Frequency dependent non-linear optical properties of molecules:

alternative solution of the coupled perturbed Hartree–Fock

equations

A. Martineza,*, A. Czajaa, P. Ottoa, J. Ladika,b

aChair of Theoretical Chemistry, Friedrich-Alexander University, Erlangen-Nurnberg, Egerlandstrasse 3, 91058 Erlangen, GermanybLaboratory of the National Foundation for Cancer Research, Egerlandstrasse 3, 91058 Erlangen, Germany

Received 3 March 2002; accepted 24 April 2002

Abstract

An alternative method to solve the coupled perturbed Hartree–Fock equations is presented. The new procedure follows a

proposition made by Langhoff, Epstein and Karplus to obtain perturbed wave functions free from arbitrary phase factors in each

order of perturbation. It is based on a different orthonormalization of the perturbed wave functions than the usual one and a

correspondent selection of the Lagrangian multipliers. In this way it is possible to incorporate the orthonormalization conditions

into the set of coupled perturbed Hartree–Fock equations. The equations are solved iteratively for each perturbation order.

Calculations of dynamic NLO-properties for CH3F, CH2F2, CHF3, and water dimer are performed to verify the method. q 2002

Elsevier Science B.V. All rights reserved.

Keywords: Non-linear optical; Coupled perturbed Hartree–Fock; Hyperpolarizabilities

1. Introduction

The potential technological applications of

materials with non-linear optical (NLO) responses

have created a great interest in the past years [1].

Further the estimation of NLO-properties is also of

interest in other fields, as for instance in the

investigation of the living cell, where strong electric

fields can occur locally, with non-negligible effect on

biomolecules with large (hyper)polarizabilities and

therefore on biochemical reactions in which these

molecules participate. Theoretical predictions can be

a desirable alternative to experimental works to

measure these properties. Particularly, quantum

mechanical calculations of different precision are

being made for systems going from isolated atoms to

infinite polymers [2–3].

At the ab initio level, some theoretical approaches

to determine static and dynamic NLO properties are

already available [4–10]. The most extended treat-

ment for the calculation of these properties in

molecules is the coupled perturbed Hartree–Fock

(CPHF) approximation [6]. The CPHF system of

equations is solved iteratively to any order of the

perturbation. Particularly detailed derivations are

made by Sekino and Bartlett [7], by Rice et al. [8],

and by Karna and Dupuis [9]. They have obtained

explicit expressions for the non-linear processes up to

0166-1280/02/$ - see front matter q 2002 Elsevier Science B.V. All rights reserved.

PII: S0 16 6 -1 28 0 (0 2) 00 2 88 -9

Journal of Molecular Structure (Theochem) 589–590 (2002) 349–358

www.elsevier.com/locate/theochem

* Corresponding author. Tel.: þ49-9131-8527773; fax: þ49-

9131-8527736.

E-mail address: alberto.martinez@pctc.chemie.uni-erlangen.de

(A. Martinez).

third order. In the solution procedure proposed in their

works they used simultaneously two sets of pertur-

bation equations for the calculation of the perturbed

density matrices in each order: the set of the ‘proper’

CPHF equations (Table 1 of Ref. [9]) and the set of the

orthonormalization equations (Table 2 of Ref. [9])

In this work we propose an alternative procedure to

solve the CPHF equations by a direct incorporation of

the orthonormalization conditions in the proper CPHF

equations. This is possible by means of a different

orthonormalization of the wave functions and a

different selection of the Lagrangian multipliers.

Such a solution is already theoretically discussed by

Langhoff et al. [6] but no general formulation has been

worked out and programmed for molecular systems.

The perturbed wave functions of each order obtained

with this method are free from arbitrary phase factors

(see Langhoff et al. [6]). This property of the

perturbed wave function is advantageous for polymer

applications. In this way it is possible to derive a new

and simple method to find the derivatives of the wave

function with respect to the quasi-momentum k by

means of a transformation of the wave function from

the k-space to the direct space and back again. These

derivatives are essential for the construction of the

dipole moment operator for polymers.

In the present work we apply our procedure to four

molecules in order to show its validity comparing the

values of m, a, and b with those reported in the

literature. We have no intention to compete with

the already available methods for molecules, but to

show that our approach gives the same results. In a

subsequent paper we extend this formalism to

periodic quasi-one-dimensional systems, where we

can take full advantage of the special properties of the

perturbed wave function.

2. Theory

2.1. The coupled perturbed Hartree–Fock equations

In this section, we give a derivation of the CPHF

equations up to third order for molecules and we

present our method for their solution. For simplicity,

we restrict our derivation to some of the first, second

and third order dynamic processes.

The electronic part of the time-dependent Schro-

dinger equation for molecules in the presence of a

monochromatic oscillating optical field can be written

as

Helð~rÞ þ HSð~r; tÞ2 i›

›t

� �cð~r; tÞ ¼ 0 ð1Þ

Here Hel is the Hamilton operator for the unperturbed

molecule and HS is the perturbation operator resulting

from the interaction of the electrons with the external

electrical field

Hel ¼X

i

21

272

i 2XA

ZA

riA

þXi,k

1

rik

!ð2Þ

HSð~r; tÞ ¼ m~FðtÞ ¼ 2m~E0 cosðvtÞ ¼ m~E0ðeþiwt þ e2iwtÞ

ð3Þ

where m is the electronic dipole moment operator

m ¼ 2XNi¼1

ðe·~riÞ ð4Þ

In these equations the summation over i runs over the

N electrons of the molecule and the summation over A

runs over the nuclei.

Using Frenkel’s variational principle

d cð~r; tÞh jHelð~rÞ þ HSð~r; tÞ2 i›

›tcð~r; tÞj i ¼ 0 ð5Þ

subject to the normalization condition

›

›tkcð~r; tÞlcð~r; tÞl ¼ 0 ð6Þ

and choosing cð~r; tÞ as a single Slater determinant

(antisymmetrized product of N time-dependent spin

orbitals), we obtain the time-dependent Hartree–Fock

equations for each of the doubly occupied molecular

orbital wi:

Fð~r; tÞ2 i›

›t

� �wið~r; tÞ ¼

Xj

1ijðtÞwjð~r; tÞ ð7Þ

Notice that in this equation and the following ones the

indices i and j run only over the occupied states, from

1 to N/2. To solve Eq. (7) we use perturbation theory,

expanding F; wi and 1ij in a power series of the

parameters l a (a ¼ x; y or z, depending on the

A. Martinez et al. / Journal of Molecular Structure (Theochem) 589–590 (2002) 349–358350

direction of the electric field):

Fð~r; tÞ ¼ Fð0Þð~rÞ þX

a

laFað1Þð~r; tÞ þXa;b

lalbFabð2Þð~r; tÞ

þXa;b;c

lalblcFabcð3Þð~r; tÞ þ · · · ð8aÞ

wið~r; tÞ ¼ wð0Þi ð~rÞ þ

Xa

lawað1Þi ð~r; tÞ þ

Xa;b

lalbwabð2Þi ð~r; tÞ

þXa;b;c

lalblcwabcð3Þi ð~r; tÞ þ · · · ð8bÞ

1ijðtÞ ¼ 1ð0Þij þ

Xa

la1að1Þij ðtÞ þ

Xa;b

lalb1abð2Þij ðtÞ

þXa;b;c

lalblc1abcð3Þij ðtÞ þ · · · ð8cÞ

The introduced wave function satisfies only the

‘intermediate’ orthonormalization condition, as

suggested by Langhoff et al. [6],Dwð0Þi ð~rÞ

wjð~r; tÞE¼Dwð0Þi ð~rÞ

wð0Þj ð~rÞ

E¼ dij ð9Þ

instead of the usual orthonormalization condition

[7–9]D~wið~r; tÞ

~wjð~r; tÞE¼Dwð0Þi ð~rÞ

wð0Þj ð~rÞ

E¼ dij ð10Þ

Where we use ~wið~r; tÞ for the perturbed wave function,

that satisfies Eq. (10) (as obtained in Refs. [7–9]) to

distinguish it from our perturbed wave function

wið~r; tÞ:Collecting terms of the same order of l (for

simplicity we take 1ð0Þij ¼ 0 for i – j; because in the

solution of the unperturbed case, that can be achieved by

means of an unitary transformation of the equation) we

obtain the following integro-differential equations (to

show the improvement that arise from taking different

orthonormalization conditions expressions of third

perturbation order are also included in this section):

Zeroth order:

Fð0Þð~rÞ2 1ð0Þi

h iwð0Þi ð~rÞ ¼ 0 ð11aÞ

First order:

Fð0Þð~rÞ2 1ð0Þi ^ v

h iw

að1Þ^i ð~rÞ

þ hað1Þð~rÞ þ vað1Þ^ð~rÞh i

wð0Þi ð~rÞ2

Xj

1að1Þ^ij w

ð0Þj ð~rÞ ¼ 0

ð11bÞ

Second order:hFð0Þð~rÞ2 1

ð0Þi ^ 2v

iw

abð2Þ^i ð~rÞ

þhhað1Þð~rÞ þ vað1Þ^ð~rÞ

iw

bð1Þ^i ð~rÞ2

Xj

1að1Þ^ij w

bð1Þ^j ð~rÞ

þhhbð1Þð~rÞ þ vbð1Þ^ð~rÞ

iw

að1Þ^i ð~rÞ2

Xj

1bð1Þ^ij w

að1Þ^j ð~rÞ

þhvabð2Þ^ð~rÞ

iwð0Þi ð~rÞ2

Xj

1abð2Þ^ij w

ð0Þj ð~rÞ ¼ 0

ð11cÞhFð0Þð~rÞ2 1

ð0Þi

iw

abð2Þ0i ð~rÞ þ

hhað1Þð~rÞ þ vað1Þþð~rÞ

i£w

bð1Þ2i ð~rÞ2

Xj

1að1Þþij w

bð1Þ2j ð~rÞ

þhhbð1Þð~rÞ þ vbð1Þ2ð~rÞ

iw

að1Þþi ð~rÞ

2X

j

1bð1Þ2ii w

að1Þþj ð~rÞ þ

hvabð2Þ0ð~rÞ

iwð0Þi ð~rÞ

2X

j

1abð2Þ0ij w

ð0Þj ð~rÞ ¼ 0

ð11dÞ

Third order:hFð0Þð~rÞ2 1

ð0Þi ^ 3v

iw

abcð3Þ^i ð~rÞ

þhhað1Þð~rÞ þ vað1Þ^ð~rÞ

iw

bcð2Þ^i ð~rÞ2

Xj

1að1Þ^ij w

bcð2Þ^j ð~rÞ

þhhbð1Þð~rÞ þ vbð1Þ^ð~rÞ

iw

acð2Þ^i ð~rÞ

2X

j

1bð1Þ^ij w

acð2Þ^j ð~rÞ þ

hhcð1Þð~rÞ þ vcð1Þ^ð~rÞ

iw

abð2Þ^i ð~rÞ

2X

j

1cð1Þ^ij w

abð2Þ^j ð~rÞ þ

hvabð2Þ^ð~rÞ

iw

cð1Þ^i ð~rÞ

2X

j

1abð2Þ^ij w

cð1Þ^j ð~rÞ þ

hvbcð2Þ^ð~rÞ

iw

að1Þ^i ð~rÞ

2X

j

1bcð2Þ^ij w

að1Þ^j ð~rÞ þ

hvacð2Þ^ð~rÞ

iw

bð1Þ^i ð~rÞ

2X

j

1acð2Þ^ij w

bð1Þ^j ð~rÞ þ

hvabcð3Þ^ð~rÞ

iwð0Þi ð~rÞ

2X

j

1abcð3Þ^ij w

ð0Þj ð~rÞ ¼ 0

ð11eÞ

A. Martinez et al. / Journal of Molecular Structure (Theochem) 589–590 (2002) 349–358 351

hFð0Þð~rÞ2 1

ð0Þi ^ v

iw

abcð3Þ^1i ð~rÞ

þhhað1Þð~rÞ þ vað1Þ^ð~rÞ

iw

bcð2Þ0i ð~rÞ2

Xj

1að1Þ^ij w

bcð2Þ0j ð~rÞ

þhhbð1Þð~rÞ þ vbð1Þ^ð~rÞ

iwacð2Þ0

i ð~rÞ2X

j

1bð1Þ^ij wacð2Þ0

j ð~rÞ

þhhcð1Þð~rÞ þ vcð1Þ7ð~rÞ

iw

abð2Þ^i ð~rÞ

2X

j

1cð1Þ7ij w

abð2Þ^j ð~rÞ þ

hvabð2Þ^ð~rÞ

iw

cð1Þ7i ð~rÞ

2X

j

1abð2Þ^ij w

cð1Þ7j ð~rÞ þ

hvbcð2Þ0ð~rÞ

iw

að1Þ^i ð~rÞ

2X

j

1bcð2Þ0ij w

að1Þ^j ð~rÞ þ

hvacð2Þ0ð~rÞ

iw

bð1Þ^i ð~rÞ

2X

j

1acð2Þ0ij w

bð1Þ^j ð~rÞ þ

hvabcð3Þ^1ð~rÞ

iwð0Þi ð~rÞ

2X

j

1abcð3Þ^1ij w

ð0Þj ð~rÞ ¼ 0 ð11fÞ

In Eqs. (11a)–(11f) we used for the wave functions of

higher order form:

wað1Þi ð~r; tÞ ¼ w

að1Þþi ð~rÞeivt þ w

að1Þ2i ð~rÞe2ivt ð12aÞ

wabð2Þi ð~r; tÞ ¼ w

abð2Þþi ð~rÞei2vt þ w

abð2Þ2i ð~rÞe2i2vt

þ wabð2Þ0i ð~rÞ ð12bÞ

wabcð3Þi ð~r; tÞ ¼ w

abcð3Þþi ð~rÞei3vt þ w

abcð3Þ2i ð~rÞe2i3vt

þ wabcð3Þþ1i ð~rÞeivt þ w

abcð3Þ21i ð~rÞe2ivt

ð12cÞ

and the operator F was partitioned as follows:

Fð0Þ ¼ 2 127 2

1 2XA

ZA

r1A

þX

j

wð0Þj ð~rÞ

D 1 2 Pi$j

rij

wð0Þj ð~rÞ

Eð13aÞ

Fað1Þ ¼ hað1Þ þ vað1Þþeivt þ vað1Þ2e2ivt

¼ 12

aðeivt þ e2ivtÞ þ vað1Þþeivt þ vað1Þ2e2ivt

ð13bÞ

Fabð2Þ ¼ vabð2Þþe2ivt þ vabð2Þ2e22ivt þ vð2Þ0 ð13cÞ

Fabcð3Þ ¼ vabcð3Þþe3ivt þ vabcð3Þ2e23ivt þ vabcð3Þþ1eivt

þ vabcð3Þ21e2ivt ð13dÞ

Here Fð0Þ is the Fock operator and Pi$j is the

permutation operator for the coordinates of electrons

i and j, respectively.

The vs are defined as:

vað1Þ^ð~rÞ ¼X

j

*wð0Þj ð~rÞ

1 2 Pi$j

rij

wað1Þ^j ð~rÞ

+

þ

*w

að1Þ7j ð~rÞ

1 2 Pi$j

rij

wð0Þj ð~rÞ

+!ð14aÞ

vabð2Þ^ð~rÞ ¼X

j

*wð0Þj ð~rÞ

1 2 Pi$j

rij

wabð2Þ^j ð~rÞ

+

þ

*w

abð2Þ7j ð~rÞ

1 2 Pi$j

rij

wð0Þj ð~rÞ

+

þ

*w

að1Þ7j ð~rÞ

1 2 Pi$j

rij

wbð1Þ^j ð~rÞ

+

þ

*w

bð1Þ7j ð~rÞ

1 2 Pi$j

rij

wað1Þ^j ð~rÞ

+!ð14bÞ

vabð2Þ0ð~rÞ ¼X

j

*wð0Þj ð~rÞ

1 2 Pi$j

rij

wabð2Þ0j ð~rÞ

+

þ

*w

abð2Þ0j ð~rÞ

1 2 Pi$j

rij

wð0Þj ð~rÞ

+

þ

*w

að1Þþj ð~rÞ

1 2 Pi$j

rij

wbð1Þþj ð~rÞ

+

þ

*w

að1Þ2j ð~rÞ

1 2 Pi$j

rij

wbð1Þ2j ð~rÞ

+!ð14cÞ

A. Martinez et al. / Journal of Molecular Structure (Theochem) 589–590 (2002) 349–358352

vabcð3Þ^ð~rÞ ¼X

j

*wð0Þj ð~rÞ

1 2 Pi$j

rij

wabcð3Þ^j ð~rÞ

+

þ

*w

abcð3Þ7j ð~rÞ

1 2 Pi$j

rij

wð0Þj ð~rÞ

+

þ

*w

bcð2Þ7j ð~rÞ

1 2 Pi$j

rij

wað1Þ^j ð~rÞ

+

þ

*w

acð2Þ7j ð~rÞ

1 2 Pi$j

rij

wbð1Þ^j ð~rÞ

+

þ

*w

abð2Þ7j ð~rÞ

1 2 Pi$j

rij

wcð1Þ^j ð~rÞ

+

þ

*w

að1Þ7j ð~rÞ

1 2 Pi$j

rij

wbcð2Þ^j ð~rÞ

+

þ

*w

bð1Þ7j ð~rÞ

1 2 Pi$j

rij

wacð2Þ^j ð~rÞ

+

þ

*w

cð1Þ7j ð~rÞ

1 2 Pi$j

rij

wabð2Þ^j ð~rÞ

+!

ð14dÞ

vabcð3Þ^1ð~rÞ ¼X

j

*wð0Þj ð~rÞ

1 2 Pi$j

rij

wabcð3Þ^1j ð~rÞ

+

þ

*w

abcð3Þ71j ð~rÞ

1 2 Pi$j

rij

wð0Þj ð~rÞ

+

þ

*w

bcð2Þ0j ð~rÞ

1 2 Pi$j

rij

wað1Þ^j ð~rÞ

+

þ

*w

acð2Þ0j ð~rÞ

1 2 Pi$j

rij

wbð1Þ^j ð~rÞ

+

þ

*w

abð2Þ7j ð~rÞ

1 2 Pi$j

rij

wcð1Þ7j ð~rÞ

+

þ

*w

að1Þ7j ð~rÞ

1 2 Pi$j

rij

wbcð2Þ0j ð~rÞ

+

þ

*w

bð1Þ7j ð~rÞ

1 2 Pi$j

rij

wacð2Þ0j ð~rÞ

+

þ

*w

cð1Þ^j ð~rÞ

1 2 Pi$j

rij

wabð2Þ^j ð~rÞ

+!

ð14eÞ

The set of Eqs. (11a)–(11f) are known as CPHF

equations.

The proposed approach differs in one point from

the already available methods: we used the inter-

mediate normalization condition Eq. (9) in place of

the usual one (Eq. (10)). Expanding Eq. (9) we obtain

for all i and j.

Zeroth order:Dwð0Þi

wð0Þj

E¼ dij ð15aÞ

First order:Dwð0Þi

wað1Þ^j

E¼ 0 ð15bÞ

Second order:Dwð0Þi

wabð2Þ^j

E¼ 0 ð15cÞD

wð0Þi

wabð2Þ0j

E¼ 0 ð15dÞ

Third order:Dwð0Þi

wabcð3Þ^j

E¼ 0 ð15eÞD

wð0Þi

wabcð3Þ^1j

E¼ 0 ð15fÞ

Notice that in the normalization equation of a given

order any perturbation contribution of lower order of

the wave function does not occur. If the usual

normalization conditions are used (Table 2 Ref. [9])

extra terms appear on the left side of Eqs. (15b)–(15d)

in the form kwað1Þ^i lwð0Þ

j l; kwabð2Þ^i lwð0Þ

j lþkwað1Þ^

i lwbð1Þ7j lþ kwbð1Þ^

i lwað1Þ7j l and kwabð2Þ0

i lwð0Þj lþ

kwað1Þ^i lwbð1Þ^

j lþ kwbð1Þ7i lwað1Þ7

j l; respectively. Seven

extra terms would appear in Eqs. (15e) and (15f),

respectively (they can be found by comparison with

the equations of Table 2 of Ref. [9]).

It is evident that for second and third order of

perturbation our expressions for the orthonormaliza-

tion conditions (Eqs. (15a)–(15f)) are simpler and

more transparent than those usually applied in the

literature [7–9].

2.2. Solution of the CPHF equations for first and

second perturbation order

The normalization conditions (Eqs. (15a)–(15f))

can be incorporated directly in the CPHF equations by

means of Lagrangian multipliers. A similar

A. Martinez et al. / Journal of Molecular Structure (Theochem) 589–590 (2002) 349–358 353

transformation could not be done if one uses the usual

normalization conditions.

The Lagrangian multipliers are obtained by

multiplying Eqs. (11a)–(11d) on the left by wð0Þj ð~rÞ

and integrating over the whole space, with the use of

Eqs.(15a)–(15d):

10i ¼ w

ð0Þi ð~rÞlFð0Þð~rÞlwð0Þ

i ð~rÞD E

ð16aÞ

1að1Þ^ij ¼ w

ð0Þj ð~rÞlvað1Þ^ð~rÞ þ hað1ÞðrÞlwð0Þ

i ð~rÞD E

ð16bÞ

1abð2Þ^ij ¼ w

ð0Þj ð~rÞlvað1Þ^ð~rÞ þ hað1ÞðrÞlwbð1Þ^

i ð~rÞD E

þ wð0Þj ð~rÞlvbð1Þ^ð~rÞ þ hbð1ÞðrÞlwað1Þ^

i ð~rÞD E

þ wð0Þj ð~rÞlvabð2Þ^ð~rÞlwð0Þ

i ð~rÞD E

ð16cÞ

1abð2Þ0ij ¼ w

ð0Þj ð~rÞlvbð1Þ2ð~rÞ þ hbð1ÞðrÞlwað1Þþ

i ð~rÞD E

þ wð0Þj ð~rÞlvað1Þþð~rÞ þ hað1ÞðrÞlwbð1Þ2

i ð~rÞD E

þ wð0Þj ð~rÞlvabð2Þ0ð~rÞlwð0Þ

i ð~rÞD E

ð16dÞ

With the determined Lagrangian multipliers we can

solve the CPHF Eqs. (11a)–(11f) expanding the N

spatial orbitals wi in the set of m atomic orbitals xp for

each order l of the perturbation:

wli ¼

Xmp¼1

Clipxp ð17Þ

Eq. (11a), the zeroth order one, is solved in the

standard form with the help of the self-consistent field

method. The equation of order l is solved using the

results of previous orders to calculate the initial values

of vl and 1l; finding the coefficients Clip iteratively

until self-consistency in the determination of vl; 1l and

Clip is reached. The coefficients are found as solution

of one system of equations of the form AiCi ¼ 2Bi for

each occupied orbital i in each iteration. The matrix Ai

contains the factor, that appear in the first term of Eqs.

(11b)–(11d) multiplying wli and the vector Bi contains

all the other terms of Eqs. (11b)–(11d) beginning with

the second one.

In the proposed way one does not need to find the

contribution of the perturbation to the virtual orbitals

of the system. In the previous way of solution [7–9],

the matrix U which they use contains the information

needed to calculate the perturbed occupied and virtual

orbitals.

The solution of the CPHF equations only fulfils the

intermediate normalization conditions. We should

transform them in each iteration step to the strict

solutions which fulfil the general normalization Eq.

(10). The transformations are [6]:

~wð0Þi ð~rÞ ¼ w

ð0Þi ð~rÞ ð18aÞ

~wað1Þ^i ð~rÞ ¼ w

að1Þ^i ð~rÞ ð18bÞ

~wabð2Þ7i ð~rÞ ¼ w

abð2Þ7i ð~rÞ2 1

2

Xj

wð0Þj ð~rÞ

£ wbð1Þ^j ð~rÞ wað1Þ7

i ð~rÞD E

þ wað1Þ^j ð~rÞ wbð1Þ7

i ð~rÞD E� �

ð18cÞ

~wabð2Þ0i ð~rÞ ¼ w

abð2Þ0i ð~rÞ2 1

2

Xj

wð0Þj ð~rÞ

£ wað1Þþj ð~rÞ wbð1Þþ

i ð~rÞD E

þ wbð1Þ2j ð~rÞ wað1Þ2

i ð~rÞD E� �

ð18dÞ

One should emphasize that in the general case our

~wið~r; tÞ; obtained using the orthonormalization con-

ditions of Eq. (9) in the perturbation procedure

followed by the transformations of Eqs. (18a)–

(18d), are not necessary identical with the ~wið~r; tÞ;that are obtained using the orthonormalization

conditions of Eq. (10) directly in the perturbation

procedure [7–9]. In contrast to the second one our

~wið~r; tÞ are free from phase factors [6] that can appear

if one uses the other way of solution [7–9]. This point

is discussed in detail by Langhoff et al. [6].

The expectation values of physical properties

obtained with both solutions (our and the one of

Refs. [7–9]) should be the same, because the phase

factors get cancelled in the calculations. However, if

one is interested in some magnitude of the perturbed

wave function of some order, like the derivatives of

the wave function with respect to the quasi-momen-

tum k in the Crystal Orbital method [11], it is very

advantageous to obtain a perturbed wave function free

from phase factors. The presence of arbitrary phase

factors could induce errors in the calculation of the

derivatives.

The corrected perturbed wave functions of Eqs.

A. Martinez et al. / Journal of Molecular Structure (Theochem) 589–590 (2002) 349–358354

(18a)–(18d) are to be used for the determination of vl

and 1l (Eqs. (14a)–(14e) and (16a)–(16d)) in each

iteration step. One has to rewrite Eqs. (14a)–(14e)

and (16a)–(16d) putting ~wlið~rÞ in place of wl

ið~rÞ:

2.3. Determination of the frequency dependent

(hyper)polarizabilities

Following the definition of Buckingham [12] for

the (hyper)polarizabilities we can obtain these

quantities from the perturbed wave function [9].

For the dynamic polarizability:

aabðvÞ ¼Xi;j

~wð0Þi lral ~wbð1Þþ

j

D Eþ ~w

bð1Þ2j lral ~wð0Þ

i

D Eð19aÞ

For the dynamic first hyperpolarizabilities:

babcð2v;v;vÞ ¼Xi;j

~wð0Þi lral ~wbcð2Þþ

j

D E

þ ~wbcð2Þ2j lral ~wð0Þ

i

D E

þ ~wcð1Þ2i lral ~wbð1Þþ

j

D E

þ ~wbð1Þ2j lral ~wcð1Þþ

i

D Eð19bÞ

babcð0;v;2vÞ ¼Xi;j

~wð0Þi lral ~wbcð2Þ0

j

D E

þ ~wbcð2Þ0j lral ~wð0Þ

i

D Eþ ~w

cð1Þþi lral ~wbð1Þþ

j

D E

þ ~wbð1Þ2i lral ~wcð1Þ2

j

D Eð19cÞ

3. Computational details and discussion

Our purpose is to demonstrate that the method of

solution of the CPHF equations presented in Section 2

(and the corresponding computational program) gives

correct results rather than to obtain very accurate

values for m, a, and b of selected molecules. Thus, we

have decided to repeat the calculations by Sekino and

Bartlett (SB) [7] for comparison.

The structures and geometrical orientations of the

molecules (CH3F, CH2F2 and CHF3) with respect to

the field are the same as the ones used by SB. The

same basis sets (double zeta plus polarization (DZP)

functions [13] with exponents zpH ¼ 0:7; zdC ¼ 0:655

and zdF ¼ 1:58 for the polarization functions on

hydrogen, carbon and fluorine, respectively) as

modified by SB were employed for all molecules.

The calculated energies, dipole moments, polariz-

abilities and first hyperpolarizabilities of these

molecules are summarized in Tables 1–3. The

Table 1

Dipole moments, polarizabilities and hyperpolarizabilities of CH3F in atomic units (a.u.)

This calculation Literature [7] GAUSSIAN98

Etotal 2139.076017 2139.075984 2139.075980

mz 20.868 20.871 (20.34%) 20.871

azzð0; 0Þ 12.608 12.580 (0.22%) 12.579

azzð2v;vÞ 12.711 12.683 (0.22%)

axxð0; 0Þ 12.429 12.410 (0.15%) 12.411

axxð2v;vÞ 12.555 12.537 (0.14%)

bzzzð0; 0; 0Þ 19.564 19.564 (0.00%) 19.562

bzzzð0;2v;vÞ 19.955 19.954 (0.01%)

bzzzð22v;v;vÞ 20.779 20.777 (0.01%)

bxxxð0; 0; 0Þ 28.110 28.117 (20.09%) 28.118

bxxxð0;2v;vÞ 28.329 28.336 (20.08%)

bxxxð22v;v;vÞ 28.795 28.802 (20.08%)

bzxxð0; 0; 0Þ 9.296 9.314 (20.19%) 29.314

bzxxð0;2v;vÞ 9.543 9.562 (20.20%)

bzxxð22v;v;vÞ 9.912 9.931 (20.19%)

A DZP basis set is used. The values in percent give the deviation of our calculated values and the ones in Ref. [7].

A. Martinez et al. / Journal of Molecular Structure (Theochem) 589–590 (2002) 349–358 355

reported calculated values of Ref. [7] and the static

values calculated with the GAUSSIAN98 program [14]

are also given. The reported dynamic (hyper)polariz-

abilities correspond to v ¼ 0:0656 a:u: (wave length

l ¼ 6943 �A).

For all molecules the agreement of the present and

the previously calculated values for all properties is

satisfactory. The greatest deviation in the polariz-

abilities occurs at CHF3 (0.53% over the value

calculated by SB for azz). The greatest deviation in

the hyperpolarizabilities occurs at CH2F2 (0.44%

under the value calculated by SB for bzzz). The

sequence of the values of bzzz, going from CH3F with

the greatest values to CHF3 with the smallest ones is

also reproduced. As expected the dispersion effect of

the dynamic values of the (hyper)polarizabilities with

Table 2

Dipole moments, polarizabilities and hyperpolarizabilities of CH2F2 in atomic units (a.u.)

This calculation Literature [7] GAUSSIAN98

Etotal 2237.965314 2237.965254 2237.965276

mz 20.916 20.920 (20.43%) 20.919

azzð0; 0Þ 12.067 12.029 (0.32%) 12.027

azzð2v;vÞ 12.160 12.122 (0.31%)

axxð0; 0Þ 12.667 12.627 (0.32%) 12.621

axxð2v;vÞ 12.752 12.712 (0.31%)

ayyð0; 0Þ 11.344 11.307 (0.33%) 11.306

ayyð2v;vÞ 11.447 11.410 (0.32%)

bzzzð0; 0; 0Þ 16.334 16.406 (20.44%) 16.407

bzzzð0;2v;vÞ 16.625 16.699 (20.44%)

bzzzð22v;v;vÞ 17.237 17.311 (20.43%)

bzxxð0; 0; 0Þ 8.278 8.290 (20.14%) 8.285

bzxxð0;2v;vÞ 8.397 8.409 (20.14%)

bzxxð22v;v;vÞ 8.708 8.719 (20.13%)

bzyyð0; 0; 0Þ 14.735 14.752 (20.12%) 14.753

bzyyð0;2v;vÞ 15.128 15.144 (20.11%)

bzyyð22v;v;vÞ 15.800 15.816 (20.10%)

A DZP basis set is used. The values in per cent give the deviation of our calculated values and the ones in Ref. [7].

Table 3

Dipole moments, polarizabilities and hyperpolarizabilities of CHF3 in atomic units (a.u.)

This calculation Literature [7] GAUSSIAN98

Etotal 2336.865779 2336.865574 2336.865600

mz 0.748 0.749 (20.13%) 0.752

azzð0; 0Þ 11.390 11.330 (0.53%) 11.338

azzð2v;vÞ 11.474 11.413 (0.53%)

axxð0; 0Þ 12.293 12.248 (0.37%) 12.241

axxð2v;vÞ 12.365 12.319 (0.37%)

bzzzð0; 0; 0Þ 223.170 223.188 (20.08%) 223.253

bzzzð0;2v;vÞ 223.656 223.672 (20.07%)

bzzzð22v;v;vÞ 224.677 224.693 (20.06%)

bxxxð0; 0; 0Þ 3.462 3.458 (0.12%) 3.454

bxxxð0;2v;vÞ 3.503 3.498 (0.14%)

bxxxð22v;v;vÞ 3.587 3.581 (0.17%)

bzxxð0; 0; 0Þ 24.264 24.271 (20.16%) 24.286

bzxxð0;2v;vÞ 24.300 24.307 (20.16%)

bzxxð22v;v;vÞ 24.476 24.483 (20.16%)

A DZP basis set is used. The values in percent give the deviation of our calculated values and the ones in Ref. [7].

A. Martinez et al. / Journal of Molecular Structure (Theochem) 589–590 (2002) 349–358356

respect to the static ones is larger when a dynamic

perturbation of higher order is involved.

In principle our calculations should give exactly

the same results as those from SB, as we already

discussed in Section 2.2. Nevertheless, differences in

the total energies and dipole moments, standard

output values of a HF–SCF calculations, are already

present in the solution of the zeroth order equations.

These small differences are consequences of the use of

Gaussian lobe orbitals in our program instead of

ordinary Gaussian functions (as seems to be the case

by SB from the comparison with the GAUSSIAN98

calculations) for the integrals calculations. Since the

(hyper)polarizabilities are higher order derivatives of

the energy with respect to the field-components, they

are more sensible to the variation introduced through

this factor. Such differences are already reported in

the literature and are not significant [15].

Additionally we have calculated the properties of

interest for a water dimer with a minimal basis set

from Clementi [16]. From the results of Table 4 we

see that in the case of this simple basis set the

description by ordinary Gaussian functions and by

Gaussian lobe orbitals is exactly the same. That is a

consequence of the lack of d-type functions, which are

more sensible as the p-functions to the use of the

Gaussian lobe representation.

The coincidence of the results demonstrate the

equivalence of the two different normalizations of the

perturbed wave functions (Eqs. (9) and (10)), for the

solution of the CPHF equations. This equivalence was

discussed by Langhoff et al. [6], but until now it was

not proven computationally.

4. Conclusions

We have developed a new way of solution for the

CPHF equations through the direct incorporation of

the normalization conditions to them. The perturbed

wave functions obtained with this approach are free

from phase factors as Langhoff et al. [6] have shown.

Detailed expressions and a computational program

were made for the calculation of the NLO-properties

Table 4

Dipole moments, polarizabilities and hyperpolarizabilities water dimer in atomic units (a.u.)

This calculation Literature [17] GAUSSIAN98

Etotal 2151.492923 2151.492959

mx 1.330 1.329

mz 1.348 1.348

azzð0; 0Þ 11.885 11.88 11.885

azzð2v;vÞ 12.009 12.01

axxð0; 0Þ 8.627 8.626

axxð2v;vÞ 8.704

ayyð0; 0Þ 0.034 0.034

ayyð2v;vÞ 0.035

axzð0; 0Þ 23.835 23.835

axzð2v;vÞ 23.872

bzzzð0; 0; 0Þ 237.490 237.49 237.488

bzzzð0;2v;vÞ 238.394 238.39

bzzzð22v;v;vÞ 240.317 240.31

bzxxð0; 0; 0Þ 7.254 7.254

bzxxð0;2v;vÞ 7.325

bzxxð22v;v;vÞ 7.452

bzyyð0; 0; 0Þ 20.144 20.144

bzyyð0;2v;vÞ 20.149

bzyyð22v;v;vÞ 20.164

bxxxð0; 0; 0Þ 228.675 228.673

bxxxð0;2v;vÞ 229.177

bxxxð22v;v;vÞ 230.232

A minimal basis set is used. The literature values are of Ref. [15].

A. Martinez et al. / Journal of Molecular Structure (Theochem) 589–590 (2002) 349–358 357

up to the second order of perturbation for molecular

systems.

Comparing our results with the values reported in

the literature for CH3F, CH2F2, CHF3, and water

dimer the validity of our approach was verified. The

very small differences between our results and those

from the previous works seem to be caused by the use

of a different function type in the basis sets. The

equivalence of the two different normalization

procedures of the perturbed wave functions was

computationally proven.

The method can be programmed for the calculation

of third order NLO-properties on molecules. It is

already successfully used for extended calculations of

dynamic NLO properties in periodic quasi-one-

dimensional systems [18].

Acknowledgments

We are very grateful to the DFG (Deutsche

Forschungsgemeinschaft Project Ot 51/9-4) for the

financial support.

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