Users Guide

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Jacek Kobus

2 Dimensional Finite Diﬀerence

Hartree-Fock Program

User’s Guide

version 2.2

Contents

1 Description of input data 3

1.1 Mandatorylabels ................................ 3

1.2 Optionallabels ................................. 7

1.3 Deprecatedlabels................................ 16

2 Examples of input data 20

3 Program’s data ﬁles 33

4 How to run the program? 33

5 Useful hints 34

1 Description of input data

x2dhf program accepts input data that consist of separate lines which contain

– a label

– a label followed by a string of characters, integer(s) and/or real number(s)

– a string of characters, integer(s) and/or real number(s)

Real numbers can be written in a ﬁxed-point or scientiﬁc notation. Note that

– labels and strings can be in upper or lower case,

– the compulsory labels must follow the order given below; the optional ones can

be inserted anywhere between the title and stop labels,

– optional parameters are enclosed in square brackets,

–rdenotes a real number, i– an integer, c– a string of characters,

– an exclamation mark or a hash placed anywhere in an input line starts a comment

and what follows “!” or “#” is ignored.

1.1 Mandatory labels

The following labels must be speciﬁed in the speciﬁed order:

TITLE

Format: title c

cis any string of up to 74 characters describing the current case. This string

is added as a header to a text ﬁle with extension dat that contains basic data

identifying a given case, i.e. atomic numbers of nuclei, grid size and the number

of electrons and orbital and exchange functions.

NUCLEI

Format: nuclei ZAZBR[c]

Set the nuclei charges and the bond length.

ZA: nuclear charge of centre A (real)

ZB: nuclear charge of centre B (real)

R: bond length (real)

c:angstrom – the internuclear separation can be given in angstrom units

if this string is included (the conversion factor 0.529177249 is used)

If |ZA−ZB|<10−6then the molecule is considered to be a homonuclear one

(see setDefaults).

CONFIG

Format: conﬁg i

i: the total charge of a system

The following cards deﬁne molecular orbitals and their occupation. Note that the

last orbital description card must contain the end label.

The possible formats are:

Format: i c

i: number of fully occupied orbitals of a given irreducible representation (ir-

rep) of the C∞vgroup; two electrons make σorbitals fully occupied and

four electrons – orbitals of other symmetries

c: symbol of the C∞virrep to which the orbitals belong (sigma, pi, delta

or phi)

Format: i c1c2

i: number of fully occupied orbitals of a given irrep of the D∞hgroup

c1: symbol of the C∞virrep to which the orbitals belong (sigma, pi, delta

or phi)

c2: symbol for the inversion symmetry of the D∞hirreps (uor g)

Use this format for a homonuclear molecule unless break card is included.

Format: i c1c2[c3[c4[c5]]]

i: number of orbitals of a given irrep of the C∞vgroup

c1: symbol for the C∞virreps to which the orbitals belong (sigma, pi, delta,

phi)

c2-c5: +,−or . (a dot); +/−denotes spin up/down electron and . denotes an

unoccupied spin orbital

Format: i c1c2c3[c4[c5[c6]]]

i: number of orbitals of a given irrep of the D∞hgroup

c1: symbol for the C∞virrep to which the orbitals belong (sigma, pi, delta,

phi)

c2: symbol for the inversion symmetry of the D∞hirrep (uor g)

c3-c6: +,−or . (a dot); +/−denotes spin up/down electron and . denotes an

unoccupied spin orbital

GRID

Format: grid NνR∞

An integer and a real deﬁne a single two-dimensional grid.

Nν: the number of grid points in νvariable

R∞: the practical inﬁnity

Nµis calculated so as to make the step size in µvariable equal to the stepsize

in νvariable. Nνand Nµhave to meet special conditions and if they are not

fulﬁlled the nearest (but smaller) appropriate values are used.

Format: grid NνNµR∞

Two integers and one real deﬁne a single two-dimensional grid.

Nν: the number of grid points in νvariable

Nµ: the number of grid points in µvariable

R∞: the practical inﬁnity

This format may be needed when interpolation between grids is attempted.

ORBPOT

Format: orbpot c

where ca character string determining the initial source of orbitals and poten-

tials. Its allowed values are:

•hydrogen – molecular orbitals are formed as a linear combination of hy-

drogenic functions on centres Aand Bas deﬁned via the lcao label. In

the case of HF or HFS calculations Coulomb (exchange) potentials are

approximated as a linear combination of Thomas-Fermi (1/r) potentials

at the two centres. If the OED method is chosen the potential function

is approximated as a linear combination of ZA/r1and ZB/r2terms and

the exchange potentials are set to zero.

•gauss – a Gaussian output is used to retrieve exponents and expansion co-

eﬃcients of (uncontracted) molecular orbitals (it is assumed that the out-

put is contained in gaussian.out and gaussian.pun ﬁles1) and Coulomb and

exchange potentials are initialized as in the hydrogen case; see prepGauss

for more details.

•gauss-c– GAUSSIAN customized output is used to retrieve exponents and

expansion coeﬃcients of molecular orbitals (it is assumed that the output

is contained in gaussianc.out ﬁle) and Coulomb and exchange potentials

are initialized as in the hydrogen case; see prepGaussCust for details.

•old – initial orbitals, Coulomb and exchange potentials are retrieved from

disk ﬁles (2dhf input.orb, 2dhf input.coul and 2dhf input.exch, respec-

tively) created in a previous run. Data deﬁning the case are retrieved

from a 2dhf input.dat textﬁle.

•qrhf – radial Hartree-Fock orbitals for the centre A and B obtained from

the qrhf program2are retrieved from disk ﬁles 1dhf centreA.orb and 1dhf-

centreB.orb, respectively, and Coulomb and exchange potentials are ini-

tialized as in the hydrogen case (see routine initHF for details).

•noexch – orbitals and Coulomb potentials are retrieved from disk ﬁles and

exchange potentials are initialized as in hydrogen case; this is useful when

going from DFT/HFS to HF calculations.

•nodat – initial orbitals and potentials are retrieved from disk ﬁles but the

content of a 2dhf input.dat ﬁle is retrieved from a 2dhf input.orb (binary)

ﬁle. Use this value when reading binary data generated by the ealier than

2.0 versions of the program.

STOP

Format: stop

This label indicates the end of input data.

1The program has been tested on output ﬁles produced by the 94 and 03 versions of the Gaussian

system of programs.

2J.Kobus and Ch. Froese Fischer, Quasi-Relativistic Hartree-Fock program for Atoms, to be published.

1.2 Optional labels

The following additional labels can be speciﬁed in any order:

BREAK

Format: break

When this label is present homonuclear molecules are calculated in C∞vsym-

metry and the D∞hsymmetry labels (uor g) are redundant.

CONV

Format: conv [i1[i2[i3]]]

Sometimes the requested accuracy of a solution is set too high and cannot be

satisﬁed on a selected grid. As a result SCF/SOR iteration process may con-

tinue in vain. To save CPU time the iterations are stopped if orbital energies

or orbital norms display no improvment over the i2and i3most recent itera-

tions, respectively (20 by default). This mechanism is activated after i1initial

iterations (600 by default).

DEBUG

Format: debug [i1[i2. . . ]i40 ]

Up to 40 diﬀerent debug ﬂags can be set at a time. If the integer ikis present

the debug ﬂag ikis set, i.e. idbg(ik) = 1 (1 ≤ik<999). These are used to

generate additional debugging information by adding the lines of the form

if (idbg(ik).eq.1) then

print *, ‘‘debugging something ...’’

...

endif

DFT

Format: dft [c1] [ c2]

c1: speciﬁes the type of DFT exchange potential to be used in Fock equations

–c1=lda – the local density approximation with the potential

VX(α) = −3

2α3

π1/3

2−2/3X

ρ1/3

where αis by default set to 2/3 (the Slater exchange potential). To

change this value use the xalpha label.

–c1=b88 – the Becke exchange potential

c2: selects the type of correlation potential to be used in Fock equations

–c2=lyp – the correlation potential of Lee, Yang and Parr

–c2=vwn – the correlation potential of Vosko, Wilk and Nusair

When the bare label is present and the method selected is HF then the exchange

contributions (LDA, B88, PW86 and PW91) and the correlation contributions (LYP

and VWN) to the total energy are calculated upon completion of the SCF iterations.

EXCHIO

Format: exchio c1c2

These parameters specify how exchange potentials are to be read/written and

manipulated (stored in random access memory). The program always keeps

all orbitals and Coulomb potentials in the memory. If computer resources

are adequate all exchange potentials can also be kept there. However, during

the relaxation of a particular orbital only a fraction of exchange potentials

is needed. Thus all exchange potentials can be kept on disk as separate ﬁles

(named fort.31, fort.32, . . . during a run) and only relevant ones are being

retrieved when necessary.

The possible values of c1and c2are in-one,in-many,out-one and out-many.

Possible combinations are

•exchio in-many out-many – read exchange potentials as separate ﬁles

and write them back as separate ﬁles

•exchio in-one out-many – read all exchange potentials in one ﬁle but

write them out as separate ﬁles

•exchio in-many out-one – read all exchange potentials separately but

write them out as a single ﬁle

•exchio in-one out-one – read and write exchange potentials in the form

of a single ﬁle (same as i2= 3); this is the default

FEFIELD

Format: feﬁeld r

r: a strength of an external static electric ﬁeld directed along the internuclear

axis (in atomic units)

FERMI

Format: fermi rArB

When this label is present the Fermi nuclear charge distribution is used. Op-

tional parameters rAand rBdeﬁne the atomic masses (in amu) of nuclei A

and B. If omitted the corresponding values are taken from the table of atomic

masses compiled by Wapstra and Audi (see blk data).

FIXORB

Format: ﬁxorb [i1[i2. . . ]i40 ]

This label is used to specify orbitals to be kept frozen during SCF/SOR process

(they are not being renormalized nor orthogonalized during the process). i1,i2,

. . . are the numbers of these orbitals as they appear on the program’s listing, i.e.

their order is reversed to that used when deﬁning the electronic conﬁguration

(see the conﬁg card). Up to 40 diﬀerent orbitals can be set at a time. Use the

bare label to keep all orbitals frozen.

FIXCOUL

Format: ﬁxcoul

If this label is present then all Coulomb potentials are kept frozen during the

SCF/SOR process.

FIXEXCH

Format: ﬁxexch

If this label is present then all exchange potentials are kept frozen during the

SCF/SOR process.

GAUSS

Format: gauss rArB

When this label is present the Gauss nuclear charge distribution is used. Op-

tional parameters rAand rBdeﬁne the atomic masses (in amu) of nuclei A

and B. If omitted, the corresponding values are taken from the table of atomic

masses compiled by Wapstra and Audi (see blk data).

HOMO

Format: homo

This label is used to impose explicitly D∞hsymmetry upon orbitals of homonu-

clear molecules in order to improve SCF/SOR convergence.

INOUT

Format: inout c1c2

The x2dhf program can be compiled to support calculation using three diﬀerent

combinations of integer/real data types: i32 (4-byte integers, 8-byte reals),

i64 (8-byte integers, 8-byte reals) and r128 (8-byte integers, 16-byte reals);

see src/Makeﬁle.am for details. Strings c1and c2determine the combination

appropriate for the format of input and output data, respectively, and each

string can be i32, i64 or r128.

In order to facilitate exchange of binary data generated on machines of diﬀer-

ent architectures or using diﬀerent compilers additional formats are available,

namely i32f, i64f or r128f which allow to export/import data in formatted

instead of unformatted form.

INTERP

Format: interp

Use this label to change the grid between separate runs of the program. Note

that only a number of grid points in one of the variables or R∞can be changed

at a time.

LCAO

Format: lcao [i]

If the source of orbitals is declared as hydrogen then this card must be present.

In such a case the initialization of each of the orbitals has to be deﬁned in terms

of the linear combination of atom centred hydrogen-like functions. For each

orbital include a card of the following format (make sure that the order of

orbitals should match the order speciﬁed under the conﬁg label):

Format: cAnAlAζAcBnBlBζBi1[i2]

where

cA– relative mixing coeﬃcient for a hydrogenic orbital on the ZAcentre

(real),

nA– its principle quantum number (integer)

lA– its orbital quantum number (integer)

ζA– the eﬀective nuclear charge if i= 1 (default) or a screening pa-

rameter if i= 2 (real)

cB– relative mixing coeﬃcient for a hydrogenic orbital on the ZB

centre (real),

nB– its principle quantum number (integer)

lB– its orbital quantum number (integer)

ζB– the eﬀective nuclear charge if i= 1 (default) or a screening pa-

rameter if i= 2 (real)

i1– set to 1 to indicate that this orbital should be initialized as a linear

combination of hydrogenic functions and not taken from a disk orbital

ﬁle (integer). When the source of orbitals is declared as hydrogen these

ﬂags are ignored. When the source of orbitals is declared as old and the

orbital and Coulomb potential data ﬁles contain fewer functions than

deﬁned by conﬁg card this ﬂag can be used to indicate which orbitals

are missing in orbital data ﬁles and require initialization. This can

be useful when generating virtual orbitals for a potential formed from

already given set of converged orbitals.

In case when the source of orbitals is declared as old and all orbitals are

already deﬁned this ﬂag can be used in DFT calculations to generate

virtual orbitals for some sort of a local potential build from a given

subset of orbitals that are kept frozen during the SCF process (see

ﬁxorb label)

i2– a number of successive over-relaxations for a given orbital (integer);

if omitted is set to 10

The mixing coeﬃcients are normalized so that |cA|+|cB|= 1

MCSOR

Format: mcsor [i1[i2] ]

Selects the MCSOR method for solving the Poisson equations for orbitals and

potentials (default) and changes the value of the MCSOR relaxation sweeps

during a single SCF cycle for orbitals (i1) and potentials (i2); by default i1=

i2= 10.

METHOD

Format: method c

Select the type of calculation.

c: HF – the Hartree-Fock method

c: DFT – the Hartree-Fock method with the Xα exchange potential (α=

2/3); see the dft label to choose another exchange or correlation potential

c: HFS – the Hartree-Fock-Slater method (Hartree-Fock with the Xα ex-

change potential) with an optimum value of the αparameter (see blk--

data.inc for details)

c: OED – One Electron Diatomic ground and excited states can be calculated

for the Coulomb potential in the prolate spheroidal coordinates (default).

It is also possible to specify the Coulomb and Krammers-Henneberger

potentials in cylindrical coordinates (see the poth3,potkh,potharm2,

potharm3 labels, respectively). When more than one orbital is speciﬁed

calculations are carried out as if in the case of a multielectron system.3

c: SCMC – the Hartree-Fock method with Xα exchange where the αparam-

eter is calculated according to the self-consistent multiplicative constant

method4

MULTIPOL

Format: multipol r[i]

r: if r > 0 multipole moment expansion coeﬃcients are recalculated when

the maximum error in orbital energy is changed by r(the default value

is 1.15; see setDefaults). When the maximum error is less than rthe

coeﬃcients are recalculated at least every i2SCF iterations as deﬁned by

the SCF label. To suppress recalculation of the coeﬃcients set rto a

negative real number. This is useful when generating potentials from a set

of ﬁxed orbitals, e.g. from GAUSSIAN orbitals.

i: number of terms in the multipole expansion used to calculate boundary

value for potentials (2 ≤i≤8 and the default is 4).

OMEGA

Format: omega ωorb [ωpot ]

One or two real numbers setting overrelaxation parameters for relaxation of

orbitals and potentials. The negative value of ωorb/ωpot indicates that its value

should be set to a near optimum value obtained from a semiempirical formula

(see initCBlocks and setomega for details).

OMEGAOPT

Format: omegaopt [ i [ rorb [rpot ] ] ]

Optional integer parameter can be set to 1 (default) or 2. In the former case

rather conservative (safe) ωvalues are chosen (this is equivalent to using the

3In this type of calculations convergence rates diﬀer greatly between orbitals. Therefore, if for a given

orbital the orbital energy threshold is reached it is being frozen.

4V.V.Karasiev and E.V.Ludenia, Self-consistent multiplicative constant method for the exchange en-

ergy in density functional theory, Phys. Rev. A 65 (2002) 062510.

omega card with the negative values of the parameters). In the latter case

somewhat better values are chosen but faster convergence is to be expected

only when good initial estimates of orbitals and potentials are available or when

calculations with ﬁxed orbitals or potentials are performed. The near-optimal ω

values obtained from a semi-empirical formula are scaled down to produce ﬁnal

values used by the program. The default values of the scaling factors for the

orbital and potential overrelaxation parameters (0.986 and 0.997, respectively)

can be changed by setting rorb and rpot to their desired values.

ORDER

Format: order [i]

An integer deﬁning the ordering of mesh points: 1 – natural column-wise, 2

– ’middle’ type of sweep (default), 3 – natural row-wise, 4 – reversed natural

column-wise (see mesh routine for details)

POTGSZ

Format: potgsz

When the OED method is chosen then this label selects a model potential due

to Green, Sellin and Zachor.5For a given atom this potential produces HF-

like orbitals but it was found useful in ﬁnding decent starting orbitals for any

molecular system.

POTGSZG

Format: potgszg

When the OED method is chosen then this label selects a model potential due

to Green, Sellin and Zachor and the Gauss nuclear charge distribution. For

a given atom this potential produces HF-like orbitals but it was found useful

in ﬁnding decent starting orbitals for any molecular system.

POTHARM2

Format: potharm2 ω

When the OED method is chosen then this label selects a two-dimensional

model potential of the form −ω2(x2+y2), where ωis a strength of the potential

(real).6.

5A.E.S.Green, D.L.Sellin and A.S.Zachor, Analytic Independent-Particle Model for Atoms, Phys. Rev.

184 (1969) 1-9.

6This parameter ωshould not be confused with the overrelaxation parameter of the SOR method.

POTHARM3

Format: potharm3 ω

When the OED method is chosen then this label selects a three-dimensional

model potential of the form −ω2(x2+y2+z2), where ωis a strength of the

potential (real).

POTH3

Format: poth3 m a V0

When the OED method is chosen then this label selects a two-dimensional

model potential of the form −V0/√a2+r2(see kh.c for details). The following

parameters can be set

m: magnatic quantum number of a state (integer)

a: width of the model potential (real)

V0: depth of the model potential (real)

In order to get the hydrogen Coulomb potential set a= 0.0 and V0= 1.0. Set

a > 0 and V > 0 to choose its smoothed variant.

POTKH

Format: potkh m ε ω [a[V0[N]]]

When the OED method is chosen then this label selects the Krammers-Henneberger

potential (see routine kh.c for details). The following parameters can be set

m: magnatic quantum number of a state being calculated (integer)

ε: laser intensity (real)

ω: laser cycle frequency (real)

a: original (before averaging over one laser cycle) soft-core potential width (a

positive real number, by default a= 1.0)

V0: original soft-core potential depth (by default V0= 1.0)

N: number of intervals in the Simpson quadrature (an integer, N= 1000 by

default)

Format: print [i1[i2. . . ]i40 ]

Up to 40 diﬀerent printing ﬂags can be set at a time. If the integer ikis

encountered the printing ﬂag ikis set, i.e. iprint(ik) = 1 (1 ≤ik<999). These

are used to generate additional printouts by adding the lines of the form

if (iprint(ik).eq.1) then

print *, ‘‘printing something ...’’

...

endif

Set

•i1= 110 to print a total radial density relative to the centre A along the

internuclear axis (−R∞≤z≤ −R/2)

•i2= 111 to print a total radial density relative to the centre B along the

internuclear axis (R/2≤z≤R∞)

See inputData for a list of available ﬂags.

PRTEVERY

Format: prtevery i1i2

Routine pmtx can be used to output two-dimensional arrays in a tabular row-

wise form with every i1-th row and i2-th column being printed (by default every

10th row and column is selected)

SCF

Format: scf [i1[i2[i3[i4[i5]]]]]

i1: maximum number of scf iterations (default 1000); to skip the scf step set

i1to a negative integer,

i2: every i2SCF iterations orbitals and potentials are saved to disk (default

20). If i2= 0 functions are saved on disk upon completion of the SCF

process. If i2<0 functions are never written to disk,

i3: if the maximum error in orbital energy is less than 10−i3than the SCF

process is terminated (10 by default),

i4: if the maximum error in orbital norm is less than 10−i4than SCF process

is terminated (10 by default),

i5: the level of output during SCF process

–i5= 1 – the orbital energy, the diﬀerence between its current and

previous value, the normalization error and the (absolute) value of the

largest overlap integral between the current orbital and all the lower

lying ones of the same symmetry (the value is zero for the lowest

orbitals of each symmetry) is printed for every orbital in every SCF

iteration

–i5= 2 – the orbital energy, the diﬀerence between its current and

previous value and the normalization error is printed for the worst

converged orbital in energy (ﬁrst line) and norm (second line) in every

SCF iteration (default)

–i5= 3 – the orbital energy, the diﬀerence between its current and

previous values and the normalization error is printed for the worst

converged orbital in energy (ﬁrst line) and norm (second line) every

i2iterations. Printing of “... multipole moment expansion coeﬃcients

(re)calculated ...” communique is suppressed

Total energy is printed every i2iterations.

SOR

Format: sor [i1[i2] ]

Selects the SOR method for solving the Poisson equations for orbitals and

potentials (default) and changes the value of the SOR relaxation sweeps during

a single SCF cycle for orbitals (i1) and potentials (i2); by default i1=i2= 10.

XALPHA

Format: xalpha α

This label allows to change the αparameter of the LDA potential (see the

HFS/DFT method); by default αis set to 2/3.

1.3 Deprecated labels

The following labels have been replaced by others but are supported for backward com-

patibility with the previous version of the input data:

INITIAL (deprecated, use ORBPOT and LCAO instead)

Format: initial i1[i2[i3] ]

i1: determine the initial source of orbitals and potentials:

–i1= 1 – molecular orbitals are formed as a linear combination of hydro-

genic functions on centres Aand B(see the description of i3for further

details); in the case of HF or HFS calculations Coulomb (exchange)

potentials are approximated as a linear combination of Thomas-Fermi

(1/r) potentials at the two centres; if method OED is chosen the po-

tential function is approximated as a linear combination of ZA/r1and

ZB/r2terms and the exchange potentials are set to zero

–i1= 2 – GAUSSIAN output is used to retrieve exponents and expan-

sion coeﬃcients of (uncontracted) molecular orbitals (it is assumed

that the output is contained in gaussian.out and gaussian.pun ﬁles)

and Coulomb and exchange potentials are initialized as in i1= 1 case;

see prepGauss for more details

–i1= 3 – GAUSSIAN customized output is used to retrieve exponents

and expansion coeﬃcients of molecular orbitals (it is assumed that the

output is contained in gaussianc.out ﬁle) and Coulomb and exchange

potentials are initialized as in i1= 1 case; see prepGaussCust for

details

–i1= 5 – initial orbitals, Coulomb and exchange potentials are retrieved

from disk ﬁles (2dhf input.orb, 2dhf input.coul and 2dhf input.exch,

respectively) created in a previous run. Data deﬁning the case are

retrieved from a 2dhf input.dat textﬁle.

–i1= 6 – orbitals and Coulomb potentials are retrieved from disk ﬁles

and exchange potentials are initialized as in i1= 1 case (convenient

when going from HFS to HF calculations)

–i1= 11 – radial Hartree-Fock orbitals for the centre A and B are

retrieved from disk ﬁles 1dhf inputA.orb and 1dhf inputB.orb, respec-

tively, and Coulomb and exchange potentials are initialized as in the

hydrogen case (see routine initHF for details).

–i1= 55 – initial orbitals and potentials are retrieved from disk ﬁles but

the content of a 2dhf input.dat ﬁle is retrieved from a 2dhf input.orb

(binary) ﬁle. Use this value when reading binary data generated by

older versions of the program.

i2: speciﬁes how exchange potentials are to be read/written and manipulated

(stored in random access memory). The program always keeps all orbitals

and Coulomb potentials in the memory. If computer resources are ade-

quate all exchange potentials can also be kept there. However, during the

relaxation of a particular orbital only a fraction of exchange potentials

is needed. Thus all exchange potentials can be kept on disk as separate

ﬁles (named fort.31, fort.32, . . . during a run) and only relevant ones are

being retrieved when necessary.

–i2= 0 – read exchange potentials as separate ﬁles and write them back

as separate ﬁles

–i2= 1 – read all exchange potentials in a ﬁle but write them out

as separate ﬁles

–i2= 2 – read all exchange potentials separately but write them out as

a single ﬁle

–i2= 3 – read and write exchange potentials in the form of a single ﬁle

(default)

i3: if i1= 1 then this parameter must be set to 1 or 2 (if omitted it is set to 1).

In such a case the initialization of each of the orbitals has to be deﬁned in

terms of the linear combination of atom centered hydrogen-like functions

For each orbital include a card of the following format (make sure that the

order of orbitals should match the order speciﬁed under the conﬁg label):

Format: cAnAlAζAcBnBlBζBi1[i2]

where

cA– relative mixing coeﬃcient for a hydrogenic orbital on the ZA

centre (real),

nA– its principle quantum number (integer)

lA– its orbital quantum number (integer)

ζA– the eﬀective nuclear charge if i3= 1 or a screening parameter

if i3= 2 (real)

cB– relative mixing coeﬃcient for a hydrogenic orbital on the ZB

centre (real),

nB– its principle quantum number (integer)

lB– its orbital quantum number (integer)

ζB– the eﬀective nuclear charge if i3= 1 or a screening parameter

if i3= 2 (real)

i1– set to 1 to freeze the orbital during SCF; otherwise set to 0

(integer)

i2– a number of successive over-relaxations for a given orbital (in-

teger); if omitted is set to 10

For other values of i1than 1 the orbital cards can be omitted but then

the i3parameter must be set to 0.

FIX

Format: ﬁx [i1[i2[i3]]]

If i1,i2or i3are set to 1 then orbitals, Coulomb potentials or exchange poten-

tials, respectively, are kept frozen during the SCF/SOR process (the respective

default values are 0, 0 and 2). If i3= 2 then exchange potentials are relaxed

only once during an SCF cycle. i2and i3cannot be set to 1 if hydrogenic

orbitals are used to initiate the orbitals.

OMEGA

Format: omega

ωorb

ωpot

Two real numbers setting over-relaxation parameters for relaxation of orbitals

and potentials. The negative value of a given parameter indicates that its value

should be set to a near optimum value obtained from a semiempirical formula

(see initCBlocks and setomega for details).

2 Examples of input data

1. 2S ground state of the Th+89 one-electron system (see examples/oed/th+89/th+89-

1s.lst).

TITLE Th+89 point/finite nucleus R = 2.5

METHOD OED

NUCLEI 90.0 0.0 2.0

CONFIG 89

1 sigma + end

GRID 169 2.5

orbpot hydrogen

lcao

1.0 1 0 90.0 0.0 1 0 1.0 0

SCF 30 10 8

omega 1.80 1.80

! fermi 232.0 0.0

STOP

2. First excited 2S state of the Th+89 one-electron system (see examples/oed/th+89/-

th+89 2s.lst).

TITLE Th+89 point/finite nucleus R = 2.5

METHOD OED

NUCLEI 90.0 0.0 2.0

CONFIG 88

1 sigma +

1 sigma + end

GRID 169 2.5

orbpot old

lcao

1.0 2 0 90.0 0.0 1 0 1.0 0

1.0 1 0 90.0 0.0 1 0 1.0 1 ! 1s orbital must be kept frozen

SCF 30 10 8

omega 1.80 1.80

!fermi 232.0 0.0

STOP

3. Hartree-Fock ground state of the beryllium atom calculated in two consecutive steps

(see examples/be/be.lst and examples/be/be-1.lst).

TITLE Be R_inf=35.0 bohr R = 2.3860 bohr

METHOD hf

NUCLEI 4.0 0.0 2.386

CONFIG 0

2 sigma end

grid 169 35.0

orbpot hydrogen

lcao

1.0 2 0 4.0 0.0 1 0 9.0 0

1.0 1 0 4.0 0.0 1 0 9.0 0

SCF 300 10 4 4 1

! note that omega for orbitals and potentials is set automatically

! omega -1.92 -1.87

stop

TITLE Be R_inf=35.0 bohr R = 2.3860 bohr

METHOD hf

NUCLEI 4.0 0.0 2.386

CONFIG 0

2 sigma end

grid 169 35.0

orbpot old

lcao

1.0 2 0 4.0 0.0 1 0 9.0 0

1.0 1 0 4.0 0.0 1 0 9.0 0

SCF 3000 10 8 8 1

stop

4. Hartree-Fock ground state energy of the hydrogen molecule (see examples/h2/h2.lst).

title H2: R = 2.0au !!!!!

method hf

!nuclei 1.0 1.0 1.4

nuclei 1.0 1.0 2.0

!homo

!break

config 0

1 sigma g end

grid 169 40.0

orbpot hydrogen

lcao

1.0 1 0 1.0 1.0 1 0 1.0 0

scf 400 10 16 16

omega 1.95 -1.87

stop

5. Hartree-Fock ground state of the BF molecule (see examples/bf/bf init2.lst).

6. HF calculations for the lowest 3Pstate of the carbon atom (see examples/c/c.lst).

title 3P C R = 2.386

method hf

nuclei 6.0 0.0 2.386

config 0

1 pi + . +

1 sigma

1 sigma end

grid 169 30.0

orbpot hydrogen

lcao

1.0 2 1 5.0 0.0 1 0 9.0 0

1.0 2 0 6.0 0.0 1 0 9.0 0

1.0 1 0 6.0 0.0 1 0 9.0 0

scf 400 20 12 12

omega 1.80 1.87

stop

7. HF calculations for the lowest 2Pstate of the C+ion (see examples/c+/c+.lst).

title 1P C+ R = 2.386

method hf

nuclei 6.0 0.0 2.386

config 1

1 pi +

1 sigma

1 sigma end

grid 169 30.0

orbpot hydrogen ! orpot old ! using 3P results

lcao

1.0 2 1 5.0 0.0 1 0 9.0 0

1.0 2 0 6.0 0.0 1 0 9.0 0

1.0 1 0 6.0 0.0 1 0 9.0 0

scf 500 20 7

!omega 1.82 1.87

stop

8. HF calculations for the lowest state of the C2molecule (see examples/c2/c2a.lst and

examples/c2/c2b.lst).

TITLE C2 R = 2.358

METHOD HF

NUCLEI 6.0 6.0 2.358

homo

!break

CONFIG 0

1 pi u

1 sigma u

1 sigma g

1 sigma u

1 sigma g end

GRID 169 40.0

orbpot gauss

! just a couple of dozen iterations to start with

SCF 50 10 10 10 1

! note the small value of overrelaxation parameter for orbitals

! often initial scf/sor iterations require slower convergence rates

omega 1.25 1.85

STOP

TITLE C2 R = 2.358

METHOD HF

NUCLEI 6.0 6.0 2.358

homo

!break

CONFIG 0

1 pi u

1 sigma u

1 sigma g

1 sigma u

1 sigma g end

GRID 169 40.0

orbpot old

SCF 500 10 10 10 1

! try a negative value for the second overrelaxation parameter to get

its more or less optimal value

omega 1.75 -1.85

STOP

9. HF calculations for the lowest state of the N2molecule (see examples/n2/n2.lst).

TITLE N2 R = 2.068

METHOD HF

NUCLEI 7.0 7.0 2.068

homo

CONFIG 0

1 pi u

1 sigma g

1 sigma u

2 sigma g

1 sigma u end

GRID 169 40.0

orbpot hydrogen

lcao

0.5 2 1 5.0 0.5 2 1 5.0 0

1.5 2 1 5.0 0.0 2 1 5.0 0

0.5 2 0 7.0 -0.5 2 0 7.0 0

0.5 2 0 7.0 0.5 2 0 7.0 0

0.5 1 0 7.0 0.5 1 0 7.0 0

0.5 1 0 7.0 -0.5 1 0 7.0 0

SCF 2000 10 10 10 2

omega 1.65 1.85

STOP

10. HF calculations for the lowest state of the F2molecule (see examples/f2/f2.lst).

title F2 R = 2.668 (1.4118449A)

method hf

nuclei 9.0 9.0 2.668

homo

config 0

1 pi g

1 pi u

1 sigma g

1 sigma u

1 sigma g

1 sigma u

1 sigma g end

grid 169 50.0

!initial 2 3 1

orbpot gauss

lcao

0.5 2 1 5.0 0.5 2 1 5.0 0

0.5 2 1 5.0 -0.5 2 1 5.0 0

0.5 2 1 6.0 0.5 2 1 6.0 0

0.5 2 0 8.0 -0.5 2 0 8.0 0

0.5 2 0 8.0 0.5 2 0 8.0 0

0.5 1 0 9.0 -0.5 1 0 9.0 0

0.5 1 0 9.0 0.5 1 0 9.0 0

scf 2000 10 10 10 1

conv 1000

omega 1.94 -1.95

stop

11. A series of HF calculations for the the F H in external static electric ﬁeld.

(a) no external ﬁeld (see examples/fh/fh-0.lst)

title FH R = 1.7328

method hf

nuclei 9.0 1.0 1.7328

config 0

1 pi

3 sigma end

grid 223 150.0

orbpot hydrogen

lcao

1.0 2 1 7.0 0.0 2 1 5.0 0

1.0 2 1 7.0 0.0 2 0 8.0 0

1.0 2 0 8.0 0.0 1 0 9.0 0

1.0 1 0 9.0 0.0 1 0 9.0 0

scf 2000 10 13 13 1

!fefield +0.0001

omega 1.90 -1.85

stop

(b) ﬁeld strength -0.0001 a.u. (see examples/fh/fh-m1.lst)

title FH R = 1.7328

method hf

nuclei 9.0 1.0 1.7328

config 0

1 pi

3 sigma end

grid 223 150.0

orbpot old

lcao

1.0 2 1 7.0 0.0 2 1 5.0 0

1.0 2 1 7.0 0.0 2 0 8.0 0

1.0 2 0 8.0 0.0 1 0 9.0 0

1.0 1 0 9.0 0.0 1 0 9.0 0

scf 2000 10 13 13 1

fefield -0.0001

omega 1.90 -1.85

stop

title FH R = 1.7328

method hf

nuclei 9.0 1.0 1.7328

config 0

1 pi

3 sigma end

grid 223 150.0

orbpot old

lcao

1.0 2 1 7.0 0.0 2 1 5.0 0

1.0 2 1 7.0 0.0 2 0 8.0 0

1.0 2 0 8.0 0.0 1 0 9.0 0

1.0 1 0 9.0 0.0 1 0 9.0 0

scf 2000 10 13 13 1

fefield -0.0002

omega 1.90 -1.85

stop

(d) ﬁeld strength +0.0001 a.u. (see examples/fh/fh-p1.lst)

title FH R = 1.7328

method hf

nuclei 9.0 1.0 1.7328

config 0

1 pi

3 sigma end

grid 223 150.0

!interp

orbpot old

lcao

1.0 2 1 7.0 0.0 2 1 5.0 0

1.0 2 1 7.0 0.0 2 0 8.0 0

1.0 2 0 8.0 0.0 1 0 9.0 0

1.0 1 0 9.0 0.0 1 0 9.0 0

scf 2000 10 13 13 1

fefield +0.0001

omega 1.90 -1.85

stop

(e) ﬁeld strength 0.0002 a.u. (see examples/fh/fh-p2.lst)

title FH R = 1.7328

method hf

nuclei 9.0 1.0 1.7328

config 0

1 pi

3 sigma end

grid 223 150.0

orbpot old

lcao

1.0 2 1 7.0 0.0 2 1 5.0 0

1.0 2 1 7.0 0.0 2 0 8.0 0

1.0 2 0 8.0 0.0 1 0 9.0 0

1.0 1 0 9.0 0.0 1 0 9.0 0

scf 2000 10 13 13 1

fefield +0.0002

omega 1.90 -1.85

stop

12. DFT calculations with LDA and LYP functionals (see examples/dft/be-1.lst and

examples/dft/be-2.lst).

title Be DFT R_inf=25.0 bohr R = 2.3860 bohr

method dft

dft b88 ! lyp

nuclei 4.0 0.0 2.386

config 0

2 sigma end

grid 169 35.0

orbpot hydrogen

lcao

1.0 2 0 4.0 0.0 1 0 9.0 0

1.0 1 0 4.0 0.0 1 0 9.0 0

scf 1000 10 8 8 1

! note that omega for potentials is set automatically

omega 1.80 -1.82

stop

TITLE Be DFT R_inf=25.0 bohr R = 2.3860 bohr

METHOD dft

dft b88 lyp

NUCLEI 4.0 0.0 2.386

CONFIG 0

2 sigma end

grid 169 35.0

orbpot old

lcao

1.0 2 0 4.0 0.0 1 0 9.0 0

1.0 1 0 4.0 0.0 1 0 9.0 0

SCF 1000 10 8 8 1

! note that omega for potentials is set automatically

omega 1.80 -1.82

stop

13. Two lowest states of the 2D hydrogenic harmonic potential (see examples/oed/h3/-

h3-1.lst and examples/oed/h3/h3-2.lst).

title Hydrogen in 2D harmonic potential: ground sigma state

method OED

nuclei 1.0 1.0 2.0

poth3 0 0.0 1.0

homo

config 0

1 sigma g + end

grid 269 50.0

!inout i32 i32f

orbpot hydrogen

lcao

1.0 1 0 1.0 1.0 1 0 1.0 0

scf 1000 10 10 10 1

omega 1.92 -1.87

stop

title Hydrogen in 2D harmonic potential: 1st excited sigma state

method OED

nuclei 1.0 1.0 2.0

poth3 0 0.0 1.0

homo

config 0

1 sigma u +

1 sigma g + end

grid 269 50.0

!inout i32 i32f

orbpot hydrogen

lcao

1.0 2 1 1.0 0.0 1 0 1.0 0

1.0 1 0 1.0 1.0 1 0 1.0 1

scf 1000 10 10 10 1

omega 1.92 -1.87

stop

14. Lowest state of the Krammers-Henneberger potential (see examples/oed/kh/kh.lst).

title Hydrogen in 2D Krammers-Henneberger potential: ground sigma state

method OED

nuclei 1.0 1.0 2.0

potkh 0 5.0 1.0 1.0 1.0 500 ! m eps w a V0 N

homo

config 0

1 sigma g + end

grid 269 50.0

initial 1 3 1

1.0 1 0 1.0 1.0 1 0 1.0 0

scf 1000 10 10 10 1

omega 1.92 -1.87

stop

15. SCMC ground state calculations for the beryllium atom (see examples/scmc/be-

scmc.lst).

TITLE Be SCMC R_inf=35.0 bohr R = 2.3860 bohr

METHOD scmc

NUCLEI 4.0 0.0 2.386

CONFIG 0

2 sigma end

GRID 91 35.0

orbpot hydrogen

lcao

1.0 2 0 4.0 0.0 1 0 9.0 0

1.0 1 0 4.0 0.0 1 0 9.0 0

SCF 1000 10 8 8 1

omega 1.82 -1.82

stop

3 Program’s data ﬁles

There are several standard names used by the program to keep track of its input and

output disk ﬁles. Normally the program writes out the data in the course of computations

and upon its completion into the following disk ﬁles:

2dhf output.dat (a text (ASCII) ﬁle) containing the title of a case, the time and date

of its commencement, the number of mesh points, the internuclear distance, the

charges of nuclei and the number of orbitals, electrons and exchange potentials (see

writeDisk for details),

2dhf output.orb (a binary ﬁle) containing molecular orbitals (in the order speciﬁed

by the input data following conﬁg label) followed by their normalization factors,

orbital energies, Lagrange multipliers and multipole moment expansion coeﬃcients

(see write),

2dhf output.coul (a binary ﬁle) containing corresponding Coulomb potentials and

2dhf output.exch (a binary ﬁle) containing all exchange potentials if the exchio [in-

one|in-many] out-one card is present

fort.31, fort.32, ... (binary ﬁles) each containing the exchange potential required

for a particular pair of orbitals if the exchio in-many [out-one|out-many] or

exchio [in-one|in-many] out-many card is present

These ﬁles can be used to restart a given case or run another with slightly modiﬁed pa-

rameters. If orbpot old card is present orbitals are retrieved from 2dhf input.orb ﬁle,

Coulomb potentials from 2dhf input.coul and exchange potentials from 2dhf input.exch

ﬁle (or fort.31,fort 32, . . . ﬁles, if exchio in-many [out-one|out-many]). Note that

there is only one set of fort ﬁles.

4 How to run the program?

In order to simplify the usage of the program, the xhf script (see tests/xhf) is provided

to facilitate handling of the disk ﬁles. The command xhf can be envoked with one, two

or three parameters. There are two basic modes of its usage:

./xhf file1 file2

runs x2dhf reading input data from file1.data ﬁle and writing text data describ-

ing the case into file2.dat ﬁle and binary data with orbitals and potentials into

file2.orb,file2.coul and file2.exch ﬁles.

./xhf file1 fil2 file3

runs x2dhf reading input data from file1.data and initial orbitals and potentials

from file2.dat,file2.orb,file2.coul and file2.exch ﬁles and writing result-

ing data into file3.dat,file3.orb,file3.coul and file3.exch ﬁles.

If, for example, be.data ﬁle contains input data for the beryllium atom (see Example 3)

then

./xhf be be-1

starts and performs the ﬁrst 300 SCF iterations. Type

./xhf be-1 be-1 be-2

to continue calculations. In order to converge the SCF process even better increase the

convergence parameters (see the scf label) and use the following command

./xhf be-1 be-2 be-1

In addition, the xhf script can be used to perform the following tasks:

./xhf stop

creats stop x2dhf ﬁle in a current directory to stop a running program

./xhf mkgauss filename

creates symbolic links gaussian.out and gaussian.pun to ﬁles filename.out and

filename.pun, respectively (see e.g. Example 5).

./xhf rmgauss

removes gaussian.out and gaussian.pun ﬁles from a current directory

./xhf clean

removes *.[dat|orb|coul|exch] ﬁles

Use ./xhf -h|help to get the complete list of available options.

5 Useful hints

1. The program should be easy to use provided you can start a calculation for a speciﬁc

system. You should not encounter any serious problems when the system contains

atoms from the ﬁrst two rows of the Periodic Table. Then even the rough hydrogenic

estimates of the orbitals should prove adequate and after the initial couple of dozen

of iterations a smooth convergence should set in.

If, however, a system contains more than 15-20 electrons the initial estimates of the

orbitals have to be good enough to avoid divergences. Then, you have to choose the

parameters of the hydrogenic orbitals carefully or perform the ﬁnite basis set cal-

culations using the Gaussian program to provide the initialization data for orbitals

(see Example 5).

One can also use HFS method to produce initial estimates of orbitals and Coulomb

potentials. For example, to start calculations for the neon atom one can use the

following input data:

This input produces good enough HFS initial estimates so that the calculations can

be continued at the HF level with the corresponding input:

One can also use HF method with some model potential, e.g. the model potential

of Green, Sellin, Zachor (label POTGSZ).

2. At the very beginning set the maximum number of SCF iterations to something

between 20 and 50 and/or impose crude convergence criteria for the orbital energy

and normalization.

3. In case of convergence problems try to perform calculations on a sparser grid. For

example, the [61 ×79/30] grid is suﬃcient to check the quality of the initial data

for the Ne2system.

4. Choose smaller values of the relaxation parameters (1.7≤ω≤1.85) to avoid

divergences in the ﬁrst few dozens of SCF iterations (rarely the values as small

as 1.2 may be needed). Subsequently the parameters should be increased to their

(near) optimum value (see the omega label and Example 8).

It is possible to set ωpot to its near-optimal value by calculating it from a semiem-

pirical formula; see the omega label. As a rule of thumb the optimal value of the

orbital relaxation parameter is somewhat smaller and, by default, is obtained by

scaling the ωpot value by 0.98 (see setDefaults).

5. How to stop the program gracefully during a lengthy calculation without killing

the process and interrupting disk read/write operations? All you have to do is to

create a (zero length) ﬁle named stop x2dhf in a working directory by typing ./xhf

stop (you can also use the Unix touch command to this end). The program stops

whenever this ﬁle is detected upon the completion of a current orbital/potential

relaxation.

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