Direct momentum-space calculations for molecules

Jouma:ofhfolecularStruciure.120(1986)343-350

343

THEOCHEM

ElswierSciencePublishersRV.,Amsterdam-_PrintedinTheNetherlands

DIRECT MOMENTUM-SPACE CALCULATIONS FOR MOLECULES G. BERTHIER’ , M. DEFRANCESCHI’, J . NAVAZA3, M. SUARD’; G. TSOUCARIS3 ‘Laboratoire de Biochimie F 75005 PARIS

Thgorique

de l’I.B.P.C.,

13 rue

Pierre

et Marie

Curie

., 1 rue Maurice Arnoux, F 92120 MONTROUGE *Laboratoire de Chimie de 1 ‘E.N.S.J.F 3 Laboratoire de Physique - Faculte de Pharmacie, Universite de Paris-Sud. rue J.B. Clement, F 97290 WATENAY MALABRY

ABSTRACT A calculation method for momentum-space wave functions, using the Fourier transforms of the Hartree-Fock equations , is presented. As shown by results on the test-systems H and H the momentum-space equations can be directly solved by numerical integ $ ation 5’echniques, includino configuration interaction effects. For a long

time,

electronic

structure

calculations

quantum-mechanical

methods have fonned

two distinct

nal

In atoms, the direct

numerical

point

loped

of view.

by Hartree

e_t al.

tions that the

symnetry

in polar

basis the

functions

The

(exponentials

: First,

system for

can be separated quently, thirties, atomic Until rical

one-electron

in the Roothaan

there

integrable

in terms

approximate from this

indeed,

but it

have to be used in order

sets

suggested

of prefixed

atomic with

orbital is

another

atomic

methods,

it

problem

comes natural and,

orbitals

conse-

as basis

both developed

the basis

ap-

coordinate

constituents

now recognized

to eliminate

under

one-electron

problems.Second,

wave function

idea,

satisfying

LCAO-SCF form 151.

is no preferential

of their

and molecular

problem,

solutions using

equa-

became more and more popular

problem,

1950

the Hartree-Fock

calculations

theory

atoms,

coupled

The valence-bond

sets

numerical

orbital

deve-

was due to the

to a one-dimension

for

however,

success

compound, where the corresponding

in three

originate

least

techniques 131 [43 before

; this

us to separate

or gaussians)

with

of molecules

to construct

functions.

years

integration

(at

of the molecular

ties

in contrast

a many-center

to us to thihk

enables

Towards 1960,

of the molecular

peculari

proach

thirty

reducing

wave function

theorem).

influence

about

of atoms which

coordinates,

of the radial DelbrUck

during

by

domains from the computatiointegration

at the SCF 1111 121 and MC-SCF level

seems to have prevailed spherical

of atcms and molecules

that

very

dependence

in the

1 arge from the

resul ts. lately,

integration

few examples

of molecular

in the ordinary

position

wave equations

solved

by direct

space were known, and they

nume-

concerned

the radial

Hartree-Fock

expansion

wave. function

of molecules

[61 171 or the corresponding

AH,,(n Iz 4) in an one-center

functrton for dihydrogen

H2 C81 and.simple

diatomic molecules C91 in prolate spheroidal coordinates. More recently, complete numerical integrations for diatomic moleaules have been presented El01 1111 1121, including

configuration

polyatomic

molecules

that the basis

interaction

in the last

1131 1141, but

seem to be unmanageable by such methods. It can be added

functions

used in LCAO- SCF and MC - SCF calculations

extended for the sake of completeness, nents should satisfy orbitals

developments

disregarding

the criteria

in order to guarantee convergence

are often

that their

on true energies

expo-

and

Cl51 C161.

MOMENTUM-SPACE SCF ANDMC-SCFFORMULATION Owing to the attractive

mathematical properties

of the Fourier

can be thought that we could overcome some of the difficulties rical integration by expressing the molecular wave equations momentumspace.

As a matter of fact,

transform,

it

preventing numein the equivalent

momentum-space has been occasionally

used,

but not with great success, for simple systems in the hope of finding rigorous solutions of the Schriidinger equations more easily, (see, e.g., 1171). Instead, two of us (J.N.

and G.T.),

using the Hartree-Fock

equation

of the H2 molecule as

an example, have shown that electronic wave functions can be computed in momentum space by starting with the Fourier transforms of the standard Quantum Chemistry methods 1181. With respect to position space, momentum-space caloulations have rather pleasant features : i) numericai wave functions, obtained without expansion in terms of basis sets ii) wave equations independent OR the existence of coordinate systems appropriate to the molecular geometry, because they have not more singularities than in one- or two-center cases iii) possibilities of electron-correlation

studies,

For a closed-shell to A nuclei tion V2

(- 2

fixed

through configuration

system of 2N electrons

in positions

interaction

or otherwise.

moving in the potential

R,, the standard Hartree-Fock

field

equations

due

in posi-

space read : - ei)

$i(r)

-

-

A

z

ai C Ir -aRal a=1

d3r’

N ej::(r’) C j=l

The corresponding

Ir

s

+

N

d3rr

C 2 j=l

+j::(r’)

cPj(r’)

[r - r’[

@i (r)

ei(r’) Oj(r) -

=

(1)

0

+I

momentum-space equations

found by Fourier transforms

are :

345

F(q) @i(P-q)

--

1

21r2

I

’

z

+j(P-q)

'ij(q)

X

2W~j(q) ~i(p - q)

C

j=l

(2)

=O

j=l

q2

where the electronic nuclei read : wij(P)

N

&

N

3 dq +& 2x2 _J q2

interaction

terms W and the “form factors”

F coming from

=J 8+:(r) cbj(r) eipmr d3r = @y(q) I$~ (q - p) d3q I A

F(q) =

(3)

C Z, eiqeRa

(4)

a=1 and the ei’s denote orbital energies. They fom equations determining the N unknown one-electron As usual in the Hartree-Fock procedure

theory,

a system of coupled functions

it is necessary

integral

9i (5).

to use an iterative

for solving

the precedi,ng momentum-space equations ; we start with N (0), from which we compute all the components of Eqs trial molecular orbitals cp. (3) (interaction terms W!?j, orbital energies e(O) etc.. .), except the first one, - ei(O)) ; in this way, we get and we divide each of thii by the quantity (p’/; the next approximation and wave functions In Eqs (3),

$14’) for the molecular

orbitals

and so on until

energies

converge.

the main calculation

which have been carried we compute functions

is a series

of convolutions

and integrals,

out here out here by the Pierce-Mustard

e(c)

for a minimumnumber of points

(pi

,

technique ej

,

Cl91 : belon-

Fk)

ging to a spherical grid centered at the origin of momentum-space. Using a Gausseach integral writes in the form Legendre technique for each coordinate,

s

f(;;) d3q=

4

C

Ai Bj Ck f (pi

, ej

(5)

) ok)

ijk

where Ai , B. , Ck are known weights corresponding

the various

coordinates.

Since,

however, contolution implies a shift in the p-space, the (c - 6) vector does not more belong to the grid chosen at the beginning, and so an interpolation procedure is required

to generate

the function

a(6 - 6) from its

values Q(s) on the

grid. This was achieved in the present work by one-dimension spline fitting. As concerns numerical integration techniques, momentum-space has an advantage : the volume where the wave functions take appreciable values is nearly independent of the size and geometry of the molecule to be studied. Only the number

of points

where the @‘s must be calculated

depends on the molecular

size.

For

the systems considered below (H2 and H3), our narrowest grid was 18 angular point and 37 radial point thick inside a sphere of radius 5 a.u. and the asymptotic

behaviour

of the @‘s is taken into account by a few outer points.

metry properties structure

The sym-

of the molecular

framework enable us to simplify the numerical a linear molecule with an inver: in the casof

of the wave function

sion center (point group Dti), the internuclear symmetry axis makes it possible to immediately integrate upon the 9 variable (i.e., a constant for C representations)

and the inversion

center yields

orbitals orbitals

according as they correspond in position space. Moreover,

the interval D,sr/Z. A momentum-space formulation

real

or pure imaginary momentum-space + or antisytnnetrical uuto symmetrical o integration overgthecf angle is limited to

can be written

down for those systems which only

require a straightforward generalization of the closed-shell Hartree-Fock theory in position space. For open-shell systems without spin intricacies, we may start either with doubly and singly occupied orbi tals, as given by the RHFRoothaan treatment

1203 or with different

orbi tals for different

spins,

according

to the

UHFtreatment suggested by one of us 1211 and by Pople and Nesbet 1221. The simplest example is a three-electron system whose occupied orbitals are automatically orthogonal and the RHFequations rather immediate in position space 1231 ; their Fourier transforms, solved below (for symmetrical H3) read :

- ell 2

al(p)

[ F(q) - W;,(q)

d3q = -L 2Tr2I q2 +- 1

w;#l) Jd3q @2(P - q)

41T2 (<

:: - W22(q) 1 cp,(P - q)

:: C F(q) - 2Wlltq)

d3q - e2) @2(p) = -$ 2ll 5 q2 + -L 4s2

J

for the doubly and singly

tW

q2

!!JJ q2

w;,(q)

occupied

I @,(P - q)

@1(p - q)

orbitals

t6b)

respectively

C241.

It is also possible to write down the equations giving the momentum-space fonctions pi of an configuration interaction by transforming the corresponding In the frame of the closed-shell multivariational equations in position space. configurational problem described

model [25]

1261, the following

by two automatically

equations,valid

orthogonal

orbi tals,are

for a two-electron obtained

:

347 a;

[ J$

(pi(P)

- 1 2s2 I

electron

are

functions

2n2 I

x$i(p-

q2

q) Nyi(q)

q2

J

q2

(P)

j#i

ai’S

9

~j (P -4)w~j(q) =ei03 Jd3q

+ aiaj

where the

3

F(q) @i(P - q) + 1

5%

(7)

.

the.coefficients

of the

to be deterniined

(i

CI expansion

and the $j’s

the

one-

= 1,2).

RESULTS AND FUTURE PROSPECTS Momentum-space Hartree-Fock le

H2 in the

standard

program was refined and u,,-

orbitals

one-electron molecular

using

we

studies

have taken

built

from 1s atomic

@I(p)

= 1’

on,

on molecu-

the computer

to compute the cg+

H2 and H3 systems. transforms

As trial

of Hiickel-like

namely

1 22 (P2 + o )

H2 sin

22 (P2 + ci )

1 + C cm (P-R)

=L

el(p)

(8)

1

(p-R)

N2

Nl

(p2 + a2)2

PI

H3 e,(p)

= i

sin

(p.R)

where R is exponent

the bond distance

and C the

linear

About 4 SCF cycle

drogen.

of H2 (or

orbitals these In the SCF level,

to cq. obtained

1 are

H - H in H2 or HS, a the value

coefficient

assigned

chosen for

to the orbital

and 8 (SCF + CI) cycles

H3) with

surmnarized in Table

’ 22 (p* + o )

N2

for

on the

the Fourier

cos (p.R)

@2(P) = f

for

to be feasible Later

and enlarged

orbitals,

Nl for

1181.

procedures

in further

Q{“),

were proved

approximation

vectorial

involved

functions orbitals

calculations

closed-shell

the

of the middle

were needed to stabilize

an accuraty

of 1 percent.

compared with

position-space

Our results

[243

calculations

1s hythe

1277

available

systems. case of molecule

He, we have performed

but also. at the MC - SCF level,

The total

energies

in position

using

E for Ii2 are close

space with

a very

large

computations one double

not only

to the corresponding atomic

basis

at the

excitation 1281.

from o + g results

In a way parallel

348

TABLE1 Total energies

of H2 and H3 at theoretical

equilibrium

H2 SCF

bond distances

d (in a.o.)

H3 SCF

Present results (momentum-space)

d 1.3834 E - 1.1343

1.4160 - 1.1529

1.684 - 1.5906

LCAOresults (positien space)

d 1.3917 E - 1.1332

1.4306 - 1.1519

1.70 - 1.5947

E281

C281

1291

to ours, numerical MC- SCF calculations based on the same model, but including up to five diexcitations in position space, g. 1141, and both achievements in position

have been presented by Laaksonen et and momentumspaces are genuine -

numerical examples of molecular MC- SCF wave functions.

Actual ly, our two-

configuration SCF energy is slightly higher than the best position it could be improved using a larger integration grid.

space value ;

In the cam!of the H3 system, we have performed only RHFcalculat+ons symmetrical form ; they give the first Fock functions

for a polyatomic

for

the

example of a pure1y numerical Hartree-

molecule

(with a slight

deviation

of the total

energy compared to the best position-space calculation 1291). Direct integnation techniques present evident advantages : they enable us to determine molecular wave functions with an accuracy comparable to the matrix calculations

using very large sets of atomic orbitals.

the polarization

effects

coming not from successive addition of basis orbitals with high angular quantum numbers, but from the iterative minimization process itself. This can be seen for instance by looking the variations of the radial functions of H3 (Fig.1). Furthermore, diffuse contributions to wave functions and electronic densities become more important when momentum-space is used, and this as concerns, for instance, COmPtOnProfile calculations The favorable aspects of momentum-space integration lecules easily

is very convenient

(Table 2). and its results

H2 and H3 lead us to wonder whether more complicated treated

in the same way. Preliminary Li2).

tomics (e.g.,

and for periodic

results

polymers (e.g.,

systems could be

for simple first-row H, linear

for modia-

system) as well

are promising. In the first case, the large orbital energies ei’s of the K shells raise difficulties as regards numerical accurary, but we can get over by the frozen position

core approximation.

space give,

the periodic dimensional

axis and continuous periodic

In the second case, the Bloch function in functions that are discrete along

by Fourier transform,

in a plane perpendicular

to the axis.

Three-

polymers should be even simpler because they have only

349

TABLE 2 Compton profiles

w ::

SCF

0.00

1.545

1.552

1.543

0.10

1.494

1.510

1.482

0.20 0.40

1.388 1.049

1.424 1.071

1.370 1.064

q

CI

0.60

0.702

0.715

0.704

1.00

0.260

0.249

0.252

1.40

0.085

0.080

0.085

x J.S. Lee J. Chem. Phys., -66, 4906 (1977) discrete wave functions 1303.

(b)

8 with respect to the (e-- trSa1 function 0 final function a0 -..-trial function 8' -.- final function 0: a) G1 ; b) e2

internuclear axis. = n/2 rd ; = R/Z rd ; = 0 rd ; = 0 rd )

360 REMERCIEMENTS Les moyens de calcul par le Conseil

Scientifique

sur

CRAY-1 utilises du C.C.V.R.

dans ce travail (Ecdle

Polytechnique

ont

@t6 attribues

- Palaiseau).

REFERENCES D.R. Hartree, Proc. Camb. Phil. Sot., 24 (1927) 89 W. Hartree, Proc. RoyFSoc., A150 (1935) 9 D. R. Hartz&and D.R. Hartree, W. Hartree and B. Swirles. Phil.Trans. Roy. Sot.. A238 (1939) 229 C. Froese, Proc. Phys. Sot., 90 (1967) 39 C.C.J. Roothaan, Rev. Mod. PhjZ., 23 (1951) 69 R.A. Buckingham, H.S.W. Massey anhS. R. Tibbs, Proc. Roy. Sot., 119 - C. Carter, Proc. Roy. Sot., A235 (1956) 321 6. Berthier, Molec. Phys.. 2v59) 225 - J. Chim. Phys., 56 (1959) 504 (For the - 2T/V ratio, read-O.982 instead of 0.742) M. Aubert, N. Bessis and G. Bessis, Phvs. Rev., A10 (1974) 51 and 61 E.A. McCullough Jr, J. Chem. Phys., 62 (1975) 39vT J. Hinze, Int. J. Quan. Chem.,S 15 p981) 69, esp. pp 84-90 See also A.D. Becke, J. Chem. Phys.,76 (1982)6037 L. Laaksonen,P. Pyykkb' and-D.Sundhox, Int. J; Quant.Chem.,-23 (1983)309 and 319 - Chem. Phys. Lett., 96 (1983) 1 L. Adamowicz and E.A. McCulloii$ Jr, J. Chem. Phys., 75 (1981) 2475 L. Laaksonen, D. Sundholm and P. Pyykkb. Chem. Phys.Ett., 105 (1984)573 B. Klahn and W. Bingel, Int. J. Quant;Chem., -11 (1977) 943 neoret. Chim.

Acta, 44 (1977)9 G. FonZS, Theoret. Chim. Acta,

59 (1981) 533 PrEE. Phys. Sot.,52

(1949)509 - R. McWeeny, ibid.,519 J. Navazaand G. Tsoucaris,Phys. Rev.,A24 (1981) 683 W.H. Pierce, Math. Tab. Aid. Camp.,-11 (T937) 244 - D. Mustard, Math. Camp., 18 (1964)578 7ZC.J. Roothaan,Rev. Mod. Phys.,32 (1960)179 G. Berthier.C. R. Acad. Sti., 23871954) 91 - J. Chim. Phys.,51 (1954)363 J.A. Pople and R.K. Nesbet,J. Em. Phys., 22 (1954)571 J. Higuchi,J. Chem. Phys., 26 (1957)151 M. Defranceschi, M. Suard an?rG.Berthier,C. R. Acad. Sci., 296 (1983)1301 Int. J. Quant.Chem.,25 (1984)863 A. Veillard,Theoret.73’iim. Acta, 4 (1966)22 A. Veillardand E. Clementi,Theor&. Chim. Acta, 7 (1967)133 M. Defrancesbtii, M. Suard and G. Berthier,C. R. AFad. Sci., 298 (1984)xxx G. Das and A.C. Wahl, J. Chem. Phys.,44 (1970)87 B. Liu, J. Chem. Phys.,58 (1983)1925 M. Defranceschiand J. DZhalle, to be published R. McWeeny and C.A. Coulson,