution of the electronic problem (in nuclear space), the
nuclear half of the Schrödinger equation is entirely non-local and requires the computation of potential energy surfaces
over large regions of configuration space. Grid-based methods, therefore, scale exponentially with the number of
degrees of freedom, and are quickly rendered useless for all but very small molecules.
For equilibrium thermal distributions, the path integral (PI) formalism provides both an elegant and computationally
feasible alternative. The equilibrium partition function can be written as a trace of the thermal, configuration-space
density matrix,
Z
Z


(10.19)
Z = tr e−β Ĥ = dx x e−β Ĥ x = dx ρ(x, x; β) .
The density matrix at inverse temperature β = (kB T )−1 is defined by the last equality. Evaluating the integrals in
Eq. (10.19) still requires computing eigenstates of Ĥ, which is generally intractable. Inserting N − 1 resolutions of the
identity, however, one obtains

 



Z
Z
Z
β
β
β
Z = dx1 dx2 · · · dxN ρ x1 , x2 ;
ρ x2 , x3 ;
· · · ρ xN , x1 ;
.
(10.20)
N
N
N
Here, the density matrices appear at an inverse temperature β/N that corresponds to multiplying the actual temperature
T by a factor of N .
The high-temperature form of the density matrix can be expressed as

 
1/2
 



i
β
mN
mN
β h
0
0 2
ρ x, x0 ;
V
(x)
+
V
(x
)
=
exp
−
(x
−
x
)
−
N
2πβ~2
2β~2
2N

(10.21)

which becomes exact as T → ∞ (a limit in which quantum mechanics converges to classical mechanics), or in other
words as β → 0 or N → ∞. Using N time slices, the partition function is therefore converted into the form
(
"
#)

N/2 Z
Z
Z
N
N
X
mN
β mN 2 X
2
Z=
dx1 dx2 · · · dxN exp −
(xi − xi+1 ) +
V (xi )
,
(10.22)
2πβ~2
N 2β 2 ~2 i=1
i=1
with the implied cyclic condition xN +1 ≡ x1 . Here, V (x) is the potential function on which the “beads” move, which
is the electronic potential generated by Q-C HEM.
Equation 10.22 has the form
Z
Z∝

e−βVeff ,

(10.23)

where the form of the effective potential Veff is evident from the integrand in Eq. (10.22). Equation (10.23) reveals
that the path-integral formulation of the quantum partition function affords a classical configurational integral for the
partition function, albeit in an extended-dimensional space The effective potential describes a classical “ring polymer”
with N beads, wherein neighboring beads are coupled by harmonic potentials that arise from the quantum nature
of the kinetic energy. The exponentially-scaling, non-local nuclear quantum mechanics problem has therefore been
mapped onto an entirely classical problem, which is amenable to standard treatments of configuration sampling. These
methods typically involve molecular dynamics or Monte Carlo sampling. Importantly, the number of extended degrees
of freedom, N , is reasonably small when the temperature is not too low: room-temperature systems involving hydrogen
atoms typically are converged using roughly N ≈ 30 beads. Therefore, fully quantum-mechanical nuclear distributions

Chapter 10: Exploring Potential Energy Surfaces: Critical Points and Molecular Dynamics

554

can be obtained at a cost only roughly 30 times that of a classical AIMD simulation. Path integral Monte Carlo (PIMC)
is activated by setting JOBTYPE = PIMC.
The single-bead (N = 1) limit of the equations above is simply classical configuration sampling. When the temperature
(controlled by the PIMC_TEMP keyword) is high, or where only heavy atoms are involved, the classical limit is often
appropriate. The path integral machinery (with a single “bead”) may be used to perform classical Boltzmann sampling.
In this case, the partition function is simply
Z
Z=

dx e−βV (x)

(10.24)

and this is what is ordinarily done in an AIMD simulation. Use of additional beads incorporates more quantummechanical delocalization, at a cost of roughly N times that of the classical AIMD simulation, and this is the primary
input variable in a PI simulation. It is controlled by the keyword PIMC_NBEADSPERATOM. The ratio of the inverse
temperature to beads (β/N ) dictates convergence with respect to the number of beads, so as the temperature is lowered,
a concomitant increase in the number of beads is required.
Integration over configuration space is performed by Metropolis Monte Carlo (MC). The number of MC steps is controlled by the PIMC_MCMAX keyword and should typically be & 105 , depending on the desired level of statistical
convergence. A warm-up run, in which the PI ring polymer is allowed to equilibrate without accumulating statistics,
can be performed using the PIMC_WARMUP_MCMAX keyword.
As in AIMD simulations, the main results of PIMC jobs in Q-C HEM are not in the job output file but are instead
output to ($QCSCRATCH/PIMC in the user’s scratch directory, thus PIMC jobs should always be run with the -save
option. The output files do contain some useful information, however, including a basic data analysis of the simulation.
Average energies (thermodynamic estimator), bond lengths (less than 5 Å), bond length standard deviations and errors
are printed at the end of the output file. The $QCSCRATCH/PIMC directory additionally contains the following files:
• BondAves: running average of bond lengths for convergence testing.
• BondBins: normalized distribution of significant bond lengths, binned within 5 standard deviations of the average
bond length.
• ChainCarts: human-readable file of configuration coordinates, likely to be used for further, external statistical
analysis. This file can get quite large, so be sure to provide enough scratch space!
• ChainView.xyz: Cartesian-formatted file for viewing the ring-polymer sampling in an external visualization program. (The sampling is performed such that the center of mass of the ring polymer system remains centered.)
• Vcorr: potential correlation function for the assessment of statistical correlations in the sampling.
In each of the above files, the first few lines contain a description of how the data are arranged.
One of the unfortunate rites of passage in PIMC usage is the realization of the ramifications of the stiff bead-bead interactions as convergence (with respect to N ) is approached. Nearing convergence—where quantum mechanical results
are correct—the length of statistical correlations grows enormously, and special sampling techniques are required to
avoid long (or non-convergent) simulations. Cartesian displacements or normal-mode displacements of the ring polymer lead to this severe stiffening. While both of these naive sampling schemes are available in Q-C HEM, they are
not recommended. Rather, the free-particle (harmonic bead-coupling) terms in the path integral action can be sampled
directly. Several schemes are available for this purpose. Q-C HEM currently adopts the simplest of these options, Levy
flights. An n-bead segment (with n < N ) of the ring polymer is chosen at random, with the length n controlled by
the PIMC_SNIP_LENGTH keyword. Between the endpoints of this segment, a free-particle path is generated by a Levy
construction, which exactly samples the free-particle part of the action. Subsequent Metropolis testing of the resulting
potential term—for which only the potential on the moved beads is required—then dictates acceptance.
Two measures of the sampling efficiency are provided in the job output file. The lifetime of the potential autocorrelation function hV0 Vτ i is provided in terms of the number of MC steps, τ . This number indicates the number
of configurations that are statically correlated. Similarly, the mean-square displacement between MC configurations is
also provided. Maximizing this number and/or minimizing the statistical lifetime leads to efficient sampling. Note that

Chapter 10: Exploring Potential Energy Surfaces: Critical Points and Molecular Dynamics

555

the optimally efficient acceptance rate may not be 50% in MC simulations. In Levy flights, the only variable controlling
acceptance and sampling efficiency is the length of the snippet. The statistical efficiency can be obtained from relatively
short runs, during which the length of the Levy snippet should be optimized by the user.

10.8.2

Job Control and Examples
PIMC_NBEADSPERATOM
Number of path integral time slices (“beads”) used on each atom of a PIMC simulation.
TYPE:
INTEGER
DEFAULT:
None.
OPTIONS:
1
Perform classical Boltzmann sampling.
>1 Perform quantum-mechanical path integral sampling.
RECOMMENDATION:
This variable controls the inherent convergence of the path integral simulation. The onebead limit represents classical sampling and the infinite-bead limit represents exact quantummechanical sampling. Using 32 beads is reasonably converged for room-temperature simulations
of molecular systems.
PIMC_TEMP
Temperature, in Kelvin (K), of path integral simulations.
TYPE:
INTEGER
DEFAULT:
None.
OPTIONS:
User-specified number of Kelvin for PIMC or classical MC simulations.
RECOMMENDATION:
None.
PIMC_MCMAX
Number of Monte Carlo steps to sample.
TYPE:
INTEGER
DEFAULT:
None.
OPTIONS:
User-specified number of steps to sample.
RECOMMENDATION:
This variable dictates the statistical convergence of MC/PIMC simulations. For converged simulations at least 105 steps is recommended.

Chapter 10: Exploring Potential Energy Surfaces: Critical Points and Molecular Dynamics
PIMC_WARMUP_MCMAX
Number of Monte Carlo steps to sample during an equilibration period of MC/PIMC simulations.
TYPE:
INTEGER
DEFAULT:
None.
OPTIONS:
User-specified number of steps to sample.
RECOMMENDATION:
Use this variable to equilibrate the molecule/ring polymer before collecting production statistics.
Usually a short run of roughly 10% of PIMC_MCMAX is sufficient.
PIMC_MOVETYPE
Selects the type of displacements used in MC/PIMC simulations.
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
0 Cartesian displacements of all beads, with occasional (1%) center-of-mass moves.
1 Normal-mode displacements of all modes, with occasional (1%) center-of-mass moves.
2 Levy flights without center-of-mass moves.
RECOMMENDATION:
Except for classical sampling (MC) or small bead-number quantum sampling (PIMC),
Levy flights should be used.
For Cartesian and normal-mode moves, the maximum
displacement is adjusted during the warm-up run to the desired acceptance rate (controlled by PIMC_ACCEPT_RATE). For Levy flights, the acceptance is solely controlled by
PIMC_SNIP_LENGTH.
PIMC_ACCEPT_RATE
Acceptance rate for MC/PIMC simulations when Cartesian or normal-mode displacements are
used.
TYPE:
INTEGER
DEFAULT:
None
OPTIONS:
0 < n < 100 User-specified rate, given as a whole-number percentage.
RECOMMENDATION:
Choose acceptance rate to maximize sampling efficiency, which is typically signified by the
mean-square displacement (printed in the job output). Note that the maximum displacement is
adjusted during the warm-up run to achieve roughly this acceptance rate.

556

Chapter 10: Exploring Potential Energy Surfaces: Critical Points and Molecular Dynamics

557

PIMC_SNIP_LENGTH
Number of “beads” to use in the Levy flight movement of the ring polymer.
TYPE:
INTEGER
DEFAULT:
None
OPTIONS:
3 ≤ n ≤ PIMC_NBEADSPERATOM User-specified length of snippet.
RECOMMENDATION:
Choose the snip length to maximize sampling efficiency. The efficiency can be estimated by the
mean-square displacement between configurations, printed at the end of the output file. This efficiency will typically, however, be a trade-off between the mean-square displacement (length of
statistical correlations) and the number of beads moved. Only the moved beads require recomputing the potential, i.e., a call to Q-C HEM for the electronic energy. (Note that the endpoints
of the snippet remain fixed during a single move, so n − 2 beads are actually moved for a snip
length of n. For 1 or 2 beads in the simulation, Cartesian moves should be used instead.)

Example 10.30 Path integral Monte Carlo simulation of H2 at room temperature
$molecule
0 1
H
H 1 0.75
$end
$rem
JOBTYPE
METHOD
BASIS
PIMC_TEMP
PIMC_NBEADSPERATOM
PIMC_WARMUP_MCMAX
PIMC_MCMAX
PIMC_MOVETYPE
PIMC_SNIP_LENGTH
$end

pimc
hf
sto-3g
298
32
10000
100000
2
10

!Equilibration run
!Production run
!Levy flights
!Moves 8 beads per MC step (10-endpts)

Example 10.31 Classical Monte Carlo simulation of a water molecule at 500K
$molecule
0 1
H
O 1 1.0
H 2 1.0 1 104.5
$end
$rem
JOBTYPE
METHOD
BASIS
AUX_BASIS
PIMC_TEMP
PIMC_NBEADSPERATOM
PIMC_WARMUP_MCMAX
PIMC_MCMAX
PIMC_MOVETYPE
PIMC_ACCEPT_RATE
$end

pimc
rimp2
cc-pvdz
rimp2-cc-pvdz
500
1
!1 bead is classical sampling
10000
!Equilibration run
100000 !Production run
0
!Cartesian displacements (ok for 1 bead)
40
!During warm-up, adjusts step size to 40% acceptance

558

Chapter 10: Exploring Potential Energy Surfaces: Critical Points and Molecular Dynamics

References and Further Reading
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Chapter 10: Exploring Potential Energy Surfaces: Critical Points and Molecular Dynamics
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Chapter 10: Exploring Potential Energy Surfaces: Critical Points and Molecular Dynamics
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560

Chapter 11

Molecular Properties and Analysis
11.1

Introduction

Q-C HEM has incorporated a number of molecular properties and wave function analysis tools:
• Population analysis for ground and excited states
• Multipole moments for ground and excited states
• Extended excited-state analysis using reduced density matrices
• Calculation of molecular intracules
• Vibrational analysis (including isotopic substitution)
• Interface to the Natural Bond Orbital (NBO) package
• Molecular orbital symmetries
• Orbital localization
• Localized orbital bonding analysis
• Data generation for one- or two-dimensional plots
• Orbital visualization using the M OL D EN and M AC M OL P LT programs
• Natural transition orbitals for excited states
• NMR shielding tensors and chemical shifts
• Molecular junctions
In addition, Chapter 13 describes energy decomposition analysis using the fragment-based absolutely-localized molecular orbital approach.

11.2

Wave Function Analysis

Q-C HEM performs a number of standard wave function analyses by default. Switching the $rem variable WAVEFUNCTION_ANALYSIS
to FALSE will prevent the calculation of all wave function analysis features (described in this section). Alternatively,
each wave function analysis feature may be controlled by its $rem variable. (The NBO program, which is interfaced
with Q-C HEM, is capable of performing more sophisticated analyses. See Section 11.3 of this manual, along with the
NBO manual, for more details.

562

Chapter 11: Molecular Properties and Analysis
WAVEFUNCTION_ANALYSIS
Controls the running of the default wave function analysis tasks.
TYPE:
LOGICAL
DEFAULT:
TRUE
OPTIONS:
TRUE
Perform default wave function analysis.
FALSE Do not perform default wave function analysis.
RECOMMENDATION:
None

Note: WAVEFUNCTION_ANALYSIS has no effect on NBO, solvent models or vibrational analyses.

11.2.1

Population Analysis

The one-electron charge density, usually written as
ρ(r) =

X

Pµν φµ (r)φν (r)

(11.1)

µν

represents the probability of finding an electron at the point r, but implies little regarding the number of electrons
associated with a given nucleus in a molecule. However, since the number of electrons N is related to the occupied
orbitals ψi by
N/2
X
2
N =2
ψa (r)
(11.2)
a

We can substitute the atomic orbital (AO) basis expansion of ψa into Eq. (11.2) to obtain
X
X
N=
Pµν Sµν =
(PS)µµ = Tr(PS)
µν

(11.3)

µ

where we interpret (PS)µµ as the number of electrons associated with φµ . If the basis functions are atom-centered, the
number of electrons associated with a given atom can be obtained by summing over all the basis functions. This leads
to the Mulliken formula for the net charge of the atom A:
X
qA = ZA −
(PS)µµ
(11.4)
µ∈A

where ZA is the atom’s nuclear charge. This is called a Mulliken population analysis. 141 Q-C HEM performs a Mulliken
population analysis by default.
POP_MULLIKEN
Controls running of Mulliken population analysis.
TYPE:
LOGICAL/INTEGER
DEFAULT:
TRUE (or 1)
OPTIONS:
FALSE (or 0) Do not calculate Mulliken populations.
TRUE (or 1)
Calculate Mulliken populations.
2
Also calculate shell populations for each occupied orbital.
−1
Calculate Mulliken charges for both the ground state and any CIS,
RPA, or TDDFT excited states.
RECOMMENDATION:
Leave as TRUE, unless excited-state charges are desired. Mulliken analysis is a trivial additional
calculation, for ground or excited states.

Chapter 11: Molecular Properties and Analysis

563

LOWDIN_POPULATION
Run Löwdin population analysis.
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
FALSE Do not calculate Löwdin populations.
TRUE
Run Löwdin population analysis.
RECOMMENDATION:
None

Although conceptually simple, Mulliken population analyses suffer from a heavy dependence on the basis set used,
as well as the possibility of producing unphysical negative numbers of electrons. An alternative is that of Löwdin
population analysis, 88 which uses the Löwdin symmetrically orthogonalized basis set (which is still atom-tagged) to
assign the electron density. This shows a reduced basis set dependence, but maintains the same essential features.
While Mulliken and Löwdin population analyses are commonly employed, and can be used to produce information
about changes in electron density and also localized spin polarizations, they should not be interpreted as oxidation
states of the atoms in the system. For such information we would recommend a bonding analysis technique (LOBA or
NBO).
A more stable alternative to Mulliken or Löwdin charges are the so-called “ChElPG” charges (“Charges from the
Electrostatic Potential on a Grid”). 25 These are the atom-centered charges that provide the best fit to the molecular
electrostatic potential, evaluated on a real-space grid outside of the van der Waals region and subject to the constraint
that the sum of the ChElPG charges must equal the molecular charge. Q-C HEM’s implementation of the ChElPG
algorithm differs slightly from the one originally algorithm described by Breneman and Wiberg, 25 in that Q-C HEM
weights the grid points with a smoothing function to ensure that the ChElPG charges vary continuously as the nuclei
are displaced. 60 (For any particular geometry, however, numerical values of the charges are quite similar to those
obtained using the original algorithm.) Note, however, that the Breneman-Wiberg approach uses a Cartesian grid and
becomes expensive for large systems. (It is especially expensive when ChElPG charges are used in QM/MM-Ewald
calculations. 64 ) For that reason, an alternative procedure based on atom-centered Lebedev grids is also available, 64
which provides very similar charges using far fewer grid points. In order to use the Lebedev grid implementation the
$rem variables CHELPG_H and CHELPG_HA must be set, which specify the number of Lebedev grid points for the
hydrogen atoms and the heavy atoms, respectively.
CHELPG
Controls the calculation of CHELPG charges.
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
FALSE Do not calculate ChElPG charges.
TRUE
Compute ChElPG charges.
RECOMMENDATION:
Set to TRUE if desired. For large molecules, there is some overhead associated with computing
ChElPG charges, especially if the number of grid points is large.

Chapter 11: Molecular Properties and Analysis

564

CHELPG_HEAD
Sets the “head space” 25 (radial extent) of the ChElPG grid.
TYPE:
INTEGER
DEFAULT:
30
OPTIONS:
N Corresponding to a head space of N/10, in Å.
RECOMMENDATION:
Use the default, which is the value recommended by Breneman and Wiberg. 25
CHELPG_DX
Sets the rectangular grid spacing for the traditional Cartesian ChElPG grid or the spacing between
concentric Lebedev shells (when the variables CHELPG_HA and CHELPG_H are specified as
well).
TYPE:
INTEGER
DEFAULT:
6
OPTIONS:
N Corresponding to a grid space of N/20, in Å.
RECOMMENDATION:
Use the default, which corresponds to the “dense grid” of Breneman and Wiberg, 25 , unless the
cost is prohibitive, in which case a larger value can be selected. Note that this default value is set
with the Cartesian grid in mind and not the Lebedev grid. In the Lebedev case, a larger value can
typically be used.
CHELPG_H
Sets the Lebedev grid to use for hydrogen atoms.
TYPE:
INTEGER
DEFAULT:
NONE
OPTIONS:
N Corresponding to a number of points in a Lebedev grid.
RECOMMENDATION:
CHELPG_H must always be less than or equal to CHELPG_HA. If it is greater, it will automatically be set to the value of CHELPG_HA.
CHELPG_HA
Sets the Lebedev grid to use for non-hydrogen atoms.
TYPE:
INTEGER
DEFAULT:
NONE
OPTIONS:
N Corresponding to a number of points in a Lebedev grid (see Section 5.5.1.
RECOMMENDATION:
None.

Finally, Hirshfeld population analysis 63 provides yet another definition of atomic charges in molecules via a Stock-

565

Chapter 11: Molecular Properties and Analysis
holder prescription. The charge on atom A, qA , is defined by
Z
ρ0 (r)
ρ(r),
qA = ZA − dr P A 0
B ρB (r)

(11.5)

where ZA is the nuclear charge of A, ρ0B is the isolated ground-state atomic density of atom B, and ρ is the molecular
density. The sum goes over all atoms in the molecule. Thus computing Hirschfeld charges requires a self-consistent
calculation of the isolated atomic densities (the promolecule) as well as the total molecule. Following SCF completion,
the Hirschfeld analysis proceeds by running another SCF calculation to obtain the atomic densities. Next numerical
quadrature is used to evaluate the integral in Eq. (11.5). Neutral ground-state atoms are used, and the choice of
appropriate reference for a charged molecule is ambiguous (such jobs will crash). As numerical integration (with
default quadrature grid) is used, charges may not sum precisely to zero. A larger XC_GRID may be used to improve the
accuracy of the integration.
The charges (and corresponding molecular dipole moments) obtained using Hirschfeld charges are typically underestimated as compared to other charge schemes or experimental data. To correct this, Marenich et al. introduced “Charge
Model 5” (CM5), 89 which employs a single set of parameters to map the Hirschfeld charges onto a more reasonable
representation of the electrostatic potential. CM5 charges generally lead to more accurate dipole moments as compared
to the original Hirschfeld charges, at negligible additional cost. CM5 is available for molecules composed of elements
H–Ca, Zn, Ge–Br, and I.
The use of neutral ground-state atoms to define the promolecular density in Hirshfeld scheme has no strict theoretical
basis and there is no unique way to construct the promolecular densities. For example, Li0 F0 , Li+ F− , or Li− F+
could each be used to construct the promolecular densities for LiF. Furthermore, the choice of appropriate reference
for a charged molecule is ambiguous, and for this reason Hirshfeld analysis is disable in Q-C HEM for any molecule
with a net charge. A solution for charged molecules is to use the iterative “Hirshfeld-I” partitioning scheme proposed
by Bultinck et al., 28,146 in which the reference state is not predefined but rather determined self-consistently, thus
eliminating the arbitrariness. The final self-consistent reference state for Hirshfeld-I partitioning usually consists of
non-integer atomic populations.
In the first iteration, the Hirshfeld-I method uses neutral atomic densities (as in the original Hirshfeld scheme), ρ0i (r)
R
with electronic population Ni0 = dr ρ0i (r) = Zi . This affords charges
!
Z
ρ0i (r)
1
qi = Zi − dr PA
ρ(r) = Zi − Ni1
(11.6)
0
i ρi (r)
on the first iteration. The new electronic population (number of electrons) for atom i is Ni1 , and is derived from the
R
promolecular populations Ni0 . One then computes new isolated atomic densities with Ni1 = dr ρ1i (r1 ) and uses them
to construct the promolecular densities in the next iteration. In general, the new weighting function for atom i in the
kth iteration is
ρk−1 (r)
k
wi,HI
(r) = P i k−1
.
(11.7)
ρi (r)
i∈A

The atomic densities ρki (r) with corresponding fractional
0,bN k c
0,dN k e
between ρi i (r) and ρi i (r) of the same atom: 28,43

electron numbers Nik are obtained by linear interpolation


 0,bN k c

 0,dN k e
ρki (r) = dNik e − Nik ρi i (r) + Nik − bNik c ρi i (r) ,

(11.8)

where bNik c and dNik e denote the integers that bracket Nik . The two atomic densities on the right side of Eq. (11.8) are
A −2
A −1
A +2
obtained from densities ρ0,Z
, ρ0,Z
, . . . , ρ0,Z
that are computed in advance. (That is, the method uses the
i
i
i
neutral atomic density along with the densities for the singly- and doubly-charged cations and anions of the element
in equation.) The Hirshfeld-I iterations are converged once the atomic populations change insignificantly between
iterations, say |Nik − Nik−1 | < 0.0005e. 28,132
The iterative Hirshfeld scheme generally affords more reasonable charges as compared to the original Hirshfeld scheme.
In LiF, for example, the original Hirshfeld scheme predicts atomic charges of ±0.57 while the iterative scheme increases

Chapter 11: Molecular Properties and Analysis

566

these charges to ±0.93. The integral in Eq. (11.6) is evaluated by numerical quadrature, and the cost of each iteration
of Hirshfeld-I is equal to the cost of computing the original Hirshfeld charges. Within Q-C HEM, Hirshfeld-I charges
are available for molecules containing only H, Li, C, N, O, F, S, and Cl. The $rem variable SYM_IGNORE must be set
to TRUE for Hirshfeld-I analysis.
HIRSHFELD
Controls running of Hirshfeld population analysis.
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
TRUE
Calculate Hirshfeld populations.
FALSE Do not calculate Hirshfeld populations.
RECOMMENDATION:
None
HIRSHFELD_READ
Switch to force reading in of isolated atomic densities.
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
TRUE
Read in isolated atomic densities from previous Hirshfeld calculation from disk.
FALSE Generate new isolated atomic densities.
RECOMMENDATION:
Use the default unless system is large. Note, atoms should be in the same order with same basis
set used as in the previous Hirshfeld calculation (although coordinates can change). The previous
calculation should be run with the -save switch.
HIRSHFELD_SPHAVG
Controls whether atomic densities should be spherically averaged in pro-molecule.
TYPE:
LOGICAL
DEFAULT:
TRUE
OPTIONS:
TRUE
Spherically average atomic densities.
FALSE Do not spherically average.
RECOMMENDATION:
Use the default.
CM5
Controls running of CM5 population analysis.
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
TRUE
Calculate CM5 populations.
FALSE Do not calculate CM5 populations.
RECOMMENDATION:
None

567

Chapter 11: Molecular Properties and Analysis
HIRSHITER
Controls running of iterative Hirshfeld population analysis.
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
TRUE
Calculate iterative Hirshfeld populations.
FALSE Do not calculate iterative Hirshfeld populations.
RECOMMENDATION:
None
HIRSHITER_THRESH
Controls the convergence criterion of iterative Hirshfeld population analysis.
TYPE:
INTEGER
DEFAULT:
5
OPTIONS:
N Corresponding to the convergence criterion of N/10000, in e.
RECOMMENDATION:
Use the default, which is the value recommended in Ref. 28

Example 11.1 Iterative Hirshfeld population analysis for F− (H2 O)
$molecule
-1 1
O
1.197566
H
1.415397
H
0.134830
F
-1.236389
$end
$rem
SYM_IGNORE
METHOD
BASIS
HIRSHITER
$end

11.2.2

-0.108087
0.827014
-0.084378
0.012239

0.000000
0.000000
0.000000
0.000000

true
B3LYP
6-31G*
true

Multipole Moments

This section discusses how to compute arbitrary electrostatic multipole moments for an entire molecule, including both
ground- and excited-state electron densities. Occasionally, however, it is useful to decompose the electronic part of
the multipole moments into contributions from individual MOs. This decomposition is especially useful for systems
containing unpaired electrons, 154 where the first-order moments hxi, hyi, and hzi characterize the centroid (mean
position) of the half-filled MO, and the second-order moments determine its radius of gyration, Rg , which characterizes
the size of the MO. Upon setting PRINT_RADII_GYRE = TRUE, Q-C HEM will print out centroids and radii of gyration
for each occupied MO and for the overall electron density. If CIS or TDDFT excited states are requested, then this
keyword will also print out the centroids and radii of gyration for each excited-state electron density.

Chapter 11: Molecular Properties and Analysis

568

PRINT_RADII_GYRE
Controls printing of MO centroids and radii of gyration.
TYPE:
LOGICAL/INTEGER
DEFAULT:
FALSE
OPTIONS:
TRUE (or 1)
Print the centroid and radius of gyration for each occupied MO and each density.
2
Print centroids and radii of gyration for the virtual MOs as well.
FALSE (or 0) Do not calculate these quantities.
RECOMMENDATION:
None

Q-C HEM can compute Cartesian multipole moments of the charge density to arbitrary order, both for the ground state
and for excited states calculated using the CIS or TDDFT methods.
MULTIPOLE_ORDER
Determines highest order of multipole moments to print if wave function analysis requested.
TYPE:
INTEGER
DEFAULT:
4
OPTIONS:
n Calculate moments to nth order.
RECOMMENDATION:
Use the default unless higher multipoles are required.
CIS_MOMENTS
Controls calculation of excited-state (CIS or TDDFT) multipole moments
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
FALSE Do not calculate excited-state moments.
TRUE
Calculate moments for each excited state.
RECOMMENDATION:
Set to TRUE if excited-state moments are desired. (This is a trivial additional calculation.) The
MULTIPOLE_ORDER controls how many multipole moments are printed.

11.2.3

Symmetry Decomposition

Q-C HEM’s default is to write the SCF wave function molecular orbital symmetries and energies to the output file. If
requested, a symmetry decomposition of the kinetic and nuclear attraction energies can also be calculated.

Chapter 11: Molecular Properties and Analysis

569

SYMMETRY_DECOMPOSITION
Determines symmetry decompositions to calculate.
TYPE:
INTEGER
DEFAULT:
1
OPTIONS:
0 No symmetry decomposition.
1 Calculate MO eigenvalues and symmetry (if available).
2 Perform symmetry decomposition of kinetic energy and nuclear attraction
matrices.
RECOMMENDATION:
None

11.2.4

Localized Orbital Bonding Analysis

Localized orbital bonding analysis 143 (LOBA) is a technique developed by Dr. Alex Thom and Eric Sundstrom at
Berkeley with Prof. Martin Head-Gordon. Inspired by the work of Rhee and Head-Gordon, 124 it makes use of the fact
that the post-SCF localized occupied orbitals of a system provide a large amount of information about the bonding in
the system.
While the canonical molecular orbitals can provide information about local reactivity and ionization energies, their
delocalized nature makes them rather uninformative when looking at the bonding in larger molecules. Localized orbitals
in contrast provide a convenient way to visualize and account for electrons. Transformations of the orbitals within the
occupied subspace do not alter the resultant density; if a density can be represented as orbitals localized on individual
atoms, then those orbitals can be regarded as non-bonding. If a maximally localized set of orbitals still requires some
to be delocalized between atoms, these can be regarded as bonding electrons. A simple example is that of He2 versus
H2 . In the former, the delocalized σg and σu canonical orbitals may be transformed into 1s orbitals on each He atom,
and there is no bond between them. This is not possible for the H2 molecule, and so we can regard there being a bond
between the atoms. In cases of multiple bonding, multiple delocalized orbitals are required.
While there are no absolute definitions of bonding and oxidation state, it has been shown that the localized orbitals
match the chemically intuitive notions of core, non-bonded, single- and double-bonded electrons, etc.. By combining
these localized orbitals with population analyses, LOBA allows the nature of the bonding within a molecule to be
quickly determined.
In addition, it has been found that by counting localized electrons, the oxidation states of transition metals can be easily
found. Owing to polarization caused by ligands, an upper threshold is applied, populations above which are regarded as
“localized” on an atom, which has been calibrated to a range of transition metals, recovering standard oxidation states
ranging from −II to VII.

570

Chapter 11: Molecular Properties and Analysis
LOBA
Specifies the methods to use for LOBA
TYPE:
INTEGER
DEFAULT:
00
OPTIONS:
ab
a
specifies the localization method
0 Perform Boys localization.
1 Perform PM localization.
2 Perform ER localization.
b
specifies the population analysis method
0 Do not perform LOBA. This is the default.
1 Use Mulliken population analysis.
2 Use Löwdin population analysis.
RECOMMENDATION:
Boys Localization is the fastest. ER will require an auxiliary basis set.
LOBA 12 provides a reasonable speed/accuracy compromise.
LOBA_THRESH
Specifies the thresholds to use for LOBA
TYPE:
INTEGER
DEFAULT:
6015
OPTIONS:
aabb aa specifies the threshold to use for localization
bb specifies the threshold to use for occupation
Both are given as percentages.
RECOMMENDATION:
Decrease bb to see the smaller contributions to orbitals. Values of aa between 40 and 75 have
been shown to given meaningful results.

On a technical note, LOBA can function of both restricted and unrestricted SCF calculations. The figures printed in the
bonding analysis count the number of electrons on each atom from that orbital (i.e., up to 1 for unrestricted or singly
occupied restricted orbitals, and up to 2 for double occupied restricted.)

11.2.5

Basic Excited-State Analysis of CIS and TDDFT Wave Functions

For CIS, TDHF, and TDDFT excited-state calculations, we have already mentioned that Mulliken population analysis
of the excited-state electron densities may be requested by setting POP_MULLIKEN = −1, and multipole moments of
the excited-state densities will be generated if CIS_MOMENTS = TRUE. Another useful decomposition for excited states
is to separate the excitation into “particle” and “hole” components, which can then be analyzed separately. 125 To do
this, we define a density matrix for the excited electron,
X
Delec
(X + Y)†ai (X + Y)ib
(11.9)
ab =
i

and a density matrix for the hole that is left behind in the occupied space:
X
Dhole
=
(X + Y)ia (X + Y)†aj
ij
a

(11.10)

571

Chapter 11: Molecular Properties and Analysis

The quantities X and Y are the transition density matrices, i.e., the components of the TDDFT eigenvector. 41 The
indices i and j denote MOs that occupied in the ground state, whereas a and b index virtual MOs. Note also that
Delec + Dhole = ∆P, the difference between the ground- and excited-state density matrices.
Upon transforming Delec and Dhole into the AO basis, one can write
X
X
∆q =
(Delec S)µµ = −
(Dhole S)µµ
µ

(11.11)

µ

where ∆q is the total charge that is transferred from the occupied space to the virtual space. For a CIS calculation, or
for TDDFT within the Tamm-Dancoff approximation, 61 ∆q = −1. For full TDDFT calculations, ∆q may be slightly
different than −1.
Comparison of Eq. (11.11) to Eq. (11.3) suggests that the quantities (Delec S) and (Dhole S) are amenable to population
analyses of precisely the same sort used to analyze the ground-state density matrix. In particular, (Delec S)µµ represents
the µth AO’s contribution to the excited electron, while (Dhole S)µµ is a contribution to the hole. The sum of these
quantities,
∆qµ = (Delec S)µµ + (Dhole S)µµ
(11.12)
represents the contribution to ∆q arising from the µth AO. For the particle/hole density matrices, both Mulliken and
Löwdin population analyses available, and are requested by setting CIS_MULLIKEN = TRUE.
CIS_MULLIKEN
Controls Mulliken and Löwdin population analyses for excited-state particle and hole density
matrices.
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
FALSE Do not perform particle/hole population analysis.
TRUE
Perform both Mulliken and Löwdin analysis of the particle and hole
density matrices for each excited state.
RECOMMENDATION:
Set to TRUE if desired. This represents a trivial additional calculation.

Although the excited-state analysis features described in this section require very little computational effort, they are
turned off by default, because they can generate a large amount of output, especially if a large number of excited states
are requested. They can be turned on individually, or collectively by setting CIS_AMPL_ANAL = TRUE. This collective
option also requests the calculation of natural transition orbitals (NTOs), which were introduced in Section 7.12.2.
(NTOs can also be requested without excited-state population analysis. Some practical aspects of calculating and
visualizing NTOs are discussed below, in Section 11.5.2.)
CIS_AMPL_ANAL
Perform additional analysis of CIS and TDDFT excitation amplitudes, including generation of
natural transition orbitals, excited-state multipole moments, and Mulliken analysis of the excited
state densities and particle/hole density matrices.
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
TRUE
Perform additional amplitude analysis.
FALSE Do not perform additional analysis.
RECOMMENDATION:
None

572

Chapter 11: Molecular Properties and Analysis
Descriptor
Leading SVs$^2$
Sum of SVs$^2$ (Omega)
E(h)
E(p)
PR_NTO
Entanglement entropy (S_HE)
Nr of entangled states (Z_HE)
Renormalized S_HE/Z_HE
 [Ang]
 [Ang]
|| [Ang]
Hole size [Ang]
Electron size [Ang]
RMS electron-hole separation [Ang]
Covariance(r_h, r_e) [Ang^2]
Correlation coefficient

Explanation
Largest NTO occupation numbers
Ω = kγ IF k2 , sum of NTO occupation numbers
P
Energy of hole NTO, EI (h) = pq αpI Fpq αqI
P
Energy of particle NTO, EI (p) = pq βpI Fpq βqI
NTO participation ratio PRNTO
P
SH|E = − i λi log2 λi
ZHE = 2SH|E
Replace λi → λi /Ω
Mean position of hole h~xh iexc
Mean position of electron h~xe iexc
Linear e/h distance d~h→e = h~xe − ~xh iexc
RMS hole size: σh = (h~xh2 iexc − h~xh i2exc )1/2
RMS elec. size: σe = (h~xe2 iexc − h~xe i2exc )1/2
2
dexc = (h|~xe − ~xh | iexc )1/2
COV (~xh , ~xe ) = h~xh · ~xe iexc − h~xh iexc · h~xe iexc
Reh = COV (~xh , ~xe ) /σh · σe

Table 11.1: Descriptors output by Q-C HEM for transition density matrix analysis. Note that squares of the SVs, which
correspond to the weights of the respective NTO pairs, are printed. Ω equals the square of the norm of the 1TDM.

11.2.6

General Excited-State Analysis

Q-Chem features a new module for extended excited-state analysis, which is interfaced to the ADC, CC/EOM-CC,
CIS, and TDDFT methods. 11,91,112–115 These analyses are based on the state, transition and difference density matrices
of the excited states; the theoretical background for such analysis is given in Chapter 7.12.
The transition-density (1TDM) based analyses include the construction and export of natural transition orbitals 90
(NTOs) and electron and hole densities, 114 the evaluation of charge transfer numbers, 112 an analysis of exciton multipole moments, 11,91,115 and quantification of electron-hole entanglement. 116 NTOs are obtained by singular value decomposition (SVD) of the 1TDM:
IF
γpq
= hΨI |p† q|ΨF i
†

γ = ασβ ,

(11.13)
(11.14)

where σ is diagonal matrix containing singular values and unitary matrices α and β contain the respective particle and
hole NTOs. Note that:
X
X
2
2
kγk2 =
γpq
=
σK
≡Ω
(11.15)
pq

K

Furthermore, the formation and export of state-averaged NTOs, and the decomposition of the excited states into transitions of state-averaged NTOs are implemented. 114 The difference and/or state densities can be exported themselves, as
well as employed to construct and export natural orbitals, natural difference orbitals, and attachment and detachment
densities. 56 Furthermore, two measures of unpaired electrons are computed. 55 In addition, a Mulliken or Löwdin population analysis and an exciton analysis can be performed based on the difference/state densities. The main descriptors of
the various analyses that are printed for each excited state are given in Tables 11.1 and 11.2. For a detailed description
with illustrative examples, see Refs. 114 and 113.
To activate any excited-state analysis STATE_ANALYSIS has to be set to TRUE. For individual analyses there is currently
only a limited amount of fine grained control. The construction and export of any type of orbitals is controlled by
MOLDEN_FORMAT to export the orbitals as M OL D EN files, and NTO_PAIRS which specifies the number of important
orbitals to print (note that the same keyword controls the number of natural orbitals, the number of natural difference
orbitals, and the number of NTOs to be printed). Setting MAKE_CUBE_FILES to TRUE triggers the construction and
export of densities in “cube file” format 59 (see Section 11.5.4 for details). Activating transition densities in $plots will

Chapter 11: Molecular Properties and Analysis
Descriptor
n_u
n_u,nl
PR_NO
p_D and p_A
PR_D and PR_A
 [Ang]
 [Ang]
|| [Ang]
Hole size [Ang]
Electron size [Ang]

573

Explanation
P
Number of unpaired electrons nu = i min(ni , 2 − ni )
P
Number of unpaired electrons nu,nl = i n2i (2 − ni )2
NO participation ratio PRNO
Promotion number pD and pA
D/A participation ratio PRD and PRA
Mean position of detachment density d~D
Mean position of attachment density d~A
Linear D/A distance d~D→A = d~A − d~D
RMS size of detachment density σD
RMS size of attachment density σA

Table 11.2: Descriptors output by Q-C HEM for difference/state density matrix analysis.

generate cube files for the transition density, the electron density, and the hole density of the respective excited states,
while activating state densities or attachment/detachment densities will generate cube files for the state density, the
difference density, the attachment density and the detachment density. Setting GUI = 2 will export data to the “.fchk”
file and switches off the generation of cube files. The population analyses are controlled by POP_MULLIKEN and
LOWDIN_POPULATION. Setting the latter to TRUE will enforce Löwdin population analysis to be employed, while by
default the Mulliken population analysis is used.
Any M OL D EN or cube files generated by the excited state analyses can be found in the directory plots in the job’s
scratch directory. Their names always start with a unique identifier of the excited state (the exact form of this human
readable identifier varies with the excited state method). The names of M OL D EN files are then followed by either
_no.mo, _ndo.mo, or _nto.mo depending on the type of orbitals they contain. In case of cube files the state
identifier is followed by _dens, _diff, _trans, _attach, _detach, _elec, or _hole for state, difference,
transition, attachment, detachment, electron, or hole densities, respectively. All cube files have the suffice .cube. In
unrestricted calculations an additional part is added to the file name before .cube which indicates α (_a) or β (_b)
spin. The only exception is the state density for which _tot or _sd are added indicating the total or spin-density parts
of the state density.
The _ctnum_atomic.om files created in the main directory serve as input for a charge transfer number analysis, as
explained, e.g., in Refs. 92,112. These files are processed by the external TheoDORE program () to create electron/hole
correlation plots and to compute fragment based descriptors.
Note: (1) In Hermitian formalisms, γ IF is a Hermitian conjugate of γ FI , but in non-Hermitian approaches, such as
coupled-cluster theory, the two are slightly different. While for quantitative interstate properties both γ IF and
γ FI are computed, the qualitative trends in exciton properties derived from (γ IF )† and γ FI are very similar. Only
one 1TDM is analyzed for EOM-CC.
(2) In spin-restricted calculations, the LIBWFA module computes NTOs for the α − alpha block of transition
density. Thus, when computing NTOs for the transitions between open-shell EOM-IP/EA states make sure
to specify correct spin states. For example, use EOM_EA_ALPHA to visualize transitions involving the extra
electron.

Chapter 11: Molecular Properties and Analysis

574

NTO_PAIRS
Controls how many hole/particle NTO pairs and frontier natural orbital pairs and natural difference orbital pairs are computed for excited states.
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
N Write N NTO/NO/NDO pairs per excited state.
RECOMMENDATION:
If activated (N > 0), a minimum of two NTO pairs will be printed for each state. Increase the
value of N if additional NTOs are desired. By default, one pair of frontier natural orbitals is
computed for N = 0.

11.3

Interface to the NBO Package

Q-C HEM incorporates the Natural Bond Orbital package (v. 5.0 and 6.0) for molecular properties and wave function
analysis. The NBO 5.0 package is invoked by setting the $rem variable NBO to TRUE and is initiated after the SCF
wave function is obtained.
Note: If switched on for a geometry optimization, the NBO package will only be invoked at the end of the last
optimization step.
Users should consult the NBO User’s Manual for options and details relating to exploitation of the features offered in
this package. The NBO 6.0 package 49,50 can be downloaded by the user from nbo6.chem.wisc.edu, and can be
invoked by: (a) setting the NBOEXE environment variable, and (b) include both NBO = TRUE and RUN_NBO6 = TRUE
in the Q-C HEM input file.
NBO analysis is also available for excited states calculated using CIS or TDDFT. Excited-state NBO analysis is still
in its infancy, and users should be aware that the convergence of the NBO search procedure may be less well-behaved
for excited states than it is for ground states, and may require specification of additional NBO parameters in the $nbo
section that is described below. Consult Ref. 150 for details and suggestions.
NBO
Controls the use of the NBO package.
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
0 Do not invoke the NBO package.
1 Do invoke the NBO package, for the ground state.
2 Invoke the NBO package for the ground state, and also each
CIS, RPA, or TDDFT excited state.
RECOMMENDATION:
None

The general format for passing options from Q-C HEM to the NBO program is shown below:
$nbo
{NBO program keywords, parameters and options}
$end

Chapter 11: Molecular Properties and Analysis

575

Note: (1) $rem variable NBO must be set to TRUE before the $nbo keyword is recognized.
(2) Q-C HEM does not support facets of the NBO package which require multiple job runs

11.4

Orbital Localization

The concept of localized orbitals has already been visited in this manual in the context of perfect-pairing and methods.
As the SCF energy is independent of the partitioning of the electron density into orbitals, there is considerable flexibility
as to how this may be done. The canonical picture, where the orbitals are eigenfunctions of the Fock operator is useful
in determining reactivity, for, through Koopmans’ theorem, the orbital energy eigenvalues give information about
the corresponding ionization energies and electron affinities. As a consequence, the HOMO and LUMO are very
informative as to the reactive sites of a molecule. In addition, in small molecules, the canonical orbitals lead us to the
chemical description of σ and π bonds.
In large molecules, however, the canonical orbitals are often very delocalized, and so information about chemical
bonding is not readily available from them. Here, orbital localization techniques can be of great value in visualizing
the bonding, as localized orbitals often correspond to the chemically intuitive orbitals which might be expected.
Q-C HEM has three post-SCF localization methods available. These can be performed separately over both occupied and
virtual spaces. The localization scheme attributed to Boys 23,24 minimizes the radial extent of the localized orbitals, i.e.,
P
2
i hii||r1 − r2 | |iii, and although is relatively fast, does not separate σ and π orbitals, leading to two ‘banana-orbitals’
in the case of a double bond. 111 Pipek-Mezey localized orbitals 111 maximize the locality of Mulliken populations,
and are of a similar cost to Boys localized orbitals, but maintain σ − π separation. Edmiston-Ruedenberg localized
P
orbitals 42 maximize the self-repulsion of the orbitals, i hii| 1r |iii. This is more computationally expensive to calculate
as it requires a two-electron property to be evaluated, but due to the work of Dr. Joe Subotnik, 136 and later Prof. YoungMin Rhee and Westin Kurlancheek with Prof. Martin Head-Gordon at Berkeley, this has been reduced to asymptotic
cubic-scaling cost (with respect to the number of occupied orbitals), via the resolution of identity approximation.
BOYSCALC
Specifies the Boys localized orbitals are to be calculated
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
0 Do not perform localize the occupied space.
1 Allow core-valence mixing in Boys localization.
2 Localize core and valence separately.
RECOMMENDATION:
None

Chapter 11: Molecular Properties and Analysis

576

ERCALC
Specifies the Edmiston-Ruedenberg localized orbitals are to be calculated
TYPE:
INTEGER
DEFAULT:
06000
OPTIONS:
aabcd
aa
specifies the convergence threshold.
If aa > 3, the threshold is set to 10−aa . The default is 6.
If aa = 1, the calculation is aborted after the guess, allowing Pipek-Mezey
orbitals to be extracted.
b
specifies the guess:
0 Boys localized orbitals. This is the default
1 Pipek-Mezey localized orbitals.
c
specifies restart options (if restarting from an ER calculation):
0 No restart. This is the default
1 Read in MOs from last ER calculation.
2 Read in MOs and RI integrals from last ER calculation.
d
specifies how to treat core orbitals
0 Do not perform ER localization. This is the default.
1 Localize core and valence together.
2 Do separate localizations on core and valence.
3 Localize only the valence electrons.
4 Use the $localize section.
RECOMMENDATION:
ERCALC 1 will usually suffice, which uses threshold 10−6 .

The $localize section may be used to specify orbitals subject to ER localization if require. It contains a list of the
orbitals to include in the localization. These may span multiple lines. If the user wishes to specify separate beta orbitals
to localize, include a zero before listing the beta orbitals, which acts as a separator, e.g.,
$localize
2 3 4 0
2 3 4 5 6
$end

11.5

Visualizing and Plotting Orbitals, Densities, and Other Volumetric Data

The free, open-source visualization program IQ MOL (www.iqmol.org) provides a graphical user interface for QC HEM that can be used as a molecular structure builder, as a tool for local or remote submission of Q-C HEM jobs, and
as a visualization tool for densities and molecular orbitals. Alternatively, Q-C HEM can generate orbital and density
data in formats suitable for plotting with various third-party visualization programs.

11.5.1

Visualizing Orbitals Using M OL D EN and M AC M OL P LT

Upon request, Q-C HEM will generate an input file for M OL D EN, a freely-available molecular visualization program. 1,126 M OL D EN can be used to view ball-and-stick molecular models (including stepwise visualization of a geometry optimization), molecular orbitals, vibrational normal modes, and vibrational spectra. M OL D EN also contains a
powerful Z-matrix editor. In conjunction with Q-C HEM, orbital visualization via M OL D EN is currently supported for

Chapter 11: Molecular Properties and Analysis

577

s, p, and d functions (pure or Cartesian), as well as pure f functions. Upon setting MOLDEN_FORMAT to TRUE, QC HEM will append a M OL D EN-formatted input file to the end of the Q-C HEM log file. As some versions of M OL D EN
have difficulty parsing the Q-C HEM log file itself, we recommend that the user cut and paste the M OL D EN-formatted
part of the Q-C HEM log file into a separate file to be read by M OL D EN.
MOLDEN_FORMAT
Requests a M OL D EN-formatted input file containing information from a Q-C HEM job.
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
TRUE Append M OL D EN input file at the end of the Q-C HEM output file.
RECOMMENDATION:
None.

M OL D EN-formatted files can also be read by M AC M OL P LT, another freely-available visualization program. 2,22 M AC M OL P LT generates orbital iso-contour surfaces much more rapidly than M OL D EN, however, within M AC M OL P LT
these surfaces are only available for Cartesian Gaussian basis functions, i.e., PURECART = 2222, which may not be the
default.
Example 11.2 Generating a M OL D EN file for molecular orbital visualization.
$molecule
0 1
O
H 1 0.95
H 1 0.95
$end

2

$rem
METHOD
BASIS
PRINT_ORBITALS
MOLDEN_FORMAT
$end

104.5

hf
cc-pvtz
true (default is to print 5 virtual orbitals)
true

For geometry optimizations and vibrational frequency calculations, one need only set MOLDEN_FORMAT to TRUE, and
the relevant geometry or normal mode information will automatically appear in the M OL D EN section of the Q-C HEM
log file.
Example 11.3 Generating a M OL D EN file to step through a geometry optimization.
$molecule
0 1
O
H 1 0.95
H 1 0.95
$end

2

$rem
JOBTYPE
METHOD
BASIS
MOLDEN_FORMAT
$end

104.5

opt
hf
6-31G*
true

Chapter 11: Molecular Properties and Analysis

11.5.2

578

Visualization of Natural Transition Orbitals

For excited states calculated using the CIS, RPA, TDDFT, EOM-CC, and ADC methods, construction of Natural Transition Orbitals (NTOs), as described in Sections 7.12.2 and 11.2.6, is requested using the $rem variable NTO_PAIRS.
This variable also determines the number of hole/particle NTO pairs that are output for each excited state and the
number of natural orbitals or natural difference orbitlas. Although the total number of hole/particle pairs is equal to
the number of occupied MOs, typically only a very small number of these pairs (often just one pair) have significant
amplitudes. (Additional large-amplitude NTOs may be encountered in cases of strong electronic coupling between
multiple chromophores. 81 )
NTO_PAIRS
Controls the writing of hole/particle NTO pairs for excited state.
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
N Write N NTO pairs per excited state.
RECOMMENDATION:
If activated (N > 0), a minimum of two NTO pairs will be printed for each state. Increase the
value of N if additional NTOs are desired.

When NTO_PAIRS > 0, Q-C HEM will generate the NTOs in M OL D EN format. The NTOs are state-specific, in the
sense that each excited state has its own NTOs, and therefore a separate M OL D EN file is required for each excited state.
These files are written to the job’s scratch directory, in a sub-directory called NTOs, so to obtain the NTOs the scratch
directory must be saved using the –save option that is described in Section 2.7. The output files in the NTOs directory
have an obvious file-naming convention. The “hole” NTOs (which are linear combinations of the occupied MOs) are
printed to the M OL D EN files in order of increasing importance, with the corresponding excitation amplitudes replacing
the canonical MO eigenvalues. (This is a formatting convention only; the excitation amplitudes are unrelated to the
MO eigenvalues.) Following the holes are the “particle” NTOs, which represent the excited electron and are linear
combinations of the virtual MOs. These are written in order of decreasing amplitude. To aid in distinguishing the two
sets within the M OL D EN files, the amplitudes of the holes are listed with negative signs, while the corresponding NTO
for the excited electron has the same amplitude with a positive sign.
Due to the manner in which the NTOs are constructed (see Section 7.12.2), NTO analysis is available only when the

579

Chapter 11: Molecular Properties and Analysis

number of virtual orbitals exceeds the number of occupied orbitals, which may not be the case for minimal basis sets.
Example 11.4 Generating M OL D EN-formatted natural transition orbitals for several excited states of uracil.
$molecule
0 1
N
-2.181263
C
-2.927088
N
-4.320029
C
-4.926706
C
-4.185901
C
-2.754591
N
-1.954845
H
-0.923072
H
-2.343008
H
-4.649401
H
-6.012020
H
-4.855603
O
-2.458932
$end
$rem
METHOD
BASIS
CIS_N_ROOTS
NTO_PAIRS
$end

11.5.3

0.068208
-1.059037
-0.911094
0.301204
1.435062
1.274555
2.338369
2.224557
3.268581
2.414197
0.301371
-1.768832
-2.200499

0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000

B3LYP
6-31+G*
3
2

Generation of Volumetric Data Using $plots

The simplest way to visualize the charge densities and molecular orbitals that Q-C HEM evaluates is via a graphical
user interface, such as those described in the preceding section. An alternative procedure, which is often useful for
generating high-quality images for publication, is to evaluate certain densities and orbitals on a user-specified grid of
points. This is accomplished by invoking the $plots option, which is itself enabled by requesting IANLTY = 200.
The format of the $plots input is documented below. It permits plotting of molecular orbitals, the SCF ground-state
density, and excited-state densities obtained from CIS, RPA or TDDFT/TDA, or TDDFT calculations. Also in connection with excited states, either transition densities, attachment/detachment densities, or natural transition orbitals (at the
same levels of theory given above) can be plotted as well.
By default, the output from the $plots command is one (or several) ASCII files in the working directory, named plot.mo,
etc.. The results then must be visualized with a third-party program capable of making 3-D plots. (Some suggestions
are given in Section 11.5.4.)

Chapter 11: Molecular Properties and Analysis

580

An example of the use of the $plots option is the following input deck:
Example 11.5 A job that evaluates the H2 HOMO and LUMO on a 1 × 1 × 15 grid, along the bond axis. The plotting
output is in an ASCII file called plot.mo, which lists for each grid point, x, y, z, and the value of each requested MO.
$molecule
0 1
H
0.0
H
0.0
$end
$rem
METHOD
BASIS
IANLTY
$end

0.0
0.0

0.35
-0.35

hf
6-31g**
200

$plots
Plot the HOMO and the LUMO on a line
1
0.0
0.0
1
0.0
0.0
15 -3.0
3.0
2
0
0
0
1
2
$end

General format for the $plots section of the Q-C HEM input deck.
$plots
One comment line
Specification of the 3-D mesh of points on 3 lines:
Nx xmin xmax
Ny ymin ymax
Nz zmin zmax
A line with 4 integers indicating how many things to plot:
NMO NRho NTrans NDA
An optional line with the integer list of MO’s to evaluate (only if NMO > 0)
MO(1) MO(2) . . . MO(NMO )
An optional line with the integer list of densities to evaluate (only if NRho > 0)
Rho(1) Rho(2) . . . Rho(NRho )
An optional line with the integer list of transition densities (only if NTrans > 0)
Trans(1) Trans(2) . . . Trans(NTrans )
An optional line with states for detachment/attachment densities (if NDA > 0)
DA(1) DA(2) . . . DA(NDA )
$end

Line 1 of the $plots keyword section is reserved for comments. Lines 2–4 list the number of one dimension points
and the range of the grid (note that coordinate ranges are in Ångstroms if INPUT_BOHR is not set, while all output
is in atomic units). Line 5 must contain 4 non-negative integers indicating the number of: molecular orbitals (NMO ),
electron densities (NRho ), transition densities and attach/detach densities (NDA ), to have mesh values calculated.
The final lines specify which MOs, electron densities, transition densities and CIS attach/detach states are to be plotted
(the line can be left blank, or removed, if the number of items to plot is zero). Molecular orbitals are numbered
1 . . . Nα , Nα + 1 . . . Nα + Nβ ; electron densities numbered where 0= ground state, 1 = first excited state, 2 = second
excited state, etc.; and attach/detach specified from state 1 → NDA .
By default, all output data are printed to files in the working directory, overwriting any existing file of the same name.
• Molecular orbital data is printed to a file called plot.mo;

581

Chapter 11: Molecular Properties and Analysis
• Densities are plotted to plots.hf ;
• Restricted unrelaxed attachment/detachment analysis is sent to plot.attach.alpha and
plot.detach.alpha;
• Unrestricted unrelaxed attachment/detachment analysis is sent to plot.attach.alpha,
plot.detach.alpha, plot.attach.beta and plot.detach.beta;
• Restricted relaxed attachments/detachment analysis is plotted in plot.attach.rlx.alpha and
plot.detach.rlx.alpha; and finally
• Unrestricted relaxed attachment/detachment analysis is sent to plot.attach.rlx.alpha,
plot.detach.rlx.alpha, plot.attach.rlx.beta and plot.detach.rlx.beta.
Output is printed in atomic units, with coordinates first followed by item value, as shown below:
x1
x2
...

y1
y1

z1
z1

a1
b1

a2
b2

...
...

aN
bN

Instead of a standard one-, two-, or three-dimensional Cartesian grid, a user may wish to plot orbitals or densities on a
set of grid points of his or her choosing. Such points are specified using a $grid input section whose format is simply
the Cartesian coordinates of all user-specified grid points:
x1
x2
...

y1
y2

z1
z2

The $plots section must still be specified as described above, but if the $grid input section is present, then these userspecified grid points will override the ones specified in the $plots section.
The Q-C HEM $plots utility allows the user to plot transition densities and detachment/attachment densities directly
from amplitudes saved from a previous calculation, without having to solve the post-SCF (CIS, RPA, TDA, or TDDFT)
equations again. To take advantage of this feature, the same Q-C HEM scratch directory must be used, and the
SKIP_CIS_RPA $rem variable must be set to TRUE. To further reduce computational time, the SCF_GUESS $rem can be
set to READ.
SKIP_CIS_RPA
Skips the solution of the CIS, RPA, TDA or TDDFT equations for wave function analysis.
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
TRUE / FALSE
RECOMMENDATION:
Set to true to speed up the generation of plot data if the same calculation has been run previously
with the scratch files saved.

11.5.3.1

New $plots input

New format for the $plots section provides readable and friendly input for generation of volumetric data. The input
section can be divided into three parts. The first part contains basic plot options which define the 3-D mesh of points.
The second part contains the selection of densities or orbitals. The advanced options are included in the last part.

582

Chapter 11: Molecular Properties and Analysis

With new plot format, there are multiple ways to define 3-D mesh points. If no plot option is given, the boundaries of
the mesh box are the maximum/minimum molecular coordinates ± 3.0 Å. The default box can be simply enlarged or
reduced by setting grid_range to a value larger or smaller than 3.0 (negative number is accepted), respectively. To
customize the mesh box, set grid_range to desired boundaries:
$plots
grid_range (-1,1)
$end

(-1,1)

(-1,1)

This defines a 2×2×2 mesh box centered at the molecular coordinate origin. Note that there is no space in the parentheses.
The number of one dimension points is the value of the box length divided by grid_spacing. The default grid
point spacing is 0.3 Å. To override the usage of grid_spacing and customize the number of 3-D points, set
grid_points to desired values.
To generate cube file (Section 11.5.4) using new plot format, just set MAKE_CUBE_FILES to TRUE in $rem section.
The new plot format is enabled by requesting PLOTS = 1.
Option
Basic plot options
grid_range

Explanation

grid_spacing
grid_points
Density/orbital selection?
alpha_molecular_orbital
beta_molecular_orbital
total_density
spin_density
transition_density
attachment_detachment_density
Advanced options
reduced_density_gradient
orbital_laplacians
average_local_ionization

boundaries† of the mesh box or increment/decrement in
the default boundaries†† in Å
grid point spacing††† in Å
Nx Ny Nz
a integer range of alpha MO’s to evaluate
a integer range of beta MO’s to evaluate
a integer range of total densities to evaluate
a integer range of spin densities to evaluate
a integer range of transition densities to evaluate
a integer range of det.-att. densities to evaluate
invoke non-covalent interaction (NCI) plot
evaluate orbital Laplacians
evaluate average local ionization energies 130 with a given
contour value of the electron density. The default is
3
0.0135e/Å (≈ 0.002e/a30 ).

†

the format: (xmin , xmax )(ymin , ymax )(zmin , zmax )
the default is 3.0 Å increment in the boundaries derived from the molecular coordinates
†††
the default is 0.3 Å; it can be overridden by option ’grid_points’
?
input format: n-m or n, indicating n-th oribtal or state; use 0 for the ground-state

††

Table 11.3: Options for new $plots input section

583

Chapter 11: Molecular Properties and Analysis

Example 11.6 Generating the cube files: the total densities of the ground and the first two excited states, the transition
and detachment/attachment densities of the first two excited states, and the 28th to 31th alpha molecular orbitals, with
customized 3-D mesh box and points.
$rem
method
basis
cis_n_roots
cis_triplets
make_cube_files
plots
$end

cis
6-31+G*
4
false
true
! triggers writing of cube files
true

$plots
grid_range (-8,8) (-8,8) (-8,8)
grid_points 40 40 40
total_density 0-2
transition_density 1-2
attachment_detachment_density 1-2
alpha_molecular_orbital
28-31
$end
$molecule
0 1
C
O
H
H
$end

-4.57074
-3.35678
-5.18272
-5.03828

2.50214
2.38653
1.79525
3.31185

-0.00000
0.00000
-0.54930
0.54930

Example 11.7 Generating the cube files of the average local ionization energies and the total density for the ground
state of aniline.
$rem
jobtype = sp
exchange = hf
basis = 6-31g*
make_cube_files = true
plots = true
$end
$plots
grid_spacing 0.1
total_density 0
average_local_ionization
$end
$molecule
0 1
H
C
C
H
C
H
C
N
H
H
C
H
C
H
$end

-2.9527248536
-1.8714921071
-1.1721238067
-1.7092440589
0.2115215040
0.7547333507
0.9165181187
2.3578744287
2.7471831094
2.7471831094
0.2115215040
0.7547333507
-1.1721238067
-1.7092440589

-0.0267579494
-0.0106828899
-0.0011274864
-0.0094706668
0.0174872124
0.0243282741
0.0252342217
0.1198186854
-0.3464272024
-0.3464272024
0.0174872124
0.0243282741
-0.0011274864
-0.0094706668

0.0000000000
0.0000000000
-1.1972702914
-2.1378190515
-1.2026764462
-2.1379453065
0.0000000000
0.0000000000
-0.8299201716
0.8299201716
1.2026764462
2.1379453065
1.1972702914
2.1378190515

Chapter 11: Molecular Properties and Analysis

11.5.4

584

Direct Generation of “Cube” Files

As an alternative to the output format discussed above, all of the $plots data may be output directly to a sub-directory
named plots in the job’s scratch directory, which must therefore be saved using the –save option described in Section 2.7. The plotting data in this sub-directory are not written in the plot.* format described above, but rather in
the form of so-called “cube” file, one for each orbital or density that is requested. The cube file format is a standard
one for volumetric data and consists of a small header followed by the orbital or density values at each grid point, in
ASCII format. (Consult Ref. 59 for the complete format specification.) Because the grid coordinates themselves are
not printed (their locations are implicit from information contained in the header), each individual cube file is much
smaller than the corresponding plot.* file would be. Cube files can be read by many standard (and freely-available)
graphics programs, including M AC M OL P LT 2,22 and VMD. 4,66 VMD, in particular, is recommended for generation of
high-quality images for publication. Cube files for the MOs and densities requested in the $plots section are requested
by setting MAKE_CUBE_FILES to TRUE, with the $plots section specified as described in Section 11.5.3.
MAKE_CUBE_FILES
Requests generation of cube files for MOs, NTOs, or NBOs.
TYPE:
LOGICAL/STRING
DEFAULT:
FALSE
OPTIONS:
FALSE Do not generate cube files.
TRUE
Generate cube files for MOs and densities.
NTOS
Generate cube files for NTOs.
NBOS
Generate cube files for NBOs.
RECOMMENDATION:
None
PLOT_SPIN_DENSITY
Requests the generation of spin densities, ρα and ρβ .
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
FALSE Do not generate spin density cube files.
TRUE
Generate spin density cube files.
RECOMMENDATION:
Set to TRUE if spin densities are desired in addition to total densities. Requires that
MAKE_CUBE_FILES be set to TRUE as well, and that one or more total densities is requested
in the $plots input section. The corresponding spin densities will then be generated also.

The following example illustrates the generation of cube files for a ground and an excited-state density, including
the corresponding spin densities. In this example, the plots sub-directory of the job’s scratch directory should contain files named dens.N.cube (total density for state N , where N = 0 or 1 represents the ground and first
excited state, respectively), dens_alpha.N.cube and dens_beta.N.cube (ρα and ρβ for each state), and

585

Chapter 11: Molecular Properties and Analysis
dens_spin.N.cube (ρα − ρβ for each state.)

Example 11.8 Generating density and spin-density cube files for the ground and first excited state of the HOO radical.
$molecule
0 2
H
1.004123
O
-0.246002
O
-1.312366
$end
$rem
PLOT_SPIN_DENSITY
MAKE_CUBE_FILES
SCF_CONVERGENCE
METHOD
BASIS
CIS_N_ROOTS
$end

-0.180454
0.596152
-0.230256

0.000000
0.000000
0.000000

true
true
8
b3lyp
6-31+G*
1

$plots
grid information and request to plot 2 densities
20 -5.0 5.0
20 -5.0 5.0
20 -5.0 5.0
0 2 0 0
0 1
$end

Cube files are also available for natural transition orbitals (Sections 7.12.2 and 11.5.2) by setting MAKE_CUBE_FILES
to NTOS, although in this case the procedure is somewhat more complicated, due to the state-specific nature of these
quantities. Cube files for the NTOs are generated only for a single excited state, whose identity is specified using
CUBEFILE_STATE. Cube files for additional states are readily obtained using a sequence of Q-C HEM jobs, in which
the second (and subsequent) jobs read in the converged ground- and excited-state information using SCF_GUESS and
SKIP_CIS_RPA.
CUBEFILE_STATE
Determines which excited state is used to generate cube files
TYPE:
INTEGER
DEFAULT:
None
OPTIONS:
n Generate cube files for the nth excited state
RECOMMENDATION:
None

An additional complication is the manner in which to specify which NTOs will be output as cube files. When
MAKE_CUBE_FILES is set to TRUE, this is specified in the $plots section, in the same way that MOs would be specified
for plotting. However, one must understand the order in which the NTOs are stored. For a system with Nα α-spin
MOs, the occupied NTOs 1, 2, . . . , Nα are stored in order of increasing amplitudes, so that the Nα ’th occupied NTO is
the most important. The virtual NTOs are stored next, in order of decreasing importance. According to this convention,
the principle NTO pair consists of the final occupied orbital and the first virtual orbital, for any particular excited state.
Thus, orbitals Nα and Nα + 1 represent the most important NTO pair, while orbitals Nα − 1 and Nα + 2 represent the

586

Chapter 11: Molecular Properties and Analysis
second most important NTO pair, etc..

Example 11.9 Generating cube files for the excitation between the principle occupied and virtual NTOs of the second
singlet excited state of uracil. Note that Nα = 29 for uracil.
$molecule
0 1
N
-2.181263
C
-2.927088
N
-4.320029
C
-4.926706
C
-4.185901
C
-2.754591
N
-1.954845
H
-0.923072
H
-2.343008
H
-4.649401
H
-6.012020
H
-4.855603
O
-2.458932
$end

0.068208
-1.059037
-0.911094
0.301204
1.435062
1.274555
2.338369
2.224557
3.268581
2.414197
0.301371
-1.768832
-2.200499

0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000

$plots
Plot the dominant NTO pair, N --> N+1
25 -5.0 5.0
25 -5.0 5.0
25 -5.0 5.0
2 0 0 0
29 30
$end
$rem
METHOD
BASIS
CIS_N_ROOTS
CIS_TRIPLETS
NTO_PAIRS
MAKE_CUBE_FILES
CUBEFILE_STATE
$end

B3LYP
6-31+G*
2
FALSE
TRUE
! calculate the NTOs
NTOS
! generate NTO cube files...
2
! ...for the 2nd excited state

Cube files for Natural Bond Orbitals (for either the ground state or any CIS, RPA, of TDDFT excited states) can be
generated in much the same way, by setting MAKE_CUBE_FILES equal to NBOS, and using CUBEFILE_STATE to select
the desired electronic state. CUBEFILE_STATE = 0 selects ground-state NBOs. The particular NBOs to be plotted are
selected using the $plots section, recognizing that Q-C HEM stores the NBOs in order of decreasing occupancies, with
all α-spin NBOs preceding any β-spin NBOs, in the case of an unrestricted SCF calculation. (For ground states, there
is typically one strongly-occupied NBO for each electron.) NBO cube files are saved to the plots sub-directory of the
job’s scratch directory. One final caveat: to get NBO cube files, the user must specify the AONBO option in the $nbo

587

Chapter 11: Molecular Properties and Analysis
section, as shown in the following example.
Example 11.10 Generating cube files for the NBOs of the first excited state of H2 O.
$rem
METHOD
BASIS
CIS_N_ROOTS
CIS_TRIPLETS
NBO
MAKE_CUBE_FILES
CUBEFILE_STATE
$end

CIS
CC-PVTZ
1
FALSE
2
! ground- and excited-state NBO
NBOS
! generate NBO cube files...
1
! ...for the first excited state

$nbo
AONBO
$end
$molecule
0 1
O
H 1 0.95
H 1 0.95
$end

2

104.5

$plots
Plot the 5 high-occupancy NBOs, one for each alpha electron
40 -8.0 8.0
40 -8.0 8.0
40 -8.0 8.0
5 0 0 0
1 2 3 4 5
$end

11.5.5

NCI Plots

Weitao Yang and co-workers 36,73 have shown that the reduced density gradient,

s(r) =

1
2(3π 2 )1/3

 ˆ
|∇ρ(r)|
ρ(r)4/3

(11.16)

provides a convenient indicator of noncovalent interactions, which are characterized by large density gradients in
regions of space where the density itself is small, leading to very large values of s(r). Q-C HEM can generate
noncovalent interactions (NCI) plots of the function s(r) in three-dimensional space. To generate these, set the
PLOT_REDUCED_DENSITY_GRAD $rem variable to TRUE. (See the nci-c8h14.in input example in $QC/samples directory.)

11.5.6

Electrostatic Potentials

Q-C HEM can evaluate electrostatic potentials on a grid of points. Electrostatic potential evaluation is controlled by the
$rem variable IGDESP, as documented below.

Chapter 11: Molecular Properties and Analysis

588

IGDESP
Controls evaluation of the electrostatic potential on a grid of points. If enabled, the output is in
an ASCII file, plot.esp, in the format x, y, z, esp for each point.
TYPE:
INTEGER
DEFAULT:
none no electrostatic potential evaluation
OPTIONS:
−2 same as the option ’-1’, plus evaluate the ESP of $external_charges$
−1 read grid input via the $plots section of the input deck
0
Generate the ESP values at all nuclear positions
+n read n grid points in bohr from the ASCII file ESPGrid
RECOMMENDATION:
None

The following example illustrates the evaluation of electrostatic potentials on a grid, note that IANLTY must also be set
to 200.
Example 11.11 A job that evaluates the electrostatic potential for H2 on a 1 by 1 by 15 grid, along the bond axis. The
output is in an ASCII file called plot.esp, which lists for each grid point, x, y, z, and the electrostatic potential.
$molecule
0 1
H
0.0
H
0.0
$end
$rem
METHOD
BASIS
IANLTY
IGDESP
$end

0.0
0.0

0.35
-0.35

hf
6-31g**
200
-1

$plots
plot the HOMO and the LUMO on a line
1
0.0
0.0
1
0.0
0.0
15 -3.0
3.0
0 0 0 0
0
$end

We can also compute the electrostatic potential for the transition density, which can be used, for example, to compute
the Coulomb coupling in excitation energy transfer.
ESP_TRANS
Controls the calculation of the electrostatic potential of the transition density
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
TRUE
compute the electrostatic potential of the excited state transition density
FALSE compute the electrostatic potential of the excited state electronic density
RECOMMENDATION:
NONE

Chapter 11: Molecular Properties and Analysis

589

The electrostatic potential is a complicated object and it is not uncommon to model it using a simplified representation
based on atomic charges. For this purpose it is well known that Mulliken charges perform very poorly. Several
definitions of ESP-derived atomic charges have been given in the literature, however, most of them rely on a leastsquares fitting of the ESP evaluated on a selection of grid points. Although these grid points are usually chosen so that
the ESP is well modeled in the “chemically important” region, it still remains that the calculated charges will change
if the molecule is rotated. Recently an efficient rotationally invariant algorithm was proposed that sought to model the
ESP not by direct fitting, but by fitting to the multipole moments. 129 By doing so it was found that the fit to the ESP was
superior to methods that relied on direct fitting of the ESP. The calculation requires the traceless form of the multipole
moments and these are also printed out during the course of the calculations. To request these multipole-derived charges
the following $rem option should be set:
MM_CHARGES
Requests the calculation of multipole-derived charges (MDCs).
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
TRUE Calculates the MDCs and also the traceless form of the multipole moments
RECOMMENDATION:
Set to TRUE if MDCs or the traceless form of the multipole moments are desired. The calculation
does not take long.

11.6

Spin and Charge Densities at the Nuclei

Gaussian basis sets violate nuclear cusp conditions. 76,108,121 This may lead to large errors in wave function at nuclei,
particularly for spin density calculations. 33 This problem can be alleviated by using an averaging operator that compute wave function density based on constraints that wave function must satisfy near Coulomb singularity. 122,123 The
derivation of operators is based on hyper virial theorem 62 and presented in Ref. 122. Application to molecular spin densities for spin-polarized 123 and DFT 149 wave functions show considerable improvement over traditional delta function
operator.
One of the simplest forms of such operators is based on the Gaussian weight function exp[−(Z/r0 )2 (r − R)2 ] that
samples the vicinity of a nucleus of charge Z located at R. The parameter r0 has to be small enough to neglect twoelectron contributions of the order O(r04 ) but large enough for meaningful averaging. The range of values between 0.15–
0.3 a.u. has been shown to be adequate, with final answer being relatively insensitive to the exact choice of r0 . 122,123
The value of r0 is chosen by RC_R0 keyword in the units of 0.001 a.u. The averaging operators are implemented for
single determinant Hartree-Fock and DFT, and correlated SSG wave functions. Spin and charge densities are printed
for all nuclei in a molecule, including ghost atoms.
RC_R0
Determines the parameter in the Gaussian weight function used to smooth the density at the
nuclei.
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
0 Corresponds the traditional delta function spin and charge densities
n corresponding to n × 10−3 a.u.
RECOMMENDATION:
We recommend value of 250 for a typical spit valence basis. For basis sets with increased flexibility in the nuclear vicinity the smaller values of r0 also yield adequate spin density.

Chapter 11: Molecular Properties and Analysis

11.7

590

Atoms in Molecules

Q-C HEM can output a file suitable for analysis with the Atoms in Molecules package (AIMPAC). The source for
AIMPAC can be freely downloaded from the web site
http://www.chemistry.mcmaster.ca/aimpac/imagemap/imagemap.htm
Users should check this site for further information about installing and running AIMPAC. The AIMPAC input file is
created by specifying a filename for the WRITE_WFN $rem.
WRITE_WFN
Specifies whether or not a wfn file is created, which is suitable for use with AIMPAC. Note that
the output to this file is currently limited to f orbitals, which is the highest angular momentum
implemented in AIMPAC.
TYPE:
STRING
DEFAULT:
(NULL) No output file is created.
OPTIONS:
filename Specifies the output file name. The suffix .wfn will
be appended to this name.
RECOMMENDATION:
None

11.8

Distributed Multipole Analysis

Distributed Multipole Analysis 133,135 (DMA) is a method to represent the electrostatic potential of a molecule in terms
of a multipole expansion around a set of points. The GDMA code 134 by Prof. Anthony Stone can be used to perform DMA analysis after Q-C HEM calculations. The GDMA program works with formatted checkpoint files (.fchk)
produced by Q-C HEM. For further information consult the documentation for the GDMA package. 134

11.9

Intracules

The many dimensions of electronic wave functions makes them difficult to analyze and interpret. It is often convenient
to reduce this large number of dimensions, yielding simpler functions that can more readily provide chemical insight.
The most familiar of these is the one-electron density ρ(r), which gives the probability of an electron being found at
the point r. Analogously, the one-electron momentum density π(p) gives the probability that an electron will have
a momentum of p. However, the wave function is reduced to the one-electron density much information is lost. In
particular, it is often desirable to retain explicit two-electron information. Intracules are two-electron distribution
functions and provide information about the relative position and momentum of electrons. A detailed account of the
different type of intracules can be found in Ref. 48. Q-C HEM’s intracule package was developed by Aaron Lee and
Nick Besley, and can compute the following intracules for or HF wave functions:
• Position intracules, P (u): describes the probability of finding two electrons separated by a distance u.
• Momentum intracules, M (v): describes the probability of finding two electrons with relative momentum v.
• Wigner intracule, W (u, v): describes the combined probability of finding two electrons separated by u and with
relative momentum v.

591

Chapter 11: Molecular Properties and Analysis

11.9.1

Position Intracules

The intracule density, I(u), represents the probability for the inter-electronic vector u = u1 − u2 :
Z
I(u) = ρ(r1 r2 ) δ(r12 − u) dr1 dr2

(11.17)

where ρ(r1 , r2 ) is the two-electron density. A simpler quantity is the spherically averaged intracule density,
Z
P (u) = I(u)dΩu ,

(11.18)

where Ωu is the angular part of v, measures the probability that two electrons are separated by a scalar distance u = |u|.
This intracule is called a position intracule. 48 If the molecular orbitals are expanded within a basis set
X
ψa (r) =
cµa φµ (r)
(11.19)
µ

The quantity P (u) can be expressed as
P (u) =

X

Γµνλσ (µνλσ)P

(11.20)

µνλσ

where Γµνλσ is the two-particle density matrix and (µνλσ)P is the position integral
Z
(µνλσ)P = φ∗µ (r) φν (r) φ∗λ (r + u)φσ (r + u) dr dΩ

(11.21)

and φµ (r), φν (r), φλ (r) and φσ (r) are basis functions. For HF wave functions, the position intracule can be decomposed into a Coulomb component,
1 X
Dµν Dλσ (µνλσ)P
(11.22)
PJ (u) =
2
µνλσ

and an exchange component,
PK (u) = −

i
1 Xh α α
β
β
Dµλ Dνσ + Dµλ
Dνσ
(µνλσ)P
2

(11.23)

µνλσ

where Dµν etc. are density matrix elements. The evaluation of P (u), PJ (u) and PK (u) within Q-C HEM has been
described in detail in Ref. 85.
Some of the moments of P (u) are physically significant, 47 for example
Z∞

u0 P (u)du

=

n(n − 1)
2

(11.24)

u0 PJ (u)du

=

n2
2

(11.25)

u2 PJ (u)du =

nQ − µ2

(11.26)

u0 PK (u)du =

−

0

Z∞
0

Z∞
0

Z∞

n
2

(11.27)

0

where n is the number of electrons and, µ is the electronic dipole moment and Q is the trace of the electronic quadrupole
moment tensor. Q-C HEM can compute both moments and derivatives of position intracules.

592

Chapter 11: Molecular Properties and Analysis

11.9.2

Momentum Intracules

¯
Analogous quantities can be defined in momentum space; I(v),
for example, represents the probability density for the
relative momentum v = p1 − p2 :
Z
¯
I(v)
= π(p1 , p2 ) δ(p12 − v)dp1 dp2
(11.28)
where π(p1 , p2 ) momentum two-electron density. Similarly, the spherically averaged intracule
Z
¯
M (v) = I(v)dΩ
v

(11.29)

where Ωv is the angular part of v, is a measure of relative momentum v = |v| and is called the momentum intracule.
The quantity M (v) can be written as
X
M (v) =
Γµνλσ (µνλσ)M
(11.30)
µνλσ

where Γµνλσ is the two-particle density matrix and (µνλσ)M is the momentum integral 19
(µνλσ)M

v2
=
2π 2

Z

φ∗µ (r)φν (r + q)φ∗λ (u + q)φσ (u)j0 (qv) dr dq du

(11.31)

The momentum integrals only possess four-fold permutational symmetry, i.e.,
(µνλσ)M = (νµλσ)M = (σλνµ)M = (λσµν)M

(11.32)

(νµλσ)M = (µνσλ)M = (λσνµ)M = (σλµν)M

(11.33)

and therefore generation of M (v) is roughly twice as expensive as P (u). Momentum intracules can also be decomposed
into Coulomb MJ (v) and exchange MK (v) components:
1 X
Dµν Dλσ (µνλσ)M
2

MJ (v) =

(11.34)

µνλσ

MK (v) = −

i
1 Xh α α
β
β
Dµλ Dνσ + Dµλ
Dνσ
(µνλσ)M
2

(11.35)

µνλσ

Again, the even-order moments are physically significant: 19
Z∞

v 0 M (v)dv =

n(n − 1)
2

(11.36)

n2
2

(11.37)

0

Z∞

u0 MJ (v)dv =

0

Z∞

v 2 PJ (v)dv = 2nET

(11.38)

0

Z∞

v 0 MK (v)dv = −

n
2

(11.39)

0

where n is the number of electrons and ET is the total electronic kinetic energy. Currently, Q-C HEM can compute
M (v), MJ (v) and MK (v) using s and p basis functions only. Moments are generated using quadrature and consequently for accurate results M (v) must be computed over a large and closely spaced v range.

593

Chapter 11: Molecular Properties and Analysis

11.9.3

Wigner Intracules

The intracules P (u) and M (v) provide a representation of an electron distribution in either position or momentum
space but neither alone can provide a complete description. For a combined position and momentum description an
intracule in phase space is required. Defining such an intracule is more difficult since there is no phase space secondorder reduced density. However, the second-order Wigner distribution, 20
Z
1
ρ2 (r1 + q1 , r1 − q1 , r2 + q2 , r2 − q2 )e−2i(p1 ·q1 +p2 ·q2 ) dq1 dq2
(11.40)
W2 (r1 , p1 , r2 , p2 ) = 6
π
can be interpreted as the probability of finding an electron at r1 with momentum p1 and another electron at r2 with
momentum p2 . [The quantity W2 (r1 , r2 , p1 , p2 is often referred to as “quasi-probability distribution” since it is not
positive everywhere.]
The Wigner distribution can be used in an analogous way to the second order reduced densities to define a combined
position and momentum intracule. This intracule is called a Wigner intracule, and is formally defined as
Z
W (u, v) = W2 (r1 , p1 , r2 , p2 )δ(r12 − u)δ(p12 − v)dr1 dr2 dp1 dp2 dΩu dΩv
(11.41)
If the orbitals are expanded in a basis set, then W (u, v) can be written as
X
W (u, v) =
Γµνλσ (µνλσ)W

(11.42)

µνλσ

where (µνλσ)W is the Wigner integral
Z Z
v2
(µνλσ)W =
φ∗µ (r)φν (r + q)φ∗λ (r + q + u)φσ (r + u)j0 (q v) dr dq dΩu
2π 2

(11.43)

Wigner integrals are similar to momentum integrals and only have four-fold permutational symmetry. Evaluating
Wigner integrals is considerably more difficult that their position or momentum counterparts. The fundamental [ssss]w
integral,
Z Z


u2 v 2
[ssss]W =
exp −α|r−A|2 −β|r+q−B|2 −γ|r+q+u−C|2 −δ|r+u−D|2 ×
2π 2
j0 (qv) dr dq dΩu
(11.44)
can be expressed as
2

[ssss]W =

2

2 2

πu2 v 2 e−(R+λ u +µ v )
2(α + δ)3/2 (β + γ)3/2

Z

e−P·u j0 (|Q + ηu|v) dΩu

(11.45)

or alternatively
2

[ssss]W =

2

2 2

2π 2 u2 v 2 e−(R+λ u +µ v
(α + δ)3/2 (β + γ)3/2

∞
) X
n=0


(2n + 1)in (P u)jn (ηuv)jn (Qv)Pn

P·Q
P Q


(11.46)

Two approaches for evaluating (µνλσ)W have been implemented in Q-C HEM, full details can be found in Ref. 153.
The first approach uses the first form of [ssss]W and used Lebedev quadrature to perform the remaining integrations
over Ωu . For high accuracy large Lebedev grids 82–84 should be used, grids of up to 5294 points are available in QC HEM. Alternatively, the second form can be adopted and the integrals evaluated by summation of a series. Currently,
both methods have been implemented within Q-C HEM for s and p basis functions only.
When computing intracules it is most efficient to locate the loop over u and/or v points within the loop over shellquartets. 34 However, for W (u, v) this requires a large amount of memory to store all the integrals arising from each
(u, v) point. Consequently, an additional scheme, in which the u and v points loop is outside the shell-quartet loop, is
available. This scheme is less efficient, but substantially reduces the memory requirements.

Chapter 11: Molecular Properties and Analysis

11.9.4

Intracule Job Control

The following $rem variables can be used to control the calculation of intracules.
INTRACULE
Controls whether intracule properties are calculated (see also the $intracule section).
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
FALSE No intracule properties.
TRUE
Evaluate intracule properties.
RECOMMENDATION:
None
WIG_MEM
Reduce memory required in the evaluation of W (u, v).
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
FALSE Do not use low memory option.
TRUE
Use low memory option.
RECOMMENDATION:
The low memory option is slower, so use the default unless memory is limited.
WIG_LEB
Use Lebedev quadrature to evaluate Wigner integrals.
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
FALSE Evaluate Wigner integrals through series summation.
TRUE
Use quadrature for Wigner integrals.
RECOMMENDATION:
None
WIG_GRID
Specify angular Lebedev grid for Wigner intracule calculations.
TYPE:
INTEGER
DEFAULT:
194
OPTIONS:
Lebedev grids up to 5810 points.
RECOMMENDATION:
Larger grids if high accuracy required.

594

Chapter 11: Molecular Properties and Analysis
N_WIG_SERIES
Sets summation limit for Wigner integrals.
TYPE:
INTEGER
DEFAULT:
10
OPTIONS:
n < 100
RECOMMENDATION:
Increase n for greater accuracy.
N_I_SERIES
Sets summation limit for series expansion evaluation of in (x).
TYPE:
INTEGER
DEFAULT:
40
OPTIONS:
n>0
RECOMMENDATION:
Lower values speed up the calculation, but may affect accuracy.
N_J_SERIES
Sets summation limit for series expansion evaluation of jn (x).
TYPE:
INTEGER
DEFAULT:
40
OPTIONS:
n>0
RECOMMENDATION:
Lower values speed up the calculation, but may affect accuracy.

595

Chapter 11: Molecular Properties and Analysis

11.9.5

596

Format for the $intracule Section

int_type

u_points
v_points
moments
derivs
accuracy

0
1
2
3
4
5
6

0–4
0–4
n

Compute P (u) only
Compute M (v) only
Compute W (u, v) only
Compute P (u), M (v) and W (u, v)
Compute P (u) and M (v)
Compute P (u) and W (u, v)
Compute M (v) and W (u, v)
Number of points, start, end.
Number of points, start, end.
Order of moments to be computed (P (u) only).
order of derivatives to be computed (P (u) only).
(10−n ) specify accuracy of intracule interpolation table (P (u) only).

Example 11.12 Compute HF/STO-3G P (u), M (v) and W (u, v) for Ne, using Lebedev quadrature with 974 point
grid.
$molecule
0 1
Ne
$end
$rem
METHOD
BASIS
INTRACULE
WIG_LEB
WIG_GRID
$end
$intracule
int_type
u_points
v_points
moments
derivs
accuracy
$end

hf
sto-3g
true
true
974

3
10
8
4
4
8

0.0
0.0

10.0
8.0

Chapter 11: Molecular Properties and Analysis

597

Example 11.13 Compute HF/6-31G W (u, v) intracules for H2 O using series summation up to n=25 and 30 terms in
the series evaluations of jn (x) and in (x).
$molecule
0 1
H1
O
H1
H2 O

r
r

H1

theta

r = 1.1
theta = 106
$end
$rem
METHOD
BASIS
INTRACULE
WIG_MEM
N_WIG_SERIES
N_I_SERIES
N_J_SERIES
$end
$intracule
int_type
u_points
v_points
$end

11.10

2
30
20

hf
6-31G
true
true
25
40
50

0.0
0.0

15.0
10.0

Harmonic Vibrational Analysis

Vibrational analysis is an extremely important tool for the quantum chemist, supplying a molecular fingerprint which
is invaluable for aiding identification of molecular species in many experimental studies. Q-C HEM includes a vibrational analysis package that can calculate vibrational frequencies and their Raman 71 and infrared activities. Vibrational
frequencies are calculated by either using an analytic Hessian (if available; see Table 10.1) or, numerical finite difference of the gradient. The default setting in Q-C HEM is to use the highest analytical derivative order available for the
requested theoretical method.
When calculating analytic frequencies at the HF and DFT levels of theory, the coupled-perturbed SCF equations must
be solved. This is the most time-consuming step in the calculation, and also consumes the most memory. The amount
of memory required is O(N 2 M ) where N is the number of basis functions, and M the number of atoms. This is
an order more memory than is required for the SCF calculation, and is often the limiting consideration when treating
larger systems analytically. Q-C HEM incorporates a new approach to this problem that avoids this memory bottleneck
by solving the CPSCF equations in segments. 78 Instead of solving for all the perturbations at once, they are divided into
several segments, and the CPSCF is applied for one segment at a time, resulting in a memory scaling of O(N 2 M/Nseg ),
where Nseg is the number of segments. This option is invoked automatically by the program.
Following a vibrational analysis, Q-C HEM computes useful statistical thermodynamic properties at standard temperature and pressure, including: zero-point vibration energy (ZPVE) and, translational, rotational and vibrational, entropies
and enthalpies.
The performance of various ab initio theories in determining vibrational frequencies has been well documented; see
Refs. 72,95,127.

Chapter 11: Molecular Properties and Analysis

11.10.1

598

Job Control

In order to carry out a frequency analysis users must at a minimum provide a molecule within the $molecule keyword
and define an appropriate level of theory within the $rem keyword using the $rem variables EXCHANGE, CORRELATION
(if required) (Chapter 4) and BASIS (Chapter 8). Since the default type of job (JOBTYPE) is a single point energy (SP)
calculation, the JOBTYPE $rem variable must be set to FREQ.
It is very important to note that a vibrational frequency analysis must be performed at a stationary point on the potential
surface that has been optimized at the same level of theory. Therefore a vibrational frequency analysis most naturally
follows a geometry optimization in the same input deck, where the molecular geometry is obtained (see examples).
Users should also be aware that the quality of the quadrature grid used in DFT calculations is more important when
calculating second derivatives. The default grid for some atoms has changed in Q-C HEM 3.0 (see Section 5.5) and for
this reason vibrational frequencies may vary slightly form previous versions. It is recommended that a grid larger than
the default grid is used when performing frequency calculations.
The standard output from a frequency analysis includes the following.
• Vibrational frequencies.
• Raman and IR activities and intensities (requires $rem DORAMAN).
• Atomic masses.
• Zero-point vibrational energy.
• Translational, rotational, and vibrational, entropies and enthalpies.
Several other $rem variables are available that control the vibrational frequency analysis. In detail, they are:
DORAMAN
Controls calculation of Raman intensities. Requires JOBTYPE to be set to FREQ
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
FALSE Do not calculate Raman intensities.
TRUE
Do calculate Raman intensities.
RECOMMENDATION:
None
VIBMAN_PRINT
Controls level of extra print out for vibrational analysis.
TYPE:
INTEGER
DEFAULT:
1
OPTIONS:
1 Standard full information print out.
If VCI is TRUE, overtones and combination bands are also printed.
3 Level 1 plus vibrational frequencies in atomic units.
4 Level 3 plus mass-weighted Hessian matrix, projected mass-weighted Hessian
matrix.
6 Level 4 plus vectors for translations and rotations projection matrix.
RECOMMENDATION:
Use the default.

Chapter 11: Molecular Properties and Analysis

599

CPSCF_NSEG
Controls the number of segments used to calculate the CPSCF equations.
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
0 Do not solve the CPSCF equations in segments.
n User-defined. Use n segments when solving the CPSCF equations.
RECOMMENDATION:
Use the default.

Example 11.14 An EDF1/6-31+G* optimization, followed by a vibrational analysis. Doing the vibrational analysis at
a stationary point is necessary for the results to be valid.
$molecule
0 1
O
C 1 co
F 2 fc
H 2 hc
co
fc
hc
fco
hco
$end

1
1

fco
hco

3

180.0

=
1.2
=
1.4
=
1.0
= 120.0
= 120.0

$rem
JOBTYPE
METHOD
BASIS
$end

opt
edf1
6-31+G*

@@@
$molecule
read
$end
$rem
JOBTYPE
METHOD
BASIS
$end

11.10.2

freq
edf1
6-31+G*

Isotopic Substitutions

By default Q-C HEM calculates vibrational frequencies using the atomic masses of the most abundant isotopes (taken
from the Handbook of Chemistry and Physics, 63rd Edition). Masses of other isotopes can be specified using the
$isotopes section and by setting the ISOTOPES $rem variable to TRUE. The format of the $isotopes section is as
follows:
$isotopes
number_of_isotope_loops tp_flag
number_of_atoms [temp pressure] (loop 1)

Chapter 11: Molecular Properties and Analysis

600

atom_number1
mass1
atom_number2
mass2
...
number_of_atoms [temp pressure] (loop 2)
atom_number1
mass1
atom_number2
mass2
...
$end

Note: Only the atoms whose masses are to be changed from the default values need to be specified. After each loop
all masses are reset to the default values. Atoms are numbered according to the order in the $molecule section.

An initial loop using the default masses is always performed first. Subsequent loops use the user-specified atomic
masses. Only those atoms whose masses are to be changed need to be included in the list, all other atoms will adopt the
default masses. The output gives a full frequency analysis for each loop. Note that the calculation of vibrational frequencies in the additional loops only involves a rescaling of the computed Hessian, and therefore takes little additional
computational time.
The first line of the $isotopes section specifies the number of substitution loops and also whether the temperature and
pressure should be modified. The tp_flag setting should be set to 0 if the default temperature and pressure are to be used
(298.18 K and 1 atm respectively), or else to 1 if they are to be altered. Note that the temperatures should be specified
in Kelvin and pressures in atmospheres.
ISOTOPES
Specifies if non-default masses are to be used in the frequency calculation.
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
FALSE Use default masses only.
TRUE
Read isotope masses from $isotopes section.
RECOMMENDATION:
None

Chapter 11: Molecular Properties and Analysis

601

Example 11.15 An EDF1/6-31+G* optimization, followed by a vibrational analysis. Doing the vibrational analysis at
a stationary point is necessary for the results to be valid.
$molecule
0 1
C
1.08900
C -1.08900
H
2.08900
H -2.08900
$end
$rem
BASIS
JOBTYPE
METHOD
$end

0.00000
0.00000
0.00000
0.00000

0.00000
0.00000
0.00000
0.00000

3-21G
opt
hf

@@@
$molecule
read
$end
$rem
BASIS
JOBTYPE
METHOD
SCF_GUESS
ISOTOPES
$end

3-21G
freq
hf
read
1

$isotopes
2
0
4
1
13.00336
2
13.00336
3
2.01410
4
2.01410
2
3
2.01410
4
2.01410
$end

11.10.3

! two loops, both at std temp and pressure
! All atoms are given non-default masses

! H’s replaced with D’s

Partial Hessian Vibrational Analysis

The computation of harmonic frequencies for systems with a very large number of atoms can become computationally
expensive. However, in many cases only a few specific vibrational modes or vibrational modes localized in a region
of the system are of interest. A typical example is the calculation of the vibrational modes of a molecule adsorbed on
a surface. In such a case, only the vibrational modes of the adsorbate are useful, and the vibrational modes associated
with the surface atoms are of less interest. If the vibrational modes of interest are only weakly coupled to the vibrational modes associated with the rest of the system, it can be appropriate to adopt a partial Hessian approach. In this
approach, 17,18 only the part of the Hessian matrix comprising the second derivatives of a subset of the atoms defined
by the user is computed. These atoms are defined in the $alist block. This results in a significant decrease in the cost
of the calculation. Physically, this approximation corresponds to assigning an infinite mass to all the atoms excluded
from the Hessian and will only yield sensible results if these atoms are not involved in the vibrational modes of interest. VPT2 and TOSH anharmonic frequencies can be computed following a partial Hessian calculation. 52 It is also
possible to include a subset of the harmonic vibrational modes with an anharmonic frequency calculation by invoking

Chapter 11: Molecular Properties and Analysis

602

the ANHAR_SEL rem. This can be useful to reduce the computational cost of an anharmonic frequency calculation or
to explore the coupling between specific vibrational modes.
Alternatively, vibrationally averaged interactions with the rest of the system can be folded into a partial Hessian calculation using vibrational subsystem analysis. 157,163 Based on an adiabatic approximation, this procedure reduces the
cost of diagonalizing the full Hessian, while providing a local probe of fragments vibrations, and providing better than
partial Hessian accuracy for the low frequency modes of large molecules. 46 Mass-effects from the rest of the system
can be vibrationally averaged or excluded within this scheme.
PHESS
Controls whether partial Hessian calculations are performed.
TYPE:
INTEGER
DEFAULT:
0 Full Hessian calculation
OPTIONS:
1 Partial Hessian calculation.
2 Vibrational subsystem analysis (massless).
3 Vibrational subsystem analysis (weighted).
RECOMMENDATION:
None
N_SOL
Specifies number of atoms included in the Hessian.
TYPE:
INTEGER
DEFAULT:
No default
OPTIONS:
User defined
RECOMMENDATION:
None
PH_FAST
Lowers integral cutoff in partial Hessian calculation is performed.
TYPE:
LOGICAL
DEFAULT:
FALSE Use default cutoffs
OPTIONS:
TRUE Lower integral cutoffs
RECOMMENDATION:
None

Chapter 11: Molecular Properties and Analysis

603

ANHAR_SEL
Select a subset of normal modes for subsequent anharmonic frequency analysis.
TYPE:
LOGICAL
DEFAULT:
FALSE Use all normal modes
OPTIONS:
TRUE Select subset of normal modes
RECOMMENDATION:
None

Example 11.16 This example shows an anharmonic frequency calculation for ethene where only the C-H stretching
modes are included in the anharmonic analysis.
$comment
ethene
restricted anharmonic frequency analysis
$end
$molecule
0 1
C
0.6665
C -0.6665
H
1.2480
H -1.2480
H -1.2480
H
1.2480
$end
$rem
JOBTYPE
METHOD
BASIS
ANHAR_SEL
N_SOL
$end
$alist
9
10
11
12
$end

0.0000
0.0000
0.9304
-0.9304
0.9304
-0.9304

0.0000
0.0000
0.0000
0.0000
0.0000
0.0000

freq
hf
sto-3g
TRUE
4

Chapter 11: Molecular Properties and Analysis

604

Example 11.17 This example shows a partial Hessian frequency calculation of the vibrational frequencies of acetylene
on a model of the C(100) surface
$comment
acetylene - C(100)
partial Hessian calculation
$end
$molecule
0 1
C 0.000
C 0.000
H 0.000
H 0.000
C 0.000
C 0.000
C 1.253
C -1.253
C 1.253
C 1.297
C -1.253
C 0.000
C -1.297
H -2.179
H -1.148
H 0.000
H 2.179
H -1.148
H -2.153
H 2.153
H 1.148
H 1.148
H 2.153
H -2.153
H 0.000
$end
$rem
JOBTYPE
METHOD
BASIS
PHESS
N_SOL
$end

0.659
-0.659
1.406
-1.406
0.786
-0.786
1.192
1.192
-1.192
0.000
-1.192
0.000
0.000
0.000
-2.156
-0.876
0.000
2.156
-1.211
-1.211
-2.156
2.156
1.211
1.211
0.876

-2.173
-2.173
-2.956
-2.956
-0.647
-0.647
0.164
0.164
0.164
1.155
0.164
2.023
1.155
1.795
0.654
2.669
1.795
0.654
-0.446
-0.446
0.654
0.654
-0.446
-0.446
2.669

freq
hf
sto-3g
TRUE
4

$alist
1
2
3
4
$end

11.10.4

Localized Mode Vibrational Analysis

The computation of harmonic frequencies leads to molecular vibrations described by coordinates which are often
highly de-localized. For larger molecules many vibrational modes can potentially contribute to a single observed
spectral band, and information about the interaction between localized chemical units can become less readily available. In certain cases, localizing vibrational modes using procedures similar to the localized orbital schemes discussed

Chapter 11: Molecular Properties and Analysis

605

previously in this manual can therefore provide a more chemically intuitive way of analysing spectral data, 67–69 interpreting two-dimensional vibrational spectra, 53 or improving calculations that go beyond the harmonic approximation. 32,51,109 It is also possible to include only a subset of the normal modes in the localization calculation by invoking
the LOCALFREQ_SELECT rem variable. This can be useful to improve convergence in larger molecules or to explore
the coupling between specific vibrational modes. These modes are defined in the $alist block. Alternatively it is possible to localize high and low frequency modes separately in a single calculation using LOCALFREQ_GROUPS and
related inputs.
LOCALFREQ
Controls whether a vibrational mode localization calculation is performed.
TYPE:
INTEGER
DEFAULT:
0 Normal mode calculation.
OPTIONS:
1 Localized mode calculation with a Pipek-Mezey like criterion.
2 Localized mode calculation with a Boys like criterion.
RECOMMENDATION:
None
LOCALFREQ_THRESH
Mode localization is considered converged when the change in the localization criterion is less
than 10−LOCALFREQ_THRESH .
TYPE:
INTEGER
DEFAULT:
6
OPTIONS:
n User-specified integer.
RECOMMENDATION:
None
LOCALFREQ_MAX_ITER
Controls the maximum number of mode localization sweeps permitted.
TYPE:
INTEGER
DEFAULT:
200
OPTIONS:
n User-specified integer.
RECOMMENDATION:
None

Chapter 11: Molecular Properties and Analysis

606

LOCALFREQ_SELECT
Select a subset of normal modes for subsequent anharmonic frequency analysis.
TYPE:
LOGICAL
DEFAULT:
FALSE Use all normal modes.
OPTIONS:
TRUE Select a subset of normal modes.
RECOMMENDATION:
None
LOCALFREQ_GROUPS
Select the number of groups of frequencies to be localized separately within a localized mode
calculation. The size of the groups are then controlled using the LOCALFREQ_GROUP1,
LOCALFREQ_GROUP2, and LOCALFREQ_GROUP3 keywords.
TYPE:
INTEGER
DEFAULT:
0 Localize all normal modes together.
OPTIONS:
1 Define one subset of modes to localize independently.
2 Define two subsets of modes to localize independently.
3 Define three subsets of modes to localize independently.
RECOMMENDATION:
None
LOCALFREQ_GROUP1
Select the number of modes to include in the first subset of modes to localize independently when
the keyword LOCALFREQ_GROUPS > 0.
TYPE:
INTEGER
DEFAULT:
NONE
OPTIONS:
n User-specified integer.
RECOMMENDATION:
Modes will be included starting with the lowest frequency mode and then in ascending energy
order up to the defined value.
LOCALFREQ_GROUP2 and LOCALFREQ_GROUP3 are defined similarly.

11.11

Anharmonic Vibrational Frequencies

Computing vibrational spectra beyond the harmonic approximation has become an active area of research owing to
the improved efficiency of computer techniques. 12,29,93,159 To calculate the exact vibrational spectrum within BornOppenheimer approximation, one has to solve the nuclear Schrödinger equation completely using numerical integration
techniques, and consider the full configuration interaction of quanta in the vibrational states. This has only been carried
out on di- or triatomic system. 30,110 The difficulty of this numerical integration arises because solving exact the nuclear
Schrödinger equation requires a complete electronic basis set, consideration of all the nuclear vibrational configuration
states, and a complete potential energy surface (PES). Simplification of the Nuclear Vibration Theory (NVT) and PES
are the doorways to accelerating the anharmonic correction calculations. There are five aspects to simplifying the
problem:

607

Chapter 11: Molecular Properties and Analysis

• Expand the potential energy surface using a Taylor series and examine the contribution from higher derivatives.
Small contributions can be eliminated, which allows for the efficient calculation of the Hamiltonian.
• Investigate the effect on the number of configurations employed in a variational calculation.
• Avoid using variational theory (due to its expensive computational cost) by using other approximations, for
example, perturbation theory.
• Obtain the PES indirectly by applying a self-consistent field procedure.
• Apply an anharmonic wave function which is more appropriate for describing the distribution of nuclear probability on an anharmonic potential energy surface.
To incorporate these simplifications, new formulae combining information from the Hessian, gradient and energy are
used as a default procedure to calculate the cubic and quartic force field of a given potential energy surface.
Here, we also briefly describe various NVT methods. In the early stage of solving the nuclear Schrödinger equation (in
the 1930s), second-order Vibrational Perturbation Theory (VPT2) was developed. 8,12,96,98,155 However, problems occur
when resonances exist in the spectrum. This becomes more problematic for larger molecules due to the greater chance
of accidental degeneracies occurring. To avoid this problem, one can do a direct integration of the secular matrix using
Vibrational Configuration Interaction (VCI) theory. 152 It is the most accurate method and also the least favored due
to its computational expense. In Q-C HEM 3.0, we introduce a new approach to treating the wave function, transitionoptimized shifted Hermite (TOSH) theory, 86 which uses first-order perturbation theory, which avoids the degeneracy
problems of VPT2, but which incorporates anharmonic effects into the wave function, thus increasing the accuracy of
the predicted anharmonic energies.

11.11.1

Vibration Configuration Interaction Theory

To solve the nuclear vibrational Schrödinger equation, one can only use direct integration procedures for diatomic
molecules. 30,110 For larger systems, a truncated version of full configuration interaction is considered to be the most
accurate approach. When one applies the variational principle to the vibrational problem, a basis function for the
nuclear wave function of the nth excited state of mode i is
m
Y
(0)
(n)
(n)
φj
(11.47)
ψi = φi
j6=i
(n)

where the φi represents the harmonic oscillator eigenfunctions for normal mode qi . This can be expressed in terms
of Hermite polynomials:
! 21
1
ωi qi2
√
ωi2
(n)
φi =
e− 2 Hn (qi ωi )
(11.48)
1
n
2
π 2 n!
With the basis function defined in Eq. (11.47), the nth wave function can be described as a linear combination of the
Hermite polynomials:
n1 X
n2 X
n3
nm
X
X
(n)
(n)
Ψ(n) =
···
cijk···m ψijk···m
(11.49)
i=0 j=0 k=0

m=0

where ni is the number of quanta in the ith mode. We propose the notation VCI(n) where n is the total number of
quanta, i.e.:
n = n1 + n2 + n3 + · · · + nm
(11.50)
To determine this expansion coefficient c(n) , we integrate the Ĥ, as in Eq. (4.1), with Ψ(n) to get the eigenvalues
(n)

c(n) = EVCI(n) = hΨ(n) |Ĥ|Ψ(n) i

(11.51)

This gives us frequencies that are corrected for anharmonicity to n quanta accuracy for a m-mode molecule. The size of
the secular matrix on the right hand of Eq. (11.51) is ((n+m)!/n!m!)2 , and the storage of this matrix can easily surpass
the memory limit of a computer. Although this method is highly accurate, we need to seek for other approximations
for computing large molecules.

608

Chapter 11: Molecular Properties and Analysis

11.11.2

Vibrational Perturbation Theory

Vibrational perturbation theory has been historically popular for calculating molecular spectroscopy. Nevertheless,
it is notorious for the inability of dealing with resonance cases. In addition, the non-standard formulas for various
symmetries of molecules forces the users to modify inputs on a case-by-case basis, 9,35,94 which narrows the accessibility
of this method. VPT applies perturbation treatments on the same Hamiltonian as in Eq. (4.1), but divides it into an
unperturbed part, Û ,

m 
X
ωi 2 2
1 ∂2
Û =
+
−
qi
(11.52)
2 ∂qi2
2
i
and a perturbed part, V̂ :
V̂ =

m
m
1 X
1 X
ηijk qi qj qk +
ηijkl qi qj qk ql
6
24
ijk=1

(11.53)

ijkl=1

One can then apply second-order perturbation theory to get the ith excited state energy:
E (i) = Û (i) + hΨ(i) |V̂ |Ψ(i) i +

X |hΨ(i) |V̂ |Ψ(j) i|2

(11.54)

Û (i) − Û (j)

j6=i

The denominator in Eq. (11.54) can be zero either because of symmetry or accidental degeneracy. Various solutions,
which depend on the type of degeneracy that occurs, have been developed which ignore the zero-denominator elements
from the Hamiltonian. 9,35,94,99 An alternative solution has been proposed by Barone, 12 which can be applied to all
molecules by changing the masses of one or more nuclei in degenerate cases. The disadvantage of this method is that
it will break the degeneracy which results in fundamental frequencies no longer retaining their correct symmetry. He
proposed
X
X
EiVPT2 =
ωj (nj + 1/2) +
xij (ni + 1/2)(nj + 1/2)
(11.55)
j

i≤j

where, if rotational coupling is ignored, the anharmonic constants xij are given by
1
xij =
4ωi ωj

11.11.3

ηiijj −

m
X
ηiik ηjjk
k

ωk2

+

m
X
k

2
2(ωi2 + ωj2 − ωk2 )ηijk
[(ωi + ωj )2 − ωk2 ] [(ωi − ωj )2 − ωk2 ]

!
(11.56)

Transition-Optimized Shifted Hermite Theory

So far, every aspect of solving the nuclear wave equation has been considered, except the wave function. Since
Schrödinger proposed his equation, the nuclear wave function has traditionally be expressed in terms of Hermite functions, which are designed for the harmonic oscillator case. Recently a modified representation has been presented. 86 To
demonstrate how this approximation works, we start with a simple example. For a diatomic molecule, the Hamiltonian
with up to quartic derivatives can be written as
Ĥ = −

1
1 ∂2
+ ω 2 q 2 + ηiii q 3 + ηiiii q 4
2 ∂q 2
2

(11.57)

and the wave function is expressed as in Eq. (11.48). Now, if we shift the center of the wave function by σ, which
is equivalent to a translation of the normal coordinate q, the shape will still remain the same, but the anharmonic
correction can now be incorporated into the wave function. For a ground vibrational state, the wave function is written
as
 ω  14 ω
2
φ(0) =
e− 2 (q−σ)
(11.58)
π
Similarly, for the first excited vibrational state, we have
φ

(1)


=

4ω 3
π

 14

ω

2

(q − σ) e 2 (q−σ)

(11.59)

609

Chapter 11: Molecular Properties and Analysis
Therefore, the energy difference between the first vibrational excited state and the ground state is
∆ETOSH = ω +

ηiii σ ηiiii σ 2
ηiiii
+
+
8ω 2
2ω
4ω

(11.60)

This is the fundamental vibrational frequency from first-order perturbation theory.
Meanwhile, We know from the first-order perturbation theory with an ordinary wave function within a QFF PES, the
energy is
ηiiii
(11.61)
∆EVPT1 = ω +
8ω 2
The differences between these two wave functions are the two extra terms arising from the shift in Eq. (11.60). To
determine the shift, we compare the energy with that from second-order perturbation theory:
∆EVPT2 = ω +

ηiiii
5ηiii 2
−
2
8ω
24ω 4

(11.62)

Since σ is a very small quantity compared with the other variables, we ignore the contribution of σ 2 and compare
∆ETOSH with ∆EVPT2 , which yields an initial guess for σ:
σ=−

5 ηiii
12 ω 3

(11.63)

Because the only difference between this approach and the ordinary wave function is the shift in the normal coordinate,
we call it “transition-optimized shifted Hermite” (TOSH) functions. 86 This approximation gives second-order accuracy
at only first-order cost.
For polyatomic molecules, we consider Eq. (11.60), and propose that the energy of the ith mode be expressed as:
∆EiTOSH = ωi +

1 X ηiijj
1 X
1 X
ηiij σij +
ηiijk σij σik
+
8ωi j ωj
2ωi j
4ωi

(11.64)

j,k

Following the same approach as for the diatomic case, by comparing this with the energy from second-order perturbation theory, we obtain the shift as
σij =

(δij − 2)(ωi + ωj )ηiij X ηkkj
−
4ωi ωj2 (2ωi + ωj )
4ωk ωj2
k

11.11.4

Job Control

The following $rem variables can be used to control the calculation of anharmonic frequencies.
ANHAR
Performing various nuclear vibrational theory (TOSH, VPT2, VCI) calculations to obtain vibrational anharmonic frequencies.
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
TRUE
Carry out the anharmonic frequency calculation.
FALSE Do harmonic frequency calculation.
RECOMMENDATION:
Since this calculation involves the third and fourth derivatives at the minimum of the
potential energy surface, it is recommended that the GEOM_OPT_TOL_DISPLACEMENT,
GEOM_OPT_TOL_GRADIENT and GEOM_OPT_TOL_ENERGY tolerances are set tighter. Note
that VPT2 calculations may fail if the system involves accidental degenerate resonances. See the
VCI $rem variable for more details about increasing the accuracy of anharmonic calculations.

(11.65)

Chapter 11: Molecular Properties and Analysis
VCI
Specifies the number of quanta involved in the VCI calculation.
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
User-defined. Maximum value is 10.
RECOMMENDATION:
The availability depends on the memory of the machine. Memory allocation for VCI calculation
is the square of 2(NVib + NVCI )/NVib NVCI with double precision. For example, a machine
with 1.5 Gb memory and for molecules with fewer than 4 atoms, VCI(10) can be carried out,
for molecule containing fewer than 5 atoms, VCI(6) can be carried out, for molecule containing
fewer than 6 atoms, VCI(5) can be carried out. For molecules containing fewer than 50 atoms,
VCI(2) is available. VCI(1) and VCI(3) usually overestimated the true energy while VCI(4)
usually gives an answer close to the converged energy.
FDIFF_DER
Controls what types of information are used to compute higher derivatives. The default uses a
combination of energy, gradient and Hessian information, which makes the force field calculation
faster.
TYPE:
INTEGER
DEFAULT:
3 for jobs where analytical 2nd derivatives are available.
0 for jobs with ECP.
OPTIONS:
0 Use energy information only.
1 Use gradient information only.
2 Use Hessian information only.
3 Use energy, gradient, and Hessian information.
RECOMMENDATION:
When the molecule is larger than benzene with small basis set, FDIFF_DER = 2 may be faster.
Note that FDIFF_DER will be set lower if analytic derivatives of the requested order are not
available. Please refers to IDERIV.
MODE_COUPLING
Number of modes coupling in the third and fourth derivatives calculation.
TYPE:
INTEGER
DEFAULT:
2 for two modes coupling.
OPTIONS:
n for n modes coupling, Maximum value is 4.
RECOMMENDATION:
Use the default.

610

Chapter 11: Molecular Properties and Analysis
IGNORE_LOW_FREQ
Low frequencies that should be treated as rotation can be ignored during
anharmonic correction calculation.
TYPE:
INTEGER
DEFAULT:
300 Corresponding to 300 cm−1 .
OPTIONS:
n Any mode with harmonic frequency less than n will be ignored.
RECOMMENDATION:
Use the default.
FDIFF_STEPSIZE_QFF
Displacement used for calculating third and fourth derivatives by finite difference.
TYPE:
INTEGER
DEFAULT:
5291 Corresponding to 0.1 bohr. For calculating third and fourth derivatives.
OPTIONS:
n Use a step size of n × 10−5 .
RECOMMENDATION:
Use the default, unless the potential surface is very flat, in which case a larger value should be
used.

611

612

Chapter 11: Molecular Properties and Analysis

Example 11.18 A four-quanta anharmonic frequency calculation on formaldehyde at the EDF2/6-31G* optimized
ground state geometry, which is obtained in the first part of the job. It is necessary to carry out the harmonic frequency
first and this will print out an approximate time for the subsequent anharmonic frequency calculation. If a FREQ job has
already been performed, the anharmonic calculation can be restarted using the saved scratch files from the harmonic
calculation.
$molecule
0 1
C
O, 1, CO
H, 1, CH, 2, A
H, 1, CH, 2, A, 3, D
CO = 1.2
CH = 1.0
A = 120.0
D = 180.0
$end
$rem
JOBTYPE
METHOD
BASIS
GEOM_OPT_TOL_DISPLACEMENT
GEOM_OPT_TOL_GRADIENT
GEOM_OPT_TOL_ENERGY
$end
@@@
$molecule
READ
$end
$rem
JOBTYPE
METHOD
BASIS
ANHAR
VCI
$end

11.12

OPT
EDF2
6-31G*
1
1
1

FREQ
EDF2
6-31G*
TRUE
4

Linear-Scaling Computation of Electric Properties

The search for new optical devices is a major field of materials sciences. Here, polarizabilities and hyperpolarizabilities
provide particularly important information on molecular systems. The response of the molecular systems in the presence of an external, monochromatic, oscillatory electric field is determined by the solution of the time-dependent SCF
(TDSCF) equations. Within the dipole approximation, the perturbation is represented as the interaction of the molecule
with a single Fourier component of the external field, E:
Ĥfield = 21 µ̂ · E(e−iωt + e+iωt )
with
µ̂ = −e

N
elec
X

r̂i .

(11.66)

(11.67)

i

Here, ω is the field frequency and µ̂ is the dipole moment operator. The TDSCF equations can be solved via standard
techniques of perturbation theory. 128 As a solution, one obtains the first-order perturbed density matrix [Px (±ω)] and
the second-order perturbed density matrices [Pxy (±ω, ±ω 0 )]. From these quantities, the following properties can be
calculated:

Chapter 11: Molecular Properties and Analysis

613



• Static polarizability: αxy (0; 0) = tr Hµx Py (ω = 0)


• Dynamic polarizability: αxy (±ω; ∓ω) = tr Hµx Py (±ω)


• Static hyperpolarizability: βxyz (0; 0, 0) = tr Hµx Pyz (ω = 0, ω = 0)


• Second harmonic generation: βxyz (∓2ω; ±ω, ±ω) = tr Hµx Pyz (±ω, ±ω)


• Electro-optical Pockels effect: βxyz (∓ω; 0, ±ω) = tr Hµx Pyz (ω = 0, ±ω)


• Optical rectification: βxyz (0; ±ω, ∓ω) = tr Hµx Pyz (±ω, ∓ω)
Here, Hµx is the matrix representation of the x component of the dipole moment.
Note that third-order properties (βxyz ) can be computed either with the equations above, which is based on a secondorder TDSCF calculation (for Pyz ), or alternatively from first-order properties using Wigner’s 2n + 1 rule. 75 The
second-order approach corresponds to MOPROP job numbers 101 and 102 (see below) whereas use of the 2n + 1 rule
corresponds to job numbers 103 and 104. Solution of the second-order TDSCF equations depends upon first-order
results and therefore convergence can be more problematic as compared to the first-order calculation. For this reason,
we recommend job numbers 103 and 104 for the calculation of first hyperpolarizabilities.
The TDSCF calculation is more time-consuming than the SCF calculation that precedes it (where the field-free, unperturbed ground state of the molecule is obtained). Q-C HEM’s implementation of the TDSCF equations is MO based
and the cost therefore formally scales asymptotically as O(N 3 ). The prefactor of the cubic-scaling step is rather small,
however, and in practice (over a wide range of molecular sizes) the calculation is dominated by the cost of contractions
with two-electron integrals, which is formally O(N 2 ) scaling but with a very large prefactor. The cost of these integral
contractions can be reduced from quadratic to O(N ) using LinK/CFMM methods (Section 4.6). 80 All derivatives are
computed analytically.
The TDSCF module in Q-C HEM is know as “MOProp", since it corresponds (formally) to time propagation of the
molecular orbitals. (For actual time propagation of the MOs, see Section 7.11.) The MOProp module has the following
features:
• LinK and CFMM support to evaluate Coulomb- and exchange-like matrices
• Analytic derivatives
• DIIS acceleration
• Both restricted and unrestricted implementations of CPSCF and TDSCF equations are available, for both HartreeFock and Kohn-Sham DFT.
• Support for LDA, GGA, and global hybrid functionals. Meta-GGA and range-separated functionals are not yet
supported, nor are functionals that contain non-local correlation (e.g., those containing VV10).

11.12.1 $fdpfreq Input Section
For dynamic response properties (i.e., ω 6= 0), various values of ω might be of interest, and it is considerably cheaper to
compute properties for multiple values of ω in a single calculation than it is to run several calculations for one frequency
each. The $fdpfreq input section is used to specify the frequencies of interest. The format is:
$fdpfreq
property
frequencies
units
$end

Chapter 11: Molecular Properties and Analysis

614

The first line is only required for third-order properties, to specify the flavor of first hyperpolarizability. The options
are
• StaticHyper (static hyperpolarizability)
• SHG (second harmonic generation)
• EOPockels (electro-optical Pockels effect)
• OptRect (optical rectification)
The second line in the $fdpfreq section contains floating-point values representing the frequencies of interest. Alternatively, for dynamic polarizabilities an equidistant sequence of frequencies can be specified by the keyword WALK (see
example below). The last line specifies the units of the input frequencies. Options are:
• au (atomic units of frequency)
• eV (frequency units, expressed in electron volts)
• Hz (frequency units, expressed in Hertz)
• nm (wavelength units, in nanometers)
• cmInv (wavenumber units, cm−1 )
Example 11.19 Static and dynamic polarizabilities, atomic units:
$fdpfreq
0.0 0.03 0.05
au
$end

Example 11.20 Series of dynamic polarizabilities, starting with 0.00 incremented by 0.01 up to 0.10:
$fdpfreq
walk 0.00 0.10 0.01
au
$end

Example 11.21 Static first hyperpolarizability, second harmonic generation and electro-optical Pockels effect, wavelength in nm:
$fdpfreq
StaticHyper SHG EOPockels
1064
nm
$end

11.12.2

Job Control for the MOProp Module

The MOProp module is invoked by specifying a job number using the MOPROP $rem variable. In addition to electric properties, this module can also compute NMR chemical shifts (MOPROP = 1); this functionality is described in
Section 11.13.

Chapter 11: Molecular Properties and Analysis
MOPROP
Specifies the job number for MOProp module.
TYPE:
INTEGER
DEFAULT:
0 Do not run the MOProp module.
OPTIONS:
1
NMR chemical shielding tensors.
2
Static polarizability.
3
Indirect nuclear spin–spin coupling tensors.
100 Dynamic polarizability.
101 First hyperpolarizability.
102 First hyperpolarizability, reading First order results from disk.
103 First hyperpolarizability using Wigner’s 2n + 1 rule.
104 First hyperpolarizability using Wigner’s 2n + 1 rule, reading
first order results from disk.
RECOMMENDATION:
None
MOPROP_PERTNUM
Set the number of perturbed densities that will to be treated together.
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
0 All at once.
n Treat the perturbed densities batch-wise.
RECOMMENDATION:
Use the default. For large systems, limiting this number may be required to avoid memory
exhaustion.
MOPROP_CONV_1ST
Sets the convergence criteria for CPSCF and 1st order TDSCF.
TYPE:
INTEGER
DEFAULT:
6
OPTIONS:
n < 10 Convergence threshold set to 10−n .
RECOMMENDATION:
None

615

Chapter 11: Molecular Properties and Analysis
MOPROP_CONV_2ND
Sets the convergence criterion for second-order TDSCF.
TYPE:
INTEGER
DEFAULT:
6
OPTIONS:
n < 10 Convergence threshold set to 10−n .
RECOMMENDATION:
None
MOPROP_MAXITER_1ST
The maximum number of iterations for CPSCF and first-order TDSCF.
TYPE:
INTEGER
DEFAULT:
50
OPTIONS:
n Set maximum number of iterations to n.
RECOMMENDATION:
Use the default.
MOPROP_MAXITER_2ND
The maximum number of iterations for second-order TDSCF.
TYPE:
INTEGER
DEFAULT:
50
OPTIONS:
n Set maximum number of iterations to n.
RECOMMENDATION:
Use the default.
MOPROP_ISSC_PRINT_REDUCED
Specifies whether the isotope-independent reduced coupling tensor K should be printed in addition to the isotope-dependent J-tensor when calculating indirect nuclear spin-spin couplings.
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
FALSE Do not print K.
TRUE
Print K.
RECOMMENDATION:
None

616

Chapter 11: Molecular Properties and Analysis
MOPROP_ISSC_SKIP_FC
Specifies whether to skip the calculation of the Fermi contact contribution to the indirect nuclear
spin-spin coupling tensor.
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
FALSE Calculate Fermi contact contribution.
TRUE
Skip Fermi contact contribution.
RECOMMENDATION:
None
MOPROP_ISSC_SKIP_SD
Specifies whether to skip the calculation of the spin-dipole contribution to the indirect nuclear
spin-spin coupling tensor.
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
FALSE Calculate spin-dipole contribution.
TRUE
Skip spin-dipole contribution.
RECOMMENDATION:
None
MOPROP_ISSC_SKIP_PSO
Specifies whether to skip the calculation of the paramagnetic spin-orbit contribution to the indirect nuclear spin-spin coupling tensor.
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
FALSE Calculate paramagnetic spin-orbit contribution.
TRUE
Skip paramagnetic spin-orbit contribution.
RECOMMENDATION:
None
MOPROP_ISSC_SKIP_DSO
Specifies whether to skip the calculation of the diamagnetic spin-orbit contribution to the indirect
nuclear spin-spin coupling tensor.
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
FALSE Calculate diamagnetic spin-orbit contribution.
TRUE
Skip diamagnetic spin-orbit contribution.
RECOMMENDATION:
None

617

Chapter 11: Molecular Properties and Analysis
MOPROP_DIIS
Controls the use of Pulay’s DIIS in solving the CPSCF equations.
TYPE:
INTEGER
DEFAULT:
5
OPTIONS:
0 Turn off DIIS.
5 Turn on DIIS.
RECOMMENDATION:
None
MOPROP_DIIS_DIM_SS
Specified the DIIS subspace dimension.
TYPE:
INTEGER
DEFAULT:
20
OPTIONS:
0 No DIIS.
n Use a subspace of dimension n.
RECOMMENDATION:
None
SAVE_LAST_GPX
Save the last G[Px ] when calculating dynamic polarizabilities in order to call the MOProp code
in a second run, via MOPROP = 102.
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
0 False
1 True
RECOMMENDATION:
None
MOPROP_RESTART
Specifies the option for restarting MOProp calculations.
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
0 Not a restart calculation.
1 Restart from a previous calculation using the same scratch directory.
RECOMMENDATION:
Need to also include "SCF_GUESS READ" and "SKIP_SCFMAN TRUE" to ensure the same
set of MOs.

618

Chapter 11: Molecular Properties and Analysis

11.12.3

619

Examples

Example 11.22 ωB97X-D/def2-SVPD static polarizability calculation for water cation, computed analytically using
the MOProp module
$rem
method
basis
scf_convergence
thresh
symmetry
sym_ignore
moprop
moprop_conv_1st
moprop_maxiter_1st
$end
$molecule
1 2
O 0.003 1.517
H 0.913 1.819
H 0.081 0.555
$end

11.13

hf
def2-svpd
11
14
false
true
2
8
200

0.000
0.000
0.000

NMR and Other Magnetic Properties

The importance of nuclear magnetic resonance (NMR) spectroscopy for modern chemistry and biochemistry cannot be
overestimated. Since there is no direct relationship between the measured NMR signals and structural properties, the
necessity for a reliable method to predict NMR chemical shifts arises and despite tremendous progress in experimental
techniques, the understanding and reliable assignment of observed experimental spectra remains often a highly difficult
task. As such, quantum chemical methods can be extremely useful, both in solution and in the solid state. 27,101,103–105
Features of Q-C HEM’s NMR package include:
• Restricted Hartree-Fock and DFT calculations of NMR chemical shifts using gauge-including atomic orbitals.
• Support of linear-scaling CFMM and LinK procedures (Section 4.6) to evaluate Coulomb- and exchange-like
matrices.
• Density matrix-based coupled-perturbed SCF approach for linear-scaling NMR calculations.
• DIIS acceleration.
• Support for basis sets up to d functions.
• Support for LDA, GGA, and global hybrid functionals. Meta-GGA and range-separated functionals are not yet
supported, nor are functionals that contain non-local correlation (e.g., those containing VV10).
Calculation of NMR chemical shifts and indirect spin-spin couplings is discussed in Section 11.13.1. Additional
magnetic properties can be computed, as described in Section 11.13.3. These include hyperfine interaction tensors
(electron spin–nuclear spin interaction) and nuclear quadrupole interactions with electric field gradients.

11.13.1

NMR Chemical Shifts and J-Couplings

NMR calculations are available at both the Hartree-Fock and DFT levels of theory. 58,140 Q-C HEM computes NMR
chemical shielding tensors using gauge-including atomic orbitals 40,54,156 (GIAOs), an approach that has proven to

620

Chapter 11: Molecular Properties and Analysis

reliable and accurate for many applications. 45,57 The shielding tensor σ is a second-order property that depends upon
the external magnetic field, B, and the spin angular momentum m for a given nucleus:
∆E = −m · (1 − σ) · B .
Using analytical derivative techniques to evaluate σ, the components of this 3 × 3 tensor are computed as
 X
 2
X
∂Pµν ∂hµν
∂ hµν
+
σij =
Pµν
∂B
∂m
∂Bi ∂mj
i
j
µν
µν

(11.68)

(11.69)

where i, j ∈ {x, y, z} indicate Cartesian components. Note that there is a separate chemical shielding tensor for each
m, that is, for each nucleus. To compute σij it is necessary to solve coupled-perturbed SCF (CPSCF) equations to
obtain the perturbed densities ∂P/∂Bi , which can be accomplished using the MO-based “MOProp” module whose
use is described below. (Use of the MOProp module to compute optical properties of molecules was discussed in
Section 11.12.) Alternatively, a linear-scaling, density matrix-based CPSCF (D-CPSCF) formulation is available, 80,105
which is described in Section 11.13.2.
In addition to chemical shifts, indirect nuclear spin-spin coupling constants, also known as scalar couplings or Jcouplings, can be computed at the SCF level. The coupling tensor JAB between atoms A and B is evaluated as the
second derivative of the electronic energy with respect to the nuclear magnetic moments m:
JAB =

∂2E
.
∂mA ∂mB

(11.70)

The indirect coupling tensor has five distinct contributions. The diamagnetic spin-orbit (DSO) contribution is calculated
as an expectation value with the ground state wave function. The other contributions are the paramagnetic spin-orbit
(PSO), spin-dipole (SD), Fermi contact (FC), and mixed SD/FC contributions. These terms require the electronic
response of the systems to the perturbation due to the magnetic nuclei. Ten distinct CPSCF equations must be solved
for each perturbing nucleus, which makes the calculation of J-coupling constants more time-consuming than that of
chemical shifts.
Some authors have recommended calculating only the Fermi contact contribution, 10 and skipping the other contributions, for 1 H-1 H coupling constants. For that purpose, Q-C HEM allows the user to skip calculation of any of the four
contributions: (FC, SD, PSO, or DSO. (The mixed SD/FC contributions is automatically calculated at no additional
cost whenever both the SD and FC contributions are computed.) See Section 11.12.2 for details. Note that omitting
any of the contributions cannot be rationalized from a theoretical point of view. Results from such calculations should
be interpreted extremely cautiously.
Note: (1) Specialized basis sets are highly recommended in any J-coupling calculation. The pcJ-n basis set family 70
has been added to the basis set library.
(2) The Hartree-Fock level of theory is not suitable to obtain J-coupling constants of any degree of reliability.
Use GGA or hybrid density functionals instead.
11.13.1.1

NMR Job Control and Examples

This section describes the use of Q-C HEM’s MO-based CPSCF code, which is contained in the “MOProp” module that
is also responsible for computing electric properties. NMR chemical shifts are requested by setting MOPROP = 1, and
J-couplings by setting JOBTYPE = ISSC. The reader is referred to to Section 11.12.2 for additional job control variables
associated with the MOProp module, as well as explanations of the ones that are invoked in the samples below. An
alternative, O(N ) density matrix-based implementation of NMR chemical shifts is also available and is described in

621

Chapter 11: Molecular Properties and Analysis
Section 11.13.2. Setting JOBTYPE = NMR invokes the density-based code, not the MO-based code.
Example 11.23 MO-based NMR calculation.
$molecule
0 1
H
C
F
F
F
$end

0.00000
1.10000
1.52324
1.52324
1.52324

0.00000
0.00000
1.22917
-0.61459
-0.61459

0.00000
0.00000
0.00000
1.06450
-1.06450

$rem
METHOD
B3LYP
BASIS
6-31G*
MOPROP
1
MOPROP_PERTNUM
0 ! do all perturbations at once
MOPROP_CONV_1ST
7 ! sets the CPSCF convergence threshold
MOPROP_DIIS_DIM_SS
4 ! no. of DIIS subspace vectors
MOPROP_MAXITER_1ST 100 ! max iterations
MOPROP_DIIS
5 ! turns on DIIS (=0 to turn off)
MOPROP_DIIS_THRESH
1
MOPROP_DIIS_SAVE
0
$end

In the following compound job, we show how to restart an NMR calculation should it exceed the maximum number
of CPSCF iterations (specified with MOPROP_MAXITER_1ST, or should the calculation run out of time on a shared
computer resource. Note that the first job is intentionally set up to exceed the maximum number of iterations, so will

622

Chapter 11: Molecular Properties and Analysis
crash. However, the calculation is restarted and completed in the second job.
Example 11.24 Illustrates how to restart an NMR calculation.
$comment
In this first job, we *intentionally* set the max number of iterations
too small, to force premature end so that we can demonstrate restart
capability in the 2nd job.
$end
$molecule
0 1
H
C
F
F
F
$end

0.00000
1.10000
1.52324
1.52324
1.52324

0.00000
0.00000
1.22917
-0.61459
-0.61459

0.00000
0.00000
0.00000
1.06450
-1.06450

$rem
METHOD
B3LYP
BASIS
6-31G*
SCF_ALGORITHM
DIIS
MOPROP
1
MOPROP_MAXITER_1ST 10
! too small, for demonstration only
GUESS_PX
1
MOPROP_DIIS_SAVE
0
! don’t hang onto the subspace vectors
$end
@@@
$molecule
0 1
H
C
F
F
F
$end

0.00000
1.10000
1.52324
1.52324
1.52324

$rem
METHOD
BASIS
SCF_GUESS
SKIP_SCFMAN
MOPROP
MOPROP_RESTART
MOPROP_MAXITER_1ST
GUESS_PX
MOPROP_DIIS_SAVE
$end

0.00000
0.00000
1.22917
-0.61459
-0.61459

0.00000
0.00000
0.00000
1.06450
-1.06450

B3LYP
6-31G*
READ
TRUE
! no need to redo the SCF
1
1
100 ! more reasonable choice
1
0

Example 11.25 J-coupling calculation: water molecule with B3LYP/cc-pVDZ

Chapter 11: Molecular Properties and Analysis

623

$molecule
0 1
O
H1 O OH
H2 O OH H1 HOH
OH = 0.947
HOH = 105.5
$end
$rem
JOBTYPE
EXCHANGE
BASIS
LIN_K
SYMMETRY
MOPROP_CONV_1ST
$end

11.13.1.2

ISSC
B3LYP
cc-pVDZ
FALSE
TRUE
6

Nucleus-Independent Chemical Shifts: Probes of Aromaticity

Unambiguous theoretical estimates of degree of aromaticity are still on high demand. The NMR chemical shift methodology offers one unique probe of aromaticity based on one defining characteristics of an aromatic system: its ability
to sustain a diatropic ring current. This leads to a response to an imposed external magnetic field with a strong (negative) shielding at the center of the ring. Schleyer et al. have employed this phenomenon to justify a new unique
probe of aromaticity. 144 They proposed the computed absolute magnetic shielding at ring centers (unweighted mean of
the heavy-atoms ring coordinates) as a new aromaticity criterion, called nucleus-independent chemical shift (NICS).
Aromatic rings show strong negative shielding at the ring center (negative NICS), while anti-aromatic systems reveal
positive NICS at the ring center. As an example, a typical NICS value for benzene is about −11.5 ppm as estimated
with Q-C HEM at the Hartree-Fock/6-31G* level. The same NICS value for benzene was also reported in Ref. 144.
The calculated NICS value for furan of −13.9 ppm with Q-C HEM is about the same as the value reported for furan in
Ref. 144. Below is one input example of how to the NICS of furan with Q-C HEM, using the ghost atom option. The
ghost atom is placed at the center of the furan ring, and the basis set assigned to it within the basis mix option must be
the basis used for hydrogen atom.

624

Chapter 11: Molecular Properties and Analysis

Example 11.26 Calculation of the NMR NICS probe of furan, HF/6-31G* level. Note the ghost atom at the center of
the ring.
$molecule
0 1
C
-0.69480
C
0.72110
C
1.11490
O
0.03140
C
-1.06600
H
2.07530
H
1.37470
H
-1.36310
H
-2.01770
GH
0.02132
$end
$rem
JOBTYPE
METHOD
BASIS
SCF_ALGORITHM
PURCAR
SEPARATE_JK
LIN_K
CFMM_ORDER
GRAIN
CFMM_PRINT
CFMMSTAT
PRINT_PATH_TIME
LINK_MAXSHELL_NUMBER
SKIP_SCFMAN
IGUESS
SCF_CONVERGENCE
ITHRSH
IPRINT
D_SCF_CONVGUIDE
D_SCF_METRIC
D_SCF_STORAGE
D_SCF_RESTART
PRINT_PATH_TIME
SYM_IGNORE
NO_REORIENT
$end

-0.62270
-0.63490
0.68300
1.50200
0.70180
1.17930
-1.49560
-1.47200
1.21450
0.32584

$basis
C 1
6-31G*
****
C 2
6-31G*
****
C 3
6-31G*
****
O 4
6-31G*
****
C 5
6-31G*
****
H 6
6-31G*
****
H 7
6-31G*
****

NMR
HF
mixed
DIIS
111
0
0
15
1
2
1
1
1
0
core
7
10
23
0
2
50
0
1
1
1

-0.00550
0.00300
0.00750
0.00230
-0.00560
0.01410
0.00550
-0.01090
-0.01040
0.00034

Chapter 11: Molecular Properties and Analysis

11.13.2

625

Linear-Scaling NMR Chemical Shift Calculations

In conventional implementations, the cost for computation of NMR chemical shifts within even the simplest quantum
chemical methods such as Hartree-Fock of DFT increases cubically with molecular size M , O(M 3 ). As such, NMR
chemical shift calculations have largely been limited to molecular systems on the order of 100 atoms, assuming no
symmetry. For larger systems it is crucial to reduce the increase of the computational effort to linear, which is possible
for systems with a nonzero HOMO/LUMO gaps and was reported for the first time by Kussmann and Ochsenfeld. 79,105
This approach incurs no loss of accuracy with respect to traditional cubic-scaling implementations, and makes feasible
NMR chemical shift calculations using Hartree-Fock or DFT approaches in molecular systems with 1,000+ atoms. For
many molecular systems the Hartree-Fock (GIAO-HF) approach provides typically an accuracy of 0.2–0.4 ppm for the
computation of 1 H NMR chemical shifts, for example. 27,101,103–105 GIAO-HF/6-31G* calculations with 1,003 atoms
and 8,593 basis functions, without symmetry, have been reported. 105 GIAO-DFT calculations are even simpler and
faster for density functionals that do not contain Hartree-Fock exchange.
The present implementation of NMR shieldings employs the LinK (linear exchange, “K”) method 100,102 for the formation of exchange contributions. 105 Since the derivative of the density matrix with respect to the magnetic field is skewsymmetric, its Coulomb-type contractions vanish. For the remaining Coulomb-type matrices the CFMM method 151 is
used. 105 In addition, a multitude of different approaches for the solution of the CPSCF equations can be selected within
Q-C HEM.
To request a NMR chemical shift calculation using the density matrix approach, set JOBTYPE to NMR in the $rem
section. Additional job-control variables can be found below.
D_CPSCF_PERTNUM
Specifies whether to do the perturbations one at a time, or all together.
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
0 Perturbed densities to be calculated all together.
1 Perturbed densities to be calculated one at a time.
RECOMMENDATION:
None
D_SCF_CONV_1
Sets the convergence criterion for the level-1 iterations. This preconditions the density for the
level-2 calculation, and does not include any two-electron integrals.
TYPE:
INTEGER
DEFAULT:
4 corresponding to a threshold of 10−4 .
OPTIONS:
n < 10 Sets convergence threshold to 10−n .
RECOMMENDATION:
The criterion for level-1 convergence must be less than or equal to the level-2 criterion, otherwise
the D-CPSCF will not converge.

Chapter 11: Molecular Properties and Analysis
D_SCF_CONV_2
Sets the convergence criterion for the level-2 iterations.
TYPE:
INTEGER
DEFAULT:
4 Corresponding to a threshold of 10−4 .
OPTIONS:
n < 10 Sets convergence threshold to 10−n .
RECOMMENDATION:
None
D_SCF_MAX_1
Sets the maximum number of level-1 iterations.
TYPE:
INTEGER
DEFAULT:
100
OPTIONS:
n User defined.
RECOMMENDATION:
Use the default.
D_SCF_MAX_2
Sets the maximum number of level-2 iterations.
TYPE:
INTEGER
DEFAULT:
30
OPTIONS:
n User defined.
RECOMMENDATION:
Use the default.
D_SCF_DIIS
Specifies the number of matrices to use in the DIIS extrapolation in the D-CPSCF.
TYPE:
INTEGER
DEFAULT:
11
OPTIONS:
n n = 0 specifies no DIIS extrapolation is to be used.
RECOMMENDATION:
Use the default.

626

627

Chapter 11: Molecular Properties and Analysis

Example 11.27 NMR chemical shifts via the D-CPSCF method, showing all input options.
$molecule
0 1
H
0.00000
C
1.10000
F
1.52324
F
1.52324
F
1.52324
$end
$rem
JOBTYPE
EXCHANGE
BASIS
D_CPSCF_PERTNUM
D_SCF_SOLVER
D_SCF_CONV_1
D_SCF_CONV_2
D_SCF_MAX_1
D_SCF_MAX_2
D_SCF_DIIS
D_SCF_ITOL
$end

11.13.3

0.00000
0.00000
1.22917
-0.61459
-0.61459

NMR
B3LYP
6-31G*
0
D-CPSCF
430 D-SCF
4
D-SCF
4
D-SCF
200 D-SCF
50
D-SCF
11
D-SCF
2
D-SCF

0.00000
0.00000
0.00000
1.06450
-1.06450

number of perturbations at once
leqs_solver
leqs_conv1
leqs_conv2
maxiter level 1
maxiter level 2
DIIS
conv. criterion

Additional Magnetic Field-Related Properties

It is now possible to calculate certain open-shell magnetic field-related properties in Q-C HEM. One is the hyperfine
interaction (HFI) tensor, describing the interaction of unpaired electron spin with an atom’s nuclear spin levels:
FC
SD
Atot
ab (N ) = Aab (N )δab + Aab (N ),

(11.71)

where the Fermi contact (FC) contribution is
AFC (N ) =

X
α 1 8π
α−β
ge gN µN
Pµν
hχµ |δ(rN )|χν i
2S 3
µν

(11.72)

and the spin-dipole (SD) contribution is
ASD
ab (N )

X
α1
α−β
=
ge gN µN
Pµν
2S
µν


χµ

2
3rN,a rN,b − δab rN
χν
5
rN


(11.73)

for a nucleus N .
Another sensitive probe of the individual nuclear environments in a molecule is the nuclear quadrupolar interaction
(NQI), arising from the interaction of a nuclei’s quadrupole moment with an applied electric field gradient (EFG),
calculated as
∂ 2 VN N
∂ 2 VeN
+
Qab (N ) =
∂XN,a ∂XN,b
∂XN,a ∂XN,b


2
X
3rN,a rN,b − δab rN
α+β
=−
Pµν
χµ
χν
5
(11.74)
rN
µν
+

X
A6=N

ZA

2
3RAN,a RAN,b − δab RAN
5
RAN

for a nucleus N . Diagonalizing the tensor gives three principal values, ordered |Q1 | ≤ |Q2 | ≤ |Q3 |, which are
components of the asymmetry parameter eta:
Q1 − Q2
η=
(11.75)
Q3

Chapter 11: Molecular Properties and Analysis

628

Both the hyperfine and EFG tensors are automatically calculated for all possible nuclei. All SCF-based methods
(HF and DFT) are available with restricted and unrestricted references. Restricted open-shell references and post-HF
methods are unavailable.

11.13.3.1

Job Control and Examples

Only one keyword is necessary in the $rem section to activate the magnetic property module.
MAGNET
Activate the magnetic property module.
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
FALSE (or 0) Don’t activate the magnetic property module.
TRUE (or 1)
Activate the magnetic property module.
RECOMMENDATION:
None.

All other options are controlled through the $magnet input section, which has the same key-value format as the $rem
section (see section 3.4). Current options are:
HYPERFINE
Activate the calculation of hyperfine interaction tensors.
INPUT SECTION: $magnet
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
FALSE (or 0) Don’t calculate hyperfine interaction tensors.
TRUE (or 1)
Calculate hyperfine interaction tensors.
RECOMMENDATION:
None. Due to the nature of the property, which requires the spin density ρα−β (r) ≡
ρα (r) − ρβ (r), this is not meaningful for restricted (RHF) references. Only UHF (not
ROHF) is available.
ELECTRIC
Activate the calculation of electric field gradient tensors.
INPUT SECTION: $magnet
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
FALSE (or 0) Don’t calculate EFG tensors and nuclear quadrupole parameters.
TRUE (or 1)
Calculate EFG tensors and nuclear quadrupole parameters.
RECOMMENDATION:
None.

629

Chapter 11: Molecular Properties and Analysis

Example 11.28 Calculating hyperfine and EFG tensors for the glycine cation.
$rem
method = hf
basis = def2-sv(p)
scf_convergence = 11
thresh = 14
symmetry = false
sym_ignore = true
magnet = true
$end
$magnet
hyperfine = true
electric = true
$end
$molecule
1 2
N
0.0000000000
C
1.4467530000
C
1.9682482963
O
1.2385450522
H
1.7988742211
H
1.7997303368
H
-0.4722340827
H
-0.5080000000
O
3.3107284257
H
3.9426948542
$end

11.14

0.0000000000
0.0000000000
0.0000000000
0.0000000000
-0.8959881458
0.8930070757
-0.0025218132
0.0766867527
-0.0000000000
-0.0000000000

0.0000000000
0.0000000000
1.4334965024
2.4218667010
-0.5223754133
-0.5235632630
0.8996536532
-0.8765335943
1.5849828121
0.7289954096

Finite-Field Calculation of (Hyper)Polarizabilities

↔
~
↔
The dipole moment vector (~
µ), polarizability tensor (α), first hyperpolarizability ( β), and higher-order hyperpolarizabilities determine the response of the system to an applied electric field:

E(F~ ) = E(0) − µ
~ (0) · F~ −

~.
1↔ ~~
1↔
α : F F − β ..F~ F~ F~ − · · · .
2!
3!

(11.76)

The various polarizability tensor elements are therefore derivatives of the energy with respect to one or more electric
fields, which might be frequency-dependent (dynamic polarizabilities) or not (static polarizabilities). The most efficient
way to compute these properties is by analytic gradient techniques, assuming that the required derivatives have been
implemented at the desired level of theory. For DFT calculations using LDA, GGAs, or global hybrid functionals the
requisite analytic gradients have been implemented and their use to compute static and dynamic (hyper)polarizabilities
is described in Section 11.12.

11.14.1

Numerical Calculation of Static Polarizabilities

Where analytic gradients are not available, static polarizabilities (only) can be computed via finite-difference in the
applied field, which is known as the finite field (FF) approach. Beginning with Q-C HEM 5.1, a sophisticated “Romberg”
approach to FF differentiation is available, which includes procedures for assessing the stability of the results with
respect to the finite-difference step size. The Romberg approach is described in Section 11.14.2. This section describes
Q-C HEM’s older approach to FF calculations based on straightforward application of small electric fields along the
appropriate Cartesian directions.

630

Chapter 11: Molecular Properties and Analysis

Dipole moments can be calculated numerically as the first derivative of the energy with respect to F~ by setting
JOBTYPE = DIPOLE and IDERIV = 0. If IDERIV is not specified explicitly, the dipole moment will be calculated
analytically, which for post-Hartree–Fock levels of theory invokes a gradient calculation in order to utilize the relaxed
wavefunction.
↔

Similarly, set JOBTYPE = POLARIZABILITY for numerical evaluation of the static polarizability tensor α. This is
performed by either first-order finite difference, taking first-order field derivatives of analytic dipole moments, or by
second-order finite difference of the energy. The latter is useful (indeed, required) for methods where analytic gradients
are not available, such as CCSD(T) for example. Note, however, that the electron cloud is formally unbound in the
presence of static electric fields and therefore a bound solution is a consequence of using a finite basis set. (With
analytic derivative techniques the perturbing field is infinitesimal so this is not an issue.) This fact, along with the
overall sensitivity of numerical derivatives to the finite-difference step size, means that care must be taken in choosing
the strength of the applied finite field.
To control the order for numerical differentiation with respect to the applied electric field, use IDERIV in the same
manner as for geometric derivatives, i.e., for polarizabilties use IDERIV = 0 for second-order finite-difference of the
energy and IDERIV = 1 for first-order finite difference of gradients. In addition, for numerical polarizabilities at the
Hartree-Fock or DFT level set RESPONSE_POLAR = -1 in order to disable the analytic polarizability code.
RESPONSE_POLAR
Control the use of analytic or numerical polarizabilities.
TYPE:
INTEGER
DEFAULT:
0 or −1 = 0 for HF or DFT, −1 for all other methods
OPTIONS:
0
Perform an analytic polarizability calculation.
−1 Perform a numeric polarizability calculation even when analytic 2nd derivatives are available.
RECOMMENDATION:
None

In finite-difference geometric derivatives the $rem variable FDIFF_STEPSIZE controls the size of the nuclear displacements but here it controls the magnitude of the electric field perturbations:
FDIFF_STEPSIZE
Displacement used for calculating derivatives by finite difference.
TYPE:
INTEGER
DEFAULT:
1 Corresponding to 1.88973 × 10−5 a.u.
OPTIONS:
n Use a step size of n times the default value.
RECOMMENDATION:
Use the default unless problems arise.

11.14.2

Romberg Finite-Field Procedure

Whereas the FF procedure described in Section 11.14.1 is a straightforward, finite-difference implementation of the
derivatives suggested in Eq. (11.76), in the Romberg procedure 38 one combines energy values obtained for a succession
of k external electric fields with amplitudes that form a geometric progression:
F (k) = ak F0 .

(11.77)

Chapter 11: Molecular Properties and Analysis

631

The FF expressions are obtained by combining truncated Taylor expansions of the energy with different amplitudes and/
or external field directions. For example, in the case of the diagonal β-tensor components the Romberg FF expression
is



!
E(F−i (k + 1) − E(Fi (k + 1)) − a E(F−i (k) − E(Fi (k))
.
(11.78)
βiii (k, 0) = 3
a(a2 − 1)(ak F0 )3
The field index ±i refers to the possible field directions, i.e., ±x, ±y or ±z. Truncation of the Taylor expansions means
that the results are contaminated by higher-order hyperpolarizabilities, and to remove this contamination, successive
“Romberg iterations” are performed using a recursive expression. For a component of a (hyper)polarizability tensor ζ,
the recursive expression is
a2n ζ(k, n − 1) − ζ(k + 1, n − 1)
ζ(k, n) =
.
(11.79)
a2n − 1
This expression leads to a triangular Romberg table enabling monitoring of the convergence of the numerical derivative. 38
As with any finite-difference procedure, the FF method for computing (hyper)polarizabilities is sensitive to the details
of numerical differentiation. The Romberg procedure allows one to find a field window, defined by its upper and
lower bounds, where the finite-difference procedure is stable. Energy values for field amplitudes below that window
suffer from too-large round-off errors, which are proportional to the energy convergence threshold. The upper bound is
imposed by the critical field amplitude corresponding to the intersection between the ground and excited-state energies.
In the Romberg procedure, this stability window defines a sub-triangle, determination of which is the primary goal in
analyzing Romberg tables.
The automatic procedure based on the analysis of field amplitude errors is implemented in scripts provided with QC HEM’s distribution. The field amplitude error is defined as the difference between ζ-values obtained for consecutive
field amplitudes at the same Romberg iteration:
k (n) = ζ(k + 1, n) − ζ(k, n) .

(11.80)

By virtue of Romberg’s recursive expression, the field error is expected to decrease with each iteration. Convergence
of the Romberg procedure can be probed using the iteration (order) error, defined as
n (k) = ζ(k, n + 1) − ζ(k, n) .

(11.81)

Automatic analysis of these quantities is described in detail in Ref. 38.
Note: (1) The automatic procedure can fail if the field window is not chosen wisely.
(2) The Romberg procedure can be performed either for one specific diagonal direction or for all Cartesian
components. In the case of the second hyperpolarizability, only the iiii, iijj components are available.
11.14.2.1

How to execute Romberg’s differentiation procedure

To perform Romberg calculations of (hyper)polarizabilities, Q-C HEM provides the following scripts in the $QC/bin/Romberg
directory:
input-Q-Chem-t-rex.sh
parse-t-rex.sh
input-Q-Chem-t-rex-3.0.f90
tddft_read.f90
eom_read.f90
T-REX-3.0.3.f90
To set up a calculation, copy these files into your home directory and compile Fortran files (*.f90). The executables should be named input-Q-Chem-t-rex-3.0, tddft_read, eom_read, and T-REX. The compilation
command is

Chapter 11: Molecular Properties and Analysis

632

gfortran file.f90 -o file
After compilation, put the binaries in the install directory (which can be the same as the directory with *.sh and
inputs), and add this directory to the $PATH variable:
bash syntax:
export PATH="$PATH":full_path_to_the_fortran_dir
csh/tsch syntax:
set path=($path full_path_to_the_fortran_dir)
(This line can be added to your .bashrc/.cshrc file for future runs.)
To run the calculation, create an input that specifies your molecule, an electronic structure method, and several additional $rem variables that (i) turn off symmetry (SYMMETRY and CC_SYMMETRY for CC/EOM calculations), (ii)
request higher-precision printing (CC_PRINT_PREC), and (iii) set up very tight convergence (SCF_CONVERGENCE,
CC_CONVERGENCE, EOM_DAVIDSON_CONVERGENCE, etc.). An example of an input file is given below.
Example 11.29 Input for Romberg calculations of ethylene molecule using B3LYP.
$molecule
0 1
C
0.00000000
H
0.94859916
H
-0.94859916
C
-0.00000000
H
-0.54409413
H
0.54409413
$end

0.00000000 -0.66880000
0.00000000 -1.19917145
0.00000000 -1.19917145
0.00000000 0.66880000
0.77704694 1.19917145
-0.77704694 1.19917145

$rem
BASIS
= sto-3g
EXCHANGE
= B3lyp
SCF_CONVERGENCE = 13
Need tight convergence for finite field calculations
SCF_MAX_CYCLES
= 200
SYMMETRY = false
All symmetries need to be turned off
CC_PRINT_PREC = 16 16 decimal points of total energies will be printed
$end

Run the script input-Q-Chem-t-rex.sh and answer the questions regarding the parameters of the geometrical
progression of field amplitudes (see example below). The script will create multiple input files for the FF calculations
based on the basic input file that you provided. After running the calculations, parse the output files using the script
parse-t-rex.sh. Then run the T-REX program. Answer the the questions to compute the dipole moment and
(hyper)polarizabilities (see example below).
Romberg differentiation is only available for methods where printing the total energy to high precision has been enabled. In the current version of Q-C HEM, CC_PRINT_PREC is implemented for the following methods: HF, DFT, MP2,
RI-MP2, MP3, CCD, CCSD, CCSD(T), QCISD, QCISD(T), TDDFT, and EOM-CCSD.
Note: When using excited-state methods such as TDDFT, CIS, and EOM-CC, state ordering may switch when the
external field is large.
With the T-REX program, you can compute the static dipole moment, polarizability, and first and second hyperpolarizabilities for one specific diagonal direction or for all Cartesian components, except that second hyperpolarizabilities
are limited to iiii and iijj components. When computing all the components, you can obtain the norm of the dipole
moment and polarizability, the Hyper-Rayleigh scattering first hyperpolarizability, the mean of the second hyperpolarizability, and other information

Chapter 11: Molecular Properties and Analysis
11.14.2.2

633

Step-by-step Example of a Finite-Field Calculation

Put the sample input file (given above) for a DFT calculation in the new directory; the name of the input file should
be input. Run the input-Q-Chem-t-rex.sh script. The following questions are asked:
Components: x=1 y=2 z=3 all_beta=4 all_alpha=5
4
File name
ethNumber of field amplitudes
5
Smallest field amplitude F_0
0.0004
a
(F_k=a^k F_0)
2.0
Number of files created
101
In this example, the answers correspond to FF calculations for F (k) = 2.0k × 0.0004 and for a geometric progression
(k = 0, 1, 2, 3, 4, 5) of external field amplitudes. The calculation is set up for all the components.
After the FF calculations (i.e., after executing Q-C HEM jobs for all generated inputs), parse the outputs using the
parse-t-rex.sh script. The energies are written in a file called prelogfile.
Run the T-REX program and answer the questions:
Components: x=1 y=2 z=3 all_beta=4
4
Number of methods
1
Number of field amplitudes k_max+1
5
Smallest field amplitude
F_0
0.0004
Step-size a
2.0
The energies are ordered in a file called logfile and the results are printed in the results file.

11.15

General Response Theory

Many of the preceding sections of chapter 11 are concerned with properties that require the solution of underlying
equations similar to those from TDDFT (see eq. (7.15)), but in the presence of a (time-dependent) perturbation:



   

A B
Σ
∆
X
V
− ωf
=
,
(11.82)
B∗ A∗
−∆∗ −Σ∗
Y
−V∗
where Σ → 0 and ∆ → 1 for canonical HF/DFT MOs. The functionality for solving these equations with a general
choice of operators representing a perturbation V is now available in Q-C HEM. Both singlet 74 and triplet 107 response
are available for a variety of operators (see table 11.4).
An additional feature of the general response module is its ability to work with non-orthogonal MOs. In a formulation
analogous to TDDFT(MI) 87 , the linear response for molecular interactions 16 , or LR(MI), method is available to solve
the linear response equations on top of ALMOs.

Chapter 11: Molecular Properties and Analysis

634

The response solver can be used with any density functional available in Q-C HEM, including range-separated functionals (e.g. CAM-B3LYP, ωB97X) and meta-GGAs (e.g. M06-2X).
There are a few limitations:
• No post-HF/correlated methods are available yet.
• Currently, only linear response is implemented.
• Only calculations on top of restricted and unrestricted (not restricted open-shell) references are implemented.
• Density functionals including non-local dispersion (e.g. VV10, ωB97M-V) are not yet available.

11.15.1

Job Control

Only one keyword is necessary in the $rem section to activate the response module. All other options are controlled
through the $response input section.
RESPONSE
Activate the general response property module.
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
FALSE (or 0) Don’t activate the general response property module.
TRUE (or 1)
Activate the general response property module.
RECOMMENDATION:
None.
ORDER
Sets the maximum order of response theory to perform.
INPUT SECTION: $response
TYPE:
STRING
DEFAULT:
LINEAR
OPTIONS:
LINEAR Perform up through linear response.
RECOMMENDATION:
None. Currently, only linear response is implemented.
SOLVER
Sets the algorithm for solving the response equations.
INPUT SECTION: $response
TYPE:
STRING
DEFAULT:
DIIS
OPTIONS:
LINEAR Iteratively solve the response equations without convergence acceleration.
DIIS
Iteratively solve the response equations using DIIS for convergence acceleration.
RECOMMENDATION:
DIIS

Chapter 11: Molecular Properties and Analysis
HAMILTONIAN
Sets the approximation used for the orbital Hessian.
INPUT SECTION: $response
TYPE:
STRING
DEFAULT:
RPA
OPTIONS:
RPA No approximations.
TDA Same as the CIS approximation.
CIS
Synonym for TDA.
RECOMMENDATION:
None.
SPIN
Does the operator access same spin (singlet) or different spin (triplet) states?
INPUT SECTION: $response
TYPE:
STRING
DEFAULT:
SINGLET
OPTIONS:
SINGLET Operator is spin-conserving.
TRIPLET Operator is not spin-conserving.
RECOMMENDATION:
None. Care must be taken as all operators in a single calculation will be forced to follow
this option.
MAXITER
Maximum number of iterations.
INPUT SECTION: $response
TYPE:
INTEGER
DEFAULT:
60
OPTIONS:
n Maximum number of iterations.
RECOMMENDATION:
Use the default value.
CONV
Convergence threshold. For the DIIS solver, this is the DIIS error norm. For the linear
solver, this is the response vector RMSD between iterations.
INPUT SECTION: $response
TYPE:
INTEGER
DEFAULT:
8
OPTIONS:
n Sets the convergence threshold to 10−n .
RECOMMENDATION:
Use the default value.

635

Chapter 11: Molecular Properties and Analysis
DIIS_START
Iteration number to start DIIS. Before this, linear iterations are performed.
INPUT SECTION: $response
TYPE:
INTEGER
DEFAULT:
1
OPTIONS:
n Iteration number to start DIIS.
RECOMMENDATION:
Use the default value.
DIIS_VECTORS
Maximum number of DIIS vectors to keep.
INPUT SECTION: $response
TYPE:
INTEGER
DEFAULT:
7
OPTIONS:
n > 0 Maximum number of DIIS vectors to keep.
RECOMMENDATION:
Use the default value.
RHF_AS_UHF
Should the response equations be solved as though an unrestricted reference is being
used?
INPUT SECTION: $response
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
TRUE
Treat an RHF wavefunction as though it were UHF.
FALSE Treat an RHF wavefunction as RHF.
RECOMMENDATION:
Use the default value. Only useful for debugging.
PRINT_LEVEL
Sets a general printing level across the response module.
INPUT SECTION: $response
TYPE:
INTEGER
DEFAULT:
2
OPTIONS:
1
Print the initial guess and the final results.
2
1 + iterations and comments.
10 Kill trees.
RECOMMENDATION:
Use the default value.

636

Chapter 11: Molecular Properties and Analysis
RUN_TYPE
Should a single response calculation be performed, or should all permutations of the orbital Hessian and excitation type be performed?
INPUT SECTION: $response
TYPE:
STRING
DEFAULT:
SINGLE
OPTIONS:
SINGLE Use only the orbital Hessian and excitation type specified in their respective keywords.
ALL
Use all permutations of RPA/TDA and singlet/triplet.
RECOMMENDATION:
Use the default value, unless a comparison between approximations and excitation types
is desired.
SAVE
Save any quantities to disk?
INPUT SECTION: $response
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
0 Don’t save any quantities to disk.
1 Save quantities in MO basis.
2 Save quantities in MO and AO bases.
RECOMMENDATION:
None.
READ
Read any quantities from disk?
INPUT SECTION: $response
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
0 Don’t read any quantities from disk.
1 Read quantities in MO basis.
2 Read quantities in AO basis.
RECOMMENDATION:
None.

637

Chapter 11: Molecular Properties and Analysis
DUMP_AO_INTEGRALS
Should AO-basis property integrals be saved to disk?
INPUT SECTION: $response
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
TRUE
Save AO-basis property integrals to disk.
FALSE Don’t save AO-basis property integrals to disk.
RECOMMENDATION:
None.
FORCE_NOT_NONORTHOGONAL
Should the canonical response equations be solved, ignoring the identity of the underlying
orbitals?
INPUT SECTION: $response
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
TRUE
FALSE
RECOMMENDATION:
Leave as false. Using the standard (canonical) response equations with non-orthogonal
MOs will give incorrect results.
FORCE_NONORTHOGONAL
Should the non-orthogonal response equations be solved, ignoring the identity of the underlying orbitals?
INPUT SECTION: $response
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
TRUE
FALSE
RECOMMENDATION:
Leave as false. When used with canonical MOs, this should give the same answer as with
the standard equations, but at greater computational cost.

638

639

Chapter 11: Molecular Properties and Analysis
FREQUENCY
Strength of one or more incident fields in atomic units. A separate response calculation
will be performed for every field strength. 0.0 corresponds to the static limit.
INPUT SECTION: $response
TYPE:
DOUBLE
DEFAULT:
0.0
OPTIONS:
l m n . . . One or more field strengths separated by spaces.
RECOMMENDATION:
None.

11.15.2 $response Section and Operator Specification
The specification of operators used in solving for response vectors is designed to be very flexible. The general form of
the $response input section is given by
$response
keyword_1 setting_1
keyword_2 setting_2
...
[operator_1_label, operator_1_origin]
[operator_2_label, operator_2_origin]
[operator_3_label, operator_3_origin]
...
$end

where the keywords are those found in section 11.15.1 (with the exception of RESPONSE).
The specification of an operator is given within a line contained by [], where the first element is a label from table 11.4,
and the second element is a label from table 11.5. Operator specifications may appear in any order. Response values
are calculated for all possible permutations of operators and their components.
For the Cartesian moment operator, a third field within [] may be specified for the order of the expansion, entered as
(i, j, k). For example, the molecular response to the moment of order (2, 5, 4) with its origin at (0.2, 0.3, 0.4) a.u. can
be found with the operator specification
[multipole, (0.2, 0.3, 0.4), (2, 5, 4)]

Table 11.4: Available operators
Operator Label
dipole or diplen
quadrupole
multipole
fermi or fc

Description
dipole (length gauge)
second moment (length gauge)
arbitrary-order Cartesian moment (length gauge)
Fermi contact

Integral
hχµ |rO |χν i
hχµ |rrT |χν i
hχµ |xi y j z k |χν i
4πge
3 hχµ |δ(rK )|χν i

spindip or sd

spin dipole

2
3rK rT
ge
K −rK
|χν i
5
2 hχµ |
rK

angmom or dipmag
dipvel

angular momentum
dipole (velocity gauge)

hχµ |LO |χν i
hχµ |∇|χν i

Chapter 11: Molecular Properties and Analysis

Table 11.5: Available operator origins
Origin Label
zero
(x, y, z)

Description
Cartesian origin, same as (0.0, 0.0, 0.0)
arbitrary point (double precision, units are bohrs)

640

641

Chapter 11: Molecular Properties and Analysis

11.15.3

Examples Including $response Section

Example 11.30 Input for calculating all components of the static (dipole) polarizability at the Cartesian origin for
tryptophan. All of the options given are defaults.
$molecule
0 1
N
-0.0699826875
C
1.3728035449
C
2.0969275417
O
3.1382490088
C
1.9529664597
H
1.8442727348
H
1.3455899915
C
3.4053646872
C
4.4845249667
N
5.6509089647
H
6.6009314349
C
5.2921619642
C
3.8942019475
C
3.2659168792
H
2.1864306677
C
4.0381762333
H
3.5696890585
C
5.4445159165
H
6.0229926396
C
6.0869576238
H
7.1656650647
H
4.5457621618
H
-0.5159777859
H
1.5420526570
H
-0.5302278747
O
1.4575846656
H
0.5990015339
$end
$rem
METHOD
BASIS
SCF_CONVERGENCE
THRESH
RESPONSE
$end
$response
ORDER
SOLVER
HAMILTONIAN
SPIN
MAXITER
CONV
DIIS_START
DIIS_VECTORS
RHF_AS_UHF
PRINT_LEVEL
RUN_TYPE
FREQUENCY
[dipole, zero]
$end

0.3321987191
0.0970713322
-0.0523593054
-0.6563684788
1.3136139853
2.2050605044
1.4594935008
1.1270611844
1.6235038050
1.2379326369
1.4112351003
0.4356274269
0.3557998019
-0.3832607567
-0.4577058843
-1.0087512639
-1.5824763141
-0.9194874753
-1.4277973542
-0.2024044961
-0.1287762497
2.2425310766
0.7478905868
-0.8143939718
-0.5823989653
0.5996887308
0.8842421241

=
=
=
=
=

hf
sto-3g
9
12
true

linear
diis
rpa
singlet
60
8
1
7
false
2
single
0.0

0.2821283177
-0.0129587739
1.3682652221
1.5380162924
-0.7956021969
-0.1801631789
-1.6885689523
-1.1918075237
-0.5598918002
-1.2284610654
-0.9028629397
-2.3131617003
-2.3263315791
-3.3431309548
-3.3815918670
-4.2870993776
-5.0755609734
-4.2519002882
-5.0130007062
-3.2767702726
-3.2458650647
0.3253979653
-0.5487661007
-0.5935463196
0.4084507634
2.4093500287
2.0047830456

642

Chapter 11: Molecular Properties and Analysis

Example 11.31 Functionally identical input for calculating all components of the static (dipole) polarizability at the
Cartesian origin for tryptophan.
$rem
jobtype
method
basis
scf_convergence
thresh
$end

=
=
=
=
=

$molecule
0 1
N
-0.0699826875
C
1.3728035449
C
2.0969275417
O
3.1382490088
C
1.9529664597
H
1.8442727348
H
1.3455899915
C
3.4053646872
C
4.4845249667
N
5.6509089647
H
6.6009314349
C
5.2921619642
C
3.8942019475
C
3.2659168792
H
2.1864306677
C
4.0381762333
H
3.5696890585
C
5.4445159165
H
6.0229926396
C
6.0869576238
H
7.1656650647
H
4.5457621618
H
-0.5159777859
H
1.5420526570
H
-0.5302278747
O
1.4575846656
H
0.5990015339
$end

polarizability
hf
sto-3g
9
12

0.3321987191
0.0970713322
-0.0523593054
-0.6563684788
1.3136139853
2.2050605044
1.4594935008
1.1270611844
1.6235038050
1.2379326369
1.4112351003
0.4356274269
0.3557998019
-0.3832607567
-0.4577058843
-1.0087512639
-1.5824763141
-0.9194874753
-1.4277973542
-0.2024044961
-0.1287762497
2.2425310766
0.7478905868
-0.8143939718
-0.5823989653
0.5996887308
0.8842421241

0.2821283177
-0.0129587739
1.3682652221
1.5380162924
-0.7956021969
-0.1801631789
-1.6885689523
-1.1918075237
-0.5598918002
-1.2284610654
-0.9028629397
-2.3131617003
-2.3263315791
-3.3431309548
-3.3815918670
-4.2870993776
-5.0755609734
-4.2519002882
-5.0130007062
-3.2767702726
-3.2458650647
0.3253979653
-0.5487661007
-0.5935463196
0.4084507634
2.4093500287
2.0047830456

11.16

Electronic Couplings for Electron- and Energy Transfer

11.16.1

Eigenstate-Based Methods

For electron transfer (ET) and excitation energy transfer (EET) processes, the electronic coupling is one of the important parameters that determine their reaction rates. For ET, Q-C HEM provides the coupling values calculated with
the generalized Mulliken-Hush (GMH), 31 fragment-charge difference (FCD), 147 Boys localization, 137 and EdmistonRuedenbeg 138 localization schemes. For EET, options include fragment-excitation difference (FED), 65 fragment-spin
difference (FSD), 160 occupied-virtual separated Boys localization, 139 or Edmiston-Ruedenberg localization. 138 In all
these schemes, a vertical excitation such as CIS, RPA or TDDFT is required, and the GMH, FCD, FED, FSD, Boys or
ER coupling values are calculated based on the excited state results.

643

Chapter 11: Molecular Properties and Analysis
11.16.1.1

Two-state approximation

Under the two-state approximation, the diabatic reactant and product states are assumed to be a linear combination of
the eigenstates. For ET, the choice of such linear combination is determined by a zero transition dipoles (GMH) or
maximum charge differences (FCD). In the latter, a 2 × 2 donor–acceptor charge difference matrix, ∆q, is defined,
with elements
Z
Z
D
A
∆qmn = qmn
− qmn
=
ρmn (r)dr −
ρmn (r)dr
r∈D

r∈A

where ρmn (r) is the matrix element of the density operator between states |mi and |ni.
For EET, a maximum excitation difference is assumed in the FED, in which a excitation difference matrix is similarly
defined with elements
Z
Z
D
A
(mn)
∆xmn = xmn − xmn =
ρex (r)dr −
ρ(mn)
(r)dr
ex
r∈D

r∈A

(mn)
ρex (r)

where
is the sum of attachment and detachment densities for transition |mi → |ni, as they correspond to the
electron and hole densities in an excitation. In the FSD, a maximum spin difference is used and the corresponding spin
difference matrix is defined with its elements as,
Z
Z
D
A
σ(mn) (r)dr
σ(mn) (r)dr −
∆smn = smn − smn =
r∈A

r∈D

where σmn (r) is the spin density, difference between α-spin and β-spin densities, for transition from |mi → |ni.
Since Q-C HEM uses a Mulliken population analysis for the integrations in Eqs. (11.83), (11.83), and (11.83), the
matrices ∆q, ∆x and ∆s are not symmetric. To obtain a pair of orthogonal states as the diabatic reactant and product
states, ∆q, ∆x and ∆s are symmetrized in Q-C HEM. Specifically,
∆q mn = (∆qmn + ∆qnm )/2

(11.83a)

∆xmn = (∆xmn + ∆xnm )/2

(11.83b)

∆smn = (∆smn + ∆snm )/2

(11.83c)

The final coupling values are obtained as listed below:
• For GMH,
VET = q

(E2 − E1 ) |~
µ12 |

(11.84)
2

(~
µ11 − µ
~ 22 )2 + 4 |~
µ12 |

• For FCD,
VET = q

(E2 − E1 )∆q 12
2

(11.85)

(∆q11 − ∆q22 )2 + 4∆q 12

• For FED,
(E2 − E1 )∆x12
VEET = q
2
(∆x11 − ∆x22 )2 + 4∆x12

(11.86)

• For FSD,
VEET = q

(E2 − E1 )∆s12
2

(11.87)

(∆s11 − ∆s22 )2 + 4∆s12

Q-C HEM provides the option to control FED, FSD, FCD and GMH calculations after a single-excitation calculation,
such as CIS, RPA, TDDFT/TDA and TDDFT. To obtain ET coupling values using GMH (FCD) scheme, one should
set $rem variables STS_GMH (STS_FCD) to be TRUE. Similarly, a FED (FSD) calculation is turned on by setting the
$rem variable STS_FED (STS_FSD) to be TRUE. In FCD, FED and FSD calculations, the donor and acceptor fragments
are defined via the $rem variables STS_DONOR and STS_ACCEPTOR. It is necessary to arrange the atomic order in

Chapter 11: Molecular Properties and Analysis

644

the $molecule section such that the atoms in the donor (acceptor) fragment is in one consecutive block. The ordering
numbers of beginning and ending atoms for the donor and acceptor blocks are included in $rem variables STS_DONOR
and STS_ACCEPTOR.
The couplings will be calculated between all choices of excited states with the same spin. In FSD, FCD and GMH
calculations, the coupling value between the excited and reference (ground) states will be included, but in FED, the
ground state is not included in the analysis. It is important to select excited states properly, according to the distribution
of charge or excitation, among other characteristics, such that the coupling obtained can properly describe the electronic
coupling of the corresponding process in the two-state approximation.
STS_GMH
Control the calculation of GMH for ET couplings.
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
FALSE Do not perform a GMH calculation.
TRUE
Include a GMH calculation.
RECOMMENDATION:
When set to true computes Mulliken-Hush electronic couplings. It yields the generalized
Mulliken-Hush couplings as well as the transition dipole moments for each pair of excited states
and for each excited state with the ground state.
STS_FCD
Control the calculation of FCD for ET couplings.
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
FALSE Do not perform an FCD calculation.
TRUE
Include an FCD calculation.
RECOMMENDATION:
None
STS_FED
Control the calculation of FED for EET couplings.
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
FALSE Do not perform a FED calculation.
TRUE
Include a FED calculation.
RECOMMENDATION:
None

Chapter 11: Molecular Properties and Analysis
STS_FSD
Control the calculation of FSD for EET couplings.
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
FALSE Do not perform a FSD calculation.
TRUE
Include a FSD calculation.
RECOMMENDATION:
For RCIS triplets, FSD and FED are equivalent. FSD will be automatically switched off and
perform a FED calculation.
STS_DONOR
Define the donor fragment.
TYPE:
STRING
DEFAULT:
0 No donor fragment is defined.
OPTIONS:
i-j Donor fragment is in the ith atom to the jth atom.
RECOMMENDATION:
Note no space between the hyphen and the numbers i and j.
STS_ACCEPTOR
Define the acceptor molecular fragment.
TYPE:
STRING
DEFAULT:
0 No acceptor fragment is defined.
OPTIONS:
i-j Acceptor fragment is in the ith atom to the jth atom.
RECOMMENDATION:
Note no space between the hyphen and the numbers i and j.
STS_MOM
Control calculation of the transition moments between excited states in the CIS and TDDFT
calculations (including SF variants).
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
FALSE Do not calculate state-to-state transition moments.
TRUE
Do calculate state-to-state transition moments.
RECOMMENDATION:
When set to true requests the state-to-state dipole transition moments for all pairs of excited
states and for each excited state with the ground state.

645

646

Chapter 11: Molecular Properties and Analysis

Example 11.32 A GMH & FCD calculation to analyze electron-transfer couplings in an ethylene and a methaniminium
cation.
$molecule
1 1
C
0.679952
N
-0.600337
H
1.210416
H
1.210416
H
-1.131897
H
-1.131897
C
-5.600337
C
-6.937337
H
-5.034682
H
-5.034682
H
-7.502992
H
-7.502992
$end
$rem
METHOD
BASIS
CIS_N_ROOTS
CIS_SINGLETS
CIS_TRIPLETS
STS_GMH
STS_FCD
STS_DONOR
STS_ACCEPTOR
MEM_STATIC
$end

0.000000
0.000000
0.940723
-0.940723
-0.866630
0.866630
0.000000
0.000000
0.927055
-0.927055
-0.927055
0.927055

0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000

CIS
6-31+G
20
true
false
true !turns on the GMH calculation
true !turns on the FCD calculation
1-6 !define the donor fragment as atoms 1-6 for FCD calc.
7-12 !define the acceptor fragment as atoms 7-12 for FCD calc.
200 !increase static memory for a CIS job with larger basis set

Example 11.33 An FED calculation to analyze excitation-energy transfer couplings in a pair of stacked ethylenes.
$molecule
0 1
C
0.670518
H
1.241372
H
1.241372
C
-0.670518
H
-1.241372
H
-1.241372
C
0.774635
H
1.323105
H
1.323105
C
-0.774635
H
-1.323105
H
-1.323105
$end
$rem
METHOD
BASIS
CIS_N_ROOTS
CIS_SINGLETS
CIS_TRIPLETS
STS_FED
STS_DONOR
STS_ACCEPTOR
$end

0.000000
0.927754
-0.927754
0.000000
-0.927754
0.927754
0.000000
0.936763
-0.936763
0.000000
-0.936763
0.936763

CIS
3-21G
20
true
false
true
1-6
7-12

0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
4.500000
4.500000
4.500000
4.500000
4.500000
4.500000

647

Chapter 11: Molecular Properties and Analysis
11.16.1.2

Multi-state treatments

When dealing with multiple charge or electronic excitation centers, diabatic states can be constructed with Boys 137
or Edmiston-Ruedenberg 138 localization. In this case, we construct diabatic states {|ΞI i} as linear combinations of
adiabatic states {|ΦI i} with a general rotation matrix U that is Nstate × Nstate in size:
|ΞI i =

NX
states

|ΦJ i Uji

I = 1 . . . Nstates

(11.88)

J=1

The adiabatic states can be produced with any method, in principle, but the Boys/ER-localized diabatization methods
have been implemented thus far only for CIS or TDDFT methods in Q-C HEM. In analogy to orbital localization,
Boys-localized diabatization corresponds to maximizing the charge separation between diabatic state centers:
NX
states

fBoys (U) = fBoys ({ΞI }) =

hΞI |~
µ|ΞI i − hΞJ |~
µ|ΞJ i

2

(11.89)

I,J=1

Here, µ
~ represents the dipole operator. ER-localized diabatization prescribes maximizing self-interaction energy:
fER (U) = fER ({ΞI }) =

NX
states

Z

~1
dR

Z

~2
dR

I=1

~ 2 )|ΞI ihΞI |ρ̂(R
~ 1 )|ΞI i
hΞI |ρ̂(R
~1 − R
~ 2|
|R

(11.90)

~ is
where the density operator at position R
~ =
ρ̂(R)

X

~ − ~r (j) )
δ(R

(11.91)

j

Here, ~r (j) represents the position of the jth electron.
These models reflect different assumptions about the interaction of our quantum system with some fictitious external
electric field/potential: (i) if we assume a fictitious field that is linear in space, we arrive at Boys localization; (ii)
if we assume a fictitious potential energy that responds linearly to the charge density of our system, we arrive at ER
localization. Note that in the two-state limit, Boys localized diabatization reduces nearly exactly to GMH. 137
As written down in Eq. (11.89), Boys localized diabatization applies only to charge transfer, not to energy transfer.
Within the context of CIS or TDDFT calculations, one can easily extend Boys localized diabatization 139 by separately
localizing the occupied and virtual components of µ
~, µ
~ occ and µ
~ virt :
fBoysOV (U) = fBoysOV ({ΞI })
=

NX
states



2

hΞI |~
µ occ |ΞI i − hΞJ |~
µ occ |ΞJ i + hΞI |~
µvirt |ΞI i − hΞJ |~
µvirt |ΞJ i

2



(11.92)

I,J=1

where
|ΞI i =

X

a
tIa
i |Φi i

(11.93)

ia

and the occupied/virtual components are defined by
X
X
X
Ja
Jb
hΞI | µ
~ |ΞJ i = δIJ
µ
~ ii −
tIa
~ ij +
tIa
~ ab
i tj µ
i ti µ
i

|

aij

{z
hΞI | µ
~

occ

|ΞJ i

(11.94)

iba

}

|

{z
virt

+ hΞI | µ
~

}
|ΞJ i

Note that when we maximize the Boys OV function, we are simply performing Boys-localized diabatization separately
on the electron attachment and detachment densities.

648

Chapter 11: Molecular Properties and Analysis

Finally, for energy transfer, it can be helpful to understand the origin of the diabatic couplings. To that end, we now
provide the ability to decompose the diabatic coupling between diabatic states into Coulomb (J), Exchange (K) and
one-electron (O) components: 148
X
X
X
X
a Qb
a Qb
a Qa
a Qb
tP
hΞP | H |ΞQ i =
tP
tP
tP
i tj (ia|jb) −
i ti Fab −
i tj Fij +
i tj (ij|ab)
ija

iab

|

{z
O

ijab

}

|

ijab

{z
J

}

|

{z
K

}

BOYS_CIS_NUMSTATE
Define how many states to mix with Boys localized diabatization. These states must be specified
in the $localized_diabatization section.
TYPE:
INTEGER
DEFAULT:
0 Do not perform Boys localized diabatization.
OPTIONS:
2 to N where N is the number of CIS states requested (CIS_N_ROOTS)
RECOMMENDATION:
It is usually not wise to mix adiabatic states that are separated by more than a few eV or a typical
reorganization energy in solvent.
ER_CIS_NUMSTATE
Define how many states to mix with ER localized diabatization. These states must be specified
in the $localized_diabatization section.
TYPE:
INTEGER
DEFAULT:
0 Do not perform ER localized diabatization.
OPTIONS:
2 to N where N is the number of CIS states requested (CIS_N_ROOTS)
RECOMMENDATION:
It is usually not wise to mix adiabatic states that are separated by more than a few eV or a typical
reorganization energy in solvent.
LOC_CIS_OV_SEPARATE
Decide whether or not to localized the “occupied” and “virtual” components of the localized diabatization function, i.e., whether to localize the electron attachments and detachments separately.
TYPE:
LOGICAL
DEFAULT:
FALSE Do not separately localize electron attachments and detachments.
OPTIONS:
TRUE
RECOMMENDATION:
If one wants to use Boys localized diabatization for energy transfer (as opposed to electron transfer) , this is a necessary option. ER is more rigorous technique, and does not require this OV
feature, but will be somewhat slower.

(11.95)

649

Chapter 11: Molecular Properties and Analysis
CIS_DIABATH_DECOMPOSE
Decide whether or not to decompose the diabatic coupling into Coulomb, exchange, and oneelectron terms.
TYPE:
LOGICAL
DEFAULT:
FALSE Do not decompose the diabatic coupling.
OPTIONS:
TRUE
RECOMMENDATION:
These decompositions are most meaningful for electronic excitation transfer processes. Currently, available only for CIS, not for TDDFT diabatic states.

Example 11.34 A calculation using ER localized diabatization to construct the diabatic Hamiltonian and couplings
between a square of singly-excited Helium atoms.
$molecule
0 1
he
0
he
0
he
0
he
0
$end

-1.0
-1.0
1.0
1.0

1.0
-1.0
-1.0
1.0

$rem
METHOD
CIS_N_ROOTS
CIS_SINGLETS
CIS_TRIPLETS
BASIS
SCF_CONVERGENCE
SYMMETRY
RPA
SYM_IGNORE
SYM_IGNORE
LOC_CIS_OV_SEPARATE
ER_CIS_NUMSTATE
CIS_DIABATh_DECOMPOSE

cis
4
false
true
6-31g**
8
false
false
true
true
false !
4
!
true
!
!

NOT localizing attachments/detachments separately.
using ER to mix 4 adiabatic states.
decompose diabatic couplings into
Coulomb, exchange, and one-electron components.

$end
$localized_diabatization
On the next line, list which excited adiabatic states we want to mix.
1 2 3 4
$end

11.16.2

Diabatic-State-Based Methods

11.16.2.1

Electronic coupling in charge transfer

A charge transfer involves a change in the electron numbers in a pair of molecular fragments. As an example, we will
use the following reaction when necessary, and a generalization to other cases is straightforward:
D− A −→ DA−

(11.96)

650

Chapter 11: Molecular Properties and Analysis

where an extra electron is localized to the donor (D) initially, and it becomes localized to the acceptor (A) in the final
state. The two-state secular equation for the initial and final electronic states can be written as


Hii − Sii E Hif − Sif E
H − ES =
=0
(11.97)
Hif − Sif E Hff − Sff E
This is very close to an eigenvalue problem except for the non-orthogonality between the initial and final states. A
standard eigenvalue form for Eq. (11.97) can be obtained by using the Löwdin transformation:
Heff = S−1/2 HS−1/2 ,

(11.98)

where the off-diagonal element of the effective Hamiltonian matrix represents the electronic coupling for the reaction,
and it is defined by
Hif − Sif (Hii + Hff )/2
V = Hifeff =
(11.99)
1 − Sif2
In a general case where the initial and final states are not normalized, the electronic coupling is written as
V =

p

Sii Sff ×

Hif − Sif (Hii /Sii + Hff /Sff )/2
Sii Sff − Sif2

(11.100)

Thus, in principle, V can be obtained when the matrix elements for the Hamiltonian H and the overlap matrix S are
calculated.
The direct coupling (DC) scheme calculates the electronic coupling values via Eq. (11.100), and it is widely used to
calculate charge transfer coupling. 26,44,106,162 In the DC scheme, the coupling matrix element is calculated directly
using charge-localized determinants (the “diabatic states” in electron transfer literature). In electron transfer systems,
it has been shown that such charge-localized states can be approximated by symmetry-broken unrestricted HartreeFock (UHF) solutions. 26,97,106 The adiabatic eigenstates are assumed to be the symmetric and anti-symmetric linear
combinations of the two symmetry-broken UHF solutions in a DC calculation. Therefore, DC couplings can be viewed
as a result of two-configuration solutions that may recover the non-dynamical correlation.
The core of the DC method is based on the corresponding orbital transformation 77 and a calculation for Slater’s determinants in Hif and Sif . 44,162 Unfortunately, the calculation of Hif is not available for DFT method because a functional
of the two densities ρi and ρf is unknown and there are no existing approximate forms for Hif . 158 To calculate charge
transfer coupling with DFT, we can use the CDFT-CI method (Section 5.13.3), the frontier molecular orbital (FMO)
approach 145,161 (Section 11.16.2.5) or a hybrid scheme – DC with CDFT wave functions (Section 11.16.2.4).

11.16.2.2

Corresponding orbital transformation

Let |Ψa i and |Ψb i be two single Slater-determinant wave functions for the initial and final states, and a and b be the
spin-orbital sets, respectively:
=

(a1 , a2 , · · · , aN )

(11.101)

b =

(b1 , b2 , · · · , bN )

(11.102)

a

Since the two sets of spin-orbitals are not orthogonal, the overlap matrix S can be defined as:
Z
S = b† a dτ.

(11.103)

We note that S is not Hermitian in general since the molecular orbitals of the initial and final states are separately
determined. To calculate the matrix elements Hab and Sab , two sets of new orthogonal spin-orbitals can be used by the
corresponding orbital transformation. 77 In this approach, each set of spin-orbitals a and b are linearly transformed,
â

= aV

(11.104)

b̂

= bU

(11.105)

651

Chapter 11: Molecular Properties and Analysis

where V and U are the left-singular and right-singular matrices, respectively, in the singular value decomposition
(SVD) of S:
S = UŝV†
(11.106)
The overlap matrix in the new basis is now diagonal
Z

Z
b̂† â = U†
b† a V = ŝ

11.16.2.3

(11.107)

Generalized density matrix

The Hamiltonian for electrons in molecules are a sum of one-electron and two-electron operators. In the following, we
derive the expressions for the one-electron operator Ω(1) and two-electron operator Ω(2) ,
Ω(1)

N
X

=

ω(i)

(11.108)

i=1

Ω(2)

N
1 X
ω(i, j)
2 i,j=1

=

(11.109)

where ω(i) and ω(i, j), for the molecular Hamiltonian, are
1
ω(i) = h(i) = − ∇2i + V (i)
2
and
ω(i, j) =

(11.110)

1
rij

(11.111)

The evaluation of matrix elements can now proceed:
Sab = hΨb |Ψa i = det(U) det(V† )

N
Y

ŝii

(11.112)

i=1

(1)

Ωab = hΨb |Ω(1) |Ψa i = det(U) det(V† )

N
X

hb̂i |ω(1)|âi i ·

i=1
(2)

Ωab = hΨb |Ω(2) |Ψa i =

N
Y

ŝjj

(11.113)

j6=i

N
N
X
Y
1
hb̂i b̂j |ω(1, 2)(1 − P12 )|âi âj i ·
ŝkk
det(U) det(V† )
2
ij

(11.114)

k6=i,j

(1)

(2)

Hab = Ωab + Ωab

(11.115)

In an atomic orbital basis set, {χ}, we can expand the molecular spin orbitals a and b,
a = χA,

â = χAV = χÂ

(11.116)

b = χB,

b̂ = χBU = χB̂

(11.117)

The one-electron terms, Eq. (11.112), can be expressed as
(1)

Ωab

=

N X
X
i

=

†
Âλi Tii B̂iσ
hχσ |ω(1)|χλ i

λσ

X

Gλσ ωσλ

(11.118)

λσ

where Tii = Sab /ŝii and define a generalized density matrix, G:
G = ÂTB̂†

(11.119)

652

Chapter 11: Molecular Properties and Analysis
Similarly, the two-electron terms, Eq. (11.114), are
(2)
Ωab

=
=

 
1 XXX
1
†
†
Âλi Âσj
Tjj B̂iµ
B̂jν
hχµ χν |ω(1, 2)|χλ χσ i
2 ij
ŝ
ii
λσ µν
X
R
GL
λµ Gσν hµν||λσi

(11.120)

λσµν

where GR and GL are generalized density matrices as defined in Eq. (11.119) except Tii in GL is replaced by 1/(2sii ).
The α- and β-spin orbitals are treated explicitly. In terms of the spatial orbitals, the one- and two-electron contributions
can be reduced to
X
X β
(1)
Ωab =
Gα
Gλσ ωσλ
(11.121)
λσ ωσλ +
λσ
(2)
Ωab

=

X

λσ
Rα
GLα
λµ Gσν (hµν|λσi

− hµν|σλi) +

λσµν

+

X
λσµν

X

Rα
GLβ
λµ Gσν hµν|λσi

λσµν
Rβ
GLα
λµ Gσν hµν|λσi

+

X

Rβ
GLβ
λµ Gσν (hµν|λσi

− hµν|σλi)

(11.122)

λσµν

The resulting one- and two-electron contributions, Eqs. (11.121) and (11.122) can be easily computed in terms of
generalized density matrices using standard one- and two-electron integral routines in Q-C HEM.

11.16.2.4

Direct coupling method for electronic coupling

It is important to obtain proper charge-localized initial and final states for the DC scheme, and this step determines the
quality of the coupling values. Q-C HEM provides three approaches to construct charge-localized states:
• The “1+1” approach
Since the system consists of donor and acceptor molecules or fragments, with a charge being localized either
donor or acceptor, it is intuitive to combine wave functions of individual donor and acceptor fragments to form a
charge-localized wave function. We call this approach “1+1” since the zeroth order wave functions are composed
of the HF wave functions of the two fragments.
For example, for the case shown in Example (11.96), we can use Q-C HEM to calculate two HF wave functions:
those of anionic donor and of neutral acceptor and they jointly form the initial state. For the final state, wave

653

Chapter 11: Molecular Properties and Analysis

functions of neutral donor and anionic acceptor are used. Then the coupling value is calculated via Eq. (11.100).
Example 11.35 To calculate the electron-transfer coupling for a pair of stacked-ethylene with “1+1” chargelocalized states
$molecule
-1 2
--1 2, 0 1
C
0.662489
H
1.227637
H
1.227637
C
-0.662489
H
-1.227637
H
-1.227637
-0 1, -1 2
C
0.720595
H
1.288664
H
1.288664
C
-0.720595
H
-1.288664
H
-1.288664
$end
$rem
JOBTYPE
METHOD
BASIS
SCF_PRINT_FRGM
SYM_IGNORE
SCF_GUESS
STS_DC
$end

0.000000
0.917083
-0.917083
0.000000
-0.917083
0.917083

0.000000
0.000000
0.000000
0.000000
0.000000
0.000000

0.000000
0.921368
-0.921368
0.000000
-0.921368
0.921368

4.5
4.5
4.5
4.5
4.5
4.5

SP
HF
6-31G(d)
FALSE
TRUE
FRAGMO
TRUE

In the $molecule subsection, the first line is for the charge and multiplicity of the whole system. The following
blocks are two inputs for the two molecular fragments (donor and acceptor). In each block the first line consists
of the charge and spin multiplicity in the initial state of the corresponding fragment, a comma, then the charge
and multiplicity in the final state. Next lines are nuclear species and their positions of the fragment. For example,
in the above example, the first block indicates that the electron donor is a doublet ethylene anion initially, and it
becomes a singlet neutral species in the final state. The second block is for another ethylene going from a singlet
neutral molecule to a doublet anion.
Note that the last three $rem variables in this example, SYM_IGNORE, SCF_GUESS and STS_DC must be set
to be the values as in the example in order to perform DC calculation with “1+1” charge-localized states. An
additional $rem variable, SCF_PRINT_FRGM is included. When it is TRUE a detailed output for the fragment HF
self-consistent field calculation is given.
• The “relaxed” approach
In “1+1” approach, the intermolecular interaction is neglected in the initial and final states, and so the final
electronic coupling can be underestimated. As a second approach, Q-C HEM can use “1+1” wave function as
an initial guess to look for the charge-localized wave function by further HF self-consistent field calculation.
This approach would ‘relax’ the wave function constructed by “1+1” method and include the intermolecular
interaction effects in the initial and final wave functions. However, this method may sometimes fail, leading to
either convergence problems or a resulting HF wave function that cannot represent the desired charge-localized
states. This is more likely to be a problem when calculations are performed with diffusive basis functions, or
when the donor and acceptor molecules are very close to each other.
To perform relaxed DC calculation, set STS_DC = RELAX.
• A hybrid scheme – constrained DFT charge-localized states
Constrained DFT (Section 5.13) can be used to obtain charge-localized states. It is recommended to set both

Chapter 11: Molecular Properties and Analysis

654

charge and spin constraints in order to generate proper charge localization. To perform DC calculation with
CDFT states, set SAVE_SUBSYSTEM = 10 and SAVE_SUBSYSTEM = 20 to save CDFT molecular orbitals in
the first two jobs of a batch jobs, and then in the third job of the batch job, set SCF_GUESS = READ and

Chapter 11: Molecular Properties and Analysis

655

STS_DC = TRUE to compute electronic coupling values.

Example 11.36 To calculate the electron-transfer coupling for a pair of stacked-ethylene with CDFT chargelocalized states
$molecule
-1 2
C 0.662489
H 1.227637
H 1.227637
C -0.662489
H -1.227637
H -1.227637
C 0.720595
H 1.288664
H 1.288664
C -0.720595
H -1.288664
H -1.288664
$end

0.000000
0.917083
-0.917083
0.000000
-0.917083
0.917083
0.000000
0.921368
-0.921368
0.000000
-0.921368
0.921368

$rem
method = wb97xd
basis = cc-pvdz
cdft = true
sym_ignore = true
save_subsystem 10
$end
$cdft
1.0
1.0 1 6
1.0
1.0 1 6
$end

s

@@@
$molecule
read
$end
$rem
method = wb97xd
basis = cc-pvdz
cdft = true
sym_ignore = true
save_subsystem 20
$end
$cdft
1.0
1.0 7 12
1.0
1.0 7 12
$end

s

@@@
$molecule
read
$end
$rem
method = hf
basis = cc-pvdz
sym_ignore = true
scf_guess = read
sts_dc = true
$end

0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
4.5
4.5
4.5
4.5
4.5
4.5

656

Chapter 11: Molecular Properties and Analysis
11.16.2.5

The frontier molecular orbital approach

The frontier molecular orbital (FMO) approach is often used with DFT to calculate ET coupling. 145,161 FMO coupling
value is essentially an off-diagonal Kohn–Sham matrix element with the overlap effect accounted
V FMO =

fDA − S (fDD + fAA ) /2
1 − S2

(11.123)

D(A)
ˆ A
ˆ
where fDA = hφD
FMO |f |φFMO i, with f being the Kohn–Sham operator of the donor-acceptor system. φFMO is the
Kohn–Sham frontier molecular orbital for the donor (acceptor) fragment, which represents one-particle scheme of a
charge transfer process.

In this approach, computations are often performed separately in the two fragments, and the off-diagonal Kohn–Sham
operator (and the overlap matrix) in the FMOs is subsequently calculated. To compute FMO couplings, Q-C HEM a
setup similar to the “1+1” approach Example 11.37 To calculate the electron-transfer coupling for a pair of stackedethylene with the FMO approach
$molecule
0 1
-0 1
C 0.662489
H 1.227637
H 1.227637
C -0.662489
H -1.227637
H -1.227637
-0 1
C 0.720595
H 1.288664
H 1.288664
C -0.720595
H -1.288664
H -1.288664
$end

0.000000
0.917083
-0.917083
0.000000
-0.917083
0.917083

0.000000
0.000000
0.000000
0.000000
0.000000
0.000000

0.000000
0.921368
-0.921368
0.000000
-0.921368
0.921368

4.5
4.5
4.5
4.5
4.5
4.5

$rem
method = lrcwpbe
omega = 370
basis = dz*
scf_print_frgm = true
sym_ignore = true
scf_guess = fragmo
sts_dc = fock
sts_trans_donor = 2-3
! use HOMO, HOMO-1 and LUMO, LUMO+1, LUMO+2 of donor
sts_trans_acceptor = 1-2 ! use HOMO and LUMO, LUMO+1 of acceptor
$end
$rem_frgm
print_orbitals = 5
$end

Note that the FMOs are not always HOMO or LUMO of fragments. We can use STS_TRANS_DONOR (and STS_TRANS_ACCEPTOR)
to select a range of occupied and virtual orbitals for FMO coupling calculations.

657

Chapter 11: Molecular Properties and Analysis

11.17

Population of Effectively Unpaired Electrons

In a stretched hydrogen molecule the two electrons that are paired at equilibrium forming a bond become un-paired
and localized on the individual H atoms. In singlet diradicals or doublet triradicals such a weak paring exists even at
equilibrium. At a single-determinant SCF level of the theory the valence electrons of a singlet system like H2 remain
perfectly paired, and one needs to include non-dynamical correlation to decouple the bond electron pair, giving rise to
a population of effectively-unpaired (“odd”, radicalized) electrons. 21,131,142 When the static correlation is strong, these
electrons remain mostly unpaired and can be described as being localized on individual atoms.
These phenomena can be properly described within wave-function formalism. Within DFT, these effects can be described by broken-symmetry approach or by using SF-TDDFT (see Section 7.3.1). Below we describe how to derive
this sort of information from pure DFT description of such low-spin open-shell systems without relying on spincontaminated solutions.
The first-order reduced density matrix (1-RDM) corresponding to a single-determinant wave function (e.g., SCF or
Kohn-Sham DFT) is idempotent:
Z
ρσ (r1 ) = γσSCF (1; 2) γσSCF (2; 1) dr2
γσSCF (1; 2) =

occ
X

(11.124)
KS
KS
ψiσ
(1)ψiσ
(2) ,

i

where ρσ (1) is the electron density of spin σ at position r1 , and γσSCF is the spin-resolved 1-RDM of a single Slater determinant. The cross product γσSCF (1; 2) γσSCF (2; 1) reflects the Hartree-Fock exchange (or Kohn-Sham exact-exchange)
governed by the HF exchange hole:
γσSCF (1; 2) γσSCF (2; 1) = ρα (1)hXσσ (1, 2)
Z
hXσσ (1, 2) dr2 = 1 .
When 1-RDM includes electron correlation, it becomes non-idempotent:
Z
Dσ (1) ≡ ρσ (1) − γσ (1; 2)γσ (2; 1) dr2 ≥ 0 .

(11.125)

(11.126)

The function Dσ (1) measures the deviation from idempotency of the correlated 1-RDM and yields the density of
effectively-unpaired (odd) electrons of spin σ at point r1 . 117,142 The formation of effectively-unpaired electrons in
singlet systems is therefore exclusively a correlation based phenomenon. Summing Dσ (1) over the spin components
gives the total density of odd electrons, and integrating the latter over space gives the mean total number of odd electrons
N̄u :
Z
X
Du (1) = 2
Dσ (1)dr1 , N̄u = Du (1)dr1 .
(11.127)
σ

The appearance of a factor of 2 in Eq. (11.127) above is required for reasons discussed in Ref. 117. In Kohn-Sham
DFT, the SCF 1-RDM is always idempotent which impedes the analysis of odd electron formation at that level of the
theory. Ref. 120 has proposed a remedy to this situation. It was noted that the correlated 1-RDM cross product entering
Eq. (11.126) reflects an effective exchange, also known as cumulant exchange. 21 The KS exact-exchange hole is itself
artificially too delocalized. However, the total exchange-correlation interaction in a finite system with strong left-right
(i.e., static) correlation is normally fairly localized, largely confined within a region of roughly atomic size. 14 The
effective exchange described with the correlated 1-RDM cross product should be fairly localized as well. With this in
mind, the following form of the correlated 1-RDM cross product was proposed: 120
γσ (1; 2) γσ (2; 1) = ρσ (1) h̄eff
Xσσ (1, 2) .

(11.128)

15
The function h̄eff
Xσσ (1; 2) is a model DFT exchange hole of Becke-Roussel (BR) form used in Becke’s B05 method.
15
The latter describes left-right static correlation effects in terms of certain effective exchange-correlation hole. The extra delocalization of the HF exchange hole alone is compensated by certain physically motivated real-space corrections
to it: 15
eff
h̄XCαα (1, 2) = h̄eff
(11.129)
Xαα (1, 2) + fc (1) h̄Xββ (1, 2) .

658

Chapter 11: Molecular Properties and Analysis

The BR exchange hole h̄eff
Xσσ is used in B05 as an auxiliary function, such that the potential from the relaxed BR hole
equals that of the exact-exchange hole. This results in relaxed normalization of the auxiliary BR hole less than or equal
to unity:
Z
eff
h̄eff
Xσσ (1; 2) dr2 = NXσ (1) ≤ 1 .

(11.130)

eff
The expression of the relaxed normalization NXσ
(r) is quite complicated, but it is possible to represent it in closed
118,119
eff
analytic form.
The smaller the relaxed normalization NXα
(1), the more delocalized the corresponding exact15
exchange hole. The α−α exchange hole is further deepened by a fraction of the β−β exchange hole, fc (1) h̄eff
Xββ (1, 2),
which gives rise to left-right static correlation. The local correlation factor fc in Eq.(11.129) governs this deepening
and hence the strength of the static correlation at each point: 15



fc (r) = min fα (r), fβ (r), 1

(11.131a)

0 ≤ fc (r) ≤ 1

(11.131b)

fα (r) =

eff
(r)
1 − NXα
.
eff
NXβ (r)

(11.131c)

Using Eqs. (11.131), (11.127), and (11.128), the density of odd electrons becomes:
eff
Dα (1) = ρα (1)(1 − NXα
(1))
eff
= ρα (1)fc (1) NXβ
(1) .

(11.132)

The final formulas for the spin-summed odd electron density and the total mean number of odd electrons read:


eff
eff
Du (1) = 4aop
nd fc (1) ρα (1) NXβ (1) + ρβ (1) NXα (1)
Z
(11.133)
N̄u = Du (r1 ) dr1 .
Here and-opp
= 0.526 is the SCF-optimized linear coefficient of the opposite-spin static correlation energy term of the
c
B05 functional. 15,119
It is informative to decompose the total mean number of odd electrons into atomic contributions. Partitioning in real
space the mean total number of odd electrons N̄u as a sum of atomic contributions, we obtain the atomic population of
odd electrons (FAr ) as:
Z
FAr =

Du (r1 ) dr1 .

(11.134)

ΩA

Here ΩA is a subregion assigned to atom A in the system. To define these atomic regions in a simple way, we use
the partitioning of the grid space into atomic subgroups within Becke’s grid-integration scheme. 13 Since the present
method does not require symmetry breaking, singlet states are calculated in restricted Kohn-Sham (RKS) manner even
at strongly stretched bonds. This way one avoids the destructive effects that the spin contamination has on FAr and
on the Kohn-Sham orbitals. The calculation of FAr can be done fully self-consistently only with the RI-B05 and RImB05 functionals. In these cases no special keywords are needed, just the corresponding EXCHANGE rem line for
these functionals. Atomic population of odd electron can be estimated also with any other functional in two steps: first
obtaining a converged SCF calculation with the chosen functional, then performing one single post-SCF iteration with
RI-B05 or RI-mB05 functionals reading the guess from a preceding calculation, as shown on the input example below:

Chapter 11: Molecular Properties and Analysis

659

Example 11.38 To calculate the odd-electron atomic population and the correlated bond order in stretched H2 , with
B3LYP/RI-mB05, and with fully SCF RI-mB05
$comment
Stretched H2: example of B3LYP calculation of the atomic population of odd electrons
with post-SCF RI-BM05 extra iteration.
$end
$molecule
0 1
H
0.
H
0.
$end

0.
0.

0.0
1.5000

$rem
SCF_GUESS
METHOD
BASIS
PURECART
THRESH
MAX_SCF_CYCLES
PRINT_INPUT
SCF_FINAL_PRINT
INCDFT
XC_GRID
SYM_IGNORE
SYMMETRY
SCF_CONVERGENCE
$end

CORE
B3LYP
G3LARGE
222
14
80
TRUE
1
FALSE
000128000302
TRUE
FALSE
9

@@@
$comment
Now one RI-B05 extra-iteration after B3LYP to generate the odd-electron atomic
population and the correlated bond order.
$end
$molecule
read
$end
$rem
SCF_GUESS
EXCHANGE
PURECART
BASIS
AUX_BASIS
THRESH
PRINT_INPUT
INCDFT
XC_GRID
SYM_IGNORE
SYMMETRY
MAX_SCF_CYCLES
SCF_CONVERGENCE
DFT_CUTOFFS
$end

READ
BM05
22222
G3LARGE
riB05-cc-pvtz
14
TRUE
FALSE
000128000302
TRUE
FALSE
0
9
0

@@@
$comment
Finally, a fully SCF run RI-B05 using the previous output as a guess.
The following input lines are obligatory here:
PURECART
22222
AUX_BASIS
riB05-cc-pvtz
DFT_CUTOFFS 0
$end

Chapter 11: Molecular Properties and Analysis

660

Mayer’s type follows in the code, using certain exact relationships between FAr , FBr , and the correlated bond order of
Mayer type BAB . Both new properties are printed at the end of the output, right after the multipoles section. It is useful
to compare the correlated bond order with Mayer’s SCF bond order. To print the latter, use SCF_FINAL_PRINT = 1.

11.18

Molecular Junctions

In molecular junctions, molecules bridge two metallic electrodes. The conductance and current-voltage relationship of
molecular junctions can be calculated using either Landauer or Non-Equilibrium Green’s Functions (NEGF). In both
cases, the Green’s function formulation is employed using a chosen level to describe the electronic structure. See Refs.
37,39 for further introduction.
In molecular junctions the current-voltage curve depends on the electron transmission function, which can be calculated using the quantum transport code developed by the Dunietz group (Kent State). The scattering-free approach,
(Landauer), provides a zero-bias limit, whilst the non-equilibrium approach, (NEGF), iteratively solves for the junction
under the effect of the finite-voltage biased system.
This quantum transport utility is invoked by setting the $rem variable TRANS_ENABLE.
TRANS_ENABLE
To invoke the molecular transport code.
TYPE:
INTEGER
DEFAULT:
0 Do not perform transport calculations (default).
OPTIONS:
1
Perform transport calculations.
−1 Print matrices needed for generating bulk model files.
RECOMMENDATION:
None

Output is provided in the Q-C HEM output file and in the following files (for closed shell system in the spin-restricted
framework):
• transmission.txt (Transmission function in the requested energy window)
• TDOS.txt (Total density of states)
• current.txt (I-V plot only for the Landauer level)
• FAmat.dat (Hamiltonian matrix for follow up calculations and analysis)
• Smat.dat (Overlap matrix for follow up calculations and analysis)
• IV-NEGF-all.txt (I-V plot obtained by NEGF method)
In the case of unrestricted spin the transmission is calculated for each spin-state as indicated by the A[B] appended to
the file names listed above (e.g. transmissionA.txt and transmissionB.txt). (The file name with the letter
AB indicates output data including both spin states (e.g. transmissionAB.txt). We note that in the closed-shell
spin-restricted case, the transmission.txt corresponds to the α spin, where the total transmission due to the spin
symmetry is twice the values included in the file. In the NEGF calculation, the above output files are placed in the
directories, Vbias1, Vbias2, · · · (the numbers in the directory names are index of bias voltage), where these output
files in each directory are of data at the given voltage.
T-Chem requires setting parameters in two transport-specific sections in the input file:
• $trans_model (molecular model regions)

661

Chapter 11: Molecular Properties and Analysis
• $trans_method (Specifies the mode to calculated the electrode self-energies)

lbasis

device region

rbasis

lgbasis

lgatom
Ag

Au

rgbasis

latom

ratom

rgatom

C

Figure 11.1: Illustration for the different regions of the molecular junction for Landauer calculation.

Figure 11.2: Illustration for the different regions of the molecular junction for NEGF calculation.
The $trans_model section provides the number of basis functions in the different regions (molecular model partitioning). There are no default values given for these parameters. The different regions are illustrated in Fig. 11.1 with a
six carbon atom chain based bridge used as an example of the Landauer calculation. The partitioning as well as the
scheme for the NEGF calculation is illustrated in Fig. 11.2. In these systems the electrodes are represented by a chain
of Au atoms. In this example of electrode wires, each single atom represents one layer.
The necessary parameters in the $trans_model section are as follows:
• trans_latom: INTEGER, atom index in the $molecule section where the central/bridge region starts (the first
atom in junction area).
• trans_ratom: INTEGER, atom index in the $molecule section where the central/bridge region ends (the last
atom in junction area).
• trans_lgatom: INTEGER, atom index where the repeat unit of the left electrode starts.
• trans_rgatom: INTEGER, atom index where the repeat unit of the right electrode ends.
See Fig. 11.1 for illustrations of the different regions, and the way the parameters define them. Atoms within numbers trans_lgatom, trans_lgatom+1, .., trans_latom-1 define the repeat unit of the left electrode (similarly for right
electrode).
Alternatively, the different regions can be provided with the atomic orbital (AO) index as follows:
• trans_lbasis: INTEGER, the number of basis functions appearing to the left of the device region (the index of
the first AO within the central region).
• trans_rbasis: INTEGER, the number of basis functions appearing to the right of the device region right electrode
(total number of basis functions minus this number equals last AO that is within the device).

Chapter 11: Molecular Properties and Analysis

662

• trans_lgbasis: INTEGER, the number of basis functions of the repeat unit of the left electrode.
• trans_rgbasis: INTEGER, the number of basis functions of the repeat unit of the right electrode.
For the NEGF calculation, trans_lbasis and trans_lgbasis must be the same number as shown in Fig. 11.2 (as for
trans_rbasis and trans_rgbasis.
Note: The assignment of trans_latom etc. has priority. If trans_latom is specified, then trans_lbasis is ignored.
Similarly for trans_latom and trans_lbasis.
For the example in Fig. 11.1, if there are a total of 18 atoms and the Au and Ag basis sets each contain 22 basis functions
per atom and the repeat unit includes a pair of Au Ag atoms, then the parameters should be given as follows (only the
first two columns are required, the rest are included for explanation):
trans_lbasis
trans_rbasis
trans_lgbasis
trans_rgbasis

88
88
44
44

No. of functions representing left electrode region (2 × 22 for Ag + 2 × 22 for Au)
No. of functions representing right electrode (2 × 22 for Ag + 2 × 22 for Au)
Size of the repeating unit of the left electrode (22 for Ag + 22 for Au)
Size of the repeating unit of the right electrode (22 for Ag + 22 for Au)

Or use the atom numbers corresponding to their position in the $molecule section:
trans_lgatom
trans_latom
trans_ratom
trans_rgatom

3
5
14
16

Third atom is used to define the repeat unit of the left electrode
Fifth atom is the first atom of the junction
Fourteenth atom is the last atom of the junction
Sixteenth atom is used to define the repeat unit of the right electrode

In this example, we have used for the same repeat unit for the left and right electrodes; this symmetry is not required.
Note: The order of atoms in the $molecule section is important and requires to following:
Repeating units (left) - Molecular Junction - Repeating units (right)
The atoms are provided first by the leftmost repeat unit with the left electrode then proceeds to the next repeat unit
up to the surface unit. Next the bridge atoms are provided followed by the right surface unit and the right electrode
region. The right electrode region starts with the surface layer and ends with the most distant layer within the bulk. The
atoms order within each electrode layer (the repeat units) must be consistent. The atoms order within the bridge region
(excluding the electrode repeat unit atoms) is arbitrary.
That is, the order of atoms in the molecule section has to adhere to the following:
1. atoms of the leftmost repeat unit
2. atoms of the next repeat unit
3. atoms of the left surface unit device
4. bridge atoms
5. atoms of the right surface unit
6. atoms of the next right electrode unit
7. atoms of the rightmost repeat unit
T-Chem allows for complete flexibility in determining the different regions of the electrode models. As a consequence,
incorrect setting of regions is not caught by the program and may produce transmission functions that are unphysical
(e.g. large values or even negative). Such errors can occur where the cluster model is partitioned (by mistake) within
the orbital space of an atom. Regions must always be defined between atomic layers. Each repeat unit atoms should be
always provided with the same internal order.
Note: At least a single repeat unit of the electrodes should be included in the bridge region. With the Landauer model,
if trans_readhs == 0, then at least one layer beyond the bridge region has to be included, an additional layer
(total of two or more) is required when trans_method != 0.

Chapter 11: Molecular Properties and Analysis
The necessary parameters in $trans_method section are listed as follows,
trans_mode
Mode of calculation.
INPUT SECTION: $trans_method
TYPE:
INTEGER
DEFAULT:
1 Landauer level.
OPTIONS:
3 A self-consistent Green’s function calculation with zero bias voltage (i.e. NEGF with zero
bias, which is used for preparation of full NEGF).
4 Full NEGF level
RECOMMENDATION:
For modes 3 and 4 SCF_ALGORITHM = NEGF must be set in in the $rem section.
trans_spin
Spin coupling scheme.
INPUT SECTION: $trans_method
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
0 For restricted spin calculations or closed-shell singlet states.
3 For unrestricted spin calculations or open-shell systems
RECOMMENDATION:
None
trans_method
Electrode surface GFs model.
INPUT SECTION: $trans_method
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
0 A wide band limit with a constant parameter trans_greens (default)
1 WBL using the Ke-Baranger-Yang TB at the Fermi energy.
2 WBL using the Lopez-Sancho TB at the Fermi energy.
3 Tight-binding (TB) following the procedure proposed by Ke-Baranger-Yang
4 TB following the procedure proposed by Lopez-Sancho (decimation).
RECOMMENDATION:
Only option 0 is available for the NEGF calculation at the current version.

663

Chapter 11: Molecular Properties and Analysis
trans_npoint
Number of grid points within the energy window of the transmission spectra calculation.
INPUT SECTION: $trans_method
TYPE:
INTEGER
DEFAULT:
300
OPTIONS:
n User-specified number of points
RECOMMENDATION:
None
trans_readhs
Flag to read the Hamiltonian and overlap matrices for the bulk model.
INPUT SECTION: $trans_method
TYPE:
INTEGER
DEFAULT:
0 Use the current Hamiltonian and overlap matrices to parse the electrode integrals.
OPTIONS:
1 Use pre-calculated electrode Hamiltonian and overlap matrices.
RECOMMENDATION:
If set to 1, the following files are requred: FAmat2l.dat and Smat2l.dat (for left
electrode model), FAmat2r.dat and Smat2r.dat (for right electrode model). If both
electrodes are of the same type, may use symbolic links of these files to the same matrices.
(For unrestricted spin model, FBmat2l.dat and FBmat2r.dat are also necessary)

Note: NEGF requires trans_readhs to be set!
trans_htype
Determines the TB property on the relevant coupling terms.
INPUT SECTION: $trans_method
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
When trans_readhs = 0:
0 All coupling integrals between the junction and electrode functions are allowed as determined at the cluster model level (no screening imposed).
1 Only coupling between neighboring repeating units of the electrode model are allowed –
all terms of the repeating units elements that are beyond the neighboring unit are set to
zero.
2 Force both the TB coupling terms and the self-energy coupling terms to the same value as
determined by the electrode model. If trans_method != 0, the inequalities trans_lbasis
≥ 2×trans_lgbasis and trans_rbasis ≥ 2×trans_rgbasis must be satisfied.
When trans_readhs = 1:
2 Available using the pre-calculated electrode data.
3 The same way as 1 but using the pre-calculated electrode data. This is only for NEGF
calculation.
RECOMMENDATION:
None

664

Chapter 11: Molecular Properties and Analysis

665

Further options are summarized below:
trans_printdos (Integer):
Controls the printout of TDOS.
0 Default, no total DOS printing.
1 A TDOS (of the junction region) will be printed to TDOS.txt (closed shell).
trans_printiv (Integer):
Controls printout of calculated current.
0 Default, no current calculated and printed
1 Current will be printed to current.txt (closed shell) or currentA/B.txt (unrestricted or
open shell).
trans_ipoints (Integer):
Number of points for current calculation.
300 Default value.
trans_adjustefermi (Integer):
Flag to adjust the Fermi energy (FE) for trans_mode = 3 or 4
0 Default, no adjustment. Fixed FE specified by trans_efermi (and trans_efermib) is used.
1 FE is chosen as midpoint of HOMO and LUMO levels
2 FE is adjusted so that charge neutrality is satisfied
3 FE is adjusted by combination way of 1 and 2, i.e. use 2 if maximum difference of density matrix in
the iteration is over 10−2 and use 3 below that.
Options 1, 2, and 3 use the same FE for α and β spins. -1, -2, -3 are the same as 1, 2, 3, respectively, but allow for
different FEs for α and β spins.
trans_nvbias (Integer):
(Only for trans_mode = 4), number of points of bias voltage.
1 Default
The bias voltage values to be calculated are defined by dividing the range between trans_vstart and trans_vmax with
this number. For example, when trans_vstart = 0.0, trans_vmax = 1.0, and trans_nvbias = 5, the voltages are 0.0,
0.25, 0.50, 0.75, and 1.0. For the case of trans_nvbias = 1, the voltage to be calculated is trans_vmax.
trans_updatedmatlr (Integer):
(Only for trans_mode = 3), flag to update L, R, LC, RC blocks (i.e. except for center block) of density matrix during
SCF with zero bias. This is to prepare consistent density matrix with modified Hamiltonian matrix forced by read-in
bulk electrode data. Note that it may make it difficult or slow to converge.
0 No update.
1 Update every iteration step (default).
2 Mix the new density matrix with ratio of trans_mixing.
The following are parameters are set to a double precision value. The allowed values are set using trans_itodfac as
follows:
trans_itodfac (Integer):
Controls the accuracy for input parameters.
100
Default, the numbers can be of 0.0x precision.
1000 The input double numbers can be of 0.00x precision, etc.
trans_vstart (Double):
(Only for trans_mode = 4). Starting voltage bias (V). The bias voltage increases from trans_vstart to trans_vmax in
the NEGF calculation.
0.0 Default
trans_devsmear (Double):
Imaginary smearing (in eV) added to the real Hamiltonian in central region retarded GF evaluation.
0.01 Default, cannot be smaller than 1/trans_itodfac

Chapter 11: Molecular Properties and Analysis

666

trans_bulksmear (Double):
Imaginary smearing (in eV) added to the real Hamiltonian in electrodes GF evaluation.
0.01 Default, cannot be smaller than 1/trans_itodfac
trans_greens (Double):
Imaginary smearing/Broadening (in eV) added to the Green’s function.
0.07 Default, cannot be smaller than 1/trans_itodfac
trans_efermi (Double):
Fermi energy of the electrode (for α spin) (in eV) used for defining energy range in calculating current for T(E).
-5.0 Default
trans_efermib (Double):
Fermi energy for β spin (in eV). If this is not given, the same value of α spin is used.
trans_vmax (Double):
Maximum voltage bias (V).
1.0 Default
trans_gridoffset (Double):
(Only for trans_mode = 4), Offset distance (in Å) to define the grid box region for bias potential energy.
5.0 Default
The box size is defined by adding the offset distance with maximum and minimum x, y and z atomic coordinates for
each direction. The bias voltage V (r) on the grid points in the box is used for calculating correction term for the Fock
matrix (i.e. hi|V |ji). Note that this box is not for correcting electrostatic potential by solving Poisson equation. The
grid box region and its grid size for the Poisson equation is given by $plots block keyword (see also example later).
The same grid size is used for both boxes.
The following are parameters for when trans_mode = 3, 4:
trans_mixing (Double):
Mixing ratio of DIIS mixing method for updating the central block of density matrix in the NEGF iteration.
1.0 Default
trans_mixhistory (Integer):
The number of NEGF iteration steps in which the history of density matrix is stocked for the DIIS method.
40 Default
trans_dehcir (Double):
Grid size, dE (in eV), for integrating the Greens function on the half circle path on imaginary plane.
1.0 Default
trans_delpart (Double):
Grid size, dE (in eV), for integrating the Greens function on the path of the linear part on imaginary plane.
0.01 Default
trans_debwin (Double):
(For trans_mode = 4). Grid size, dE (in eV), for integration on the non-equilibrium term.
0.01 Default
trans_numres (Integer):
The number of poles at Fermi energy enclosed by closed contour on the imaginary plane.
100 Default
trans_peconv (Integer):
The convergence criteria of the iteration of the Poisson equation. The threshold is 10−n hartree of maximum energy
difference over the all grid points.
9 Default

Chapter 11: Molecular Properties and Analysis

667

trans_pemaxite (Integer):
Maximum iteration number of the Poisson equation.
1000 Default
trans_readesp (Integer):
Flag of read-in electrostatic potential energy. The data is printed in "ReadInESP/" directory with option of -1 or 0, and
read from the same directory name with option of 0 or 1.
-1 Print the read-in ESP data in "ReadInESP/" directory at the first step and stop calculation.
0
Print the read-in ESP data in "ReadInESP/" at the first step and continue calculation using it (default).
1
Read the pre-calculated read-in ESP data from "ReadInESP/" and continue calculation.
trans_restart (Integer):
Flag to restart reading the density matrix files, DAmat.dat (and DBmat.dat) from the "TransRestart/" directory.
0 No restart (default).
1 Read file of density matrix.
Note: The default energy window for transmission and current calculations is defined as:
trans_emin = trans_efermi - trans_vmax/2 and trans_emax = trans_efermi + trans_vmax/2
if trans_emin and trans_emax are not given.
The trans_emin and trans_emax values can be set to determine the energy window for calculating the transmission
function. If specified, these values will override the window defined by trans_efermi and trans_vmax values.
If higher accuracy parameters are set, trans_itodfac must be increased. For example for three digits accuracy (e.g.
trans_efermi = -5.341), then trans_itodfac = 1000 or higher must be used. (This parameter applies to all double
precision parameters.)
For trans_readhs = 1, the following parameters to parse precalculated Hamiltonian and overlap matrices have to be
provided:
• trans_totorb2 (Integer):
Total number of basis functions in the electrode models (if set, then same size is assumed for both electrodes)
(no default value).
• trans_totorb2l: Integer number, total number of basis functions in left electrode model (no default value).
• trans_totorb2r: Integer number, total number of basis functions in right electrode model (no default value).
• trans_startpoint: Integer number, start point (basis number) for reading the TB integrals. Note, the basis number, that is, index number of basis function starts from 0. (if set, then same size is assumed for both electrodes)
(default value 0).
• trans_startpointl: given as integer number, left start point (basis number) for reading the TB integrals. Note,
the basis number, that is, index number of basis function starts from 0. (default value 0).
• trans_startpointr: given as integer number, right start point (basis number) for reading the TB integrals. Note,
the basis number, that is, index number of basis function starts from 0. (no default value).

Chapter 11: Molecular Properties and Analysis
As an example of the Landauer calculation, the sample Q-C HEM input is given below.
Example 11.39 Quantum transport Landauer calculation applied to C6 between two gold electrodes.
$molecule
0 1
Ag -11.0
Au
-8.3
Ag
-5.6
Au
-2.9
Ag
-0.2
Au
2.5
C
4.8
C
6.5
C
8.2
C
9.9
C
11.6
C
13.3
Au
15.6
Ag
18.3
Au
21.0
Ag
23.7
Au
26.4
Ag
29.1
$end

0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

$rem
METHOD
BASIS
ECP
GEOM_OPT_MAXCYC
INCDFT
MEM_STATIC
MAX_SCF_CYCLES
MEM_TOTAL
MOLDEN_FORMAT
SCF_CONVERGENCE
SCF_ALGORITHM
TRANS_ENABLE
$end

B3LYP
lanl2dz
lanl2dz
200
FALSE
8000
400
32000
TRUE
10
diis
1

$trans-method
trans_spin
trans_npoints
trans_method
trans_readhs
trans_printdos
trans_efermi
trans_vmax
$end

0
300
0
0
1
-6.50
4.00

$trans-model
trans_lgatom
trans_latom
trans_ratom
trans_rgatom
$end

3
5
14
16

A sample for unrestricted spin calculation can be found in the $QC/samples/tchem directory.
For NEGF calculations, note the followings concerning input etc:
• SCF_ALGORITHM = NEGF in $rem is necessary for NEGF calculation.

668

Chapter 11: Molecular Properties and Analysis

669

• The same numbers for trans_lgbasis and trans_lbasis, and also for trans_rgbasis and trans_rbasis must be
used.
• Only WBL method for evaluating self-energy is available for the NEGF in the current version (trans_method =
0)
• trans_htype = 3 and trans_readhs = 1 are required
• SYM_IGNORE = TRUE to keep the original input coordinates.
• MEM_TOTAL and MEM_STATIC may need to be set to ensure enough memory is available.
• The bias voltage of +V /2 is added on the left electrode, and −V /2 on the right electrode, where the bias potential
slope is along x-axis, i.e. the system structure must be built along x direction.
• In the $plots section (when defining a grid box for the Poisson equation solving), the grid box region must cover
all atoms except for the left and right electrode parts defined in $trans_model. All integer index flags in $plots
can be 0.
• For calculations on bulk electrode, the same structure with the same orientation in xyz-Cartesian coordinate
system (but different size) must be used so as to reproduce the same overlap matrix at the used part.
• For better calculations, update of electrode relevant (non-center) blocks in the density matrix (trans_updatedmatlr
= 1) and Fermi energy to satisfy charge neutrality (trans_adjustefermi = 2 or 3) are recommended at zero bias
(trans_opt = 3) before performing a full NEGF calculation. However, these options may make it difficult or
slow to reach convergence.
• The criterion for convergence in the NEGF iterations is the maximum difference in density matrix elements, and
is not a energy threshold.
NEGF calculations depend on the following pre-calculated properties:
Step 1: Pre-calculations:
A: Hamiltonian and overlap matrices for the left and right bulk electrodes (required)
B: A converged junction electronic state by standard DFT (recommended)
C: Electrostatic potential of large electrode region (optional)
Step 2: Self-consistent Greens function calculation with zero bias:
Non zero bias cases should be calculated using density matrices calculated with zero bias voltage to obtain
converged density matrix. It is also recommend to evaluate the Fermi energy level within the NEGF scheme.
Step 3: NEGF calculations should be obtained by increasing the bias sequentially.

Chapter 11: Molecular Properties and Analysis

670

Examples of these steps applied to C2 between two aluminium electrodes are given below.
Example 11.40 Step 1-A of the NEGF calculation, the pre-calculation of the bulk electrode. Flag keyword of printing
matrices must be set (i.e. trans_enable != 0).
$molecule
0 1
Al
-15.04250
Al
-12.30750
Al
-9.572500
Al
-6.837500
Al
-4.102500
Al
-1.367500
Al
1.367500
Al
4.102500
Al
6.837500
Al
9.572500
Al
12.30750
Al
15.04250
$end
$rem
UNRESTRICTED
SYM_IGNORE
MAX_SCF_CYCLES
EXCHANGE
CORRELATION
ECP
BASIS
SCF_CONVERGENCE
$end

0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0

true
true
500
hf
none
hwmb
hwmb
4

@@@
$molecule
read
$end
$rem
UNRESTRICTED
SYM_IGNORE
MAX_SCF_CYCLES
EXCHANGE
CORRELATION
ECP
BASIS
SCF_GUESS
SCF_GUESS_MIX
SCF_CONVERGENCE
$end

true
true
500
b3lyp
none
hwmb
hwmb
read
3
4

0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0

Chapter 11: Molecular Properties and Analysis

671

@@@
$molecule
read
$end
$rem
UNRESTRICTED
SYM_IGNORE
MAX_SCF_CYCLES
EXCHANGE
CORRELATION
ECP
BASIS
SCF_GUESS
SCF_GUESS_MIX
SCF_CONVERGENCE
TRANS_ENABLE
$end

true
true
500
b3lyp
none
hwmb
hwmb
read
3
7
-1

$trans-method
trans_spin
$end

2

Example 11.41 Step 1-B of the NEGF calculation - the pre-calculation by the standard DFT. The same molecular
structure with the step 2 must be used
$molecule
0 1
Al -13.615
Al -10.880
Al
-8.145
Al
-5.410
Al
-2.675
C
-0.627
C
0.627
Al
2.675
Al
5.410
Al
8.145
Al
10.880
Al
13.615
$end

0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0

$rem
UNRESTRICTED
SYM_IGNORE
MAX_SCF_CYCLES
EXCHANGE
CORRELATION
BASIS
ECP
SCF_CONVERGENCE
$end

0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0

true
true
400
b3lyp
none
hwmb
hwmb
5

Chapter 11: Molecular Properties and Analysis

672

@@@
$molecule
read
$end
$rem
UNRESTRICTED
SYM_IGNORE
EXCHANGE
CORRELATION
BASIS
ECP
MAX_SCF_CYCLES
SCF_CONVERGENCE
SCF_GUESS
SCF_GUESS_MIX
$end

true
true
b3lyp
none
hwmb
hwmb
400
6
read
3

Example 11.42 Step 2 of the NEGF calculation. As the preparation of this step, followings are necessary:
1. FAmat2l.dat, FAmat2r.dat, Smat2l.dat, and Smat2r.dat (also FBmat2l.dat and FBmat2r.dat
if the calculation is spin-unrestricted ) must be placed in the same directory of the Q-Chem input file by coping
or linking the output files of the step 1-A.
2. Restart directory of the standard DFT obtained in the step 1-B must be copied to here.
3. (Optional) Read-in electrostatic potential data in "ReadInESP/" directory must be placed if this option is used
(for trans_readesp = 1). This option can provide more bulk electrode electrostatic environment as the boundary
condition of Poisson equation solving (see the sample files in $QC/samples/tchem/) for more details).
$molecule
read
$end
$rem
JOBTYPE
UNRESTRICTED
SYM_IGNORE
MAXSCF
EXCHANGE
CORRELATION
BASIS
ECP
SCF_CONVERGENCE
SCF_ALGORITHM
SCF_GUESS
MEM_TOTAL
MEM_STATIC
TRANS_ENABLE
$end

sp
true
true
500
b3lyp
none
hwmb
hwmb
4
negf
read
16000
4000
1

$plots
For NEGF (for Poisson equation)
190
-9.5
9.5
80
-4.0
4.0
80
-4.0
4.0
0 0 0 0
0
$end

Chapter 11: Molecular Properties and Analysis

673

$trans-method
trans_opt
3
trans_spin
2
trans_npoints
500
trans_method
0
trans_printdos
1
trans_printiv
1
trans_adjustefermi 1
trans_vmax
1.0
trans_emin
-6.5
trans_emax
-2.5
trans_mixing
0.1
trans_mixhistory
50
trans_dehcir
1.0
trans_delpart
0.01
trans_numres
100
trans_peconv
8
trans_pemaxite
1000
trans_updatedmatlr 0
trans_readesp
0
trans_htype
3
trans_readhs
1
trans_totorb2
48
trans_startpointl 16
trans_startpointr 32
$end
$trans-model
trans_lbasis
trans_rbasis
trans_lgbasis
trans_rgbasis
$end

8
8
8
8

Example 11.43 Step 3 of the NEGF calculation. As the preparation of this step, followings are necessary:
1. In the same way as step 2, FAmat2l.dat, FAmat2r.dat, Smat2l.dat, and Smat2r.dat (also FBmat2l.dat
and FBmat2r.dat for spin-unrestricted calculations) must be placed.
2. Restart directory for Q-Chem generated in the step 2 must be copied to here (only coordinates are used).
3. Restart directory for density matrix "TransRestart/" must be created and DAmat.dat (and DBmat.dat for
spin-unrestricted) generated in the step 2 must be copied or linked in the directory.
4. Read-in electrostatic potential data in "ReadInESP/" directory used in the step 2 must be copied to here.
5. Put Fermi energy obtained in Step 2 (recommended).
$molecule
read
$end
$rem
UNRESTRICTED
MAXSCF
SYM_IGNORE
EXCHANGE
ECP
SCF_CONVERGENCE
SCF_ALGORITHM
MEM_TOTAL
MEM_STATIC
TRANS_ENABLE
$end

true
500
true
b3lyp
hwmb
4
negf
16000
4000
1

Chapter 11: Molecular Properties and Analysis
$plots
For NEGF calculation
190
-9.5
9.5
80
-4.0
4.0
80
-4.0
4.0
0 0 0 0
0
$end
$trans-method
trans_opt
4
trans_spin
2
trans_npoints
500
trans_method
0
trans_printdos
1
trans_printiv
1
trans_adjustefermi 0
trans_efermi
-4.421836
trans_vmax
0.5
trans_emin
-6.5
trans_emax
-2.5
trans_mixing
0.2
trans_mixhistory
50
trans_dehcir
1.0
trans_delpart
0.01
trans_debwin
0.01
trans_numres
100
trans_peconv
8
trans_pemaxite
1000
trans_gridoffset
4.0
trans_updatedmatlr 0
trans_nvbias
6
trans_restart
1
trans_readesp
1
trans_htype
3
trans_readhs
1
trans_totorb2
48
trans_startpointl 16
trans_startpointr 32
$end
$trans-model
trans_lbasis
trans_rbasis
trans_lgbasis
trans_rgbasis
$end

8
8
8
8

674

Chapter 11: Molecular Properties and Analysis

675

References and Further Reading
[1] The M OL D EN program may be freely downloaded from www.cmbi.ru.nl/molden/molden.html.
[2] M AC M OL P LT may be downloaded from https://brettbode.github.io/wxmacmolplt.
[3] NBO 5.0 manual: www.chem.wisc.edu/~nbo5.
[4] The VMD program may be downloaded from www.ks.uiuc.edu/Research/vmd.
[5] Ground-State Methods (Chapters 4 and 6).
[6] Excited-State Calculations (Chapter 7).
[7] Basis Sets (Chapter 8).
[8] A. Adel and D. M. Dennison. Phys. Rev., 43:716, 1933. DOI: 10.1103/PhysRev.43.716.
[9] W. D. Allen, Y. Yamaguchi, A. G. Csázár, D. A. Clabo, Jr., R. B. Remington, and H. F. Schaefer III. Chem.
Phys., 145:427, 1990. DOI: 10.1016/0301-0104(90)87051-C.
[10] T. Bally and P. R. Rablen. J. Org. Chem, 76:4818, 2011. DOI: 10.1021/jo200513q.
[11] S. A. Bäppler, F. Plasser, M. Wormit, and A. Dreuw. Phys. Rev. A, 90:052521, 2014. DOI: 10.1103/PhysRevA.90.052521.
[12] V. Barone. J. Chem. Phys., 122:014108, 2005. DOI: 10.1063/1.1824881.
[13] A. D. Becke. J. Chem. Phys., 88:2547, 1988. DOI: 10.1063/1.454033.
[14] A. D. Becke. J. Chem. Phys., 119:2972, 2003. DOI: 10.1063/1.1589733.
[15] A. D. Becke and E. R. Johnson. J. Chem. Phys., 122:154104, 2005. DOI: 10.1063/1.1884601.
[16] E. J. Berquist and D. S. Lambrecht. A first principles approach for partitioning linear response properties into
additive and cooperative contributions. (preprint).
[17] N. A. Besley and J. A. Bryan. J. Phys. Chem. C, 112:4308, 2008. DOI: 10.1021/jp076167x.
[18] N. A. Besley and K. A. Metcalf. J. Chem. Phys., 126:035101, 2007. DOI: 10.1063/1.2426344.
[19] N. A. Besley, A. M. Lee, and P. M. W. Gill. Mol. Phys., 100:1763, 2002. DOI: 10.1080/00268970110111779.
[20] N. A. Besley, D. P. O’Neill, and P. M. W. Gill. J. Chem. Phys., 118:2033, 2003. DOI: 10.1063/1.1532311.
[21] R. C. Bochicchio. J. Mol. Struct. (Theochem), 429:229, 1998. DOI: 10.1016/S0166-1280(97)00357-6.
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Chapter 12

Molecules in Complex Environments:
Solvent Models, QM/MM and QM/EFP
Features, Density Embedding
12.1

Introduction

Q-C HEM has incorporated a number of methods for complex systems such as molecules in solutions, proteins, polymers, molecular clusters, etc., summarized as follows:
• Implicit solvation models;
• QM/MM tools;
• EFP and QM/EFP approach (polarizable electrostatic embedding); and
• Density embedding methods.

12.2

Chemical Solvent Models

Ab initio quantum chemistry makes possible the study of gas-phase molecular properties from first principles. In liquid
solution, however, these properties may change significantly, especially in polar solvents. Although it is possible to
model solvation effects by including explicit solvent molecules in the quantum-chemical calculation (e.g. a supermolecular cluster calculation, averaged over different configurations of the molecules in the first solvation shell), such
calculations are very computationally demanding. Furthermore, cluster calculations typically do not afford accurate
solvation energies, owing to the importance of long-range electrostatic interactions. Accurate prediction of solvation
free energies is, however, crucial for modeling of chemical reactions and ligand/receptor interactions in solution.
Q-C HEM contains several different implicit solvent models, which differ greatly in their level of sophistication. These
are generally known as self-consistent reaction field (SCRF) models, because the continuum solvent establishes a
“reaction field” (additional terms in the solute Hamiltonian) that depends upon the solute electron density, and must
therefore be updated self-consistently during the iterative convergence of the wave function. The simplest and oldest of
these models that is available in Q-C HEM is the Kirkwood-Onsager model, 57,58,89 in which the solute molecule is placed
inside of a spherical cavity and its electrostatic potential is represented in terms of a single-center multipole expansion.
More sophisticated models, which use a molecule-shaped cavity and the full molecular electrostatic potential, include
the conductor-like screening model 62 (COSMO) and the closely related conductor-like PCM (C-PCM), 6,28,115 along

682

Chapter 12: Molecules in Complex Environments

Model

Kirkwood-Onsager
Langevin Dipoles
C-PCM
SS(V)PE/
IEF-PCM
COSMO
Isodensity SS(V)PE
SM8
SM12
SMD

Cavity
Construction

Discretization

spherical
atomic spheres
(user-definable)
atomic spheres
(user-definable)
atomic spheres
(user-definable)
predefined
atomic spheres
isodensity contour
predefined
atomic spheres

point charges
dipoles in
3-d space
point charges or
smooth Gaussians
point charges or
smooth Gaussians

predefined
atomic spheres
predefined
atomic spheres

generalized
Born
point charges

NonElectrostatic
Terms?
no

Supported
Basis
Sets
all

no

all

userspecified
userspecified

all
all

point charges

none

all

point charges
generalized
Born

none

all
6-31G*
6-31+G*
6-31+G**

automatic
automatic

all

automatic

all

Table 12.1: Summary of implicit solvation models available in Q-C HEM, indicating how the solute cavity is constructed and discretized, whether non-electrostatic terms are (or can be) included, which basis sets are available for
use with each model, and whether analytic first and second derivatives are available for optimizations and frequency
calculations.
with the “surface and simulation of volume polarization for electrostatics” [SS(V)PE] model. 20 The latter is also known
as the “integral equation formalism” (IEF-PCM). 17,18
The C-PCM and IEF-PCM/SS(V)PE are examples of what are called “apparent surface charge” SCRF models, although the term polarizable continuum models (PCMs), as popularized by Tomasi and coworkers, 114 is now used
almost universally to refer to this class of solvation models. Q-C HEM employs a Switching/Gaussian or “SWIG” implementation of these PCMs. 42,67–70 This approach resolves a long-standing—though little-publicized—problem with
standard PCMs, namely, that the boundary-element methods used to discretize the solute/continuum interface may lead
to discontinuities in the potential energy surface for the solute molecule. These discontinuities inhibit convergence of
geometry optimizations, introduce serious artifacts in vibrational frequency calculations, and make ab initio molecular
dynamics calculations virtually impossible. 67,68 In contrast, Q-C HEM’s SWIG PCMs afford potential energy surfaces
that are rigorously continuous and smooth. Unlike earlier attempts to obtain smooth PCMs, the SWIG approach largely
preserves the properties of the underlying integral-equation solvent models, so that solvation energies and molecular
surface areas are hardly affected by the smoothing procedure.
Other solvent models available in Q-C HEM include the “Langevin dipoles” model; 35,36 as well as versions 8 and 12
of the SMx models, and the SMD model, developed at the University of Minnesota. 80,81,84 SM8 and SM12 are based
upon the generalized Born method for electrostatics, augmented with atomic surface tensions intended to capture nonelectrostatic effects (cavitation, dispersion, exchange repulsion, and changes in solvent structure). Empirical corrections
of this sort are also available for the PCMs mentioned above, but within SM8 and SM12 these parameters have been
optimized to reproduce experimental solvation energies. SMD (where the “D” is for “density") combines IEF-PCM
with the non-electrostatic corrections, but because the electrostatics is based on the density rather than atomic point
charges, it is supported for arbitrary basis sets whereas SM8 and SM12 are not.
Table 12.1 summarizes the implicit solvent models that are available in Q-C HEM. Solvent models are invoked via the
SOLVENT_METHOD keyword, as shown below. Additional details about each particular solvent model can be found in
the sections that follow. In general, these methods are available for any SCF level of electronic structure theory, though
in the case of SM8 only certain basis sets are supported. Post-Hartree–Fock calculations can be performed by first
running an SCF + PCM job, in which case the correlated wave function will employ MOs and Hartree-Fock energy
levels that are polarized by the solvent.

683

Chapter 12: Molecules in Complex Environments

Energy Derivatives
SCF energy gradient
SCF energy Hessian
CIS/TDDFT energy gradient
CIS/TDDFT energy Hessian
MP2 & DH-DFT energy
derivatives
Coupled cluster methods

C-PCM
yes
yes
yes
yes

SS(V)PE/
IEF-PCM
yes
no
no
no

COSMO
yes
yes

SM8

SM12

yes
no
no
no
— unsupported —
— unsupported —

SMD
yes
no

— unsupported —
— unsupported —

Table 12.2: Summary of analytic energy gradient and Hessian available with implicit solvent models.
Table 12.2 summarizes the analytical energy gradient and Hessian available with implicit solvent models. For unsupported methods, finite difference methods may be used for performing geometry optimizations and frequency calculations.
Note: The job-control format for specifying implicit solvent models changed significantly starting in Q-C HEM version 4.2.1. This change was made in an attempt to simply and unify the input notation for a large number of
different models.
SOLVENT_METHOD
Sets the preferred solvent method.
TYPE:
STRING
DEFAULT:
0
OPTIONS:
0
Do not use a solvation model.
ONSAGER
Use the Kirkwood-Onsager model (Section 12.2.1).
PCM
Use an apparent surface charge, polarizable continuum model
(Section 12.2.2).
ISOSVP
Use the isodensity implementation of the SS(V)PE model
(Section 12.2.5).
COSMO
Use COSMO (similar to C-PCM but with an outlying charge
correction; 5,61 see Section 12.2.7).
SM8
Use version 8 of the Cramer-Truhlar SMx model (Section 12.2.8.1).
SM12
Use version 12 of the SMx model (Section 12.2.8.2).
SMD
Use SMD (Section 12.2.8.3).
CHEM_SOL Use the Langevin Dipoles model (Section 12.2.9).
RECOMMENDATION:
Consult the literature. PCM is a collective name for a family of models and additional input
options may be required in this case, in order to fully specify the model. (See Section 12.2.2.)
Several versions of SM12 are available as well, as discussed in Section 12.2.8.2.

Before going into detail about each of these models, a few potential points of confusion warrant mention, with regards
to nomenclature. First, “PCM” refers to a family of models that includes C-PCM and SS(V)PE/IEF-PCM (the latter
two being completely equivalent 18 ). One or the other of these models can be selected by additional job control variables
in a $pcm input section, as described in Section 12.2.2. COSMO is very similar to C-PCM but includes a correction for
that part of the solute’s electron density that penetrates beyond the cavity (the so-called “outlying charge”). 5,61 This is
discussed in Section 12.2.7.
Two implementations of the SS(V)PE model are also available. The PCM implementation (which is requested by
setting SOLVENT_METHOD = PCM) uses a solute cavity constructed from atom-centered spheres, as with most other
PCMs. On the other hand, setting SOLVENT_METHOD = ISOSVP requests an SS(V)PE calculation in which the solute

Chapter 12: Molecules in Complex Environments

684

cavity is defined by an isocontour of the solute’s own electron density, as advocated by Chipman. 20–22 This is an appealing, one-parameter cavity construction, although it is unclear that this construction alone is superior in its accuracy
to carefully-parameterized atomic radii, 7 at least not without additional, non-electrostatic terms included, 91–94 which
are available in Q-C HEM’s implementation of the isodensity version of SS(V)PE (Section 12.2.6). Moreover, analytic
energy gradients are not available for the isodensity cavity construction, whereas they are available when the cavity is
constructed from atom-centered spheres. One additional subtlety, which is discussed in detail in Ref. 69, is the fact that
the PCM implementation of the equation for the SS(V)PE surface charges [Eq. (12.2)] uses an asymmetric K matrix.
In contrast, Chipman’s isodensity implementation uses a symmetrized K matrix. Although the symmetrized version is
somewhat more computationally efficient when the number of surface charges is large, the asymmetric version is better
justified, theoretically. 69 (This admittedly technical point is clarified in Section 12.2.2 and in particular in Table 12.3.)
Regarding the accuracy of these models for solvation free energies (∆G298 ), SM8 achieves sub-kcal/mol accuracy for
neutral molecules, based on comparison to a large database of experimental values, although average errors for ions
are more like 4 kcal/mol. 30 To achieve comparable accuracy with IEF-PCM/SS(V)PE, non-electrostatic terms must be
included. 64,91,93 The SM12 model does not improve upon SM8 in any statistical sense, 84 but does lift one important
restriction on the level of electronic structure that can be combined with these models. Specifically, the Generalized
Born model used in SM8 is based on a variant of Mulliken-style atomic charges, and is therefore parameterized only
for a few small basis sets, e.g., 6-31G*. SM12, on the other hand, uses a variety of charge schemes that are stable with
respect to basis-set expansion, and can therefore be combined with any level of electronic structure theory for the solute.
Like IEF-PCM, the SMD model is also applicable to any basis sets, and its accuracy is comparable to SM8 and SM12. 81
Quantitative fluid-phase thermodynamics can also be obtained using Klamt’s COSMO-RS approach, 59,65 where RS
stands for “real solvent”. The COSMO-RS approach is not included in Q-C HEM and requires the COSMOtherm
program, which is licensed separately through COSMOlogic, 1 but Q-C HEM can write the input files that are need by
COSMOtherm.
The following sections provide more details regarding theory and job control for the various implicit solvent models
that are available in Q-C HEM. In addition, recent review articles are available for PCM methods, 114 SMx, 30 and
COSMO. 60 Formal relationships between various PCMs have been discussed in Refs. 21,69.

12.2.1

Kirkwood-Onsager Model

The simplest implicit solvation model available in Q-C HEM is the Kirkwood-Onsager model, 57,58,89 wherein the solute
is placed inside of a spherical cavity that is surrounded by a homogeneous dielectric medium. This model is characterized by two parameters: the cavity radius, a, and the solvent dielectric constant, ε. The former is typically calculated
according to
a = (3Vm /4πNA )1/3
(12.1)
where Vm is the solute’s molar volume, usually obtained from experiment (molecular weight or density 120 ), and NA
is Avogadro’s number. It is also common to add 0.5 Å to the value of a in Eq. (12.1) in order to account for the first
solvation shell. 123 Alternatively, a is sometimes selected as the maximum distance between the solute center of mass
and the solute atoms, plus the relevant van der Waals radii. A third option is to set 2a (the cavity diameter) equal to the
largest solute–solvent internuclear distance, plus the van der Waals radii of the relevant atoms. Unfortunately, solvation
energies are typically quite sensitive to the choice of a (and to the construction of the solute cavity, more generally).
Unlike older versions of the Kirkwood-Onsager model, in which the solute’s electron distribution was described entirely
in terms of its dipole moment, Q-C HEM’s version can use multipoles of arbitrarily high order, including the Born
(monopole) term for charged solutes, 10 in order to describe the solute’s electrostatic potential. The solute–continuum
electrostatic interaction energy is then computed using analytic expressions for the interaction of the point multipoles
with a dielectric continuum.
Energies and analytic gradients for the Kirkwood-Onsager solvent model are available for Hartree-Fock, DFT, and
CCSD calculations. It is often advisable to perform a gas-phase calculation of the solute molecule first, which can
serve as the initial guess for a subsequent Kirkwood-Onsager implicit solvent calculation.

Chapter 12: Molecules in Complex Environments

685

The Kirkwood-Onsager SCRF is requested by setting SOLVENT_METHOD = ONSAGER in the $rem section (along
with normal job control variables for an energy or gradient calculation), and furthermore specifying several additional
options in a $solvent input section, as described below. Of these, the keyword CavityRadius is required. The $rem
variable CC_SAVEAMPL may save some time for CCSD calculations using the Kirkwood-Onsager model.
Note: SCRF and CCSD combo works only in CCMAN (with CCMAN2 = FALSE).
Note: The following three job control variables belong only in the $solvent section. Do not place them in the $rem
section. As with other parts of the Q-C HEM input file, this input section is not case-sensitive.
CavityRadius
Sets the radius of the spherical solute cavity.
INPUT SECTION: $solvent
TYPE:
FLOAT
DEFAULT:
No default.
OPTIONS:
a Desired cavity radius, in Ångstroms.
RECOMMENDATION:
Use Eq. (12.1).
Dielectric
Sets the dielectric constant of the solvent continuum.
INPUT SECTION: $solvent
TYPE:
FLOAT
DEFAULT:
78.39
OPTIONS:
ε Use a (dimensionless) value of ε.
RECOMMENDATION:
As per required solvent; the default corresponds to water at 25◦ C.
MultipoleOrder
Determines the order to which the multipole expansion of the solute charge density is
carried out.
INPUT SECTION: $solvent
TYPE:
INTEGER
DEFAULT:
15
OPTIONS:
` Include up to `th order multipoles.
RECOMMENDATION:
Use the default. The multipole expansion is usually converged by order ` = 15.

Example 12.1 Onsager model applied at the Hartree-Fock level to H2 O in acetonitrile

686

Chapter 12: Molecules in Complex Environments
$molecule
0 1
O
0.00000000
H
-0.75908339
H
0.75908339
$end

0.00000000
0.00000000
0.00000000

$rem
METHOD
BASIS
SOLVENT_METHOD
$end

HF
6-31g**
Onsager

$solvent
CavityRadius
Dielectric
MultipoleOrder
$end

1.8
35.9
15

!
!
!

0.11722303
-0.46889211
-0.46889211

1.8 Angstrom Solute Radius
Acetonitrile
this is the default value

Example 12.2 Kirkwood-Onsager SCRF applied to hydrogen fluoride in water, performing a gas-phase calculation
first.
$molecule
0 1
H
F
$end

0.000000
0.000000

$rem
METHOD
BASIS
$end

0.000000
0.000000

-0.862674
0.043813

HF
6-31G*

@@@
$molecule
0 1
H
F
$end

0.000000
0.000000

$rem
JOBTYPE
METHOD
BASIS
SOLVENT_METHOD
SCF_GUESS
$end
$solvent
CavityRadius
$end

12.2.2

0.000000
0.000000

-0.862674
0.043813

FORCE
HF
6-31G*
ONSAGER
READ ! read vacuum solution as a guess

2.5

Polarizable Continuum Models

Clearly, the Kirkwood-Onsager model is inappropriate if the solute is very non-spherical. Nowadays, a more general
class of “apparent surface charge” SCRF solvation models are much more popular, to the extent that the generic
term “polarizable continuum model” (PCM) is typically used to denote these methods. 114 Apparent surface charge
PCMs improve upon the Kirkwood-Onsager model in two ways. Most importantly, they provide a much more realistic
description of molecular shape, typically by constructing the “solute cavity” (i.e., the interface between the atomistic

687

Chapter 12: Molecules in Complex Environments
Model
COSMO
C-PCM
IEF-PCM
SS(V)PE

Literature
Refs.
62
6,115
18,20
20,22

Matrix K

Matrix R

S
S
S − (fε /2π)DAS


S − (fε /4π) DAS + SAD†

−fε
−fε

−fε 1
−fε 1


1
1 − 2π
DA


1
1 − 2π
DA

Scalar fε
(ε − 1)/(ε + 1/2)
(ε − 1)/ε
(ε − 1)/(ε + 1)
(ε − 1)/(ε + 1)

Table 12.3: Definition of the matrices in Eq. (12.2) for the various PCMs that are available in Q-C HEM. The matrix
S consists of Coulomb interactions between the cavity charges and D is the discretized version of the matrix that
generates the outward-pointing normal electric field vector. (See Refs. 21,22,42 for detailed definitions.) The matrix
A is diagonal and contains the surface areas of the cavity discretization elements, and 1 is a unit matrix. At the level
of Eq. (12.2), COSMO and C-PCM differ only in the dielectric screening factor fε , although COSMO includes an
additional outlying charge correction that goes beyond Eq. (12.2). 5,61
region and the dielectric continuum) from a union of atom-centered spheres, an aspect of the model that is discussed
in Section 12.2.2.2. In addition, the exact electron density of the solute (rather than a multipole expansion) is used to
polarize the continuum. Electrostatic interactions between the solute and the continuum manifest as an induced charge
density on the cavity surface, which is discretized into point charges for practical calculations. The surface charges are
determined based upon the solute’s electrostatic potential at the cavity surface, hence the surface charges and the solute
wave function must be determined self-consistently.

12.2.2.1

Formal Theory and Discussion of Different Models

The PCM literature has a long history 114 and there are several different models in widespread use; connections between
these models have not always been appreciated. 18,20,21,69 Chipman 20,21 has shown how various PCMs can be formulated
within a common theoretical framework; see Ref. 42 for a pedagogical introduction. The PCM takes the form of a set
of linear equations,
Kq = Rv ,
(12.2)
in which the induced charges qi at the cavity surface discretization points [organized into a vector q in Eq. (12.2)] are
computed from the values vi of the solute’s electrostatic potential at those same discretization points. The form of the
matrices K and R depends upon the particular PCM in question. These matrices are given in Table 12.3 for the PCMs
that are available in Q-C HEM.
The oldest PCM is the so-called D-PCM model of Tomasi and coworkers, 87 but unlike the models listed in Table 12.3,
D-PCM requires explicit evaluation of the electric field normal to the cavity surface, This is undesirable, as evaluation
of the electric field is both more expensive and more prone to numerical problems as compared to evaluation of the
electrostatic potential. Moreover, the dependence on the electric field can be formally eliminated at the level of the
integral equation whose discretized form is given in Eq. (12.2). 20 As such, D-PCM is essentially obsolete, and the
PCMs available in Q-C HEM require only the evaluation of the electrostatic potential, not the electric field.
The simplest PCM that continues to enjoy widespread use is the Conductor-Like Screening Model (COSMO) introduced by Klamt and Schüürmann. 62 Truong and Stefanovich 115 later implemented the same model with a slightly
different dielectric scaling factor (fε in Table 12.3), and called this modification GCOSMO. The latter was implemented within the PCM formalism by Barone and Cossi et al., 6,28 who called the model C-PCM (for “conductor-like”
PCM). In each case, the dielectric screening factor has the form
fε =

ε−1
,
ε+x

(12.3)

where Klamt and Schüürmann proposed x = 1/2 but x = 0 was used in GCOSMO and C-PCM. The latter value is
the correct choice for a single charge in a spherical cavity (i.e., the Born ion model), although Klamt and coworkers
suggest that x = 1/2 is a better compromise, given that the Kirkwood-Onsager analytical result is x = `/(` + 1) for an
`th-order multipole centered in a spherical cavity. 5,62 The distinction is irrelevant in high-dielectric solvents; the x = 0

Chapter 12: Molecules in Complex Environments

688

and x = 1/2 values of fε differ by only 0.6% for water at 25◦ C, for example. Truong 115 argues that x = 0 does a
better job of preserving Gauss’ Law in low-dielectric solvents, but more accurate solvation energies (at least for neutral
molecules, as compared to experiment) are sometimes obtained using x = 1/2 (Ref. 6). This result is likely highly
sensitive to cavity construction, and in any case, both versions are available in Q-C HEM.
Whereas the original COSMO model introduced by Klamt and Schüürmann 62 corresponds to Eq. (12.2) with K and
R as defined in Table 12.3, Klamt and coworkers later introduced a correction for outlying charge that goes beyond
Eq. (12.2). 5,61 Klamt now consistently refers to this updated model as “COSMO”, 60 and we shall adopt this nomenclature as well. COSMO, with the outlying charge correction, is available in Q-C HEM and is described in Section 12.2.7.
In contrast, C-PCM consists entirely of Eq. (12.2) with matrices K and R as defined in Table 12.3, although it is
possible to modify the dielectric screening factor to use the x = 1/2 value (as in COSMO) rather than the x = 0 value.
Additional non-electrostatic terms can be added at the user’s discretion, as discussed below, but there is no explicit
outlying charge correction in C-PCM. These and other fine-tuning details for PCM jobs are controllable via the $pcm
input section that is described in Section 12.2.3.
As compared to C-PCM, a more sophisticated treatment of continuum electrostatic interactions is afforded by the “surface and simulation of volume polarization for electrostatics” [SS(V)PE] approach. 20 Formally speaking, this model
provides an exact treatment of the surface polarization (i.e., the surface charge induced by the solute charge that is
contained within the solute cavity, which induces a surface polarization owing to the discontinuous change in dielectric constant across the cavity boundary) but also an approximate treatment of the volume polarization (arising from
the aforementioned outlying charge). The “SS(V)PE” terminology is Chipman’s notation, 20 but this model is formally equivalent, at the level of integral equations, to the “integral equation formalism” (IEF-PCM) that was developed
originally by Cancès et al.. 17,113 Some difference do arise when the integral equations are discretized to form finitedimensional matrix equations, 69 and it should be noted from Table 12.3 that SS(V)PE uses a symmetrized form of
the K matrix as compared to IEF-PCM. The asymmetric IEF-PCM is the recommended approach, 69 although only
the symmetrized version is available in the isodensity implementation of SS(V)PE that is discussed in Section 12.2.5.
As with the obsolete D-PCM approach, the original version of IEF-PCM explicitly required evaluation of the normal
electric field at the cavity surface, but it was later shown that this dependence could be eliminated to afford the version
described in Table 12.3. 18,20 This version requires only the electrostatic potential, and is thus preferred, and it is this
version that we designate as IEF-PCM. The C-PCM model becomes equivalent to SS(V)PE in the limit ε → ∞, 20,69
which means that C-PCM must somehow include an implicit correction for volume polarization, even if this was not
by design. 61 For ε & 50, numerical calculations reveal that there is essentially no difference between SS(V)PE and
C-PCM results. 69 Since C-PCM is less computationally involved as compared to SS(V)PE, it is the PCM of choice
in high-dielectric solvents. The computational savings relative to SS(V)PE may be particularly significant for large
QM/MM/PCM jobs. For a more detailed discussion of the history of these models, see the lengthy and comprehensive
review by Tomasi et al.. 114 For a briefer discussion of the connections between these models, see Refs. 21,42,69.

12.2.2.2

Cavity Construction and Discretization

Construction of the cavity surface is a crucial aspect of PCMs, as computed properties are quite sensitive to the details
of the cavity construction. Most cavity constructions are based on a union of atom-centered spheres (see Fig. 12.1), but
there are yet several different constructions whose nomenclature is occasionally confused in the literature. Simplest and
most common is the van der Waals (vdW) surface consisting of a union of atom-centered spheres. Traditionally, 8,112 and
by default in Q-C HEM, the atomic radii are taken to be 1.2 times larger than vdW radii extracted from crystallographic
data, originally by Bondi (and thus sometimes called “Bondi radii”). 9 This 20% augmentation is intended to mimic the
fact that solvent molecules cannot approach all the way to the vdW radius of the solute atoms, though it’s not altogether
clear that this is an optimal value. (The default scaling factor in Q-C HEM is 1.2 but can be modified by the user.) An
alternative to scaling the atomic radii is to add a certain fixed increment to each, representing the approximate size of
a solvent molecule (e.g., 1.4 Å for water) and leading to what is known as the solvent accessible surface (SAS). From
another point of view, the SAS represents the surface defined by the center of a spherical solvent molecule as it rolls
over the vdW surface, as suggested in Fig. 12.1. Both the vdW surface and the SAS possess cusps where the atomic
spheres intersect, although these become less pronounced as the atomic radii are scaled or augmented. These cusps
are eliminated in what is known as the solvent-accessible surface (SES), sometimes called the Connolly surface or the

689

Chapter 12: Molecules in Complex Environments
van der Waals
surface

re-entrant
surface

solventaccessible
surface

solvent
probe

Figure 12.1: Illustration of various solute cavity surface definitions for PCMs. 70 The union of atomic van der Waals
spheres (shown in gray) defines the van der Waals (vdW) surface, in black. Note that actual vdW radii from the
literature are sometimes scaled in constructing the vdW surface. If a probe sphere (representing the assumed size of a
solvent molecule) is rolled over the van der Waals surface, then its center point traces out the solvent accessible surface
(SAS), shown in green; the SAS is equivalent to a vdW surface where the atomic radii are increases by the radius of
the probe sphere. Finally, one can use the probe sphere to smooth out the sharp crevasses in the vdW surface using the
re-entrant surface elements shown in red, resulting in the solvent-excluded surface (SES).

“molecular surface". The SES uses the surface of the probe sphere at points where it is simultaneously tangent to two
or more atomic spheres to define elements of a “re-entrant surface” that smoothly connects the atomic (or “contact”)
surface. 70
Having chosen a model for the cavity surface, this surface is discretized using atom-centered Lebedev grids 71–73 of
the same sort that are used to perform the numerical integrations in DFT. (Discretization of the re-entrant facets of the
SES is somewhat more complicated but similar in spirit. 70 ) Surface charges qi are located at these grid points and the
Lebedev quadrature weights can be used to define the surface area associated with each discretization point. 67
A long-standing (though not well-publicized) problem with the aforementioned discretization procedure is that it fails
to afford continuous potential energy surfaces as the solute atoms are displaced, because certain surface grid points
may emerge from, or disappear within, the solute cavity, as the atomic spheres that define the cavity are moved. This
undesirable behavior can inhibit convergence of geometry optimizations and, in certain cases, lead to very large errors in vibrational frequency calculations. 67 It is also a fundamental hindrance to molecular dynamics calculations. 68
Building upon earlier work by York and Karplus, 125 Lange and Herbert 67,68,70 developed a general scheme for implementing apparent surface charge PCMs in a manner that affords smooth potential energy surfaces, even for ab initio
molecular dynamics simulations involving bond breaking. 42,68 Notably, this approach is faithful to the properties of the
underlying integral equation theory on which the PCMs are based, in the sense that the smoothing procedure does not
significantly perturb solvation energies or cavity surface areas. 67,68 The smooth discretization procedure combines a
switching function with Gaussian blurring of the cavity surface charge density, and is thus known as the “Switching/
Gaussian” (SWIG) implementation of the PCM.
Both single-point energies and analytic energy gradients are available for SWIG PCMs, when the solute is described
using molecular mechanics or an SCF (Hartree-Fock or DFT) electronic structure model, except that for the SES
cavity model only single-point energies are available. Analytic Hessians are available for the C-PCM model only. (As
usual, vibrational frequencies for other models will be computed, if requested, by finite difference of analytic energy
gradients.) Single-point energy calculations using correlated wave functions can be performed in conjunction with these
solvent models, in which case the correlated wave function calculation will use Hartree-Fock molecular orbitals that
are polarized in the presence of the continuum dielectric solvent (i.e., there is no post-Hartree–Fock PCM correction).

Chapter 12: Molecules in Complex Environments

690

Researchers who use these PCMs are asked to cite Refs. 68,69, which provide the details of Q-C HEM’s implementation,
and Ref. 70 if the SES is used. (We point the reader in particular to Ref. 68, which provides an assessment of the
discretization errors that can be anticipated using various PCMs and Lebedev grids; default grid values in Q-C HEM
were established based on these tests.) When publishing results based on PCM calculations, it is essential to specify
both the precise model that is used (see Table 12.3) as well as how the cavity was constructed.
For example, “Bondi radii multiplied by 1.2”, which is the Q-C HEM default, except for hydrogen, where the factor is
reduced to 1.1, 99 as per usual. Radii for main-group elements that were not provided by Bondi are taken from Ref. 79.
Absent details such as these, PCM calculations will be difficult to reproduce in other electronic structure programs.

12.2.2.3

Nonequilibrium Solvation for Vertical Excitation, Ionization and Emission

In vertical excitation or ionization, the solute undergoes a sudden change in its charge distribution. Various microscopic motions of the solvent have characteristic times to reach certain polarization response, and fast part of the
solvent response (electrons) can follow such a dynamic process while the remaining degrees of freedom (nuclei) remain unchanged as in the initial state. Such splitting of the solvent response gives rise to nonequilibrium solvation. In
the literature, two different approaches have been developed for describing nonequilibrium solvent effects: the linear
response (LR) approach 14,26 and the state-specific (SS) approach. 15,25,48,112 Both are implemented in Q-C HEM, 127 ,at
the SCF level for vertical ionization and at the corresponding level (CIS, TDDFT or ADC, see Section 7.8.7) for vertical excitation. A brief introduction to these methods is given below, and users of the nonequilibrium PCM features
are asked to cite Refs. 127 and 85. State-specific solvent-field equilibration for long-lived excited states to compute
e.g. emission energies is implemented for the ADC-suite of methods as described in section 7.8.7. Users of this
equilibrium-solvation PCM please cite and be referred to Ref. 86.
The LR approach considers the solvation effects as a coupling between a pair of transitions, one for solute and the
other for solvent. The transition frequencies when the interaction between the solute and solvent is turned on may
be determined by considering such an interaction as a perturbation. In the framework of TDDFT, the solvent/solute
interaction is given by 45


Z
Z
Z
Z
1
0
ω 0 = dr dr0 dr00 dr000 ρtr∗ (r)
+
g
(r,
r
)
XC
|r − r0 |


(12.4)
1
00 000
tr 000
+
g
(r
,
r
)
ρ
(r
)
,
× χ∗ (r0 , r00 , ω)
XC
|r00 − r000 |
where χ is the charge density response function of the solvent and ρtr (r) is the solute’s transition density. This term
accounts for a dynamical correction to the transition energy so that it is related to the response of the solvent to the
charge density of the solute oscillating at the solute transition frequency (ω). Within a PCM, only classical Coulomb
interactions are taken into account, and Eq. (12.4) becomes
Z
Z
Z
Z
ρtr∗ (r)
ρtr (r0 )
0
(12.5)
ωPCM
= dr ds
ds0 dr0 Q(s, s0 , ε) 0
,
|r − s|
|s − r0 |
where Q is PCM solvent response operator for a generic dielectric constant, ε. The integral of Q and the potential of
the density ρtr gives the surface charge density for the solvent polarization.
The state-specific (SS) approach takes into account the capability of a part of the solvent degrees of freedom to respond
instantaneously to changes in the solute wave function upon excitation. Such an effect is not accounted for in the LR
approach. In SS, a generic solvated-solute excited state Ψi is obtained as a solution of a nonlinear Schrödinger equation


Ĥ vac + V̂0slow + V̂ifast |Ψi i = EiSS |Ψi i
(12.6)
that depends upon the solute’s charge distribution. Here Ĥ vac is the usual Hamiltonian for the solute in vacuum and the
reaction field operator V̂i generates the electrostatic potential of the apparent surface charge density (Section 12.2.2.1),
corresponding to slow and fast polarization response. The solute is polarized self-consistently with respect to the
solvent’s reaction field. In case of vertical ionization rather than excitation, both the ionized and non-ionized states can

Chapter 12: Molecules in Complex Environments

691

be treated within a ground-state formalism. For vertical excitations, self-consistent SS models have been developed for
various excited-state methods, 48,82 including both CIS and TDDFT.
In a linear dielectric medium, the solvent polarization is governed by the electric susceptibility, χ = [ε(ω) − 1]/4π,
where ε(ω) is the frequency-dependent permittivity, In case of very fast vertical transitions, the dielectric response is
ruled by the optical dielectric constant, εopt = n2 , where n is the solvent’s index of refraction. In both LR and SS,
the fast part of the solvent’s degrees of freedom is in equilibrium with the solute density change. Within PCM, the fast
solvent polarization charges for the SS excited state i can be obtained by solving the following equation: 25


Kεopt qifast,SS = Rεopt vi + v(qslow
) .
(12.7)
0
Here qfast,SS is the discretized fast surface charge. The dielectric constants in the matrices K and R (Section 12.2.2.1)
are replaced with the optical dielectric constant, and vi is the potential of the solute’s excited state density, ρi . The
quantity v(qslow
) is the potential of the slow part of the apparent surface charges in the ground state, which are given
0
by


ε − εopt
slow
q0 =
q0 .
(12.8)
ε−1
For LR-PCM, the solvent polarization is subjected to the first-order changes to the electron density (TDDFT linear
density response), and thus Eq. (12.7) becomes
Kεopt qfast,LR
= Rεopt v(ρtr
i ).
i

(12.9)

The LR approach for CIS/TDDFT excitations and the self-consistent SS method (using the ground-state SCF) for
vertical ionizations are available in Q-C HEM. The self-consistent SS method for vertical excitations is not available,
because this method is problematic in the vicinity of (near-) degeneracies between excited states, such as in the vicinity
of a conical intersection. The fundamental problem in the SS approach is that each wave function Ψi is an eigenfunction
of a different Hamiltonian, since Eq. (12.6) depend upon the specific state of interest. To avoid the ordering and the
non-orthogonality problems, we compute the vertical excitation energy using a first-order, perturbative approximation
to the SS approach, 16,19 in what we have termed the “ptSS” method. 85 The zeroth-order excited-state wave function
can be calculated using various excited-state methods (currently available for CIS and TDDFT in Q-C HEM) with
solvent-relaxed molecular orbitals obtained from a ground-state PCM calculation. As mentioned previously, LR and
SS describe different solvent relaxation features in nonequilibrium solvation. In the perturbation scheme, we can
calculate the LR contribution using the zeroth-order transition density, in what we have called the "ptLR" approach.
The combination of ptSS and ptLR yields quantitatively good solvatochromatic shifts in combination with TDDFT but
not with the correlated variants of ADC, for which the pure ptSS approach was shown to be superior. 85,127
The LR and SS approaches can also be used in the study of photon emission processes. 49 An emission process can
be treated as a vertical excitation at a stationary point on the excited-state potential surface. The basic requirement
therefore is to prepare the solvent-relaxed geometry for the excited-state of interest. TDDFT/C-PCM analytic gradients
and Hessian are available.
Section 7.3.5 for computational details regarding excited-state geometry optimization with PCM. An emission process
is slightly more complicated than the absorption case. Two scenarios are discussed in literature, depending on the lifetime of an excited state in question. In the limiting case of ultra-fast excited state decay, when only fast solvent degrees
of freedom are expected to be equilibrated with the excited-state density. In this limit, the emission energy can be computed exactly in the same way as the vertical excitation energy. In this case, excited state geometry optimization should
be performed in the nonequilibrium limit. The other limit is that of long-lived excited state, e.g., strongly fluorescent
species and phosphorescence. In the long-lived case, excited state geometry optimization should be performed with the
solvent equilibrium limit. Thus, the excited state should be computed using an equilibrium LR or SS approach, and the
ground state is calculated using nonequilibrium self-consistent SS approach. The latter approach is implemented for
the ADC-based methods as described in Section 7.8.7.

12.2.3

PCM Job Control

A PCM calculation is requested by setting SOLVENT_METHOD = PCM in the $rem section. As mentioned above, there
are a variety of different theoretical models that fall within the PCM family, so additional fine-tuning may be required,

Chapter 12: Molecules in Complex Environments

692

as described below.

12.2.3.1 $pcm section
Most PCM job control is accomplished via options specified in the $pcm input section, which allows the user to specify
which flavor of PCM will be used, which algorithm will be used to solve the PCM equations, and other options. The
format of the $pcm section is analogous to that of the $rem section:
$pcm

$end



Note: The following job control variables belong only in the $pcm section. Do not place them in the $rem section.
Theory
Specifies the which polarizable continuum model will be used.
INPUT SECTION: $pcm
TYPE:
STRING
DEFAULT:
CPCM
OPTIONS:
CPCM
Conductor-like PCM with fε = (ε − 1)/ε.
COSMO Original conductor-like screening model with fε = (ε − 1)/(ε + 1/2).
IEFPCM IEF-PCM with an asymmetric K matrix.
SSVPE
SS(V)PE model, equivalent to IEF-PCM with a symmetric K matrix.
RECOMMENDATION:
The IEF-PCM/SS(V)PE model is more sophisticated model than either C-PCM or
COSMO, and probably more appropriate for low-dielectric solvents, but it is also more
computationally demanding. In high-dielectric solvents there is little difference between
these models. Note that the keyword COSMO in this context simply affects the dielectric
screening factor fε ; to obtain the outlying charge correction suggested by Klamt, 5,61 one
should use SOLVENT_METHOD = COSMO rather than SOLVENT_METHOD = PCM. (See
Section 12.2.7.)

Chapter 12: Molecules in Complex Environments

693

Method
Specifies which surface discretization method will be used.
INPUT SECTION: $pcm
TYPE:
STRING
DEFAULT:
SWIG
OPTIONS:
SWIG
Switching/Gaussian method
ISWIG
“Improved” Switching/Gaussian method with an alternative switching function
Spherical Use a single, fixed sphere for the cavity surface.
Fixed
Use discretization point charges instead of smooth Gaussians.
RECOMMENDATION:
Use of SWIG is recommended only because it is slightly more efficient than the switching
function of ISWIG. On the other hand, ISWIG offers some conceptually more appealing
features and may be superior in certain cases. Consult Refs. 68,69 for a discussion of
these differences. The Fixed option uses the Variable Tesserae Number (VTN) algorithm
of Li and Jensen, 74 with Lebedev grid points. VTN uses point charges with no switching
function or Gaussian blurring, and is therefore subject to discontinuities in geometry optimizations. It is not recommended, except to make contact with other calculations in the
literature.
SwitchThresh
Threshold for discarding grid points on the cavity surface.
INPUT SECTION: $pcm
TYPE:
INTEGER
DEFAULT:
8
OPTIONS:
n Discard grid points when the switching function is less than 10−n .
RECOMMENDATION:
Use the default, which is found to avoid discontinuities within machine precision. Increasing n reduces the cost of PCM calculations but can introduce discontinuities in the
potential energy surface.

Construction of the solute cavity is an important part of the model and users should consult the literature in this capacity,
especially with regard to the radii used for the atomic spheres. The default values provided in Q-C HEM correspond
to the consensus choice that has emerged over several decades, namely, to use vdW radii scaled by a factor of 1.2.
The most widely-used set of vdW radii are those determined from crystallographic data by Bondi, 9 although the radius
for hydrogen was later adjusted to 1.1 Å, 99 and radii for those main-group elements not addressed by Bondi were
provided later. 79 This extended set of vdW is used by default in Q-C HEM, and for simplicity we call these “Bondi
radii” regardless of whether they come from Bondi’s original paper or the later work. Alternatively, atomic radii from
the Universal Force Field (UFF) are available. 96 The main appeal of UFF radii is that they are defined for all atoms
of the periodic table, though the quality of these radii for PCM applications is unclear. Finally, the user may specify
his or her own radii for cavity construction using a $van_der_waals input section, the format for which is described
in Section 12.2.9. No scaling factor is applied to user-defined radii. Note that R = 0 is allowed for a particular
atomic radius, in which case the atom in question is not used to construct the cavity surface. This feature facilitates the
construction of “united atom” cavities, 7 in which the hydrogen atoms do not get their own spheres and the heavy-atom
radii are increased to compensate Finally, since the solvent molecules should not be able to penetrate all the way to the
atomic vdW radii of the solute, it is traditional either to scale the atomic radii (vdW surface construction) or else to
augment them with an assumed radius of a spherical solvent molecule (SAS construction), but not both.

Chapter 12: Molecules in Complex Environments
Radii
Specifies which set of atomic van der Waals radii will be used to define the solute cavity.
INPUT SECTION: $pcm
TYPE:
STRING
DEFAULT:
BONDI
OPTIONS:
BONDI Use the (extended) set of Bondi radii.
FF
Use Lennard-Jones radii from a molecular mechanics force field.
UFF
Use radii form the Universal Force Field.
READ
Read the atomic radii from a $van_der_waals input section.
RECOMMENDATION:
Bondi radii are widely used. The FF option requires the user to specify an MM force field
using the FORCE_FIELD $rem variable, and also to define the atom types in the $molecule
section (see Section 12.3). This is not required for UFF radii.
vdwScale
Scaling factor for the atomic van der Waals radii used to define the solute cavity.
INPUT SECTION: $pcm
TYPE:
FLOAT
DEFAULT:
1.2
OPTIONS:
f Use a scaling factor of f > 0.
RECOMMENDATION:
The default value is widely used in PCM calculations, although a value of 1.0 might be
appropriate if using a solvent-accessible surface.
SASradius
Form a “solvent accessible” surface with the given solvent probe radius.
INPUT SECTION: $pcm
TYPE:
FLOAT
DEFAULT:
0.0
OPTIONS:
r Use a solvent probe radius of r, in Å.
RECOMMENDATION:
The solvent probe radius is added to the scaled van der Waals radii of the solute atoms. A
common solvent probe radius for water is 1.4 Å, but the user should consult the literature
regarding the use of solvent-accessible surfaces.

694

Chapter 12: Molecules in Complex Environments

695

SurfaceType
Selects the solute cavity surface construction.
INPUT SECTION: $pcm
TYPE:
STRING
DEFAULT:
VDW_SAS
OPTIONS:
VDW_SAS van der Waals or solvent-accessible surface
SES
solvent-excluded surface
RECOMMENDATION:
The vdW surface and the SAS are each comprised simply of atomic spheres and thus
share a common option; the only difference is the specification of a solvent probe radius,
SASradius. For a true vdW surface, the probe radius should be zero (which is the default), whereas for the SAS the atomic radii are traditionally not scaled, hence vdwScale
should be set to zero (which is not the default). For the SES, only SWIG discretization is
available, but this can be used with any set of (scaled or unscaled) atomic radii, or with
radii that are augmented by SASradius.

Historically, discretization of the cavity surface has involved “tessellation” methods that divide the cavity surface
area into finite polygonal “tesserae”. (The GEPOL algorithm 90 is perhaps the most widely-used tessellation scheme.)
Tessellation methods, however, suffer not only from discontinuities in the cavity surface area and solvation energy as a
function of the nuclear coordinates, but in addition they lead to analytic energy gradients that are complicated to derive
and implement. To avoid these problems, Q-C HEM’s SWIG PCM implementation 67–69 uses Lebedev grids to discretize
the atomic spheres. These are atom-centered grids with icosahedral symmetry, and may consist of anywhere from 26 to
5294 grid points per atomic sphere. The default values used by Q-C HEM were selected based on extensive numerical
tests. 68,69 The default for MM atoms (in MM/PCM or QM/MM/PCM jobs) is N = 110 Lebedev points per atomic
sphere, whereas the default for QM atoms is N = 302. (This represents a change relative to Q-C HEM versions earlier
than 4.2.1, where the default for QM atoms was N = 590.) These default values exhibit good rotational invariance
and absolute solvation energies that, in most cases, lie within ∼0.5–1.0 kcal/mol of the N → ∞ limit, 68 although
exceptions (especially where charged solutes are involved) can be found. 69
Note: The acceptable values for the number of Lebedev points per sphere are N = 26, 50, 110, 194, 302, 434, 590,
770, 974, 1202, 1454, 1730, 2030, 2354, 2702, 3074, 3470, 3890, 4334, 4802, or 5294.
HPoints
The number of Lebedev grid points to be placed on H atoms in the QM system.
INPUT SECTION: $pcm
TYPE:
INTEGER
DEFAULT:
302
OPTIONS:
Acceptable values are listed above.
RECOMMENDATION:
Use the default for geometry optimizations. For absolute solvation energies, the user may
want to examine convergence with respect to N .

Chapter 12: Molecules in Complex Environments

696

HeavyPoints
The number of Lebedev grid points to be placed non-hydrogen atoms in the QM system.
INPUT SECTION: $pcm
TYPE:
INTEGER
DEFAULT:
302
OPTIONS:
Acceptable values are listed above.
RECOMMENDATION:
Use the default for geometry optimizations. For absolute solvation energies, the user may
want to examine convergence with respect to N .
MMHPoints
The number of Lebedev grid points to be placed on H atoms in the MM subsystem.
INPUT SECTION: $pcm
TYPE:
INTEGER
DEFAULT:
110
OPTIONS:
Acceptable values are listed above.
RECOMMENDATION:
Use the default for geometry optimizations. For absolute solvation energies, the user may
want to examine convergence with respect to N . This option applies only to MM/PCM
or QM/MM/PCM calculations.
MMHeavyPoints
The number of Lebedev grid points to be placed on non-hydrogen atoms in the MM
subsystem.
INPUT SECTION: $pcm
TYPE:
INTEGER
DEFAULT:
110
OPTIONS:
Acceptable values are listed above.
RECOMMENDATION:
Use the default for geometry optimizations. For absolute solvation energies, the user may
want to examine convergence with respect to N . This option applies only to MM/PCM
or QM/MM/PCM calculations.

Especially for complicated molecules, the user may want to visualize the cavity surface. This can be accomplished
by setting PrintLevel ≥ 2, which will trigger the generation of several .PQR files that describe the cavity surface.
(These are written to the Q-C HEM output file.) The .PQR format is similar to the common .PDB (Protein Data Bank)
format, but also contains charge and radius information for each atom. One of the output .PQR files contains the
charges computed in the PCM calculation and radii (in Å) that are half of the square root of the surface area represented
by each surface grid point. Thus, in examining this representation of the surface, larger discretization points are
associated with larger surface areas. A second .PQR file contains the solute’s electrostatic potential (in atomic units),
in place of the charge information, and uses uniform radii for the grid points. These .PQR files can be visualized using
various third-party software, including the freely-available Visual Molecular Dynamics (VMD) program, 2,46 which is
particularly useful for coloring the .PQR surface grid points according to their charge, and sizing them according to
their contribution to the molecular surface area. (Examples of such visualizations can be found in Ref. 67.)

Chapter 12: Molecules in Complex Environments

697

PrintLevel
Controls the printing level during PCM calculations.
INPUT SECTION: $pcm
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
0 Prints PCM energy and basic surface grid information. Minimal additional printing.
1 Level 0 plus PCM solute-solvent interaction energy components and Gauss’ Law error.
2 Level 1 plus surface grid switching parameters and a .PQR file for visualization of
the cavity surface apparent surface charges.
3 Level 2 plus a .PQR file for visualization of the electrostatic potential at the surface
grid created by the converged solute.
4 Level 3 plus additional surface grid information, electrostatic potential and apparent
surface charges on each SCF cycle.
5 Level 4 plus extensive debugging information.
RECOMMENDATION:
Use the default unless further information is desired.

Finally, note that setting Method to Spherical in the $pcm input selection requests the construction of a solute cavity
consisting of a single, fixed sphere. This is generally not recommended but is occasionally useful for making contact
with the results of Born models in the literature, or the Kirkwood-Onsager model discussed in Section 12.2.1. In this
case, the cavity radius and its center must also be specified in the $pcm section. The keyword HeavyPoints controls
the number of Lebedev grid points used to discretize the surface.
CavityRadius
Specifies the solute cavity radius.
INPUT SECTION: $pcm
TYPE:
FLOAT
DEFAULT:
None
OPTIONS:
R Use a radius of R, in Ångstroms.
RECOMMENDATION:
None.
CavityCenter
Specifies the center of the spherical solute cavity.
INPUT SECTION: $pcm
TYPE:
FLOAT
DEFAULT:
0.0 0.0 0.0
OPTIONS:
x y z Coordinates of the cavity center, in Ångstroms.
RECOMMENDATION:
The format is CavityCenter followed by three floating-point values, delineated by spaces.
Uses the same coordinate system as the $molecule section.

698

Chapter 12: Molecules in Complex Environments
12.2.3.2

Examples

The following example shows a very basic PCM job. The solvent dielectric is specified in the $solvent section, which
is described below.
Example 12.3 A basic example of using the PCMs: optimization of trifluoroethanol in water.
$rem
JOBTYPE
BASIS
METHOD
SOLVENT_METHOD
$end

OPT
6-31G*
B3LYP
PCM

$pcm
Theory
Method
Solver
HeavyPoints
HPoints
Radii
vdwScale
$end

CPCM
SWIG
Inversion
194
194
Bondi
1.2

$solvent
Dielectric 78.39
$end
$molecule
0 1
C
-0.245826
C
0.244003
O
0.862012
F
0.776783
F
-0.858739
F
-1.108290
H
-0.587975
H
0.963047
H
0.191283
$end

-0.351674
0.376569
-0.527016
-0.909300
0.511576
-1.303001
0.878499
1.147195
-1.098089

-0.019873
1.241371
2.143243
-0.666009
-0.827287
0.339419
1.736246
0.961639
2.489052

The next example uses a single spherical cavity and should be compared to the Kirkwood-Onsager job, Example 12.1

699

Chapter 12: Molecules in Complex Environments
on page 685.
Example 12.4 PCM with a single spherical cavity, applied to H2 O in acetonitrile
$molecule
0 1
O
0.00000000
H
-0.75908339
H
0.75908339
$end
$rem
METHOD
BASIS
SOLVENT_METHOD
$end
$pcm
method
HeavyPoints
CavityRadius
CavityCenter
Theory
$end
$solvent
Dielectric
$end

0.00000000
0.00000000
0.00000000

0.11722303
-0.46889211
-0.46889211

HF
6-31g**
pcm

spherical
! single spherical cavity with 590 discretization points
590
1.8
! Solute Radius, in Angstrom
0.0 0.0 0.0 ! Will be at center of Standard Nuclear Orientation
SSVPE

35.9

!

Acetonitrile

Finally, we consider an example of a united-atom cavity. Note that a user-defined van der Waals radius is supplied only
for carbon, so the hydrogen radius is taken to be zero and thus the hydrogen atoms are not used to construct the cavity

Chapter 12: Molecules in Complex Environments

700

surface. (As mentioned above, the format for the $van_der_waals input section is discussion in Section 12.2.9).
Example 12.5 United-atom cavity construction for ethylene.
$comment
Benzene (in benzene), with a united-atom cavity construction
R = 2.28 A for carbon, R = 0 for hydrogen
$end
$molecule
0 1
C
1.38620
C
0.69310
C
-0.69310
C
-1.38620
C
-0.69310
C
0.69310
H
2.46180
H
1.23090
H
-1.23090
H
-2.46180
H
-1.23090
H
1.23090
$end

0.000000
1.200484
1.200484
0.000000
-1.200484
-1.200484
0.000000
2.131981
2.131981
0.000000
-2.131981
-2.131981

$rem
EXCHANGE
BASIS
SOLVENT_METHOD
$end

hf
6-31G*
pcm

$pcm
theory
method
printlevel
radii
$end

iefpcm
swig
1
read

0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000

! this is a synonym for ssvpe

$solvent
dielectric 2.27
$end
$van_der_waals
1
6
2.28
1
0.00
$end

12.2.3.3 $solvent section
The solvent for PCM calculations is specified using the $solvent section, as documented below. In addition, the $solvent
section can be used to incorporate non-electrostatic interaction terms into the solvation energy. (The Theory keyword
in the $pcm section specifies only how the electrostatic interactions are handled.) The general form of the $solvent
input section is shown below. The $solvent section was used above to specify parameters for the Kirkwood-Onsager
SCRF model, and will be used again below to specify the solvent for SMx calculations (Section 12.2.8); in each
case, the particular options that can be listed in the $solvent section depend upon the value of the $rem variable
SOLVENT_METHOD.
$solvent
NonEls



Chapter 12: Molecules in Complex Environments

701

NSolventAtoms 
SolventAtom    
SolventAtom    
. . .
 
. . .
$end

The keyword SolventAtom requires multiple parameters, whereas all other keywords require only a single parameter.
In addition to any (optional) non-electrostatic parameters, the $solvent section is also used to specify the solvent’s
dielectric constant. If non-electrostatic interactions are ignored, then this is the only keyword that is necessary in the
$solvent section. For nonequilibrium TDDFT/C-PCM calculations (Section 12.2.2.3), the optical dielectric constant
should be specified in the $solvent section as well.
Dielectric
The (static) dielectric constant of the PCM solvent.
INPUT SECTION: $solvent
TYPE:
FLOAT
DEFAULT:
78.39
OPTIONS:
ε Use a dielectric constant of ε > 0.
RECOMMENDATION:
The static (i.e., zero-frequency) dielectric constant is what is usually called “the” dielectric
constant. The default corresponds to water at 25◦ C.
OpticalDielectric
The optical dielectric constant of the PCM solvent.
INPUT SECTION: $solvent
TYPE:
FLOAT
DEFAULT:
1.78
OPTIONS:
ε∞ Use an optical dielectric constant of ε∞ > 0.
RECOMMENDATION:
The default corresponds to water at 25◦ C. Note that ε∞ = n2 , where n is the solvent’s
index of refraction.

The non-electrostatic interactions currently available in Q-C HEM are based on the work of Cossi et al., 27 and are
computed outside of the SCF procedure used to determine the electrostatic interactions. The non-electrostatic energy is
highly dependent on the input parameters and can be extremely sensitive to the radii chosen to define the solute cavity.
Accordingly, the inclusion of non-electrostatic interactions is highly empirical and probably needs to be considered on
a case-by-case basis. Following Ref. 27, the cavitation energy is computed using the same solute cavity that is used
to compute the electrostatic energy, whereas the dispersion/repulsion energy is computed using a solvent-accessible
surface.
The following keywords (in the $solvent section) are used to define non-electrostatic parameters for PCM calculations.

Chapter 12: Molecules in Complex Environments
NonEls
Specifies what type of non-electrostatic contributions to include.
INPUT SECTION: $solvent
TYPE:
STRING
DEFAULT:
None
OPTIONS:
Cav
Cavitation energy
Buck
Buckingham dispersion and repulsion energy from atomic number
LJ
Lennard-Jones dispersion and repulsion energy from force field
BuckCav Buck + Cav
LJCav
LJ + Cav
RECOMMENDATION:
A very limited set of parameters for the Buckingham potential is available at present.
NSolventAtoms
The number of different types of atoms.
INPUT SECTION: $solvent
TYPE:
INTEGER
DEFAULT:
None
OPTIONS:
N Specifies that there are N different types of atoms.
RECOMMENDATION:
This keyword is necessary when NonEls = Buck, LJ, BuckCav, or LJCav. Methanol
(CH3 OH), for example, has three types of atoms (C, H, and O).
SolventAtom
Specifies a unique solvent atom.
INPUT SECTION: $solvent
TYPE:
Various
DEFAULT:
None.
OPTIONS:
Input (TYPE)
Description
Number1 (INTEGER): The atomic number of the atom
Number2 (INTEGER): How many of this atom are in a solvent molecule
Number3 (INTEGER): Force field atom type
SASradius (FLOAT):
Probe radius (in Å) for defining the solvent accessible surface
RECOMMENDATION:
If not using LJ or LJCav, Number3 should be set to 0. The SolventAtom keyword is
necessary when NonEls = Buck, LJ, BuckCav, or LJCav.

702

Chapter 12: Molecules in Complex Environments
Temperature
Specifies the solvent temperature.
INPUT SECTION: $solvent
TYPE:
FLOAT
DEFAULT:
300.0
OPTIONS:
T Use a temperature of T , in Kelvin.
RECOMMENDATION:
Used only for the cavitation energy.
Pressure
Specifies the solvent pressure.
INPUT SECTION: $solvent
TYPE:
FLOAT
DEFAULT:
1.0
OPTIONS:
P Use a pressure of P , in bar.
RECOMMENDATION:
Used only for the cavitation energy.
SolventRho
Specifies the solvent number density
INPUT SECTION: $solvent
TYPE:
FLOAT
DEFAULT:
Determined for water, based on temperature.
OPTIONS:
ρ Use a density of ρ, in molecules/Å3 .
RECOMMENDATION:
Used only for the cavitation energy.
SolventRadius
The radius of a solvent molecule of the PCM solvent.
INPUT SECTION: $solvent
TYPE:
FLOAT
DEFAULT:
None
OPTIONS:
r Use a radius of r, in Å.
RECOMMENDATION:
Used only for the cavitation energy.

703

704

Chapter 12: Molecules in Complex Environments
The following example illustrates the use of the non-electrostatic interactions.

Example 12.6 Optimization of trifluoroethanol in water using both electrostatic and non-electrostatic PCM interactions. OPLSAA parameters are used in the Lennard-Jones potential for dispersion and repulsion.
$rem
JOBTYPE
BASIS
METHOD
SOLVENT_METHOD
FORCE_FIELD
$end
$pcm
Theory
Method
Solver
HeavyPoints
HPoints
Radii
vdwScale
$end

OPT
6-31G*
B3LYP
PCM
OPLSAA

CPCM
SWIG
Inversion
194
194
Bondi
1.2

$solvent
NonEls
NSolventAtoms
SolventAtom
SolventAtom
SolventRadius
Temperature
Pressure
SolventRho
Dielectric
$end
$molecule
0 1
C
-0.245826
C
0.244003
O
0.862012
F
0.776783
F
-0.858739
F
-1.108290
H
-0.587975
H
0.963047
H
0.191283
$end

LJCav
2
8 1 186 1.30
1 2 187 0.01
1.35
298.15
1.0
0.03333
78.39

12.2.3.4

-0.351674
0.376569
-0.527016
-0.909300
0.511576
-1.303001
0.878499
1.147195
-1.098089

-0.019873
1.241371
2.143243
-0.666009
-0.827287
0.339419
1.736246
0.961639
2.489052

23
22
24
26
26
26
27
27
25

Job control and Examples for Non-Equilibrium Solvation

The OpticalDielectric keyword in $solvent is always needed. The LR energy is automatically calculated while the
CIS/TDDFT calculations are performed with PCM, but it is turned off while the perturbation scheme is employed.

Chapter 12: Molecules in Complex Environments
ChargeSeparation
Partition fast and slow charges in solvent equilibrium state
INPUT SECTION: $pcm
TYPE:
STRING
DEFAULT:
No default.
OPTIONS:
Marcus Do slow-fast charge separation in the ground state.
Excited Do slow-fast charge separation in an excited-state.
RECOMMENDATION:
Charge separation is used in conjunction with the StateSpecific keyword in $pcm.
StateSpecific
Specifies which the state-specific method will be used.
INPUT SECTION: $pcm
TYPE:
Various
DEFAULT:
No default.
OPTIONS:
Marcus Run self-consistent SS method in the ground-state with a given slow polarization charges.
Perturb Perform ptSS and ptLR for vertical excitations.
i
The ith excited-state used for charge separation (for emission).
RECOMMENDATION:

NoneqGrad
Control whether perform excited state geometry optimization in equilibrium or nonequilibrium.
INPUT SECTION: $pcm
TYPE:
NONE
DEFAULT:
No default.
OPTIONS:
RECOMMENDATION:
Specify it for nonequilibrium optimization otherwise equilibrium geometry optimization
will be performed.

705

Chapter 12: Molecules in Complex Environments

Example 12.7 LR-TDDFT/C-PCM low-lying vertical excitation energy
$molecule
0 1
C
0
O
0
$end

0
0

0.0
1.21

$rem
EXCHANGE
CIS_N_ROOTS
CIS_SINGLETS
CIS_TRIPLETS
RPA
BASIS
SOLVENT_METHOD
$end
$pcm
Theory
Method
Solver
Radii
$end

B3LYP
10
TRUE
TRUE
TRUE
6-31+G*
PCM

CPCM
SWIG
Inversion
Bondi

$solvent
Dielectric
OpticalDielectric
$end

78.39
1.777849

706

707

Chapter 12: Molecules in Complex Environments

Example 12.8 PCM solvation effects on the vertical excitation energies of planar DMABN using the ptSS and ptLR
methods.
$molecule
0 1
C
0.000046
C
1.210027
C
1.214640
C
0.000164
C
-1.214349
C
-1.209753
H
2.151949
H
2.164371
H
-2.164082
H
-2.151763
C
-0.000227
N
-0.000475
N
0.000053
C
-1.258656
H
-1.041042
H
-1.860897
H
-1.859247
C
1.258563
H
1.860651
H
1.859362
H
1.040664
$end

-0.000398
0.000379
-0.000065
-0.000616
-0.001557
-0.001846
0.001377
0.000481
-0.002008
-0.002287
0.001061
0.002405
-0.000156
0.001284
0.001615
-0.885647
0.889133
-0.000660
0.886208
-0.888604
-0.000097

$rem
EXCHANGE
OMEGA
BASIS
CIS_N_ROOTS
RPA
CIS_SINGLETS
CIS_TRIPLETS
CIS_DYNAMIC_MEM
CIS_RELAXED_DENSITY
USE_NEW_FUNCTIONAL
SOLVENT_METHOD
PCM_PRINT
$end
$pcm
NonEquilibrium
Theory
StateSpecific
$end
$solvent
Dielectric
OpticalDielectric
$end

1.904953
1.186051
-0.194515
-0.933832
-0.194687
1.185775
1.722018
-0.709640
-0.709781
1.721615
3.325302
4.484321
-2.297372
-3.036994
-4.102376
-2.811117
-2.810237
-3.037285
-2.810755
-2.811461
-4.102609

LRC-wPBEPBE
260
6-31G*
10
2
1
0
TRUE
TRUE
TRUE
PCM
1

IEFPCM
Perturb

35.688000
1.806874

! Acetonitrile

Chapter 12: Molecules in Complex Environments

Example 12.9 Aqueous phenol ionization using state-specific nonequilibrium PCM
$molecule
0 1
C -0.189057 -1.215927 -0.000922
H -0.709319 -2.157526 -0.001587
C
1.194584 -1.155381 -0.000067
H
1.762373 -2.070036 -0.000230
C
1.848872
0.069673
0.000936
H
2.923593
0.111621
0.001593
C
1.103041
1.238842
0.001235
H
1.595604
2.196052
0.002078
C -0.283047
1.185547
0.000344
H -0.862269
2.095160
0.000376
C -0.929565 -0.042566 -0.000765
O -2.287040 -0.159171 -0.001759
H -2.663814
0.725029
0.001075
$end
$rem
EXCHANGE
wPBE
CORRELATION
PBE
LRC_DFT
1
OMEGA
300
BASIS
6-31+G*
SCF_CONVERGENCE
8
SOLVENT_METHOD
PCM
PCM_PRINT
1
$end
$pcm
NonEquilibrium
$end
$solvent
Dielectric
78.39
OpticalDielectric
1.777849
$end
@@@
$molecule
1 2
read
$end
$rem
EXCHANGE
CORRELATION
LRC_DFT
OMEGA
BASIS
SCF_CONVERGENCE
SOLVENT_METHOD
PCM_PRINT
SCF_GUESS
$end
$pcm
StateSpecific
$end
$solvent
Dielectric
OpticalDielectric
$end

wPBE
PBE
1
300
6-31+G*
8
PCM
1
READ

Marcus

78.39
1.777849

708

709

Chapter 12: Molecules in Complex Environments

Example 12.10 PCM solvation effects on the emission energy of twisted DMABN in acetonitrile.
$molecule
0 1
C
-0.000002
C
1.230178
C
1.244862
C
0.000003
C
-1.244845
C
-1.230160
H
2.171128
H
2.180670
H
-2.180657
H
-2.171112
C
-0.000023
N
-0.000011
N
-0.000008
C
0.000001
H
0.000005
H
-0.882908
H
0.882911
C
-0.000009
H
0.882917
H
-0.882901
H
-0.000038
$end

-0.000003
0.000003
0.000001
-0.000013
-0.000000
0.000000
0.000013
0.000009
0.000006
0.000006
-0.000002
-0.000001
-0.000005
1.253881
1.099076
1.818975
1.818967
-1.253886
-1.818966
-1.818992
-1.099071

$rem
EXCHANGE
OMEGA
BASIS
CIS_N_ROOTS
RPA
CIS_SINGLETS
CIS_TRIPLETS
CIS_DYNAMIC_MEM
CIS_RELAXED_DENSITY
USE_NEW_FUNCTIONAL
SOLVENT_METHOD
PCM_PRINT
$end

1.894447
1.160765
-0.200360
-0.917901
-0.200354
1.160753
1.704064
-0.750140
-0.750120
1.704037
3.293509
4.459415
-2.309193
-3.021235
-4.098660
-2.705815
-2.705805
-3.021246
-2.705849
-2.705803
-4.098670

LRC-wPBEPBE
280
6-31G*
10
2
1
0
TRUE
TRUE
TRUE
PCM
1

$pcm
ChargeSeparation
StateSpecific
$end

Excited
1

$solvent
Dielectric
OpticalDielectric
$end

35.688000
1.806874

@@@
$molecule
READ
$end
$rem
EXCHANGE
OMEGA
BASIS
SCF_GUESS
SCF_CONVERGENCE
SOLVENT_METHOD
PCM_PRINT
$end

LRC-wPBEPBE
280
6-31G*
READ
8
PCM
1

! Acetonitrile

Chapter 12: Molecules in Complex Environments

12.2.4

710

Linear-Scaling QM/MM/PCM Calculations

Recall that PCM electrostatics calculations require the solution of the set of linear equations given in Eq. (12.2), to
determine the vector q of apparent surface charges. The precise forms of the matrices K and R depend upon the
particular PCM (Table 12.3), but in any case they have dimension Ngrid × Ngrid , where Ngrid is the number of Lebedev
grid points used to discretize the cavity surface. Construction of the matrix K−1 R affords a numerically exact solution
3
2
to Eq. (12.2), whose cost scales as O(Ngrid
) in CPU time and O(Ngrid
) in memory. This cost is exacerbated by
smooth PCMs, which discard fewer interior grid points so that Ngrid tends to be larger, for a given solute, as compared
to traditional discretization schemes. For QM solutes, the cost of inverting K is usually negligible relative to the cost
of the electronic structure calculation, but for the large values of Ngrid that are encountered in MM/PCM or QM/MM/
3
PCM jobs, the O(Ngrid
) cost of inverting K is often prohibitively expensive.
To avoid this bottleneck, Lange and Herbert 42 have developed an iterative conjugate gradient (CG) solver for Eq. (12.2)
2
whose cost scales as O(Ngrid
) in CPU time and O(Ngrid ) in memory. A number of other cost-saving options are
available, including efficient pre-conditioners and matrix factorizations that speed up convergence of the CG iterations,
and a fast multipole algorithm for computing the electrostatic interactions. 76 Together, these features lend themselves to
a solution of Eq. (12.2) whose cost scales as O(Ngrid ) in both memory and CPU time, for sufficiently large systems. 42
Currently, these options are available only for C-PCM, not for SS(V)PE/IEF-PCM.
Listed below are job control variables for the CG solver, which should be specified within the $pcm input section.
Researchers who use this feature are asked to cite the original SWIG PCM references 68,69 as well as the reference for
the CG solver. 42
Solver
Specifies the algorithm used to solve the PCM equations.
INPUT SECTION: $pcm
TYPE:
STRING
DEFAULT:
INVERSION
OPTIONS:
INVERSION Direct matrix inversion
CG
Iterative conjugate gradient
RECOMMENDATION:
Matrix inversion is faster for small solutes because it needs to be performed only once in
a single-point calculation. However, the CG solver (which must be applied at each SCF
iteration) is recommended for large MM/PCM or QM/MM/PCM calculations.
CGThresh
The threshold for convergence of the conjugate gradient solver.
INPUT SECTION: $pcm
TYPE:
INTEGER
DEFAULT:
6
OPTIONS:
n Conjugate gradient converges when the maximum residual is less than 10−n .
RECOMMENDATION:
The default typically affords PCM energies on par with the precision of matrix inversion
for small systems. For systems that have difficulty with SCF convergence, one should
increase n or try the matrix inversion solver. For well-behaved or very large systems, a
smaller n might be permissible.

Chapter 12: Molecules in Complex Environments
DComp
Controls decomposition of matrices to reduce the matrix norm for the CG Solver.
INPUT SECTION: $pcm
TYPE:
INTEGER
DEFAULT:
1
OPTIONS:
0 Turns off matrix decomposition
1 Turns on matrix decomposition
3 Option 1 plus only stores upper half of matrix and enhances gradient evaluation
RECOMMENDATION:
None
PreCond
Controls the use of the pre-conditioner for the CG solver.
INPUT SECTION: $pcm
TYPE:
None
DEFAULT:
Off
OPTIONS:
No options. Specify the keyword to enable pre-conditioning.
RECOMMENDATION:
A Jacobi block-diagonal pre-conditioner is applied during the conjugate gradient algorithm to improve the rate of convergence. This reduces the number of CG iterations,
at the expense of some overhead. Pre-conditioning is generally recommended for large
systems.
NoMatrix
Specifies whether PCM matrices should be explicitly constructed and stored.
INPUT SECTION: $pcm
TYPE:
None
DEFAULT:
Off
OPTIONS:
No options. Specify the keyword to avoid explicit construction of PCM matrices.
RECOMMENDATION:
2
Storing the PCM matrices requires O(Ngrid
) memory. If this is prohibitive, the NoMatrix
option forgoes explicit construction of the PCM matrices, and instead constructs the matrix elements as needed, reducing the memory requirement to O(Ngrid ) at the expense of
additional computation.

711

Chapter 12: Molecules in Complex Environments
UseMultipole
Controls the use of the adaptive fast multipole method in the CG solver.
INPUT SECTION: $pcm
TYPE:
None
DEFAULT:
Off
OPTIONS:
No options. Specify the keyword in order to enable the fast multipole method.
RECOMMENDATION:
The fast multipole approach formally reduces the CPU time to O(Ngrid ), but is only beneficial for spatially extended systems with several thousand cavity grid points. Requires
the use of NoMatrix.
MultipoleOrder
Specifies the highest multipole order to use in the FMM.
INPUT SECTION: $pcm
TYPE:
INTEGER
DEFAULT:
4
OPTIONS:
n The highest order multipole in the multipole expansion.
RECOMMENDATION:
Increasing the multipole order improves accuracy but also adds more computational expense. The default yields satisfactory performance in common QM/MM/PCM applications.
Theta
The multipole acceptance criterion.
INPUT SECTION: $pcm
TYPE:
FLOAT
DEFAULT:
0.6
OPTIONS:
n A number between zero and one.
RECOMMENDATION:
The default is recommended for general usage. This variable determines when the use
of a multipole expansion is valid. For a given grid point and box center in the FMM, a
multipole expansion is accepted when r/d ≤ Theta, where d is the distance from the grid
point to the box center and r is the radius of the box. Setting Theta to one will accept all
multipole expansions, whereas setting it to zero will accept none. If not accepted, the grid
point’s interaction with each point inside the box is computed explicitly. A low Theta is
more accurate but also more expensive than a higher Theta.

712

Chapter 12: Molecules in Complex Environments

713

NBox
The FMM boxing threshold.
INPUT SECTION: $pcm
TYPE:
INTEGER
DEFAULT:
100
OPTIONS:
n The maximum number of grid points for a leaf box.
RECOMMENDATION:
The default is recommended. This option is for advanced users only. The adaptive FMM
boxing algorithm divides space into smaller and smaller boxes until each box has less
than or equal to NBox grid points. Modification of the threshold can lead to speedup or
slowdown depending on the molecular system and other FMM variables.
A sample input file for the linear-scaling QM/MM/PCM methodology can be found in the $QC/samples directory,
under the name QMMMPCM_crambin.in. This sample involves a QM/MM description of a protein (crambin) in
which a single tyrosine side chain is taken to be the QM region. The entire protein is immersed in a dielectric using
C-PCM with SWIG discretization.

12.2.5

Isodensity Implementation of SS(V)PE

12.2.5.1

Basic Job Control

As discussed above, results obtained various types of PCMs are quite sensitive to the details of the cavity construction.
Q-C HEM’s implementation of PCMs, using Lebedev grids, simplifies this construction somewhat, but leaves the radii
of the atomic spheres as empirical parameters (albeit ones for which widely-used default values are provided). An
alternative implementation of the SS(V)PE solvation model is also available, 22 which attempts to further eliminate
empiricism associated with cavity construction by taking the cavity surface to be a specified iso-contour of the solute’s
electron density. [We call this the isodensity implementation of SS(V)PE in Table 12.3, and it is based on Chipman’s
“symmetrized” form of the K matrix. 22,69 ] In this case, the cavity surface is discretized by projecting a single-center
Lebedev grid onto the iso-contour surface. Unlike the PCM implementation discussed in Section 12.2.2, for which
point-group symmetry is disabled, this implementation of SS(V)PE supports full symmetry for all Abelian point groups.
The larger and/or the less spherical the solute molecule is, the more points are needed to get satisfactory precision in the
results. Further experience will be required to develop detailed recommendations for this parameter. Values as small
as 110 points are usually sufficient for diatomic or triatomic molecules. The default value of 1202 points is adequate to
converge the energy within 0.1 kcal/mol for solutes the size of mono-substituted benzenes.
Energy gradients are also not available for this implementation of SS(V)PE, although they are available for the implementation described in Section 12.2.2 in which the cavity is constructed from atom-centered spheres. As with the
PCMs discussed in that section, the solute may be described using Hartree-Fock theory or DFT; post-Hartree–Fock
correlated wave functions can also take advantage of molecular orbitals that are polarized using SS(V)PE. Researchers
who use the isodensity SS(V)PE feature are asked to cite Ref. 20.
In related work, Pomogaeva and Chipman 91–94 recently introduced a “composite method for implicit representation of
solvent” (CMIRS) that is based on SS(V)PE electrostatics but adds non-electrostatic terms. This model is available in
Q-C HEM 126 and is discussed in Section 12.2.6. In its current implementation, CMIRS requires an isodensity SS(V)PE
calculation, However, the current implementation computes the non-electrostatic interactions using the cavity and the
solute’s charge density generated from the isodensity SS(V)PE. To use the CMIRS model, an isodensity SS(V)PE
calculation must be requested (as described below), and the IDEFESR must be set to 1 in the $svp input section (see
Section 12.2.5.2).
An isodensity SS(V)PE calculation is requested by setting SOLVENT_METHOD = ISOSVP in the $rem section, in
addition to normal job control variables for a single-point energy calculation. Whereas the other solvation models

Chapter 12: Molecules in Complex Environments

714

described in this chapter use specialized input sections (e.g., $pcm) in lieu of a slew of $rem variables, the isodensity
SS(V)PE code is an interface between Q-C HEM and a code written by Chipman, 22 so some $rem variables are used for
job control of isodensity SS(V)PE calculations. These are listed below.
SVP_MEMORY
Specifies the amount of memory for use by the solvation module.
TYPE:
INTEGER
DEFAULT:
125
OPTIONS:
n corresponds to the amount of memory in Mb.
RECOMMENDATION:
The default should be fine for medium size molecules with the default Lebedev grid, only increase if needed.
SVP_PATH
Specifies whether to run a gas phase computation prior to performing the solvation procedure.
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
0 runs a gas-phase calculation and after
convergence runs the SS(V)PE computation.
1 does not run a gas-phase calculation.
RECOMMENDATION:
Running the gas-phase calculation provides a good guess to start the solvation stage and provides
a more complete set of solvated properties.
SVP_CHARGE_CONV
Determines the convergence value for the charges on the cavity. When the change in charges
fall below this value, if the electron density is converged, then the calculation is considered
converged.
TYPE:
INTEGER
DEFAULT:
7
OPTIONS:
n Convergence threshold set to 10−n .
RECOMMENDATION:
The default value unless convergence problems arise.
SVP_CAVITY_CONV
Determines the convergence value of the iterative isodensity cavity procedure.
TYPE:
INTEGER
DEFAULT:
10
OPTIONS:
n Convergence threshold set to 10−n .
RECOMMENDATION:
The default value unless convergence problems arise.

Chapter 12: Molecules in Complex Environments

715

SVP_GUESS
Specifies how and if the solvation module will use a given guess for the charges and cavity points.
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
0 No guessing.
1 Read a guess from a previous Q-C HEM solvation computation.
2 Use a guess specified by the $svpirf section from the input
RECOMMENDATION:
It is helpful to also set SCF_GUESS to READ when using a guess from a previous Q-C HEM run.

This last $rem variable requires specification of a $svpirf input section, the format for which is the following:
$svpirf
<# point>     
<# point>   
$end

12.2.5.2

The $svp Input Section

More refined control over SS(V)PE jobs is obtained using a $svp input section. These are read directly by Chipman’s
SS(V)PE solvation module and therefore must be specified in the context of a FORTRAN namelist. The format is as
follows:
$svp
=, =,...
=
$end

For example, the section may look like this:
$svp
RHOISO=0.001, DIELST=78.39, NPTLEB=110
$end

The following keywords are supported in the $svp section:
DIELST
The static dielectric constant.
INPUT SECTION: $svp
TYPE:
FLOAT
DEFAULT:
78.39
OPTIONS:
real number specifying the constant.
RECOMMENDATION:
The default value 78.39 is appropriate for water solvent.

Chapter 12: Molecules in Complex Environments
IDEFESR
Specifies whether to request a CMIRS calculation.
INPUT SECTION: $svp
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
0 do not invoke a CMIRS calculation.
1 do invoke a CMIRS calculation.
RECOMMENDATION:

ISHAPE
A flag to set the shape of the cavity surface.
INPUT SECTION: $svp
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
0 use the electronic isodensity surface.
1 use a spherical cavity surface.
RECOMMENDATION:
Use the default surface.
RHOISO
Value of the electronic isodensity contour used to specify the cavity surface. (Only relevant for ISHAPE = 0.)
INPUT SECTION: $svp
TYPE:
FLOAT
DEFAULT:
0.001
OPTIONS:
Real number specifying the density in electrons/bohr3 .
RECOMMENDATION:
The default value is optimal for most situations. Increasing the value produces a smaller
cavity which ordinarily increases the magnitude of the solvation energy.
RADSPH
Sphere radius used to specify the cavity surface (Only relevant for ISHAPE=1.)
INPUT SECTION: $svp
TYPE:
FLOAT
DEFAULT:
Half the distance between the outermost atoms plus 1.4 Ångstroms.
OPTIONS:
Real number specifying the radius in bohr (if positive) or in Ångstroms (if negative).
RECOMMENDATION:
Make sure that the cavity radius is larger than the length of the molecule.

716

Chapter 12: Molecules in Complex Environments
INTCAV
A flag to select the surface integration method.
INPUT SECTION: $svp
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
0 Single center Lebedev integration.
1 Single center spherical polar integration.
RECOMMENDATION:
The Lebedev integration is by far the more efficient.
NPTLEB
The number of points used in the Lebedev grid for the single-center surface integration.
(Only relevant if INTCAV = 0.)
INPUT SECTION: $svp
TYPE:
INTEGER
DEFAULT:
1202
OPTIONS:
Valid choices are: 6, 18, 26, 38, 50, 86, 110, 146, 170, 194, 302, 350, 434, 590, 770,
974, 1202, 1454, 1730, 2030, 2354, 2702, 3074, 3470, 3890, 4334,
4802, or 5294.
RECOMMENDATION:
The default value has been found adequate to obtain the energy to within 0.1 kcal/mol for
solutes the size of mono-substituted benzenes.
NPTTHE, NPTPHI
The number of (θ,φ) points used for single-centered surface integration (relevant only if
INTCAV = 1.)
INPUT SECTION: $svp
TYPE:
INTEGER
DEFAULT:
8,16
OPTIONS:
θ,φ specifying the number of points.
RECOMMENDATION:
These should be multiples of 2 and 4 respectively, to provide symmetry sufficient for all
Abelian point groups. Defaults are too small for all but the tiniest and simplest solutes.

717

Chapter 12: Molecules in Complex Environments

718

LINEQ
Flag to select the method for solving the linear equations that determine the apparent point
charges on the cavity surface.
INPUT SECTION: $svp
TYPE:
INTEGER
DEFAULT:
1
OPTIONS:
0 use LU decomposition in memory if space permits, else switch to LINEQ = 2
1 use conjugate gradient iterations in memory if space permits, else use LINEQ = 2
2 use conjugate gradient iterations with the system matrix stored externally on disk.
RECOMMENDATION:
The default should be sufficient in most cases.
CVGLIN
Convergence criterion for solving linear equations by the conjugate gradient iterative
method (relevant if LINEQ = 1 or 2.)
INPUT SECTION: $svp
TYPE:
FLOAT
DEFAULT:
1.0E-7
OPTIONS:
Real number specifying the actual criterion.
RECOMMENDATION:
The default value should be used unless convergence problems arise.

Note that the single-center surface integration approach that is used to find the isodensity surface may fail for certain
very non-spherical solute molecules. The program will automatically check for this, aborting with a warning message if
necessary. The single-center approach succeeds only for what is called a “star surface”, meaning that an observer sitting
at the center has an unobstructed view of the entire surface. Said another way, for a star surface any ray emanating out
from the center will pass through the surface only once. Some cases of failure may be fixed by simply moving to a new
center with the ITRNGR parameter described below. But some surfaces are inherently non-star surfaces and cannot be
treated with this program until more sophisticated surface integration approaches are developed and implemented.

Chapter 12: Molecules in Complex Environments
ITRNGR
Translation of the cavity surface integration grid.
INPUT SECTION: $svp
TYPE:
INTEGER
DEFAULT:
2
OPTIONS:
0 No translation (i.e., center of the cavity at the origin
of the atomic coordinate system)
1 Translate to the center of nuclear mass.
2 Translate to the center of nuclear charge.
3 Translate to the midpoint of the outermost atoms.
4 Translate to midpoint of the outermost non-hydrogen atoms.
5 Translate to user-specified coordinates in Bohr.
6 Translate to user-specified coordinates in Ångstroms.
RECOMMENDATION:
The default value is recommended unless the single-center integrations procedure fails.
TRANX, TRANY, TRANZ
x, y, and z value of user-specified translation (only relevant if ITRNGR is set to 5 or 6).
INPUT SECTION: $svp
TYPE:
FLOAT
DEFAULT:
0, 0, 0
OPTIONS:
x, y, and z relative to the origin in the appropriate units.
RECOMMENDATION:
None.
IROTGR
Rotation of the cavity surface integration grid.
INPUT SECTION: $svp
TYPE:
INTEGER
DEFAULT:
2
OPTIONS:
0 No rotation.
1 Rotate initial xyz axes of the integration grid to coincide
with principal moments of nuclear inertia (relevant if ITRNGR = 1)
2 Rotate initial xyz axes of integration grid to coincide with
principal moments of nuclear charge (relevant if ITRNGR = 2)
3 Rotate initial xyz axes of the integration grid through user-specified
Euler angles as defined by Wilson, Decius, and Cross.
RECOMMENDATION:
The default is recommended unless the knowledgeable user has good reason otherwise.

719

Chapter 12: Molecules in Complex Environments

720

ROTTHE ROTPHI ROTCHI
Euler angles (θ, φ, χ) in degrees for user-specified rotation of the cavity surface (relevant
if IROTGR = 3).
INPUT SECTION: $svp
TYPE:
FLOAT
DEFAULT:
0,0,0
OPTIONS:
θ, φ, χ in degrees
RECOMMENDATION:
None.
IOPPRD
Specifies the choice of system operator form.
INPUT SECTION: $svp
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
0 Symmetric form.
1 Non-symmetric form.
RECOMMENDATION:
The default uses more memory but is generally more efficient, we recommend its use
unless there is shortage of memory available.

By default, Q-C HEM will check the validity of the single-center expansion by searching for the isodensity surface in
two different ways: first, working inwards from a large distance, and next by working outwards from the origin. If the
same result is obtained (within tolerances) using both procedures, then the cavity is accepted. If the two results do not
agree, then the program exits with an error message indicating that the inner isodensity surface is found to be too far
from the outer isodensity surface.
Some molecules, for example C60 , can have a hole in the middle. Such molecules have two different “legal” isodensity
surfaces, a small inner one inside the “hole”, and a large outer one that is the desired surface for solvation. In such
cases, the cavity check described in the preceding paragraph causes the program to exit. To avoid this, one can consider
turning off the cavity check that works out from the origin, leaving only the outer cavity determined by working in
from large distances.
ICVICK
Specifies whether to perform cavity check
INPUT SECTION: $svp
TYPE:
INTEGER
DEFAULT:
1
OPTIONS:
0 no cavity check, use only the outer cavity
1 cavity check, generating both the inner and outer cavities and compare.
RECOMMENDATION:
Consider turning off cavity check only if the molecule has a hole and if a star (outer)
surface is expected.

721

Chapter 12: Molecules in Complex Environments
Solvent
Benzene
Cyclohexane
DMSO
CH3 CN
Water

SolvRho
0.0421
0.0396
0.05279
0.03764
0.05

A
−0.00522
−0.00938
−0.00951
−0.008178
−0.006736

B
0.01294
0.03184
0.044791
0.045278
0.032698

C

D

Gamma

−1249.6

−162.07
−0.33914
−21.405

4.1
1.3
3.7

Table 12.4: Optimized CMIRS parameters (from Ref. 126) using RHOISO = 0.0010 a.u. and, for ions, the proton
reference ∆G value from Ref. 56.

12.2.6

Composite Method for Implicit Representation of Solvent (CMIRS)

Whereas PCMs, including sophisticated ones like SS(V)PE and IEF-PCM, account for long-range solute–solvent interactions, an accurate model for free energies of solvation must also include a treatment of short-range, non-electrostatic
interactions. Various models decompose these interactions in different ways, but usually the non-electrostatic terms
attempt to model all or most of the following: solute–solvent dispersion (van der Waals) interactions, Pauli (exchange)
repulsion between solute and solvent, the work associated with forming the solute cavity within the dielectric medium
(the so-called “cavitation energy”), hydrogen-bonding and other specific interactions due to the molecular structure of
the solvent, and changes in the structure (and therefore the entropy) of the neat solvent upon introduction of the solute
Pomogaeva and Chipman 91–94 have introduced an implicit solvation model that attempts to model these non-electrostatic
interactions alongside a PCM-style treatment of the bulk electrostatics. They call this approach the “composite method
for implicit representation of solvent” (CMIRS), and it consists first of a self-consistent treatment of solute–continuum
electrostatics using the SS(V)PE model (Section 12.2.5). To this electrostatics calculation, CMIRS adds a solute–
solvent dispersion term that is modeled upon the non-local VV09 van der Waals dispersion density functional, 117 a
Pauli repulsion contribution that depends upon the tail of the solute’s electron density that extends beyond the solute
cavity, and a hydrogen-bonding correction based on the maximum and minimum values of the normal component of
the electric field generated by the solute at the cavity surface. The Gibbs free energy of solvation is thus modeled as
∆GCMIRS (ρs ) =∆GSS(V)PE (ρs ) + ∆GDEFESR (ρs ) ,

(12.10)

where ∆GSS(V)PE is the continuum electrostatics contribution from the SS(V)PE model, which is based on a solute
cavity defined as an isocontour ρs of the solute’s charge density. The second term contains the short-range dispersion,
exchange, and "field-extremum short-range" (DEFESR) interactions:
∆GDEFESR =∆Gdisp + ∆Gexch + ∆GFESR .

(12.11)

These terms are evaluated only once, using a converged charge density for the solute from a SS(V)PE calculation.
The CMIRS approach was implemented in Q-C HEM by Zhi-Qiang You and John Herbert. 126 In the course of this
work, a serious error was discovered in the the original implementation of ∆Gdisp by Pomogaeva and Chipman, in the
GAMESS program. Although reparameterization of a corrected version of the model leads to only small changes in the
overall error statistics across a large database of experimental free energies of solvation, the apportionment between
energy components in Eq. (12.11) changes significantly. 126 By request of Dan Chipman, the Q-C HEM implementation
is termed “CMIRS v. 1.1", reserving v. 1.0 for the original GAMESS implementation of Pomogaeva and Chipman.
The CMIRS model is independently parametrized for each solvent of interest, but uses no more than five empirical
parameters per solvent. It is presently available in Q-C HEM for water, acetonitrile, dimethyl sulfoxide, benzene,
and cyclohexane. Error statistics for ∆G compare very favorably to those of the SMx models that are described in
Section 12.2.8, e.g., mean unsigned errors < 0.7 kcal/mol in benzene and cyclohexane and < 1.5 kcal/mol in water.
The latter statistic includes challenging ionic solutes; errors for charge-neutral aqueous solutes are smaller still. 126
The current implementation of CMIRS in Q-C HEM computes the electrostatic energy using Chipman’s isodensity
SS(V)PE module. The resulting isodensity cavity and the solute charge density are then employed in the calculation
of the DEFESR interactions. To request a CMIRS calculation, users must set IDEFESR = 1 in an isodensity SS(V)PE

722

Chapter 12: Molecules in Complex Environments
Solvent
Benzene
Cyclohexane
DMSO
CH3 CN
Water

SolvRho
0.0421
0.0396
0.05279
0.03764
0.05

A
−0.00572
−0.00721
−0.002523
−0.003805
−0.006496

B
0.01116
0.05618
0.011757
0.03223
0.050833

C

D

Gamma

−566.7

−817.93
−0.44492
−30.503

4.3
1.2
3.2

Table 12.5: Optimized CMIRS parameters (from Ref. 126) using RHOISO = 0.0005 a.u. and, for ions, the proton
reference ∆G value from Ref. 56.

calculation (see Section 12.2.5.2). The solvent-dependent empirical parameters A, B, C, D, Gamma in the CMIRS
model need to be specified in the $pcm_nonels section. Three additional parameters are also required. One is the
damping parameter Delta in the dispersion equation. We recommend the parameter fixed at δ = 7 a.u. (about 3.7 Å), an
optimized value that only considers dispersion at intermolecular distances larger than van der Waals contact distance.
The second is solvent’s average electron density SolvRho. The last one is the number of Gauss–Laguerre points
GauLag_N for the integration over the solvent region in the exchange equation. We recommend 40 grid points for
efficient integration with accuracy. Optimized parameters for the supported solvents are listed in Tables 12.4 and 12.5
for two different values of ρs . 126

723

Chapter 12: Molecules in Complex Environments

Example 12.11 CMIRS calculation for methane in the water solvent.
$rem
EXCHANGE
BASIS
SCF_CONVERGENCE
MEM_TOTAL
MEM_STATIC
SYM_IGNORE
XC_GRID
$end

=
=
=
=
=
=
=

$molecule
0 1
C
0.000000
H
0.748558
H
0.506094
H
-0.627326
H
-0.627326
$end

b3lyp5
6-31+g*
8
4000
400
true
000096001202

0.000005
0.801458
-0.972902
0.085706
0.085706

0.000000
0.000000
0.000000
0.895425
-0.895425

@@@
$rem
EXCHANGE
BASIS
SCF_CONVERGENCE
MAX_SCF_CYCLES
SOLVENT_METHOD
SCF_GUESS
PCM_PRINT
MEM_TOTAL
MEM_STATIC
SVP_MEMORY
SYM_IGNORE
XC_GRID
$end

=
=
=
=
=
=
=
=
=
=
=
=

b3lyp5
6-31+g*
8
100
isosvp
read
1
4000
400
1000
true
000096001202

$molecule
read
$end
$svp
RHOISO=0.001, DIELST=78.36, NPTLEB=1202,
ITRNGR=2, IROTGR=2, IPNRF=1, IDEFESR=1
$end
$pcm_nonels
A
-0.006736
B
0.032698
C
-1249.6
D
-21.405
Delta
7.0
Gamma
3.7
SolvRho 0.05
GauLag_N 40
$end

724

Chapter 12: Molecules in Complex Environments

12.2.7

COSMO

According to Table 12.3, COSMO and C-PCM appear to differ only in the dielectric screening factor, fε in Eq. (12.3).
Indeed, surface charges in either model are computed according to
q = −fε S−1 v ,

(12.12)

and as discussed in Section 12.2.3 the user has the option to choose either the original value suggested by Klamt, 61,62
fε = (ε − 1)/(ε + 1/2), or else fε = (ε − 1)/ε as in, e.g., Refs. 28,69,115. More importantly, however, COSMO
differs from C-PCM in that the former includes a correction for outlying charge that goes beyond Eq. (12.12), whereas
C-PCM consists of nothing more than induced surface charges computed (self-consistently) according to Eq. (12.12)
Upon solution of Eq. (12.12), the outlying charge correction in COSMO 5,61 is obtained by first defining a larger cavity
that is likely to contain essentially all of the solute’s electron density; in practice, this typically means using atomic
radii of 1.95 R, where R denotes the original atomic van der Waals radius that was used to compute q. (Note that
unlike the PCMs described in Sections 12.2.2 and 12.2.3, where the atomic radii have default values but a high degree
of user-controllability is allowed, the COSMO atomic radii are parameterized for this model and are fixed.) A new set
of charges, q0 = −fε (S0 )−1 v0 , is then computed on this larger cavity surface, and the charges on the original cavity
surface are adjusted to new values, q00 = q + q0 . Finally, a corrected electrostatic potential on the original surface is
computed according to v00 = −fε Sq00 . It is this potential that is used to compute the solute–continuum electrostatic
P
interaction (polarization energy), Gpol = 21 i qi00 vi00 . (For comparison, when the C-PCM approach described in
P
Section 12.2.2 is used, the electrostatic polarization energy is Gpol = 21 i qi vi , computed using the original surface
charges q and surface electrostatic potential v.) With this outlying charge correction, Q-C HEM’s implementation of
COSMO resembles the one in T URBOMOLE. 100
A COSMO calculation is requested by setting SOLVENT_METHOD = COSMO in the $rem section, in addition to normal
job control variables. The keyword Dielectric in the $solvent section is used to set the solvent’s static dielectric
constant, as described above for other solvation models. COSMO calculations can also be used as a starting point
for COSMO-RS calculations, 63,65 where “RS” stands for “real solvent”. The COSMO-RS approach is not included
in Q-C HEM and requires the COSMOtherm program, which is licensed separately through COSMOlogic. 1 Q-C HEM
users who are interested in COSMOtherm can request special versions of Q-C HEM for the generation of σ-surface
files that are needed by COSMOtherm.

12.2.8

SM8, SM12, and SMD Models

The SMx models developed by Cramer, Truhlar, and coworkers at the University of Minnesota are a class of implicit
solvation models that are designed to be “universal” in the sense that they can be applied to any solvent for which
a small of descriptors is known. 30 (Note that the x in SMx is simply a version number; SM8, SM12, and SMD are
available in Q-C HEM.) In particular, the solvent descriptors are:
• dielectric constant
• refractive index
• bulk surface tension
• acidity on the Abraham scale
• basicity on the Abraham scale
• carbon aromaticity, which equals the fraction of non-hydrogenic solvent atoms that are aromatic carbon atoms
• electronegative halogenicity, which equals the fraction of non-hydrogenic solvent atoms that are F, Cl, or Br).
These models consist of a generalized Born treatment of continuum electrostatic interactions, along with non-electrostatic
interactions that are parameterized in terms of atomic surface tensions. The non-electrostatic interactions include cavitation, dispersion, and changes in solvent structure, and the treatment of these non-electrostatic effects is crucial to
obtaining accurate (free) energies of solvation.

Chapter 12: Molecules in Complex Environments
12.2.8.1

725

The SM8 Model

The SM8 model is described in detail in Ref. 80. It may be employed in conjunction with density functional theory
(with any density functional available in Q-C HEM) or with Hartree-Fock theory, but is intended for use only with the
6-31G*, 6-31+G*, and 6-31+G** basis sets, for reasons discussed below.
Bulk (continuum) electrostatic interactions in SM8 are described in terms of a generalized Born (GB) SCRF, using
a solute cavity constructed from atom-centered spheres. For the atoms H, C, N, O, F, Si, P, S, Cl, and Br, atomic
radii have been specifically optimized for use with SM8, whereas for other atoms the Bondi radius is used, 9 or else a
value of 2.0 Å for atoms not included in Bondi’s paper. Geometry-dependent radii are computed from these “intrinsic”
Coulomb radii via a de-screening approximation. 80
In addition to GB electrostatics, there are several other contributions to the SM8 standard-state free energy of solvation.
The first of these is called the electronic-nuclear-polarization (ENP) energy, or simply the electronic polarization (EP)
energy if the solute geometry is assumed to be identical in the gas and solution phases. Another contribution to the
free energy of solvation comes from short-range interactions with solvent molecules in the first solvation shell, and is
sometimes called the cavitation/dispersion/solvent-structure (CDS) term. The CDS contribution to the solvation energy
is a sum of terms that are each proportional (with geometry-dependent proportionality constants called atomic surface
tensions) to the solvent-accessible surface areas (SASAs) of the individual solute atoms. The SASA of the solute
molecule is the area of a surface generated by the center of a spherical effective solvent molecule rolling on the van
der Waals surface of the solute molecule, as in the solvent-accessible surface that was mentioned in Section 12.2.2.
The SASA is computed using the Analytic Surface Area (ASA) algorithm of Ref. 77 and Bondi’s values for the van
der Waals radii, 9 or else a value of 2.0 Å if no Bondi radius is available. (Note that, as in the case of non-electrostatic
interactions in PCMs, this means that a different molecular surface is used for the bulk electrostatics as compared to the
non-electrostatic interactions.) The solvent probe radius used to generate the SASAs is set to 0.40 Å for all solvents.
Note that the solvent-structure part of the CDS term includes many aspects of solvent structure that are not described
by bulk electrostatics, for example, hydrogen bonding, exchange repulsion, and the variation of the effective dielectric
constant in the first solvation shell, relative to its bulk value. The semi-empirical nature of the CDS term also makes up
for errors due to (i) assuming fixed and model-dependent values of the intrinsic Coulomb radii, and (ii) any systematic
errors in the description of the solute–solvent electrostatic interactions using the GB approximation.
The final component of the SM8 solvation free energy is the concentration component. This is zero if the standard-state
concentration of the solute is the same in the gas and solution phases (e.g., if it is 1 mole/liter in the gas phase as well
as in the solution). Otherwise, this correction can be computed using ideal gas formulas, as discussed below.
SM8 does not require the user to assign molecular mechanics atom types to atoms or groups; all atomic surface tensions
in the theory are unique and continuous functions of the solute geometry, defined by the model and calculated internally
within Q-C HEM. In principle, SM8 can be used with any level of electronic structure theory so long as accurate partial
charges can be computed, but Q-C HEM’s implementation of SM8 specifically uses self-consistently polarized Charge
Model 4 (CM4) class IV charges. 55 CM4 charges are obtained from Löwdin population analysis charges, via a mapping
whose parameters depend on the basis set (and only on the basis set, not on the density functional or anything else).
Basis sets supported for SM8 calculations in Q-C HEM are:
• 6-31G*
• 6-31+G*
• 6-31+G**
The charge mapping parameters are given in Ref. 55. Other basis sets should not be used in SM8 calculations.
The SM8 solvation free energy is output at T = 298 K for a standard-state concentration of 1 M in both the gas and
solution phase. However, solvation free energies in the literature are often tabulated using a standard state of P = 1 atm
for the gase. To convert 1 M-to1 M solvation free energies at 298 K to a standard state consisting of P = 1 atm for the
gas and a 1 M concentration in solution, add +1.89 kcal/mol to the computed solvation free energy.

726

Chapter 12: Molecules in Complex Environments

Solution-phase geometry optimizations can be carried out, but basis sets that use spherical harmonic d functions, or
angular momentum higher than d (f , g, etc.) are not supported. Since, by definition, the 6-31G*, 6-31+G*, and 631+G** basis sets have Cartesian d shells, they are examples of basis sets that may be used for geometry optimization
with SM8. Solution-phase Hessian calculations can be carried out by numerical differentiation of analytical energy
gradients or by double differentiation of energies, although the former procedure is both more stable and more economical. The analytic gradients of SM8 are based on the analytical derivatives of the polarization free energy and the
analytical derivatives of the CDS terms derived in Ref. 131.
An SM8 calculation is requested by setting SOLVENT_METHOD = SM8 in the $rem section, along with other jobcontrol variables appropriate for a Hartree-Fock or DFT calculation. Additional variables for SMx calculations should
be listed in a $smx input section; for SM8, the only additional variable that is required is the name of the solvent:
Solvent
Sets the SMx solvent
INPUT SECTION: $smx
TYPE:
STRING
DEFAULT:
water
OPTIONS:
Any name from the list of solvents given below.
RECOMMENDATION:
NONE

1,1,1-trichloroethane
1,1,2-trichloroethane
1,1-dichloroethane
1,2,4-trimethylbenzene
1,4-dioxane
1-bromo-2-methylpropane
1-bromopentane
1-bromopropane
1-butanol
1-chloropentane
1-chloropropane
1-decanol
1-fluorooctane
1-heptanol
1-hexanol
1-hexene
1-hexyne
1-iodobutane
1-iodopentene
1-iodopropane
1-nitropropane
1-nonanol
1-octanol
1-pentanol
1-pentene
1-pentyne
1-propanol
2,2,20trifluoroethanol
2,2,4-trimethylpentane

bromoethane
bromooctane
butanal
butanoicacid
butanone
butanonitrile
butylethanoate
butylamine
butylbenzene
carbon disulfide
carbon tetrachloride
chlorobenzene
chlorotoluene
cis-1,2-dimethylcyclohexane
decalin
cyclohexane
cyclohexanone
cyclopentane
cyclopentanol
cyclopentanone
decane
dibromomethane
dibutyl ether
dichloromethane
diethyl ether
diethylsulfide
diethylamine
diiodomethane
dimethyldisulfide

m-ethylbenzoate
m-ethylethanoate
m-ethylmethanoate
m-ethylphenylketone
m-ethylpropanoate
m-ethylbutanoate
m-ethylcyclohexane
m-ethylformamide
m-xylene
heptane
hexadecane
hexane
nitrobenzene
nitroethane
nitromethane
methylaniline
nonane
octane
pentane
o-chlorotoluene
o-cresol
o-dichlorobenzene
o-nitrotoluene
o-xylene
pentadecane
pentanal
pentanoic acid
pentylethanoate
pentylamine

727

Chapter 12: Molecules in Complex Environments
2,4-dimethylpentane
2,4-dimethylpyridine
2,6-dimethylpyridine
2-bromopropane
2-chlorobutane
2-heptanone
2-hexanone
2-methylpentane
2-methylpyridine
2-nitropropane
2-octanone
2-pentanone
2-propanol
2-propen-1-ol
3-methylpyridine
3-pentanone
4-heptanone
4-methyl-2-pentanone
4-methylpyridine
5-nonanone
acetic acid
acetone
acetonitrile
aniline
anisole
benzaldehyde
benzene
benzonitrile
benzyl alcohol
bromobenzene

dimethylacetamide
dimethylformamide
dimethylpyridine
dimethyl sulfoxide
dipropylamine
dodecane
E-1,2-dichloroethene
E-2-pentene
ethanethiol
ethanol
ethylethanoate
ethylmethanoate
ethylphenyl ether
ethylbenzene
ethylene glycol
fluorobenzene
formamide
formic acid
hexadecyliodide
hexanoic acid
iodobenzene
iodoethane
iodomethane
isobutanol
isopropyl ether
isopropylbenzene
isopropyltoluene
m-cresol
mesitylene
methanol

perfluorobenzene
phenyl ether
propanal
propanoic acid
propanonitrile
propylethanoate
propylamine
p-xylene
pyridine
pyrrolidine
sec-butanol
t-butanol
t-butylbenzene
tetrachloroethene
tetrahydrofuran
tetrahyrothiophenedioxide
tetralin
thiophene
thiophenol
toluene
trans-decalin
tribromomethane
tributylphosphate
trichloroethene
trichloromethane
triethylamine
undecane
water
Z-1,2-dichloroethene
other

The final choice, Solvent = “other”, requires an additional free-format file called “solvent_data” that should contain
the float-point values of the following solvent descriptors:
Dielec
SolN
SolA
SolB
SolG
SolC
SolH

dielectric constant, ε, of the solvent
index of refraction at optical frequencies at 293 K, nD
20
P
Abraham’s hydrogen bond acidity, α2H
P
Abraham’s hydrogen bond basicity, β2H
γ = γm /γ 0 (default is 0.0), where γm is the macroscopic surface tension at air/solvent
interface at 298 K, and γ 0 is 1 cal mol−1 Å−2 (1 dyne/cm = 1.43932 cal mol−1 Å−2 )
aromaticity, φ : the fraction of non-hydrogenic solvent atoms that are aromatic
carbon atoms
electronegative “halogenicity”, ψ : the fraction of non-hydrogenic solvent atoms that are
F, Cl or Br

For a desired solvent, these values can be derived from experiment or from interpolation or extrapolation of data
available for other solvents. Solvent parameters for common organic solvents are tabulated in the Minnesota Solvent
Descriptor Database. The latest version of this database is available at:
http://comp.chem.umn.edu/solvation/mnsddb.pdf
The SM8 test suite contains the following representative examples:

Chapter 12: Molecules in Complex Environments

728

• single-point solvation energy and analytical gradient calculation for 2,2-dichloroethenyl dimethyl phosphate in
water at the M06-2X/6-31G* level;
• single-point solvation energy calculation for 2,2-dichloroethenyl dimethyl phosphate in benzene at the M06-2X/
6-31G* level;
• single-point solvation energy calculation for 2,2-dichloroethenyl dimethyl phosphate in ethanol at the M06-2X/
6-31G* level;
• single-point solvation energy calculation for 5-fluorouracil in water at the M06/6-31+G* level;
• single-point solvation energy calculation for 5-fluorouracil in octanol at the M06-L/6-31+G* level;
• single-point solvation energy and analytical gradient calculation for 5-fluorouracil in fluorobenzene at the M06-HF/
6-31+G** level;
• geometry optimization for protonated methanol CH3 OH+
2 in water at the B3LYP/6-31G* level;
• finite-difference frequency (with analytical gradient) calculation for protonated methanol CH3 OH+
2 in water at
the B3LYP/6-31G* level.
Users who wish to calculate solubilities can calculate them from the free energies of solvation by the method described
in Ref. 111. The present model can also be used with confidence to calculate partition coefficients (e.g., Henry’s Law
constants, octanol/water partition coefficients, etc.) by the method described in Ref. 29.
The user should note that the free energies of solvation calculated by the SM8 model in the current version of Q-C HEM
are all what may be called equilibrium free energies of solvation. The nonequilibrium algorithm required for vertical
excitation energies 75 is not yet available in Q-C HEM. (Nonequilibrium versions of PCMs are available instead; see
Section 12.2.2.3.)

12.2.8.2

The SM12 Model

The SM12 model 84 is also available in Q-C HEM. Similar to SM8, it employs (a) the generalized Born approximation
for the bulk electrostatic contribution to the free energy of solvation, and (b) the same formulas (with re-optimized
parameters) for CDS contributions. SM12 holds several advantages over SM8, and perhaps foremost among these
is that it uses CM5 charges, 83 which are based on Hirshfeld population analysis, or else charges derived from the
electrostatic potential, 11,104 for the bulk electrostatics term. These charges are stable with respect to extension of the
basis set, and thus SM12 can be used with larger basis sets whereas SM8 is limited to 6-31G*, 6-31+G*, and 6-31+G**,
due to instabilities in the Löwdin charges in larger basis sets. In addition, SM12 is parameterized using a more diverse
training set as compared to SM8, and is defined for the entire periodic table. However, the SM12 analytic gradient is
not available in Q-C HEM at present.
An SM12 calculation is requested by setting SOLVENT_METHOD = SM12 in the $rem section. The manner in which
the electrostatic term is computed is controlled by the Charges keyword in the $smx input section.

Chapter 12: Molecules in Complex Environments

729

Charges
Sets the type of atomic charges for the SM12 electrostatic term.
INPUT SECTION: $smx
TYPE:
STRING
DEFAULT:
CM5
OPTIONS:
CM5
Charge Model 5 charges 83
MK
Merz-Singh-Kollman charges 104
CHELPG ChElPG charges 11
RECOMMENDATION:
None. Merz-Singh-Kollman and ChElPG charges are fit to reproduce the molecular electrostatic potential on the van der Waals surface or on a cubic grid, respectively, whereas
CM5 is an empirical model based on Hirshfeld population analysis.

Example 12.12 SM12CM5 calculation of the solvation free energy of water in the 1-octanol solvent.
$molecule
0 1
O
0.000000
H
0.758502
H -0.758502
$end

0.125787
-0.503148
-0.503148

$rem
SCF_GUESS
METHOD
BASIS
SYM_IGNORE
SOLVENT_METHOD
$end

0.000000
0.000000
0.000000

core
b3lyp
6-31G*
true
sm12

$smx
solvent 1octanol
charges chelpg
$end

12.2.8.3

The SMD Model

The SMD model 81 is also available in Q-C HEM. Within this model, the electrostatic contribution to the free energy
solvation is described via the IEF-PCM model, where the CDS contributions follow the formulas as SM8 and SM12
with the parameters re-optimized to be compatible with the IEF-PCM electrostatics. Relative to SM8 or SM12, where
the electrostatic interactions are defined in terms of atomic point charges that are sensitive to the choice of basis set
(and therefore only certain basis sets are supported for use with these models), SMD can be used with any basis set.
An SMD energy or gradient calculation is requested by setting SOLVENT_METHOD = SMD in the $rem section. While
Q-C HEM users can vary the parameters for the IEF-PCM part of the SMD calculation, this should be done with caution
because a modified IEF-PCM electrostatics might be less compatible with CDS parameters and thus lead to less accurate

730

Chapter 12: Molecules in Complex Environments
results.
Example 12.13 SMD force calculation for tetrahydrofuran (THF) in the pentane solvent.
$molecule
0 1
C
-0.361658
C
-1.331098
O
-0.592574
C
0.798089
C
0.964682
H
-0.625676
H
-0.333229
H
-1.697529
H
-2.193412
H
1.130199
H
1.348524
H
1.050613
H
1.843065
$end
$rem
JOBTYPE
METHOD
BASIS
SOLVENT_METHOD
$end

-0.986967
0.144597
1.354183
1.070899
-0.396154
-1.925862
-1.158101
0.068518
0.181620
1.238399
1.754318
-0.489646
-0.855802

0.222366
-0.108363
0.036738
0.136509
-0.256319
-0.267011
1.302753
-1.140448
0.562129
1.169839
-0.514697
-1.343151
0.199659

force
b3lyp
6-31G*
smd

$smx
solvent pentane
$end

12.2.9

Langevin Dipoles Model

Q-C HEM provides the option to calculate molecular properties in aqueous solution and the magnitudes of the hydration
free energies by the Langevin dipoles (LD) solvation model developed by Jan Florián and Arieh Warshel 35,36 at the
University of Southern California. In this model, a solute molecule is surrounded by a sphere of point dipoles, with
centers on a cubic lattice. Each of these “Langevin” dipoles changes its size and orientation in the electrostatic field of
the solute and the other Langevin dipoles. The electrostatic field from the solute is determined rigorously by the integration of its charge density, whereas for dipole–dipole interactions, a 12 Å cutoff is used. The Q-C HEM/C HEM S OL 1.0
implementation of the LD model is fully self-consistent in that the molecular quantum mechanical calculation takes
into account solute–solvent interactions. Further details on the implementation and parameterization of this model can
be found in the literature. 35,36
The results of C HEM S OL calculations are printed in the standard output file. Below is a part of the output for a calculation on the methoxide anion (corresponding to the sample input given later on, and the sample file in the $QC/samples
directory).
Energy Component
LD Electrostatic energy
Hydrophobic energy
van der Waals energy
Bulk correction
Solvation free energy, ∆G(ILD)

Value / kcal mol−1
−86.14
0.28
−1.95
−10.07
−97.87

Table 12.7: Results of the iterative Langevin Dipoles (ILD) solvation model, for aqueous methoxide.

Chapter 12: Molecules in Complex Environments

731

The total hydration free energy, ∆G(ILD) is calculated as a sum of several contributions. Note that the electrostatic part
of ∆G is calculated by using the linear-response approximation 35 and contains contributions from the polarization of
the solute charge distribution due to its interaction with the solvent. This results from the self-consistent implementation
of the Langevin dipoles model within Q-C HEM.
To perform an LD calculation in Q-C HEM, specify normal job-control variables for a Hartree-Fock or DFT calculation,
and set SOLVENT_METHOD = CHEM_SOL in the $rem section. Additional fine-tuning is accomplished using a set of
keywords in a $chem_sol input section. The remainder of this section summarizes these keywords.
EField
Determines how the solute charge distribution is approximated in evaluating the electrostatic field of the solute.
INPUT SECTION: $chem_sol
TYPE:
INTEGER
DEFAULT:
1
OPTIONS:
1 Exact solute charge distribution is used.
0 Solute charge distribution is approximated by Mulliken atomic charges.
RECOMMENDATION:
None. The Mulliken-based procedure is faster but less rigorous.
NGrids
Sets the number of grids used to calculate the average hydration free energy.
INPUT SECTION: $chem_sol
TYPE:
INTEGER
DEFAULT:
5 ∆Ghydr will be averaged over 5 different grids.
OPTIONS:
n Use n different grids.
RECOMMENDATION:
None. The maximum allowed value of n is 20.
Print
Controls printing in the C HEM S OL part of the Q-C HEM output file.
INPUT SECTION: $chem_sol
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
0 Limited printout
1 Full printout
RECOMMENDATION:
None.

Chapter 12: Molecules in Complex Environments

732

ReadRadii
Read user-defined atomic radii from section $van_der_waals.
INPUT SECTION: $chem_sol
TYPE:
None
DEFAULT:
Off
OPTIONS:
No options. Specify the keyword to use user-defined atomic radii.
RECOMMENDATION:
None.

Accurate calculations of hydration free energies require a judicious choice of the solute–solvent boundary in terms
of atom-type dependent parameters. The default atomic van der Waals radii available in Q-C HEM were chosen to
provide reasonable hydration free energies for most solutes and basis sets. These parameters basically coincide with
the C HEM S OL 2.0 radii given in Ref. 36. The only difference between the Q-C HEM and C HEM S OL 2.0 atomic radii
stems from the fact that Q-C HEM parameter set uses radii for carbon and oxygen that are independent of the atom’s
hybridization state. User-defined atomic radii can be specified by declaring the option ReadRadii in the $chem_sol
input section, and then placing the radii in the $van_der_waals section. Two different (and mutually exclusive) formats
can be used, as shown below.
$van_der_waals
1
atomic_number
...
$end

vdW_radius

$van_der_waals
2
sequential_atom_number
...
$end

vdW_radius

The purpose of the second format is to permit the user to customize the radius of specific atoms, in the order that they
appear in the $molecule section, rather than simply by atomic numbers as in format 1. The radii of atoms that are not
listed in the $van_der_waals input will be assigned default values. The atomic radii that were used in the calculation

Chapter 12: Molecules in Complex Environments

733

are printed in the C HEM S OL part of the output file in the column denoted rp. All radii should be given in Ångstroms.
Example 12.14 A Langevin dipoles calculation on the methoxide anion. A customized value is specified for the radius
of the C atom.
$molecule
-1 1
C
0.0000
O
0.0000
H
0.0000
H
0.8782
H -0.8782
$end

0.0000
0.0000
1.0140
-0.5070
-0.5070

$rem
METHOD
BASIS
SCF_CONVERGENCE
SOLVENT_METHOD
$end

-0.5274
0.7831
-1.0335
-1.0335
-1.0335

hf
6-31G
6
Chem_Sol

$chem_sol
ReadRadii
$end
$van_der_waals
2
1 2.5
$end

12.2.10

Poisson Boundary Conditions

Each of the implicit solvation models described above is designed for isotropic, bulk solvation—conditions that can
be qualitatively described using a scalar dielectric constant, ε. For an anisotropic environment, such as a liquid/vapor
interface, a more general approach is to solve Poisson’s equation with a spatially-varying dielectric function, ε(r).
Such an approach can be used, for example, to model the air/water interface by describing certain regions of space
using ε = 1 (air) and other regions using ε = 78 (water). 23,24 The atomistic region is described at the SCF level.
Construction of the dielectric function ε(r) is based on a solute cavity, somewhat analogous to the solute cavity in a
PCM calculation but where the dielectric changes smoothly (rather than abruptly) from ε = 1 inside the cavity (in the
atomistic QM region) to ε = εsolvent outside, in the continuum solvent. Options for modeling a liquid/vapor interface
are also available; see Section 12.2.10.3. In contrast to PCMs, which must approximate the “volume polarization” due
to penetration of the tails of the QM charge density beyond the solute cavity, this effect is described exactly under
Poisson boundary conditions. The price to be paid for lifting this approximation is that charge densities must be
discretized and integrated over three-dimensional space rather than simply a two-dimensional solute cavity surface.
Q-C HEM’s implementation of Poisson boundary conditions was introduced in Ref. 24 and refined in Ref. 23. Users of
this methodology are asked to cite at least Ref. 23.

12.2.10.1

Theory

The most general formulation of Poisson’s equation, for a dielectric function rather than a dielectric constant, is
h
i
ˆ · ε(r)∇ϕ
ˆ tot (r) = −4πρsol (r) .
∇
(12.13)
The solute’s charge density will be separated into nuclear and electronic components,
ρsol (r) = ρnuc (r) + ρelec (r) .

(12.14)

734

Chapter 12: Molecules in Complex Environments

The total electrostatic potential ϕtot is comprised of the solute’s electrostatic potential ϕelec + ϕnuc , which is obtained
from the density in Eq. (12.14), plus the polarization potential induced in the medium:
ϕtot (r) = ϕelec (r) + ϕnuc (r) + ϕpol (r) .

(12.15)

To compute ρelec , the electronic contribution to the electrostatic potential in vacuum is first evaluated on a Cartesian
grid,
NX
basis
D
E
1
gν (r) .
(12.16)
ϕelec (ri ) =
Pµν gµ (r)
|r − ri |
µν
Here P is the one-electron density matrix, gµ and gν are atom-centered Gaussian basis functions, and the integration is
over r. The electronic part of the solute’s charge density is then obtained from
ρelec (r) = −

1 ˆ2
∇ ϕelec (r) ,
4π

(12.17)

where in practice the Laplacian of ϕelec is evaluated using an eighth-order finite difference scheme. 23 The nuclear
charge density ρnuc is described classically,
ρnuc (r) = −

NX
atoms

ZA δ (r − RA ) ,

(12.18)

A

but to avoid numerical problems the nuclear charges are smeared out over one Cartesian grid voxel in practice. Operationally, this is achieved by adding ZA /dV to the grid point nearest RA , where dV is the volume of a Cartesian voxel.
The exact nuclear electrostatic potential in vacuum is computed according to
Natoms
ZA
1 X
.
ϕnuc (r) = −
4π
|r − RA |

(12.19)

A

The Dirac delta function in Eq. (12.18) has dimensions of (volume)−1 and therefore incorporates the aforementioned
factor of dV −1
Having computed the electronic and nuclear charge densities and the electrostatic potentials in vacuum, one has in
hand an exact solution of Eq. (12.13) for the case where ε(r) = 1. The induced polarization charge density and
corresponding electrostatic potential are then obtained iteratively, following closely the procedure outlined in Refs. 33
and 4. Equation (12.13) is rewritten to appear as Poisson’s equation in vacuum with an additional source charge density:
"
#
ˆ ln ε(r) · ∇ϕ
ˆ tot (r)
ρ
(r)
∇
sol
ˆ 2 ϕtot (r) = −4π
∇
+
ε(r)
4π
(12.20)


= −4π ρsol (r) + ρpol (r) ,
where the polarization charge density is

ρpol (r) = ρiter (r) +

1 − ε(r)
ε(r)


ρsol (r) .

(12.21)

The iterative charge density ρiter (r) takes the form
ˆ ln ε(r) · ∇ϕ
ˆ (k) (r)
∇
tot
(12.22)
4π
at iteration k + 1. This function is nonzero only in transition regions where the dielectric is being interpolated from
vacuum to solvent. Note also that the second term in Eq. (12.21) vanishes wherever ε(r) = 1. The quantity ρiter (r) is
(k)
(k+1)
(0)
iterated to convergence by performing updates ρiter (r) → ρiter (r) using Eq. (12.22), taking ϕtot (r) to be the solute’s
electrostatic potential in vacuum. Each time ρiter (r) is updated, a new polarization charge density is generated and
Eq. (12.20) is solved to obtain a new total electrostatic potential. To stabilize the iterative updates of ρiter (r), we use a
damping procedure

η ˆ
(k+1)
ˆ (k) (r) + (1 − η)ρ(k) (r)
ρiter (r) =
∇ ln ε(r) · ∇ϕ
tot
iter
4π
(12.23)
(k+1)
(k)
= ηρiter + (1 − η)ρiter (r)
(k+1)

ρiter (r) =

735

Chapter 12: Molecules in Complex Environments
rather than using Eq. (12.22) as written. Following Refs. 33 and 4, we use η = 0.6.

Once the total charge density and electrostatic potential are iterated to self-consistency, the free energy of solvation,
Z
1
Gpol =
dr ϕpol (r) ρsol (r) ,
(12.24)
2
is computed, where
ϕpol (r) = ϕtot (r) − ϕsol (r) .

(12.25)

Since ϕtot and ρtot are computed self-consistently at each SCF iteration, the polarization effects must be incorporated
into the Fock matrix. This is accomplished by adding the correction term ∆F̂ = δGpol /δρelec . The matrix form of this
correction in the atomic orbital basis is
Z
∆Fµν = dr ϕpol (r) gµ (r) gν (r) .
(12.26)
For additional details and a full description of the algorithm, see Ref. 23.

12.2.10.2

Nonequilibrium Solvation for Vertical Ionization

Nonequilibrium solvation within the framework of PCMs is discussed in Section 12.2.2.3, and that methodology 85,127
has been adapted for vertical ionization processes in conjunction with Poisson boundary conditions. 23,24 A state-specific
approach has been implemented that requires solution of a nonlinear Schrödinger equation, Eq. (12.6). For the reference
state, corresponding to i = 0 in Eq. (12.6), this is equivalent to solving Eq. (12.13) for the induced polarization potential
and computing the solvation free energy and Fock matrix correction according to Eqs. (12.24) and (12.26), respectively.
For the ionized state, denoted by a subscript i, the solvent polarization is partitioned into fast and slow components.
Within the Marcus partitioning scheme, 127 Eq. (12.20) becomes 23
−

1 ˆ2
fast
∇ ϕtot,i (r) = ρsol,i (r) + ρslow
pol,0 (r) + ρpol,i (r) ,
4π

(12.27)

where the fast polarization charge density of the ionized state, ρfast
pol,i , is given by
ρfast
pol,i (r)


= ρiter,i (r) +

1 − εopt (r)
εopt (r)





ρsol,i (r) + ρslow
pol,0 (r)

(12.28)

and the fast polarization potential, ϕfast
pol,i , is
slow
ϕfast
pol,i (r) = ϕtot,i (r) − ϕsol,i (r) − ϕpol,0 (r) .

(12.29)

Subscripts i and 0 refer to the ionized state and the reference state, respectively. The quantity εopt in Eq. (12.28) is the
solvent’s optical dielectric constant, and ρiter,i is computed using Eq. (12.23) but with a dielectric function constructed
using εopt and with ϕfast
tot,i substituted in place of ϕtot . For the Marcus partitioning scheme, the slow component of the
induced solvent polarization charge density is computed as the three-dimensional (volume charge rather than surface
charge) analogue of Eq. (12.8).

12.2.10.3

Job Control for the Poisson Equation Solver

A calculation using Poisson boundary conditions is requested by setting SOLVE_PEQ = TRUE in the $rem section. For
computational efficiency, solvent effects from the Poisson equation solver are not incorporated until the error in the
vacuum SCF calculation falls below a specified threshold, taken to be 10−3 a.u. by default and controllable using the
$rem variable PEQ_SWITCH.
Note: When SOLVE_PEQ = TRUE, the keywords SYM_IGNORE and NO_REORIENT are also set to TRUE. Poisson’s
equation is discretized on a three-dimensional grid and the energy need not be rotationally invariant unless the
grid spacing is sufficiently small.

Chapter 12: Molecules in Complex Environments

736

SOLVE_PEQ
Perform a solvation free energy calculation on a Cartesian grid using Poisson equation boundary
conditions.
TYPE:
STRING
DEFAULT:
False
OPTIONS:
TRUE/FALSE
RECOMMENDATION:
None.
PEQ_SWITCH
Inclusion of solvent effects begins when the SCF error falls below 10−PEQ_SWITCH .
TYPE:
INTEGER
DEFAULT:
3
OPTIONS:
n Corresponding to 10−n
RECOMMENDATION:
Use the default unless solvent effects need to be incorporated earlier in the SCF procedure.

Further job control for the Poisson equation solver (PEqS) routines is set up in two sections: $peqs_grid and $peqs.
The former is used to define the Cartesian discretization grid, as follows:
$peqs_grid
DimX   
DimY   
DimZ   
$end

The Cartesian grid must be orthorhombic and Nx, Ny, and Nz refer to the number of grid points for the x, y, and z
axes.
Note: A fourth-order multigrid algorithm is used to accelerate convergence of the conjugate gradient PEqS routine
that solves Eq. (12.20). 23 This requires that the number of grid points be odd, with the additional constraint
that the quantities Nx − 1, Ny − 1, and Nz − 1 must all be divisible by 8.
Values in the last two columns of the $peqs_grid section specify the minimum and maximum values of x, y, and
z, in units of Ångstroms. The length of the grid in the x direction is Lx = xmax − xmin and the grid spacing is
∆x = Lx /(Nx − 1). The volume element discussed in the context of Eq. (12.18) is dV = ∆x∆y∆z.
The $peqs section controls other aspects of a PEqS calculation, including construction of the solute cavity and other
required parameters for interfacial or nonequilibrium solvation. The format is:
$peqs

$end



Available keywords are described below.

Chapter 12: Molecules in Complex Environments
PEQMaxIter
Sets the maximum number of iterations used in the conjugate gradient solver.
INPUT SECTION: $peqs
TYPE:
INTEGER
DEFAULT:
500
OPTIONS:
User-defined.
RECOMMENDATION:
Use the default unless the calculation fails to converge.
PEQSolverThresh
The electrostatic potential is considered converged when the error falls below
10−PEQSolverThresh .
INPUT SECTION: $peqs
TYPE:
INTEGER
DEFAULT:
5
OPTIONS:
n Corresponding to 10−n
RECOMMENDATION:
Use the default unless a tighter convergence criterion is desired, at greater computational
cost.
PolarIterScale
Specifies the mixing parameter η that is used in Eq. (12.23) to stabilize iterative solution
for the polarization charge density.
INPUT SECTION: $peqs
TYPE:
FLOAT
DEFAULT:
0.6
OPTIONS:
η Desired value of the mixing parameter (unit-less).
RECOMMENDATION:
Use the default, which was tested in Refs. 33 and 4, unless the calculation proves difficult
to converge.
BatchSize
Evaluate electrostatic potential integrals in batches over small parts of the Cartesian grid,
rather than computing all integrals in a single batch.
INPUT SECTION: $peqs
TYPE:
INTEGER
DEFAULT:
5000
OPTIONS:
n Corresponding to a batch size of n grid points.
RECOMMENDATION:
For large grids (≥ 106 grid points), a batch size n ≈ Nx Ny Nz /5 is recommended.

737

738

Chapter 12: Molecules in Complex Environments
SolventDielectric
Sets the dielectric value for the solvent.
INPUT SECTION: $peqs
TYPE:
FLOAT
DEFAULT:
78.39
OPTIONS:
ε Desired value of ε (dimensionless).
RECOMMENDATION:
The default value corresponds to water at 25◦ C.
OpticalDielectric
Sets the optical dielectric value of the solvent.
INPUT SECTION: $peqs
TYPE:
FLOAT
DEFAULT:
1.7778
OPTIONS:
εopt Desired value of εopt (dimensionless).
RECOMMENDATION:
The optical dielectric is equal to the square of the index of refraction. The default value
corresponds to water at 25◦ C.
SoluteCavity
Specifies the type of solute cavity, which determines the form of the dielectric function
ε(r).
INPUT SECTION: $peqs
TYPE:
STRING
DEFAULT:
Vacuum
OPTIONS:
Vacuum
Perform calculation in vacuum, ε(r) ≡ 1.
RigidVDW Create a cavity using spherically symmetric error functions.
Spherical
Create a single spherical cavity around the solute.
RECOMMENDATION:
None.

In conventional PCMs the solute cavity is a rigid two-dimensional surface constructed from a union of atomic spheres,
the radii of which are generally take to be equal to the van der Waals radius (rvdW ) scaled by a factor of 1.2; see
Section 12.2.3. 42 This cavity is rigid in the sense that once the atomic coordinates are specified, it remains unchanged
during the SCF cycles. (The isodensity cavity discussed in Section 12.2.5 is an exception, but is not available for
the PEqS method.) Furthermore, there is an abrupt and discontinuous change in the dielectric constant at the solute/
continuum boundary. A three-dimensional analogue of this cavity, using continuous and differentiable sphericallysymmetric error functions, has been implemented for PEqS calculations, following the procedure in Ref. 33. The
dielectric function, which depends parameterically on the nuclear positions RA , is
(
)
atoms

 NY
ε(r; {RA }) = εsolvent − 1
h(dA , ∆; |r − RA |) ,
(12.30)
A

Chapter 12: Molecules in Complex Environments
where




|r − RA | − dA
1
1 + erf
.
h(dA , ∆; |r − RA |) =
2
∆

739

(12.31)

Equation (12.30) smoothly interpolates between ε = 1 and ε = εsolvent , over a length scale of ≈ 4∆. The value of ∆ is
specified using the RigidScale keyword. The quantity dA in Eqs. (12.30) and (12.31) sets radius of the atomic sphere
for atom A. By default, dA = 1.2 rvdW but this can be controlled as described below.
RigidScale
Sets the length scale on which the error function employed in the rigid vdW cavity construction interpolates the dielectric value from vacuum to solvent.
INPUT SECTION: $peqs
TYPE:
FLOAT
DEFAULT:
0.265 Å
OPTIONS:
∆ Specifies the desired value (in Å); see Eq. (12.31).
RECOMMENDATION:
Use the default value, which was tuned in Ref. 33 so that errors in small-molecule solvation energies were ≈ 1 kcal/mol.
VDWType
Specifies details for the rigid vdW cavity construction.
INPUT SECTION: $peqs
TYPE:
STRING
DEFAULT:
Scaled
OPTIONS:
Unscaled dA = rvdW
Scaled
dA = rvdW × scale
Shifted
dA = (rvdW + shif t) × scale
RECOMMENDATION:
None. The values of scale and shif t are set with the VDWScale and VDWShift keywords.
VDWScale
Sets the empirical scale factor applied to the atomic van der Waals radius.
INPUT SECTION: $peqs
TYPE:
FLOAT
DEFAULT:
1.2
OPTIONS:
scale Specifies the desired dimensionless scaling factor for the atomic vdW radii.
RECOMMENDATION:
Use the default value.

Chapter 12: Molecules in Complex Environments

740

VDWShift
Adjusts the center of the spherically-symmetric error functions when constructing the
rigid vdW cavity.
INPUT SECTION: $peqs
TYPE:
FLOAT
DEFAULT:
0.0 Ångstroms
OPTIONS:
shif t Specifies the desired shift (in Å).
RECOMMENDATION:
None. If VDWType is set to Shifted, the vdW scale factor is set to 1.0 by default. This
can be adjusted using the VDWScale keyword.

Setting the SoluteCavity keyword to Spherical requests the construction of a spherical cavity around the solute, and
the dielectric is smoothly interpolated from vacuum to solvent using a hyperbolic tangent function: 24
ε(r) =


o
1n
(εsolvent + 1) + (εsolvent − 1) tanh α(r − rmid ) .
2

(12.32)

Here, rmid is the distance where the dielectric assumes the value ε(rmid ) = (εsolvent + 1)/2. The value of rmid is taken
to be the sum of the sphere radius, R, and half of the interpolation length, L: rmid = R + L/2. The sphere radius
and interpolation length are controlled with the SphereRadius and InterpolLength keywords described below. The
parameter α controls the sharpness of the switching process, with α = 4/L by default.
SphereRadius
Sets the radius of the spherical solute cavity.
INPUT SECTION: $peqs
TYPE:
FLOAT
DEFAULT:
No default.
OPTIONS:
R Desired spherical cavity radius (in Å).
RECOMMENDATION:
None. See the Supporting Information of Ref. 24 for more information.
InterpolLength
Sets the length scale on which the dielectric is smoothly interpolated from vacuum to
solvent.
INPUT SECTION: $peqs
TYPE:
FLOAT
DEFAULT:
No default.
OPTIONS:
L Desired interpolation length (in Å).
RECOMMENDATION:
None.

Chapter 12: Molecules in Complex Environments

741

InterpolScale
For a given interpolation length (L), InterpolScale (α) sets the sharpness of the dielectric
transition from vacuum to solvent.
INPUT SECTION: $peqs
TYPE:
FLOAT
DEFAULT:
α = 4/L
OPTIONS:
α Desired interpolation scale factor (in Å−1 ).
RECOMMENDATION:
Use the default unless a broader (smaller α) or narrower (larger α) transition region is
desired.
Interface
Perform a solvation calculation at a solvent/vacuum interface.
INPUT SECTION: $peqs
TYPE:
STRING
DEFAULT:
False
OPTIONS:
True Modify the dielectric function to simulate an interface between solvent and vacuum.
False Perform a solvation calculation in bulk solvent.
RECOMMENDATION:
The user will also need to specify the length scale on which the dielectric is smoothly
interpolated from the bulk solvent value to vacuum, and the location of the Gibbs dividing
surface (see below).

By setting Interface = True, the dielectric function on the Cartesian grid is further modified to mimic a solvent/vacuum
interface. First, the solute cavity is constructed as specified by the SoluteCavity keyword as discussed above. Then,
the dielectric function is smoothly interpolated in the z direction across the Gibbs dividing surface (z ≡ zGDS ) using
the following hyperbolic tangent switching function: 23,24
o

1n
ε(z) =
(εsolvent + 1) + (1 − εsolvent ) tanh β(z − zGDS ) .
(12.33)
2
This interpolates the dielectric function over the interface length Linterface , centered at the Gibbs dividing surface,
z ≡ zGDS . Both of these values are controlled by the keywords InterfaceLength and GibbsDS, respectively. Similar
to the parameter α used in the spherical cavity construction, the parameter β = 4/Linterface controls the sharpness of
the interpolation across the Gibbs dividing surface.
InterfaceLength
Sets the length scale over which the dielectric function is smoothly transitioned from bulk
solvent to vacuum in the z direction
INPUT SECTION: $peqs
TYPE:
FLOAT
DEFAULT:
None
OPTIONS:
Linterface Desired interface length (in Å).
RECOMMENDATION:
This sets the value β = 4/Linterface in Eq. (12.33). See the Supporting Information of
Ref. 24 for a full description of the solvent/vacuum interface construction.

Chapter 12: Molecules in Complex Environments
GibbsDS
Sets the location of the Gibbs dividing surface, in the z direction.
INPUT SECTION: $peqs
TYPE:
FLOAT
DEFAULT:
None
OPTIONS:
zGDS Desired location of the Gibbs dividing surface (in Å).
RECOMMENDATION:
Consult the literature. One such way to determine this value is to compute a density profile
of the solvent as a function of z and set this location to be where the solvent density has
decreased to 50% of the bulk value. Usually zGDS ≈ Linterface /2.
NonequilJob
Obtain the nonequilibrium free energy of solvation for a vertical ionization process.
INPUT SECTION: $peqs
TYPE:
STRING
DEFAULT:
False
OPTIONS:
True Compute the nonequilibrium free energy for a vertical ionization process.
False Compute the equilibrium solvation free energy.
RECOMMENDATION:
None.
NonequilPartition
Specifies the manner in which the solvent response is partitioned into fast and slow components.
INPUT SECTION: $peqs
TYPE:
STRING
DEFAULT:
Marcus
OPTIONS:
Marcus Employ the Marcus partitioning scheme.
Pekar
Employ the Pekar partitioning scheme.
RECOMMENDATION:
Use the default. Although the fast and slow solvation responses are different between the
two approaches, the total solvation free energy is the same,127 but the Pekar scheme is
computationally more expensive than the Marcus scheme.

742

743

Chapter 12: Molecules in Complex Environments
NonequilState
Specifies the state of interest for a nonequilibrium vertical ionization.
INPUT SECTION: $peqs
TYPE:
STRING
DEFAULT:
Reference
OPTIONS:
Reference The reference (initial) state, from which an electron will be removed.
Ionized
The final (ionized) state.
RECOMMENDATION:
None. Both values will be needed in a compound input job, in order to compute the
nonequilibrium response to vertical ionization; see Example 17.

12.2.10.4

Examples

The following example computes the solvation free energy of a water molecule immersed in water. The Cartesian grid
is cubic with a side length of 15.0 Å and 73 grid points in each direction. A rigid vdW cavity is used based on scaled
vdW radii for the atomic spheres. The dielectric is not set, so takes the default value of 78.39.
Example 12.15 Free energy of solvation of water in water.
$rem
SCF_CONVERGENCE
THRESH
EXCHANGE
BASIS
SOLVE_PEQ
PEQ_SWITCH
$end
$molecule
0 1
O
0.053004
H
0.003424
H
-0.844842
$end

5
14
wB97X-V
6-31+G*
TRUE
0

-0.020947
0.185855
0.146674

-0.034784
0.910594
-0.358413

$peqs
SOLUTECAVITY RIGIDVDW
$end
$peqs_grid
DimX 73 -7.50 7.50
DimY 73 -7.50 7.50
DimZ 73 -7.50 7.50
$end

The next example illustrates calculation of the solvation free energy of a water molecule at a water/vacuum interface,
with the Gibbs dividing surface placed at z = 0.50 Å. The length of the interface is set to Linterface = 2.75 Å and the
dielectric is interpolated from bulk solvent to vacuum in the positive z direction across the Gibbs dividing surface. The

744

Chapter 12: Molecules in Complex Environments
Cartesian grid, solute cavity, and solvent dielectric are the same as in the previous example.
Example 12.16 Free energy of solvation of water at a water/vacuum interface.
$rem
SCF_CONVERGENCE
THRESH
EXCHANGE
BASIS
SOLVE_PEQ
PEQ_SWITCH
$end
$molecule
0 1
O
0.053004
H
0.003424
H
-0.844842
$end

5
14
wB97X-V
6-31+G*
TRUE
0

-0.020947
0.185855
0.146674

-0.034784
0.910594
-0.358413

$peqs
SOLUTECAVITY RIGIDVDW
INTERFACE TRUE
INTERFACELENGTH 2.75
GIBBSDS 0.50
INTERFACEDIRECTION POSITIVE
$end
$peqs_grid
DimX 73 -7.50 7.50
DimY 73 -7.50 7.50
DimZ 73 -7.50 7.50
$end

The final example illustrates a nonequilibrium (NonEquilJob = True) solvation calculation for the vertical ionization
of H2 O− in bulk water. This is a compound job that first calculates the equilibrium solvation free energy of the
anionic state (NonEquilState = Reference), then computes the nonequilibrium energy correction for the ionized state
(NonEquilState = Ionized). The two jobs are separated by “@@@”. The Cartesian grid and solvent dielectric are the
same as the previous examples, but the solute cavity is chosen to be spherical with a radius of 2.0 Å. The Marcus
scheme (NonEquilPartition = Marcus) is used to partition the solvent response into fast and slow components. Since
this is the default method, the NonEquilPartition keyword is omitted.
Example 12.17 Nonequilibrium free energy of solvation for the vertical ionization of H2 O−

745

Chapter 12: Molecules in Complex Environments
$rem
SCF_CONVERGENCE
THRESH
EXCHANGE
BASIS
SOLVE_PEQ
PEQ_SWITCH
$end
$molecule
-1 2
O
0.053004
H
0.003424
H
-0.844842
$end

5
14
HF
6-31++G*
TRUE
0

-0.020947
0.185855
0.146674

-0.034784
0.910594
-0.358413

$peqs
SOLUTECAVITY SPHERICAL
SPHERERADIUS 2.00
NONEQUILJOB TRUE
NONEQUILSTATE REFERENCE
$end
$peqs_grid
DimX 73 -7.50 7.50
DimY 73 -7.50 7.50
DimZ 73 -7.50 7.50
$end
@@@
$rem
SCF_CONVERGENCE
THRESH
EXCHANGE
BASIS
SOLVE_PEQ
PEQ_SWITCH
$end
$molecule
0 1
O
0.053004
H
0.003424
H
-0.844842
$end

5
14
HF
6-31++G*
TRUE
0

-0.020947
0.185855
0.146674

$peqs
SOLUTECAVITY SPHERICAL
SPHERERADIUS 2.00
NONEQUILJOB TRUE
NONEQUILSTATE IONIZED
$end
$peqs_grid
DimX 73 -7.50 7.50
DimY 73 -7.50 7.50
DimZ 73 -7.50 7.50
$end

-0.034784
0.910594
-0.358413

Chapter 12: Molecules in Complex Environments

12.3

746

Stand-Alone QM/MM Calculations

Q-C HEM can perform hybrid quantum mechanics/molecular mechanics (QM/MM) calculations either as a stand-alone
program, which is described in this section, or in conjunction with the C HARMM package. 47 See Section 12.4 for a
description of a latter approach.

12.3.1

Available QM/MM Methods and Features

Three modes of operation are available:
• MM calculations only (no QM)
• QM/MM calculations using a two-layer ONIOM model with mechanical embedding
• QM/MM calculations using the Janus model for electronic embedding
Q-C HEM can carry out purely MM calculations, wherein the entire molecular system is described by a MM force
field and no electronic structure calculation is performed. The MM force fields available at present are AMBER, 119
CHARMM, 37 and OPLSAA. 52
As implemented in Q-C HEM, the ONIOM model 116 is a mechanical embedding scheme that partitions a molecular
system into two subsystems (layers): an MM subsystem and a QM subsystem. The total energy of an ONIOM system
is given by
QM
MM
MM
Etotal = Etotal
− EQM
+ EQM
(12.34)
MM
MM
is the MM energy of the QM
where Etotal
is the MM energy of the total system (i.e., QM + MM subsystems), EQM
QM
subsystem, and EQM is the QM energy of the QM subsystem. MM energies are computed via a specified MM force
field, and QM energies are computed via a specified electronic structure calculation.

The advantage of the ONIOM model is its simplicity, which allows for straightforward application to a wide variety
of systems. A disadvantage of this approach, however, is that QM subsystem does not interact directly with the MM
MM
subsystem. Instead, such interactions are incorporated indirectly, in the Etotal
contribution to the total energy. As a
result, the QM electron density is not polarized by the electrostatic charges of the MM subsystem.
If the QM/MM interface partitions the two subsystems across a chemical bond, a link atom (hydrogen) must be introduced to act as a cap for the QM subsystem. Currently, Q-C HEM supports only carbon link atoms, of atom type 26, 35,
and 47 in the CHARMM27 force field.
The Janus model 102 is an electronic embedding scheme that also partitions the system into MM and QM subsystems,
but is more versatile than the ONIOM model. The Janus model in Q-C HEM is based upon the “YinYang atom” model
of Shao and Kong. 103 In this approach, the total energy of the system is simply the sum of the subsystem energies,
Etotal = EMM + EQM

(12.35)

The MM subsystem energy, EMM , includes van der Waals interactions between QM and MM atoms but not QM/
MM Coulomb interactions. Rather, EQM includes the direct Coulomb potential between QM atoms and MM atoms
as external charges during the QM calculation, thus allowing the QM electron density to be polarized by the MM
atoms. Because of this, Janus is particularly well suited (as compared to ONIOM) for carrying out excited-state QM/
MM calculations, for excited states of a QM model system embedded within the electrostatic environment of the MM
system. Within a Janus calculation, Q-C HEM first computes EMM with the specified force field and then computes
EQM with the specified electronic structure theory.
When the Janus QM/MM partition cuts across a chemical bond, a YinYang atom 103 is automatically introduced by
Q-C HEM. This atom acts as a hydrogen cap in the QM calculation, yet also participates in MM interactions. To retain
charge neutrality of the total system, the YinYang atom has a single electron and a modified nuclear charge in the QM

747

Chapter 12: Molecules in Complex Environments

calculation, equal to qnuclear = 1 + qMM (i.e., the charge of a proton plus the charge on the YinYang atom in the MM
subsystem).
Because this modified charge will affect the bond containing the YinYang atom, an additional repulsive Coulomb
potential is applied between the YinYang atom and its connecting QM atom to maintain a desirable bond length. The
additional repulsive Coulomb energy is added to EMM . The YinYang atom can be an atom of any kind, but it is highly
recommended to use carbon atoms as YinYang atoms.
Q-C HEM’s stand-alone QM/MM capabilities also include the following features:
• Analytic QM/MM gradients are available for QM subsystems described with density functional theory (DFT)
or Hartree-Fock (HF) electronic structure theory, allowing for geometry optimizations and QM/MM molecular
dynamics.
• Single-point QM/MM energy evaluations are available for QM subsystems described with most post-HF correlated wave functions.
• Single-point QM/MM calculations are available for excited states of the QM subsystem, where the latter may
be described using CIS, TDDFT, or correlated wave function models. Analytic gradients for excited states are
available for QM/MM calculations if the QM subsystem is described using CIS.
• Single-point MM or QM/MM energy evaluations and analytic gradients are available using periodic boundary
conditions with Ewald summation.
• Implicit solvation for both Janus QM/MM calculations as well as MM-only calculations is available using the
Polarizable Continuum Models (PCMs) discussed in Section 12.2.2.
• Gaussian blurring of MM external charges is available for Janus QM/MM calculations.
• User-defined MM atoms types, MM parameters, and force fields.

12.3.2

Using the Stand-Alone QM/MM Features

12.3.2.1 $molecule section
To perform QM/MM calculations, the user must assign MM atom types for each atom in the $molecule section. The
format for this specification is modeled upon that used by the T INKER molecular modeling package, 97 although the
T INKER program is not required to perform QM/MM calculations using Q-C HEM. Force field parameters and MM
atom type numbers used within Q-C HEM are identical to those used T INKER for the AMBER99, CHARMM27, and
OPLSAA force fields, and the format of the force field parameters files is also the same.
The $molecule section must use Cartesian coordinates to define the molecular geometry for internal QM/MM calculations; the Z-matrix format is not valid. MM atom types are specified in the $molecule section immediately after the
Cartesian coordinates on a line so that the general format for the $molecule section is
$molecule
 
    
. . .
$end

For example, one can define a TIP3P water molecule using AMBER99 atom types, as follows:
$molecule
0 1
O
H

-0.790909
-1.628044

1.149780
1.245320

0.907453
1.376372

2001
2002

748

Chapter 12: Molecules in Complex Environments
H
$end

-0.669346

1.913705

0.331002

2002

When the input is specified as above, Q-C HEM will determine the MM bond connectivity based on the distances
between atoms; if two atoms are sufficiently close, they are considered to be bonded. Occasionally this approach can
lead to problems when non-bonded atoms are in close proximity of one another, in which case Q-C HEM might classify
them as bonded regardless of whether the appropriate MM bond parameters are available. To avoid such a scenario,
the user can specify the bonds explicitly by setting the $rem variable USER_CONNECT = TRUE, in which case the
$molecule section must have the following format
$molecule
 
        
. . .
$end

Each  is the index of an atom to which  is bonded. Four bonds must be specified for each atom,
even if that atom is connected to fewer than four other atoms. (For non-existent bonds, use zero as a placeholder.)
Currently, Q-C HEM supports no more than four MM bonds per atom.
After setting USER_CONNECT = TRUE, a TIP3P water molecule in the AMBER99 force field could be specified as
follows:
$molecule
0 1
O
H
H
$end

-0.790909
-1.628044
-0.669346

1.149780
1.245320
1.913705

0.907453
1.376372
0.331002

2001
2002
2002

2 3 0 0
1 0 0 0
1 0 0 0

Explicitly defining the bonds in this way is highly recommended.

12.3.2.2 $force_field_params section
In many cases, all atoms types (within both the QM and MM subsystems) will be defined by a given force field. In
certain cases, however, a particular atom type may not be defined in a given force field. For example, a QM/MM
calculation on the propoxide anion might consist of a QM subsystem containing an alkoxide functional group, for
which MM parameters do not exist. Even though the alkoxide moiety is described using quantum mechanics, van der
Waals parameters are nominally required for atoms within the QM subsystem, which interact with the MM atoms via
Lennard-Jones-type interactions.
In such cases, there are four possible options, the choice of which is left to the user’s discretion:
1. Use a similar MM atom type as a substitute for the missing atom type.
2. Ignore the interactions associated with the missing atom type.
3. Define a new MM atom type and associated parameters.
4. Define a new force field.
These options should be applied with care. Option 1 involves selecting an atom type that closely resembles the undefined MM atom. For example, the oxygen atom of an alkoxide moiety could perhaps use the MM atom type corresponding to the oxygen atom of a neutral hydroxyl group. Alternatively, the atom type could be ignored altogether
(option 2) by specifying MM atom type 0 (zero). Setting the atom type to zero should be accompanied with setting all

749

Chapter 12: Molecules in Complex Environments

four explicit bond connections to placeholders if USER_CONNECT = TRUE. An atom type of zero will cause all MM
energies involving that atom to be zero.
The third option in the list above requires the user to specify a $force_field_params section in the Q-C HEM input file.
This input section can be used to add new MM atom type definitions to one of Q-C HEM’s built-in force fields. At a
minimum, the user must specify the atomic charge and two Lennard-Jones parameters (radius and well depth, ), for
each new MM atom type. Bond, angle, and torsion parameters for stretches, bends, and torsions involving the new
atom type may also be specified, if desired. The format for the $force_field_params input section is
$force_field_params
NumAtomTypes 
AtomType -1   
AtomType -2   
. . .
AtomType -n   
Bond    
. . .
Angle     
. . .
Torsion       
. . .
$end

The first line in this input section specifies how many new MM atom types appear in this section (). These are
specified on the following lines labeled with the AtomType tag. The atom type numbers are required to be negative
and to appear in the order −1, −2, −3, . . . , −n. The $molecule section for a water molecule, with user-defined MM
parameters for both oxygen and hydrogen, might appear as follows:
$molecule
0 1
O
H
H
$end

-0.790909
-1.628044
-0.669346

1.149780
1.245320
1.913705

0.907453
1.376372
0.331002

-1
-2
-2

2 3 0 0
1 0 0 0
1 0 0 0

The remainder of each AtomType line in the $force_field_params section consists of a charge (in elementary charge
units), a Lennard-Jones radius (in Å), and a Lennard-Jones well depth (, in kcal/mol).
Each (optional) Bond line in the $force_field_params section defines bond-stretching parameters for a bond that contains a new MM atom type. The bond may consist of both atoms  and  defined an AtomType line, or else
 may be defined with an AtomType line and  defined as a regular atom type for the force field. In the latter
case, the label for  should be the number of its general van der Waals type. For example, the atom type for a TIP3P
oxygen in AMBER99 is 2001, but its van der Waals type is 21, so the latter would be specified in the Bond line. The
remaining entries of each Bond line are the harmonic force constant, in kcal/mol/Å2 , and the equilibrium distance, in
Å.
Similar to the Bond lines, each (optional) Angle line consists of one or more new atom types along with existing van
der Waals types. The central atom of the angle is . The harmonic force constant (in units of kcal/mol/degree) and
equilibrium bond angle (in degrees) are the final entries in each Bond line.
Each (optional) Torsion line consists of one or more new MM atom types along with regular van der Waals types.
The connectivity of the four atoms that constitute the dihedral angle is –––, and the torsional potential
energy function is
Etorsion (θ) = ktorsion [1 + cos(mθ − φ)]
(12.36)
The force constant (ktorsion ) is specified in kcal/mol and the phase angle (φ) in degrees. The multiplicity (m) is an
integer.

Chapter 12: Molecules in Complex Environments
12.3.2.3

750

User-Defined Force Fields

Option 4 in the list on page 748 is the most versatile, and allows the user to define a completely new force field. This
option is selected by setting FORCE_FIELD = READ, which tells Q-C HEM to read force field parameters from a text
file whose name is specified in the $force_field_params section as follows:
$force_field_params
Filename 
$end

Here,  is the full (absolute) path and name of the file that Q-C HEM will attempt to read for the
MM force field. For example, if the user has a file named MyForceField.prm that resides in the path /Users/me/parameters
then this would be specified as
$force_field_params
Filename /Users/me/parameters/MyForceField.prm
$end

Within the force field file, the user should first declare various rules that the force field will use, including how van der
Waals interactions will be treated, scaling of certain interactions, and the type of improper torsion potential. The rules
are declared in the file as follows:
RadiusRule 
EpsilonRule 
RadiusSize 
ImptorType 
vdw-14-scale 
chg-14-scale 
torsion-scale 

Currently, only a Lennard-Jones potential is available for van der Waals interactions. RadiusRule and EpsilonRule
control how to average σ and , respectively, between atoms A and B in their Lennard-Jones potential. The options available for both of these rules are Arithmetic [e.g., σAB = (σA + σB )/2] or Geometric [e.g., σAB =
(σA σB )1/2 ]. RadiusSize has options Radius or Diameter, which specify whether the parameter σ is the van
der Waals radius or diameter in the Lennard-Jones potential.
ImptorType controls the type of potential to be used for improper torsion (out-of-plane bending) energies, and has
two options: Trigonometric or Harmonic. These options are described in more detail below.
The scaling rules takes a floating point argument . The vdw-14-scale and chg-14-scale rules only affect
van der Waals and Coulomb interactions, respectively, between atoms that are separated by three consecutive bonds
(atoms 1 and 4 in the chain of bonds). These interaction energies will be scaled by . Similarly, torsion-scale
scales dihedral angle torsion energies.
After declaring the force field rules, the number of MM atom types and van der Waals types in the force field must be
specified using:
NAtom 
Nvdw 

where  is a positive integer.
Next, the atom types, van der Waals types, bonds, angles, dihedral angle torsion, improper torsions, and Urey-Bradley
parameters can be declared in the following format:
Atom 1   
Atom 2   
. . .

751

Chapter 12: Molecules in Complex Environments
Atom    
. . .
vdw 1   
vdw 2   
. . .
vdw    
. . .
Bond    
. . .
Angle     
. . .
Torsion       
. . .
Improper       
. . .
UreyBrad     

The parameters provided in the force field parameter file correspond to a basic MM energy functional of the form
EMM = ECoul + EvdW + Ebond + Eangle + Etorsion + Eimptor + EUreyBrad

(12.37)

Coulomb and van der Waals interactions are computed for all non-bonded pairs of atoms that are at least three consecutive bonds apart (i.e., 1–4 pairs and more distant pairs). The Coulomb energy between atom types 1 and 2 is
simply
q1 q2
ECoul = fscale
(12.38)
r12
where q1 and q2 are the respective charges on the atoms (specified with  in elementary charge units) and
r12 is the distance between the two atoms. For 1–4 pairs, fscale is defined with chg-14-scale but is unity for all
other valid pairs. The van der Waals energy between two atoms with van der Waals types a and b, and separated by
distance rab , is given by a “6-12” Lennard-Jones potential:
"
12

6 #
σab
σab
EvdW (rab ) = fscale ab
−2
(12.39)
rab
rab
Here, fscale is the scaling factor for 1–4 interactions defined with vdw-14-scale and is unity for other valid interactions. The quantities ab and σab are the averages of the parameters of atoms a and b as defined with EpsilonRule
and RadiusRule, respectively (see above). The units of  are Å , and the units of  are kcal/mol.
Hereafter, we refer to atoms’ van der Waals types with a, b, c, ... and atoms’ charges with 1,2, 3, ....
The bond energy is a harmonic potential,
Ebond (rab ) = kbond (rab − req )2

(12.40)

where kbond is provided by  in kcal/mol/Å2 and req by  in Å.
Note that  and  in the Bond definition correspond to the van der Waals type indices from the vdw definitions,
not the Atom indices.
The bending potential between two adjacent bonds connecting three different atoms (--) is also taken to
be harmonic,
Eangle (θabc ) = kangle (θabc − θeq )2
(12.41)
Here, kangle is provided by  in kcal/mol/degrees and θeq by  in
degrees. Again, , , and  correspond to van der Waals types defined with vdw.
The energy dependence of the --- dihedral torsion angle, where , , , and  are van
der Waals types, is defined by
X
Etorsion (θabcd ) = fscale
kabcd [1 + cos(mθabcd − φ)]
(12.42)
m

752

Chapter 12: Molecules in Complex Environments

Here, fscale is the scaling factor defined by torsion-scale. The force constant kabcd is defined with  in kcal/mol, and the phase angle φ is defined with  in degrees. The summation is
over multiplicities, m, and Q-C HEM supports up to three different values of m per dihedral angle. The force constants
and phase angles may depend on m, so if more than one multiplicity is used, then    should be specified for each multiplicity. For example, to specify a dihedral torsion
between van der Waals types 2–1–1–2, with multiplicities m = 2 and m = 3, we might have:
Torsion

2

1

1

2

2.500

180.0

2 1.500 60.0 3

Improper torsion angle energies for four atoms ---, where  is the central atom, can be computed
in one of two ways, as controlled by the ImptorType rule. If ImptorType is set to Trigonometric, then the
improper torsion energy has a functional form similar to that used for dihedral angle torsions:
Eimptor (θabcd ) =

kabcd
[1 + cos(m θabcd − φ)]
Nequiv

(12.43)

Here, θabcd is the out-of-plane angle of atom , in degrees, and kabcd is the force constant defined with , in kcal/mol. The phase φ and multiplicity m need to be specified in the Improper declaration, although
the definition of an improper torsion suggests that these values should be set to φ = 0 and m = 2. The quantity Nequiv
accounts for the number of equivalent permutations of atoms , , and , so that the improper torsion angle is
only computed once. If ImptorType is set to Harmonic, then in place of Eq. (12.43), the following energy function
is used:
kabcd 2
Eimptor (θabcd ) =
θ
(12.44)
Nequiv abcd
The Urey-Bradley energy, which accounts for a non-bonded interaction between atoms  and  that are separated
by two bonds (i.e., a 1-3 interaction through --), is given by
EUreyBrad (rac ) = kabc (rac − req )2

(12.45)

The distance in Å between atoms  and  is rac , the equilibrium distance req is provided by  in Å, and the force constant kabc is provided by  in kcal/mol/Å2 .
A short example of a valid text-only file defining a force field for a flexible TIP3P water could be as follows:
//-- Force Field Example --//
// -- Rules -- //
RadiusRule Geometric
RadiusSize Radius
EpsilonRule Geometric
ImptorType Trigonometric
vdw-14-scale 1.0
chg-14-scale 0.8
torsion-scale 0.5
// -- Number of atoms and vdw to expect -- //
NAtom 2
Nvdw 2
// -- Atoms -- //
Atom 1 -0.8340 2
Atom 2 0.4170 1

TIP3P Oxygen
TIP3P Hydrogen

// -- vdw -- //
vdw 1 0.0000 0.0000 H parameters
vdw 2 1.7682 0.1521 O parameters

Chapter 12: Molecules in Complex Environments

753

// -- Bond -- //
Bond 1 2 553.0 0.9572
// -- Angle -- //
Angle 1 2 1 100.0

104.52

Lines that do not begin with one of the keywords will be ignored, and have been used here as comments.

12.3.2.4 $qm_atoms and $forceman sections
For QM/MM calculations (but not for purely MM calculations) the user must specify the QM subsystem using a
$qm_atoms input section, which assumes the following format:
$qm_atoms
  . . .
. . .

$end

Multiple indices can appear on a single line and the input can be split across multiple lines. Each index is an integer
corresponding to one of the atoms in the $molecule section, beginning at 1 for the first atom in the $molecule section.
Link atoms for the ONIOM model and YinYang atoms for the Janus model are not specified in the $qm_atoms section,
as these are inserted automatically whenever a bond connects a QM atom and an MM atom.
Q-Chem 4.2.2 and later versions also support, for example
$qm_atoms
18:31 35
$end

which specifies 15 QM atoms (atoms 18 through 31; atom 35).
For Janus QM/MM calculations, there are several ways of dealing with van der Waals interactions between the QM and
MM atoms. By default, van der Waals interactions are computed for all QM–MM and MM–MM atom pairs but not for
QM–QM atom pairs. In some cases, the user may prefer not to neglect the van der Waals interactions between QM–QM
atoms, or the user may prefer to neglect any van der Waals interaction that involves a QM atom. Q-C HEM allows the
user this control via two options in the $forceman section. To turn on QM–QM atom van der Waals interactions, the
user should include the following in their input:
$forceman
QM-QMvdw
$end

Similarly, to turn off all van der Waals interactions with QM atoms, the following should be included:
$forceman
NoQM-QMorQM-MMvdw
$end

12.3.2.5

Periodic Boundary Conditions

Periodic boundary conditions (using Ewald summation for the long-range Coulomb interactions) can be used in conjunction with both MM-only calculations and QM/MM calculations. The approach is based off of the work of Nam
et al. 88 and (independently) Riccardi et al., 98 as implemented in both the A MBER 118 and C HARMM 12,98 programs.

754

Chapter 12: Molecules in Complex Environments

These approaches use Mulliken charges to represent the periodic images of the wave function, and while suitable for
semi-empirical calculations with minimal basis sets, instabilities in the Mulliken charges for extended basis sets lead
to SCF convergence failure in the QM/MM-Ewald calculations. 44 The implementation in Q-C HEM thus allows for the
use of ChElPG charges to represent the image wave functions, affording an algorithm that is stable in extended basis
sets. 43,44
The efficiency of the Ewald summation is governed by the parameter, α, that controls the partition of the Coulomb
potential into short- and long-range components, and in the QM/MM-Ewald method there are separate values of α for
the QM and MM portions of the calculations. Improper selection of αMM and/or αQM can greatly increase the computational time, and the choices that are optimal for MM calculations need not be optimal for QM/MM calculations. 44
The cost of the MM Ewald summation scales as O(Nrecip Natoms ), where Nrecip is the number of reciprocal-space
lattice vectors that is used for the k-space sum. The QM portion of the calculation scales as O(Nrecip NQM Natoms ),
where NQM is the number of QM atoms (whereas Natoms = NQM + NMM is the total number of atoms). The MM
Ewald parameter is thus selected to minimize the amount of work that is done in real space. The optimal value, which
−1
is typically αMM ≈ 0.5 Å , can be found by solving the equation 44

where

αMM = 2C/L

(12.46)

h

i1/2
C = −ln 10−SCF_CONVERGENCE

(12.47)

and L is the length of the simulation cell. (Only cubic simulation cells are available at present.) In contrast, the
parameter αQM should be selected to minimize the total number of vectors in both real and reciprocal space. The
optimal value, which is often αQM ≈ 0.1 Å

−1

, is determined by solving the equation 44

3
2CL3 αQM

π 3/2

+

2
αQM
L2

π 1/2

− αQM L − 2C = 0 .

(12.48)

To perform an MM- or QM/MM-Ewald job, one must set MM_SUBTRACTIVE = TRUE and EWALD_ON = TRUE in the
$rem section, but otherwise job control is largely done through the $forceman section. The following variables must be
set for every type of Ewald calculation.
• The keyword Ewald will turn on Ewald summation.
• The keyword alpha should be followed by a value for the MM Ewald parameter and then the QM Ewald parameter. (The latter must be set even for MM-only jobs.)
• Box_length specifies the side length of the cubic simulation cell, in Å.
The following parameters are optional for further job control.
• Dielectric specifies a dielectric constant for the surrounding medium, which appears in the “dipole term” of
Ewald summation (Edipole in Ref. 44). If no value is set, the dielectric constant is set to infinity, corresponding
to “tin foil" boundary conditions.
• The keyword Ewald_SCF_thresh_on, followed by a real number, causes Q-C HEM to wait until the DIIS error falls below the specified value before adding the Ewald correction to the Fock matrix, thus obviating the
sometimes-costly Ewald correction in early SCF cycles. (The default value is 1.0, which turns on Ewald summation immediately in most cases)
A short example of a $forceman section using Ewald summation could be as follows:
$forceman
ewald
alpha
box_length

0.35
15.00

0.1

Chapter 12: Molecules in Complex Environments
dielectric
mm_read_scratch
ewald_scf_thresh_on
$end

12.3.3

755

88.0
0.0001

Additional Job Control Variables

A QM/MM job is requested by setting the $rem variables QM_MM_INTERFACE and FORCE_FIELD. Also required are
a $qm_atoms input section and appropriate modifications to the $molecule section, as described above. Additional job
control variables are detailed here.
QM_MM_INTERFACE
Enables internal QM/MM calculations.
TYPE:
STRING
DEFAULT:
NONE
OPTIONS:
MM
Molecular mechanics calculation (i.e., no QM region)
ONIOM QM/MM calculation using two-layer mechanical embedding
JANUS
QM/MM calculation using electronic embedding
RECOMMENDATION:
The ONIOM model and Janus models are described above. Choosing MM leads to no electronic
structure calculation. However, when using MM, one still needs to define the $rem variables
BASIS and EXCHANGE in order for Q-C HEM to proceed smoothly.
FORCE_FIELD
Specifies the force field for MM energies in QM/MM calculations.
TYPE:
STRING
DEFAULT:
NONE
OPTIONS:
AMBER99
AMBER99 force field
CHARMM27 CHARMM27 force field
OPLSAA
OPLSAA force field
RECOMMENDATION:
None.
CHARGE_CHARGE_REPULSION
The repulsive Coulomb interaction parameter for YinYang atoms.
TYPE:
INTEGER
DEFAULT:
550
OPTIONS:
n Use Q = n × 10−3
RECOMMENDATION:
The repulsive Coulomb potential maintains bond lengths involving YinYang atoms with the potential V (r) = Q/r. The default is parameterized for carbon atoms.

Chapter 12: Molecules in Complex Environments
GAUSSIAN_BLUR
Enables the use of Gaussian-delocalized external charges in a QM/MM calculation.
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
TRUE
Delocalizes external charges with Gaussian functions.
FALSE Point charges
RECOMMENDATION:
None
GAUSS_BLUR_WIDTH
Delocalization width for external MM Gaussian charges in a Janus calculations.
TYPE:
INTEGER
DEFAULT:
NONE
OPTIONS:
n Use a width of n × 10−4 Å.
RECOMMENDATION:
Blur all MM external charges in a QM/MM calculation with the specified width. Gaussian blurring is currently incompatible with PCM calculations. Values of 1.0–2.0 Å are recommended in
Ref. 31.
MODEL_SYSTEM_CHARGE
Specifies the QM subsystem charge if different from the $molecule section.
TYPE:
INTEGER
DEFAULT:
NONE
OPTIONS:
n The charge of the QM subsystem.
RECOMMENDATION:
This option only needs to be used if the QM subsystem (model system) has a charge that is
different from the total system charge.
MODEL_SYSTEM_MULT
Specifies the QM subsystem multiplicity if different from the $molecule section.
TYPE:
INTEGER
DEFAULT:
NONE
OPTIONS:
n The multiplicity of the QM subsystem.
RECOMMENDATION:
This option only needs to be used if the QM subsystem (model system) has a multiplicity that is
different from the total system multiplicity. ONIOM calculations must be closed shell.

756

Chapter 12: Molecules in Complex Environments

757

USER_CONNECT
Enables explicitly defined bonds.
TYPE:
STRING
DEFAULT:
FALSE
OPTIONS:
TRUE
Bond connectivity is read from the $molecule section
FALSE Bond connectivity is determined by atom proximity
RECOMMENDATION:
Set to TRUE if bond connectivity is known, in which case this connectivity must be specified in
the $molecule section. This greatly accelerates MM calculations.
MM_SUBTRACTIVE
Specifies whether a subtractive scheme is used in the ECoul , Eq. (12.38), portion of the calculation.
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
FALSE Only pairs that are not 1-2, 1-3, or 1-4 pairs are used.
TRUE
All pairs are calculated, and then the pairs that are double counted (1-2, 1-3, and 1-4) are subtracted out.
RECOMMENDATION:
When running QM/MM or MM calculations there is not recommendation. When running a QM/
MM Ewald calculation the value must be set to TRUE.

12.3.4

QM/MM Examples

• QM/MM Example 1
Features of this job:
– Geometry optimization using ONIOM mechanical embedding.
– MM region (water 1) described using OPLSAA.
– QM region (water 2) described using PBE0/6-31G*.
– $molecule input section contains user-defined MM bonds. A zero is used as a placeholder if there are no
more connections.

758

Chapter 12: Molecules in Complex Environments

Example 12.18 ONIOM optimization of water dimer.
$rem
METHOD
BASIS
QM_MM_INTERFACE
FORCE_FIELD
USER_CONNECT
JOBTYPE
MOLDEN_FORMAT
$end

pbe0
6-31G*
oniom
oplsaa
true
opt
true

$qm_atoms
4 5 6
$end
$molecule
0 1
O
-0.790909
H
-1.628044
H
-0.669346
O
1.178001
H
0.870001
H
0.472696
$end

1.149780
1.245320
1.913705
-0.686227
-1.337091
-0.008397

0.907453
1.376372
0.331002
0.841306
1.468215
0.851892

186
187
187
186
187
187

2
1
1
5
4
4

3
0
0
6
0
0

0
0
0
0
0
0

0
0
0
0
0
0

• QM/MM Example 2
Features of this job:
– Janus electronic embedding with a YingYang link atom (the glycosidic carbon at the C10 position of the
deoxyribose).
– MM region (deoxyribose) is described using AMBER99.
– QM region (adenine) is described using HF/6-31G*.
– The first 5 electronically excited states are computed with CIS. MM energy interactions between a QM
atom and an MM atom (e.g., van der Waals interactions, as well as angles involving a single QM atom) are
assumed to be the same in the excited states as in the ground state.
– $molecule input section contains user-defined MM bonds.
– Gaussian-blurred charges are used on all MM atoms, with a width set to 1.5 Å.

759

Chapter 12: Molecules in Complex Environments

Example 12.19 Excited-state single-point QM/MM calculation on deoxyadenosine.
$rem
METHOD
BASIS
QM_MM_INTERFACE
USER_CONNECT
FORCE_FIELD
GAUSSIAN_BLUR
GAUSS_BLUR_WIDTH
CIS_N_ROOTS
CIS_TRIPLETS
MOLDEN_FORMAT
PRINT_ORBITALS
$end

cis
6-31G*
janus
true
amber99
true
15000
5
false
true
true

$qm_atoms
18 19 20 21 22 23 24 25 26 27 28 29 30 31
$end
$molecule
0 1
O
0.000000
C
0.000000
C
1.427423
O
1.924453
C
2.866758
C
2.435730
C
2.832568
O
3.554295
H
-0.918053
H
-0.520597
H
-0.520597
H
1.435560
H
3.838325
H
1.936098
H
2.031585
H
3.838626
H
4.214443
N
2.474231
C
1.538394
N
1.421481
C
2.344666
C
2.704630
N
2.152150
N
3.660941
C
4.205243
N
3.949915
C
2.991769
H
0.951862
H
2.449361
H
1.442640
H
4.963977
$end

0.000000
0.000000
0.000000
-1.372676
-1.556753
0.816736
-0.159062
1.211441
0.000000
-0.885828
0.885828
0.337148
-1.808062
1.681209
-0.217259
0.075227
1.727289
-2.760890
-2.869204
-4.070993
-4.815233
-6.167666
-7.057611
-6.579606
-5.691308
-4.402308
-4.014545
-2.033257
-8.012246
-6.767115
-6.079842

0.000000
1.440000
1.962363
1.980293
0.934073
1.151710
0.042099
1.932365
-0.280677
1.803849
1.803849
2.998879
1.359516
0.714498
-0.694882
-0.305832
1.463640
0.168322
-0.826353
-1.308051
-0.582836
-0.619591
-1.455273
0.239638
1.066416
1.191662
0.323275
-1.177884
-1.436882
-2.097307
1.729564

1244
1118
1121
1123
1124
1126
1128
1249
1245
1119
1120
1122
1125
1127
1129
1130
1250
1132
1136
1135
1134
1140
1142
1139
1138
1137
1133
1145
1143
1144
1141

2
1
2
3
4
3
5
6
1
2
2
3
5
6
7
7
8
5
18
19
20
21
22
22
24
25
18
19
23
23
25

9
3
4
5
7
7
6
17
0
0
0
0
0
0
0
0
0
19
20
21
22
23
29
25
26
27
21
0
0
0
0

0
10
6
0
13
8
15
0
0
0
0
0
0
0
0
0
0
27
28
0
27
24
30
0
31
0
26
0
0
0
0

0
11
12
0
18
14
16
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

• QM/MM Example 3
Features of this job:
– An MM-only calculation. BASIS and EXCHANGE need to be defined, in order to prevent a crash, but no
electronic structure calculation is actually performed.

760

Chapter 12: Molecules in Complex Environments

– All atom types and MM interactions are defined in $force_field_params using the CHARMM27 force field.
Atomic charges, equilibrium bond distances, and equilibrium angles have been extracted from a HF/631G* calculation, but the force constants and van der Waals parameters are fictitious values invented for
this example.
– Molecular dynamics is propagated for 10 steps within a microcanonical ensemble (NVE), which is the only
ensemble available at present. Initial velocities are sampled from a Boltzmann distribution at 400 K.
Example 12.20 MM molecular dynamics with user-defined MM parameters.
$rem
BASIS
METHOD
QM_MM_INTERFACE
FORCE_FIELD
USER_CONNECT
JOBTYPE
TIME_STEP
AIMD_STEPS
AIMD_INIT_VELOC
AIMD_TEMP
$end
$molecule
-2 1
C
0.803090
C
-0.803090
H
1.386121
H
-1.386121
H
-1.386121
H
1.386121
$end

sto-3g
hf
MM
charmm27
true
aimd
42
10
thermal
400

0.000000
0.000000
0.930755
-0.930755
0.930755
-0.930755

0.000000
0.000000
0.000000
0.000000
0.000000
0.000000

-1
-1
-2
-2
-2
-2

2
1
1
2
2
1

3
4
0
0
0
0

6
5
0
0
0
0

0
0
0
0
0
0

$force_field_params
NumAtomTypes 2
AtomType -1 -0.687157 2.0000 0.1100
AtomType -2 -0.156422 1.3200 0.0220
Bond
-1 -1 250.00 1.606180
Bond
-1 -2 300.00 1.098286
Angle
-2 -1 -2 50.00 115.870
Angle
-2 -1 -1 80.00 122.065
Torsion -2 -1 -1 -2 2.500 180.0 2
$end

Further examples of QM/MM calculations can be found in the $QC/samples directory, including a QM/MM/PCM
example, QMMMPCM_crambin.in. This calculation consists of a protein molecule (crambin) described using a force
field, but with one tyrosine side chain described using electronic structure theory. The entire QM/MM system is placed
within an implicit solvent model, of the sort described in Section 12.2.2.

12.4

Q-CHEM/CHARMM Interface

Q-C HEM can be used a QM back-end for QM/MM calculations using C HARMM package. 47 In this case, both software
packages are required to perform the calculations, but all the code required for communication between the programs
is incorporated in the released versions. Stand-alone QM/MM calculations are described in Section 12.3.
QM/MM jobs that use the C HARMM interface are controlled using the following $rem keywords:

Chapter 12: Molecules in Complex Environments
QM_MM
Turns on the Q-C HEM/C HARMM interface.
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
TRUE
Do QM/MM calculation through the Q-C HEM/C HARMM interface.
FALSE Turn this feature off.
RECOMMENDATION:
Use the default unless running calculations with C HARMM.
QMMM_PRINT
Controls the amount of output printed from a QM/MM job.
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
TRUE
Limit molecule, point charge, and analysis printing.
FALSE Normal printing.
RECOMMENDATION:
Use the default unless running calculations with C HARMM.
QMMM_CHARGES
Controls the printing of QM charges to file.
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
TRUE
Writes a charges.dat file with the Mulliken charges from the QM region.
FALSE No file written.
RECOMMENDATION:
Use the default unless running calculations with C HARMM where charges on the QM region need
to be saved.
IGDEFIELD
Triggers the calculation of the electrostatic potential and/or the electric field at the positions of
the MM charges.
TYPE:
INTEGER
DEFAULT:
UNDEFINED
OPTIONS:
O Computes ESP.
1 Computes ESP and EFIELD.
2 Computes EFIELD.
RECOMMENDATION:
Must use this $rem when IGDESP is specified.

761

Chapter 12: Molecules in Complex Environments
GEOM_PRINT
Controls the amount of geometric information printed at each step.
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
TRUE
Prints out all geometric information; bond distances, angles, torsions.
FALSE Normal printing of distance matrix.
RECOMMENDATION:
Use if you want to be able to quickly examine geometric parameters at the beginning and end of
optimizations. Only prints in the beginning of single point energy calculations.
QMMM_FULL_HESSIAN
Trigger the evaluation of the full QM/MM Hessian.
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
TRUE
Evaluates full Hessian.
FALSE Hessian for QM-QM block only.
RECOMMENDATION:
None
LINK_ATOM_PROJECTION
Controls whether to perform a link-atom projection
TYPE:
LOGICAL
DEFAULT:
TRUE
OPTIONS:
TRUE
Performs the projection
FALSE No projection
RECOMMENDATION:
Necessary in a full QM/MM Hessian evaluation on a system with link atoms
HESS_AND_GRAD
Enables the evaluation of both analytical gradient and Hessian in a single job
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
TRUE
Evaluates both gradient and Hessian.
FALSE Evaluates Hessian only.
RECOMMENDATION:
Use only in a frequency (and thus Hessian) evaluation.

762

763

Chapter 12: Molecules in Complex Environments
GAUSSIAN_BLUR
Enables the use of Gaussian-delocalized external charges in a QM/MM calculation.
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
TRUE
Delocalizes external charges with Gaussian functions.
FALSE Point charges
RECOMMENDATION:
None
SKIP_CHARGE_SELF_INTERACT
Ignores the electrostatic interactions among external charges in a QM/MM calculation.
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
TRUE
No electrostatic interactions among external charges.
FALSE Computes the electrostatic interactions among external charges.
RECOMMENDATION:
None

Example 12.21 Do a basic QM/MM optimization of the water dimer. You need C HARMM to do this but this is the
Q-C HEM file that is needed to test the QM/MM functionality. These are the bare necessities for a Q-C HEM/C HARMM
QM/MM calculation.
$molecule
0 1
O
-0.91126
H
-1.75684
H
-0.55929
$end

$rem
METHOD
BASIS
QM_MM
JOBTYPE
$end

1.09227
1.51867
1.74495

hf
cc-pvdz
true
force

$external_charges
1.20426
-0.64330
1.01723
-1.36906
0.43830
-0.06644
$end

1.02007
1.28260
0.36940

!
!
!
!

HF Exchange
Correlation Consistent Basis
Turn on QM/MM calculation
Need this for QM/MM optimizations

0.79922
1.39217
0.91277

-0.83400
0.41700
0.41700

The Q-C HEM/C HARMM interface is unique in that:
• The external point charges can be replaced with Gaussian-delocalized charges with a finite width. 31 This is an
empirical way to include the delocalized character of the electron density of atoms in the MM region. This can
be important for the electrostatic interaction of the QM region with nearby atoms in the MM region.
• We allow the evaluation of the full QM/MM Hessian. 124 When link atoms are inserted to saturate the QM region,
all Hessian elements associated with link atoms are automatically projected onto their QM and MM host atoms.

Chapter 12: Molecules in Complex Environments

764

• For systems with a large number of MM atoms, one can define blocks consisting of multiple MM atoms (i.e.,
mobile blocks) and efficiently evaluate the corresponding mobile-block Hessian (MBH) for normal mode analysis.

12.5

Effective Fragment Potential Method

The Effective Fragment Potential (EFP) method is a computationally inexpensive way of modeling intermolecular
interactions in non-covalently bound systems. The EFP approach can be viewed as a QM/MM scheme with no empirical
parameters. Originally EFP was developed by Mark Gordon’s group, 32,40 and was implemented in GAMESS. 101 A
review of the EFP theory and applications can be found in Ref. 39,41. A related approach, also based on distributed
multipoles, is called XPol; it is described in Section 13.11.
A new implementation of the EFP method based on the libefp library by Dr. Ilya Kaliman (see https://libefp.
github.io) has been added to Q-C HEM. 53,54 The new EFP module is called EFPMAN2. EFPMAN2 can run
calculations in parallel on shared memory multi-core computers and clusters of computers. EFPMAN2 is interfaced
with the CCMAN and CCMAN2 modules to allow coupled cluster and EOM-CC calculations with EFP. CIS and
TDDFT calculations with EFP are also available.

12.5.1

Theoretical Background

The total energy of the system consists of the interaction energy of the effective fragments (E ef−ef ) and the energy of
the ab initio (i.e., QM) region in the field of the fragments. The former includes electrostatics, polarization, dispersion
and exchange-repulsion contributions (the charge-transfer term, which might be important for description of the ionic
and highly polar species, is omitted in the current implementation):
E ef -ef = Eelec + Epol + Edisp + Eex-rep .

(12.49)

The QM-EF interactions are computed as follows. The electrostatics and polarization parts of the EFP potential contribute to the quantum Hamiltonian via one-electron terms,
0
Hpq
= Hpq + hp|V̂ elec + V̂ pol |qi

(12.50)

whereas dispersion and exchange-repulsion QM-EF interactions are treated as additive corrections to the total energy.
The electrostatic component of the EFP energy accounts for Coulomb interactions. In molecular systems with hydrogen bonds or polar molecules, this is the leading contribution to the total intermolecular interaction energy. 13 An accurate representation of the electrostatic potential is achieved by using multipole expansion (obtained from the Stone’s
distributed multipole analysis) around atomic centers and bond midpoints (i.e., the points with high electronic density) and truncating this expansion at octupoles. 32,40,108,109 The fragment-fragment electrostatic interactions consist of
charge-charge, charge-dipole, charge-quadrupole, charge-octupole, dipole-dipole, dipole-quadrupole, and quadrupolequadrupole terms, as well as terms describing interactions of electronic multipoles with the nuclei and nuclear repulsion
energy.
Electrostatic interaction between an effective fragment and the QM part is described by perturbation V̂ elec of the ab
initio Hamiltonian (see Eq. (12.50)). The perturbation enters the one-electron part of the Hamiltonian as a sum of
contributions from the expansion points of the effective fragments. Contribution from each expansion point consists of
four terms originating from the electrostatic potential of the corresponding multipole (charge, dipole, quadrupole, and
octupole).
The multipole representation of the electrostatic density of a fragment breaks down when the fragments are too close.
The multipole interactions become too repulsive due to significant overlap of the electronic densities and the chargepenetration effect. The magnitude of the charge-penetration effect is usually around 15% of the total electrostatic
energy in polar systems, however, it can be as large as 200% in systems with weak electrostatic interactions. 106 To
account for the charge-penetration effect, the simple exponential damping of the charge-charge term is used. 38,106 The

765

Chapter 12: Molecules in Complex Environments

charge-charge screened energy between the expansion points k and l is given by the following expression, where αk
and αl are the damping parameters associated with the corresponding expansion points:

ch-ch
Ekl

or

=
=



1 − (1 + αk Rkl /2)e−αk Rkl q k q l /Rkl , if αk = αl


α2
α2
1 − 2 l 2 e−αk Rkl − 2 k 2 e−αl Rkl q k q l /Rkl , if αk 6= αl
αl − αk
αk − αl

(12.51)
(12.52)

Damping parameters are included in the potential of each fragment, but QM-EFP electrostatic interactions are currently
calculated without damping corrections.
Alternatively, one can obtain the short-range charge-penetration energy using the spherical Gaussian overlap (SGO)
approximation: 107

 21 2
1
Skl
pen
Ekl = −2
(12.53)
−2ln|Skl |
Rkl
where Skl is the overlap integral between localized MOs k and l, calculated for the exchange-repulsion term, Eq. (12.63).
This charge-penetration energy is calculated and printed separately from the rest of the electrostatic energy. Using
overlap-based damping generally results in a more balanced description of intermolecular interactions and is recommended.
Polarization accounts for the intramolecular charge redistribution in response to external electric field. It is the major component of many-body interactions responsible for cooperative molecular behavior. EFP employs distributed
polarizabilities placed at the centers of valence LMOs. Unlike the isotropic total molecular polarizability tensor, the
distributed polarizability tensors are anisotropic.
The polarization energy of a system consisting of an ab initio and effective fragment regions is computed as 32
E pol = −

1 X k mult,k
1 X k ai,elec,k
µ (F
+ F ai,nuc,k ) +
µ̄ F
2
2
k

(12.54)

k

where µk and µ̄k are the induced dipole and the conjugated induced dipole at the distributed point k; F mult,k is the
external field due to static multipoles and nuclei of other fragments, and F ai,elec,k and F ai,nuc,k are the fields due to
the electronic density and nuclei of the ab initio part, respectively.
The induced dipoles at each polarizability point k are computed as
µk = αk F total,k

(12.55)

where αk is the distributed polarizability tensor at k. The total field F total,k comprises from the static field and the
field due to other induced dipoles, Fkind , as well as the field due to nuclei and electronic density of the ab initio region:
F ai,total,k = F mult,k + F ind,k + F ai,elec,k + F ai,nuc,k

(12.56)

As follows from the above equation, the induced dipoles on a particular fragment depend on the values of the induced
dipoles of all other fragments. Moreover, the induced dipoles on the effective fragments depend on the ab initio
electronic density, which, in turn, is affected by the field created by these induced dipoles through a one electron
contribution to the Hamiltonian:
x,y,z
1 X X (µka + µ̄ka )a
(12.57)
V̂ pol = −
2
R3
a
k

where R and a are the distance and its Cartesian components between an electron and the polarizability point k. In
sum, the total polarization energy is computed self-consistently using a two level iterative procedure. The objectives
of the higher and lower levels are to converge the wave function and induced dipoles for a given fixed wave function,
respectively. In the absence of the ab initio region, the induced dipoles of the EF system are iterated until self-consistent
with each other.

766

Chapter 12: Molecules in Complex Environments

Self-consistent treatment of polarization accounts for many-body interaction effects. Polarization energy between EFP
fragments is augmented by gaussian-like damping functions with default parameter α = β = 0.6, applied to electric
field F : 107
F = F0 f damp
(12.58)
p
p
(12.59)
f damp = 1.0 − exp(− αβr2 )(1 + αβr2 )
Dispersion provides a leading contribution to van der Waals and π-stacking interactions. The dispersion interaction is
expressed as the inverse R dependence:
X
E disp =
C6 R−6
(12.60)
n

where coefficients C6 are derived from the frequency-dependent distributed polarizabilities with expansion points located at the LMO centroids, i.e., at the same centers as the polarization expansion points. The higher-order dispersion
terms (induced dipole-induced quadrupole, induced quadrupole/induced quadrupole, etc.) are approximated as 1/3 of
the C6 term. 3
For small distances between effective fragments, dispersion interactions are corrected for charge penetration and electronic density overlap effect either with the Tang-Toennies damping formula 110 with parameter b = 1.5,
!
6
X
(bR)k
kl
−bR
C6 → 1 − e
C6kl ,
(12.61)
k!
k=0

or else using interfragment overlap (so-called overlap-based damping): 107
2
C6kl → 1 − Skl
1 − 2 log |Skl | + 2 log2 |Skl |





C6kl

(12.62)

QM-EFP dispersion interactions are currently disabled.
Exchange-repulsion originates from the Pauli exclusion principle, which states that the wave function of two identical
fermions must be anti-symmetric. In traditional classical force fields, exchange-repulsion is introduced as a positive
(repulsive) term, e.g., R−12 in the Lennard-Jones potential. In contrast, EFP uses a wave function-based formalism to
account for this inherently quantum effect. Exchange-repulsion is the only non-classical component of EFP and the
only one that is repulsive.
The exchange-repulsion interaction is derived as an expansion in the intermolecular overlap, truncated at the quadratic
term, 50,51 which requires that each effective fragment carries a basis set that is used to calculate overlap and kinetic
one-electron integrals for each interacting pair of fragments. The exchange-repulsion contribution from each pair of
localized orbitals i and j belonging to fragments A and B, respectively, is:
r
2
−2 ln |Sij | Sij
exch
(12.63)
Eij
= −4
π
Rij
!
X
X
A
B
−2Sij
Fik Skj +
Fjl Sil − 2Tij
(12.64)
k∈A

l∈B

!
2
+2Sij

X
J∈B

−1
−ZJ RiJ

+2

X
l∈B

−1
Ril

+

X
I∈A

−1
−ZI RIj

+2

X

−1
Rkj

−

−1
Rij

(12.65)

k∈A

where i, j, k and l are the LMOs, I and J are the nuclei, S and T are the intermolecular overlap and kinetic energy
integrals, and F is the Fock matrix element.
exch
The expression for the Eij
involves overlap and kinetic energy integrals between pairs of localized orbitals. In
addition, since Eq. (12.63) is derived within an infinite basis set approximation, it requires a reasonably large basis
set to be accurate [6-31+G* is considered to be the smallest acceptable basis set, 6-311++G(3df,2p) is recommended].
These factors make exchange-repulsion the most computationally expensive part of the EFP energy calculations of
moderately sized systems.

767

Chapter 12: Molecules in Complex Environments

Large systems require additional considerations. Since total exchange-repulsion energy is given by a sum of terms in
Eq. (12.63) over all the fragment pairs, its computational cost formally scales as O(N 2 ) with the number of effective
fragments N . However, exchange-repulsion is a short-range interaction; the overlap and kinetic energy integrals decay
exponentially with the inter-fragment distance. Therefore, by employing a distance-based screening, the number of
overlap and kinetic energy integrals scales as O(N ). Consequently, for large systems exchange-repulsion may become
less computationally expensive than the long-range components of EFP (such as Coulomb interactions).
The QM-EFP exchange-repulsion energy is currently disabled.

12.5.2

Excited-State Calculations with EFP

Interface of EFP with EOM-CCSD (both via CCMAN and CCMAN2), CIS, CIS(D), and TDDFT has been developed. 66,105 In the EOM-CCSD/EFP calculations, the reference-state CCSD equations for the T cluster amplitudes are
solved with the HF Hamiltonian modified by the electrostatic and polarization contributions due to the effective fragments, Eq. (12.50). In the coupled-cluster calculation, the induced dipoles of the fragments are frozen at their HF
values.
The transformed Hamiltonian H̄ effectively includes Coulomb and polarization contributions from the EFP part. As H̄
is diagonalized in an EOM calculation, the induced dipoles of the effective fragments are frozen at their reference state
value, i.e., the EOM equations are solved with a constant response of the EFP environment. To account for solvent
response to electron rearrangement in the EOM target states (i.e., excitation or ionization), a perturbative non-iterative
correction is computed for each EOM root as follows. The one-electron density of the target EOM state (excited or
ionized) is calculated and used to re-polarize the environment, i.e., to recalculate the induced dipoles of the EFP part in
the field of an EOM state. These dipoles are used to compute the polarization energy corresponding to this state.
The total energy of the excited state with inclusion of the perturbative response of the EFP polarization is:
EOM/EFP

EIP

= EEOM + ∆Epol

(12.66)

where EEOM is the energy found from EOM-CCSD procedure and ∆Epol has the following form:
∆Epol

=

x,y,z
1 X Xh
−(µkex,a − µkgr,a )(Famult,k + Fanuc,k )
2
a

(12.67)

k

ai,k
ai,k
ai,k
+(µ̃kex,a Fex,a
− µ̃kgr,a Fgr,a
) − (µkex,a − µkgr,a + µ̃kex,a − µ̃kgr,a )Fex,a

i

(12.68)

ai
ai
and Fex
are the fields due to the reference (HF) state and excited-state electronic densities, respectively. µkgr
where Fgr
k
and µ̃gr are the induced dipole and conjugated induced dipole at the distributed polarizability point k consistent with
the reference-state density, while µkex and µ̃kex are the induced dipoles corresponding to the excited state density.

The first two terms in Eq. (12.67) provide a difference of the polarization energy of the QM/EFP system in the excited
and ground electronic states; the last term is the leading correction to the interaction of the ground-state-optimized
induced dipoles with the wave function of the excited state.
The EOM states have both direct and indirect polarization contributions. The indirect term comes from the orbital
relaxation of the solute in the field due to induced dipoles of the solvent. The direct term given by Eq. (12.67) is the
response of the polarizable environment to the change in solute’s electronic density upon excitation. Note that the direct
polarization contribution can be very large (tenths of eV) in EOM-IP/EFP since the electronic densities of the neutral
and the ionized species are very different.
An important advantage of the perturbative EOM/EFP scheme is that it does not compromise multi-state nature of
EOM and that the electronic wave functions of the target states remain orthogonal to each other since they are obtained
with the same (reference-state) field of the polarizable environment. For example, transition properties between these
states can be calculated.
EOM-CC/EFP scheme works with any type of the EOM excitation operator Rk currently supported in Q-C HEM, i.e.,
spin-flipping (SF), excitation energies (EE), ionization potential (IP), electron affinity (EA) (see Section 7.7.13 for

768

Chapter 12: Molecules in Complex Environments

details). However, direct polarization correction requires calculation of one-electron density of the excited state, and
will be computed only for the methods with implemented one-electron properties.
Implementation of CIS/EFP, CIS(D)/EFP, and TDDFT/EFP methods is similar to the implementation of EOM/EFP.
Polarization correction as in Eq. 12.67 is calculated and added to the CIS or TDDFT excitation energies.

12.5.3

Extension to Macromolecules: Fragmented EFP Scheme

Macromolecules such as proteins or DNA present a large number of electronic structure problems (photochemistry,
redox chemistry, reactivity) that can be described within QM/EFP framework. EFP has been extended to deal with
such complex systems via the so-called fragmented EFP scheme (fEFP). The current Q-C HEM implementation allows
one to (i) compute interaction energy between a ligand and a macromolecule (both represented by EFP) and (ii) to
calculate the excitation energies, ionization potentials, electronic affinities of a QM moiety interacting with a fEFP
macromolecule using QM/EFP scheme (see Section 12.5.2). In the present implementation, the ligand cannot be
covalently bound to the macromolecule.
There are multiple ways to cut a large molecule into units depending on the position of the cut between two covalently
bound residues. An obvious way to cut a protein is to cut through peptide bonds such that each fragment represents
one amino acid. Alternatively, one can cut bonds between two atoms of the same nature (carbonyl and carbon-α or
carbon-α and the first carbon of the side chain). The user can choose the most appropriate way to cut.
Consider a protein (P ) consisting of N amino acids, A1 A2 . . . AN , and is split into N fragments (Ai ). The fragments
can be saturated by either Hydrogen Link Atom 104 (HLA) or by mono-valent groups of atoms from the neighboring
fragment(s), called Cap Link Atom (CLA) hereafter. If fragments are capped using the HLA scheme, the hydrogen is
located along the peptide bond axis and at the distance corresponding to the equilibrium bond length of a CH bond:
P = A1 H +

N
−1
X

HAi H + HAN

(12.69)

i=2

In the CLA scheme, the cap has exactly the same geometry as the respective neighboring group. If the cuts are made
through peptide bonds (one fragment is one amino acid), the caps (C i ) are either an aldehyde to saturate the -N(H) end
of the fragment, or an amine to saturate the -C(=O) extremity of the fragment.
P = A1 C 2 +

N
−1
X

C i−1 Ai C i+1 + C N −1 AN

(12.70)

i=2

Q-C HEM provides a two-step script, prefefp.pl, located in $QC/bin which takes a PDB file and breaks it into
capped fragments in the GAMESS format, such that the EFP parameters for these capped fragments can be generated, as
explained in Section 12.5.7. As the EFP parameters are generated for each capped fragment, the neighboring fragments
have duplicated parameter points (overlapping areas) in both the HLA and CLA schemes due to the overlapping caps.
Since multipole expansion points and polarizability expansion points are computed on each capped residue by the standard procedure, the multipole (and damping terms) and polarizabilities need to be removed (C ∅ ) from the overlapping
areas.
Equations (12.69) and (12.70) become:
P = A1 C ∅ +

N
−1
X

C ∅ Ai C ∅ + C ∅ AN

(12.71)

i=2

The details concerning this removing procedure are presented in Section 12.5.7.
Once these duplicate parameters are removed from the EFP parameters of the capped fragments, the EFP-EFP and
QM-EFP calculations can be conducted as usual.
Currently, fEFP includes electrostatic and polarization contributions, which appear in EFP(ligand)/fEFP(macromolecule)
and in QM/fEFP calculations (note that the QM part is not covalently bound to the macromolecule). Consequently, the

769

Chapter 12: Molecules in Complex Environments
total interaction energy (E tot ) between a ligand (L) and a protein (P ) divided into fragments is:
E tot (P − L) = E elec (P − L) + E pol (P − L)

(12.72)

The electrostatics is an additive term; its contribution to fragment-fragment and ligand-fragment interaction is computed
as follows:
N


X
E elec (P − L) =
(12.73)
E elec C ∅ Ai=1 C ∅ − L
i

The polarization contribution in an EFP system (no QM) is:
E pol (P − L) = −

1 X k mult,k 1 X k mult,k
µ F
+
µ F
2
2
k∈P,L

(12.74)

k∈P

The first term is the polarization energy obtained upon convergence of the induced dipoles of the ligand (µkefp (L)) and
all fragments (µkfefp (Ai )). The system is thus fully polarized, all fragments (Ai or L) are polarizing each other until
self-consistency.
X
µkefp (L) =
αk (F mult,k + F ind,k )
(12.75)
k∈Ai

µkf ef p (Ai )

=

X X

αk (F mult,k + F ind,k )

(12.76)

j6=i k∈L,Aj

The second term of Eq. (12.74) is the polarization of the protein by itself; this value has to be subtracted once the
induced dipoles (Eq. 12.75) converged.
The LA scheme is available to perform QM/fEFP job. In this situation the fEFP has to include a macromolecule
(covalent bond between fragments). This scheme is not able yet to perform QM/fEFP/EFP in which a macromolecule
and solvent molecules would be described at the EFP level of theory.
In addition to the HLA and CLA schemes, Q-C HEM also features Molecular Fragmentation with Conjugated Caps approach (MFCC) which avoids the issue of overlapping of saturated fragments and was developed in 2003 by Zhang. 129,130
The MFCC procedure consists of a summation over the interactions between a ligand and capped residues (CLA
scheme) and a subtraction over the interactions of merged caps (C i+1 C i−1 ), the so-called “concaps”, with the ligand.
N − 1 concap fragments are actually used to subtract the overlapping effect.
P = A1 C 2 +

N
−1
X

C i−1 Ai C i+1 + C N −1 AN −

i=2

N
−1
X

C i+1 C i−1

(12.77)

i

In this scheme the contributions due to overlapping caps simply cancel out and the EFP parameters do not need any
modifications, in contrast to the HLA or CLA procedures. However, the number of parameters that need to be generated
is larger (N capped fragments + N − 1 concaps).
The MFCC electrostatic interaction energy is given as the sum of the interaction energy between each capped fragment
(C i−1 Ai C i+1 ) and the ligand minus the interaction energy between each concap (C i−1 C i+1 ) and the ligand:
E

elec

(P − L) =

N
X
i

E

elec

C

i−1

Ai C

i+1

−1

 NX


−L −
E elec C i−1 C i+1 − L

(12.78)

i

The main advantage of MFCC is that the multipole expansion obtained on each capped residue or concap are kept
during the E elec (P − L) calculation. In the present implementation, there are no polarization contributions. The
MFCC scheme is not yet available for QM/fEFP.

12.5.4

Running EFP Jobs

The current version supports single point calculations in systems consisting of (i) ab initio and EFP regions (QM/
MM); or (ii) EFP region only. The ab initio region can be described by conventional quantum methods like HF,

Chapter 12: Molecules in Complex Environments

770

DFT, or correlated methods including methods for the excited states [CIS, CIS(D), TDDFT, EOM-CCSD methods].
Theoretical details on the interface of EFP with EOM-CCSD and CIS(D) can be found in Refs. 105 and 66.
Note: EFP provides both implicit (through orbital response) and explicit (as instantaneous response of the polarizable
EFP fragments) corrections to the electronic excited states. EFP-modified excitation energies are printed in the
property section of the output.
Electrostatic, polarization, exchange-repulsion, and dispersion contributions are calculated between EFs; only electrostatic and polarization terms are evaluated between ab initio and EF regions.
The ab initio region is specified by regular Q-C HEM input using $molecule and $rem sections. In calculations with no
QM part, the $molecule section should contain a dummy atom (for example, helium).
Positions of EFs are specified in the $efp_fragments section. Two geometry formats are available for fragments, Euler
angle format and XYZ format. In Euler angle format, each line in this section contains the information on an individual
fragment: fragment’s name and position, specified by center-of-mass coordinates (x, y, z) and the Euler rotation angles
(α, β, γ) relative to the fragment frame, i.e., the coordinates of the standard fragment provided in the fragment library.
The XYZ format is identical to the coordinate format used in GAMESS; i.e., the name of the fragment is provided on
the first line followed by three lines specifying names and x,y,z coordinates of first three atoms of the fragment.
When using EFPMAN2 you can also specify positions and orientations of the fragments using (x, y, z) coordinates of
the first three atoms of the fragment. To enable this feature use EFP_COORD_XYZ keyword. The format of input goes
as follows: for each fragment the first line should contain the name of the fragment and the following three lines should
specify coordinates of the three first atoms belonging to a fragment. The sample input can be found in the examples
section below.

12.5.5

Library of Fragments

The effective fragments are rigid and their potentials are generated from a set of ab initio calculations on each unique
isolated fragment. The EFP includes: (i) multipoles (produced by the Stone’s Distributed Multipolar Analysis) for
Coulomb and polarization terms; (ii) static polarizability tensors centered at localized molecular orbital (LMO) centroids (obtained from coupled-perturbed Hartree-Fock calculations), which are used for calculations of polarization;
(iii) dynamic polarizability tensors centered on the LMOs that are generated by time-dependent HF calculations and
used for calculations of dispersion; and (iv) the Fock matrix, basis set, and localized orbitals needed for the exchangerepulsion term. Additionally, the EF potential contains coordinates of atoms, coordinates of the points of multi-polar
expansion (typically, atoms and bond mid-points), coordinates of the LMO centroids, electrostatic and polarization
screening parameters, and atomic labels of the EF atoms.
Q-C HEM provides a library of standard fragments with precomputed effective fragment potentials. Currently the
library includes common organic solvents, nucleobases, and molecules from S22 and S66 datasets for non-covalent
interactions; see Table 12.8. EFP potentials in GAMESS format are supported by new EFPMAN2 module. They are
stored in $QCAUX/fraglib directory.
Note: The fragments from Q-C HEM fragment library have _L added to their names to distinguish them from userdefined fragments.
The parameters for the standard fragments were computed as follows. The geometries of the solvent molecules were optimized by MP2/cc-pVTZ; geometries of nucleobases were optimized with RI-MP2/cc-pVTZ. Geometries of molecules
from S22 and S66 datasets are discussed in Ref. 34. The EFP parameters were obtained in GAMESS. To generate the
electrostatic multipoles and electrostatic screening parameters, analytic DMA procedure was used, with 6-31+G* basis
for non-aromatic compounds and 6-31G* for aromatic compounds and nucleobases. The rest of the potential, i.e., static
and dynamic polarizability tensors, wave function, Fock matrix, etc., were obtained using 6-311++G(3df,2p) basis set.

771

Chapter 12: Molecules in Complex Environments

Table 12.8: Standard fragments available in Q-C HEM
acetone
acetonitrile
adenine
ammonia
benzene
carbon tetrachloride
cytosine C1
cytosine C2a
cytosine C2b
cytosine C3a
cytosine C3b
dichloromethane
dimethyl sulfoxide
guanine enol N7
guanine enol N9
guanine enol N9RN7
guanine keton N7
guanine keton N9
methane
methanol
phenol
thymine
toluene
water
acetamide, S66, gas phase
acetamide, S66, H-bonded dimer
acetic acid, S66, gas phase
acetic acid, S66, H-bonded dimer
adenine, S22 stack dimer
adenine, S22 WC dimer
2-aminopyridine, S22
cyclopentane, S66
ethylene
acetylene
formic acid, S22 H-bonded dimer
formamide, S22 dimer
hydrogen cyanide
indole, S22
methylamine, S66
neopentane, S66
O2
pentane, S66
peptide, S66
pyrazine
pyridine, S66
2-pyridoxine, S22
thymine, S22 stack dimer
thymine, S22 WC dimer
uracil, S66, gas phase
uracil, S66, H-bonded dimer

ACETONE_L
ACETONITRILE_L
ADENINE_L
AMMONIA_L
BENZENE_L
CCL4_L
CYTOSINE_C1_L
CYTOSINE_C2A_L
CYTOSINE_C2B_L
CYTOSINE_C3A_L
CYTOSINE_C3B_L
DCM_L
DMSO_L
GUANINE_EN7_L
GUANINE_EN9_L
GUANINE_EN9RN7_L
GUANINE_KN7_L
GUANINE_KN9_L
METHANE_L
METHANOL_L
PHENOL_L
THYMINE_L
TOLUENE_L
WATER_L
ACETAMIDE_L
ACETAMIDE_HB_L
ACETICAC_L
ACETICAC_HB_L
ADENINE_L
ADENINE_WC_L
AMINOPYRIDINE_L
CPENTANE_L
ETHENE_L
ETHYNE_L
FORMICAC_HB_L
FORMID_L
HCN_L
INDOLE_L
MENH2_L
NEOPENTANE_L
O2_L
PENTANE_L
PEPTIDE_L
PYRAZINE_L
PYRIDINE_L
PYRIDOXINE_L
THYMINE_L
THYMINE_WC_L
URACIL_L
URACIL_HB_L

Chapter 12: Molecules in Complex Environments

12.5.6

772

Calculation of User-Defined EFP Potentials

User-defined EFP parameters can be generated in MAKEFP job in GAMESS (see the GAMESS manual for details).
The EFP potential generation begins by determining an accurate structure for the fragment (EFP is the frozen-geometry
potential, so the fragment geometry will remain the same in all subsequent calculations). We recommend MP2/ccPVTZ level of theory.

12.5.6.1

Generating EFP Parameters in GAMESS

EFP parameters can be generated in GAMESS using MAKEFP job (RUNTYP=MAKEFP). For EFP parameters calculations, 6-311++G(3df,2p) basis set is recommended. Original Stone’s distributed multipole analysis (bigexp=0 in the
group $stone is recommended for non-aromatic compound; optionally, one may decrease the basis set to 6-31G* or
6-31+G* for generation of electrostatic multipoles and screening parameters. (To prepare such a “mixed" potential,
one has to run two separate MAKEFP calculations in larger and smaller bases, and combine the corresponding parts of
the potential). In aromatic compounds, one must either use numerical grid for generation of multipoles (bigexp=4.0)
or use 6-31G* basis with standard analytic DMA, which is recommended. The MAKEFP job produces (usually in the
scratch directory) the .efp file containing all the necessary EFP parameters. See the GAMESS manual for further details.
Below are examples of the RUNTYP=MAKEFP GAMESS input file for water and benzene.
GAMESS input example for water.
$contrl units=angs local=boys runtyp=makefp coord=cart icut=11 $end
$system timlim=99999 mwords=200 $end
$scf soscf=.f. diis=.t. conv=1.0d-06 $end
$basis gbasis=n311 ngauss=6 npfunc=2 ndfunc=3 nffunc=1
diffs=.t. diffsp=.t. $end
$stone
bigexp=0.0
$end
$damp ifttyp(1)=3,2 iftfix(1)=1,1 thrsh=500.0 $end
$dampgs
h3=h2
bo31=bo21
$end
$data
water h2o (geometry: mp2/cc-pvtz)
c1
o1 8.0 0.0000 0.0000 0.1187
h2 1.0 0.0000 0.7532 -0.4749
h3 1.0 0.0000 -0.7532 -0.4749
$end
GAMESS input example for benzene.
$contrl units=bohr local=boys runtyp=makefp coord=cart icut=11 $end
$system timlim=99999 mwords=200 $end
$scf soscf=.f. diis=.t. conv=1.0d-06 $end
$basis gbasis=n311 ngauss=6 npfunc=2 ndfunc=3 nffunc=1
diffs=.t. diffsp=.t. $end
$stone
bigexp=4.0

Chapter 12: Molecules in Complex Environments

773

$end
$damp ifttyp(1)=3,2 iftfix(1)=1,1 thrsh=500.0 $end
$dampgs
c6=c5
c2=c1
c3=c1
c4=c1
c5=c1
c6=c1
h8=h7
h9=h7
h10=h7
h11=h7
h12=h7
bo32=bo21
bo43=bo21
bo54=bo21
bo61=bo21
bo65=bo21
bo82=bo71
bo93=bo71
bo104=bo71
bo115=bo71
bo126=bo71
$end
$data
benzene c6h6 (geometry: mp2/cc-pvtz)
c1
c1
6.0
1.3168
-2.2807
0.0000
c2
6.0
2.6336
0.0000
0.0000
c3
6.0
1.3168
2.2807
0.0000
c4
6.0
-1.3168
2.2807
0.0000
c5
6.0
-2.6336
-0.0000
0.0000
c6
6.0
-1.3168
-2.2807
0.0000
h7
1.0
2.3386
-4.0506
0.0000
h8
1.0
4.6772
0.0000
0.0000
h9
1.0
2.3386
4.0506
0.0000
h10
1.0
-2.3386
4.0506
0.0000
h11
1.0
-4.6772
0.0000
0.0000
h12
1.0
-2.3386
-4.0506
0.0000
$end

12.5.7

fEFP Input Structure

A two-step script, prefefp.pl located in $QC/bin, allows users to break molecular structures from a PDB file into
the capped fragments in the GAMESS format, such that parameters for fEFP calculations can be generated.
To use the prefefp.pl scripts you need a PDB file, a MAP file, and a directory with all your .efp parameter files.
Run the following commands to: (1) obtain the N input file generating the N EFP parameters for the N capped
fragments, and (2) create the EFP input file in XYZ format.
perl prefp.pl 1  

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