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Interaction-induced Properties

and

Perturbation Theory

presented by
Jesfis Juanés i Timoneda

has been accepted towards fulfillment
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.0

Major professor

Dam_the_22nd_n£_September 1986.

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INTERACTION-INDUCED PROPERTIES
AND
PERTURBATION THEORY

By

Jesus Juanés i Timoneda

A DISSERTATION

Submitted to Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Chemistry

1986

ABSTRACT
INTERACTION-INDUCED PROPERTIES

AND
PERTURBATION THEORY

By

I O 0
Jesus Juanos 1 Timoneda

The perturbation expansions obtained from Lowdin’s projection-operator
formalism are derived in a new way, using Kato’s formulation of
perturbation theory. This formulation does not involve the symbolic use
of the inverse of singular operators. Kato’s approach provides a
convenient algebraic alternative to diagrammatic techniques for obtaining
eigenvalues and eigenvectors. Different normalization criteria imposable
on the wave function are easily visualized in terms of the operator that
yields the perturbed state vector when it acts upon the unperturbed wave
function.

We use a label-free exchange perturbation method to calculate the
dipole moment of interacting Re and H atoms as function of internuclear
separation. In the label-free formalism, the unperturbed Hamiltonian and
perturbation terms are constructed so that each is invariant with respect
to exchange of electrons between the interacting atoms; then a Rayleigh-
Schrodinger perturbation expansion with a fully antisymmetrized set of
zeroth—order wave functions yields the interaction energy and collision-
induced properties. Good agreement with ab initio results for the He...H
dipole is obtained when a long-range dispersion contribution is added to

the first-order overlap and exchange contributions.

Jesus Juanos i Timoneda

We calculate the dipole moment for the quartet state of H3 in its
linear and right triangular configurations and we find that the first-
order overlap-exchange and the long-range contributions are about the same
order of magnitude at intermediate range. We prove that a system of n
atoms in any spatial configuration has zero dipole moment when the
electrons are described by s-type functions (Slater or Gaussian) and the
dipole moment is calculated as the expectation value of the corresponding
operator with an unperturbed, but fully antisymmetrized wave function.

An analytic expression for the damped pair dipole moment is calculated
in terms of Clebsch-Gordan coefficients and reduced matrix elements. We
show explicitly with the first nonlocal hyperpolarizability how
susceptibility densities may be cast in a form without secular
divergences. The theory of generalized functions is applied to calculate

the limits in the expressions for susceptibility densities and tensors.

ACKNOWLEDGMENTS

This dissertation is an account of the research perfOrmed under
Professor Katharine L.C. Hunt’s direction. I would like to express my
appreciation and gratefulness for her guidance and advice. Also, I
acknowledge the financial support of the Donors of the Petroleum Research
Fund, Standard Oil ($0810) and "la Caixa de Barcelona".

The computations involved in this research have been carried out at a
time of expansion of the computational facilites in the Chemistry

Department. My thanks to those who helped me in making efficient use of

them.

TABLE OF CONTENTS

LIST OF TABLES

LIST OF FIGURES

CHAPTER 1.

CHAPTER 2.

CHAPTER 3.

CHAPTER 4.

CHAPTER 5.

INTRODUCTION

DERIVATION OF PERTURBATION EXPANSIONS

2.1. Introduction

2.2. On Lowdin’s studies in perturbation theory

2.3. Theoretical Analysis

2.4. Discussion

2.5. Relationships among perturbation methods

INTERACTION-INDUCED PAIR DIPOLE MOMENT

3.1. Introduction

3.2. Exchange perturbation theory in label-free form

3.3. Comparison with other exchange perturbation theories

3.3.1. Polarization and exchange contributions to
interaction energies and pair properties
3.3.2. Density-matrix perturbation theory

3.4. Computational methods

3.5. Results and discussion

INTERACTION-INDUCED TRIPLET DIPOLE MOMENT

4.1. Introduction

4.2. Dipole moment calculations

4.3. Results and discussions

OVERLAP DAMPED PAIR DIPOLE MOMENT

5.1. Introduction

5.2. Generalized functions and charge-susceptibility

iii

densities
5.3. Nonlocal polarizability and hyperpolarizability
calculation
5.4. Damped dispersion dipole moment calculation
APPENDIX A
APPENDIX B
APPENDIX C
APPENDIX D
APPENDIX E
APPENDIX F

REFERENCES

iv

LIST OF TABLES

TABLE CHAPTER
1 3 Orbital exponents and contraction coefficients of

the Gaussian basis (103 6p) on each atom.

FIGURE

1

CHAPTER

3

LIST OF FIGURES

Collision-induced dipole of He...H as a function of
the internuclear separation R in the range from 4.0
to 11.0 a.u. Curve A shows the ab initio results
obtained by Meyer, and Curve B the ab initio
results of Ulrich gt a1. Curves C and D have been
obtained by adding the long-range dispersion dipole
to the exchange-overlap dipole computed form the
energy derivative (Eq. 13). Curve C shows the
dipole in the Gaussian basis, and D the dipole in
the Slater basis. Curves E and F have been
obtained similarly, but the exchange-overlap dipole
has been computed as an expectation value from Eq.
12; Curve E shows results from the Gaussian basis,
F from the Slater basis. Curve G is a plot on the
leading term of the dispersion dipole, D7R-7. The

ordinate is scaled logarithmically and the dipole

p(R) is given in a.u.

Collision-induced dipole of He...H as a function of
internuclear separation R in the range from 0 to
4.0 a.u. Curve A shows the ab initio results of

Ulrich gt a1. Curve B and C show the lowest-order

vi

exchange-overlap dipole, obtained from Eq. 13, B in
Slater basis and C in Gaussion basis. Curve D
shows the exchange—overlap dipole computed as an
expectation value with the zeroth—order
wavefunction; results from the Gaussian and Slater
bases superimpose on this scale. The dipole p(R)

is given in a.u.

Ratio of the correction term Apz to the exchange-
overlap dipole for He...H, plotted as a function of
the internuclear separation R (in a.u.). The
exchange-overlap dipole has been computed as the
derivative of the first-order interaction energy
with respect to an applied field. Curve A shows
the results from the Slater-basis calculation, and

Curve B the results from the Gaussian basis.

Spatial configuration of three identical nuclei (3,
b, c). e=a, "/2, "/3 for the linear, right-triangle

and isosceles—triangle configurations.

H dispersion dipole moment. X component. Right-

3

triangle configuration.

H3 dispersion dipole moment. Z component. Right~

vii

triangle configuration.

H3 overlap-exchange dipole moment. X component.

Right-triangle configuration.

H3 overlap-exchange dipole moment. 2 component.

Right-triangle configuration.

H3 overlap-exchange dipole moment. Linear

configuration.

H3 dispersion dipole moment. Linear configuration.

\llll

CHAPTER 1.
INTRODUCTION.

A major part of this work is devoted to the analysis of interactions
between atoms or molecules at separations such that the overlap is not
negligible. From this analysis we derive methods to calculate
interaction-induced properties. These properties allow us to predict the
behavior of matter interacting with radiation; alternatively the
calculated properties may be compared with those computed from
experimental data. We have also worked on several theoretical aspects of
the methods used in our study.

Perturbation theory is the framework within which most of our work has
been performed. The next chapter is devoted to a theoretical analysis of
perturbation theory. 0n the basis of an entirely algebraic formalism, we
present formulae to calculate corrections to the perturbed wave function
and eigenvalue to any order. Several normalization criteria are
associated with the way in which different projection operators are
manipulated. We use a reduced resolvent for which no problem about
singularities arises. This powerful technique allows us to establish
consistency with other formalisms and developments in perturbation theory.
We conclude the second chapter by presenting a unification of several
different formalisms, with common principles.

In chapter 3 we use a special form of perturbation theory, yi§.,
exchange perturbation theory in label-free form, to calculate interaction-
induced dipole moments. In particular, the collision-induced dipole
moment of a diaton is computed. We compare our method with other
perturbation theories and we analyze an approximation employing additive

long- and short-range contributions to the pair dipole moment. Since the

f0

Hellmann—Feynman theorem is not satisfied at the level of approximation
used in our calculations, the dipole moment is calculated both as the
expectation value of the corresponding operator and as the field
derivative of the energy. The results that each strategy yields and
comparison with accurate ab initio calculations are used to assess the
reliability of different methods in computing pair dipoles. Our
computations have been performed with Gaussisan~ and Slater-type orbitals.
The work with Slater-type orbitals required us to augment the contents of
the standard tables in order to calculate one of the integrals over these
functions.

The same perturbation method as in Chapter 3 is used in Chapter 4. We
derive an expression for the overlap-exchange contribution to the triplet
dipole moment. The significance of the results of this chapter for
studies of many-body effects is twofold: We prove that some methods used
to calculate interaction-induced properties are not reliable. Second, we
show that the results for the three-body long-range and exchange-overlap
contributions to the pair dipole of quartet H3 are comparable in order of
magnitude at intermediate range. This result is known to hold also for
the energy.

The exchange—overlap dipole (upon which Chapters 3 and 4 focus)
vasishes when the separation becomes very large. The dispersion dipole
moment as calculated in the region where overlap and exchange effects are
negligible diverges when the separation approaches zero. In chapter 5 we
calculate the overlap damped dispersion dipole moment, which remains
finite. Also, we present a general treatment of susceptibility densities

and tensors of potential use in the study of nonlinear phenomena. We

treat the problem of secular divergencies, and we give an analytic
expression for the first nonlocal hyperpolarizability density, needed to

compute several damped properties, including the dispersion dipole.

CHAPTER 2

2.1. - INTRODUCTION

Perturbation techniques are widely used ([1,2] and refs. therein) in
the study of intermolecular interactions and their effects on super-
molecular system properties. The wave operator that generates the perturbed
state vector from the unperturbed wave function is important within this
context, and in a general treatment of self-consistent field theory [3].

Lowdin has developed a projection operator formalism [3,5] that gives
the wave operator in terms of a reduced resolvent operator T. Within
Lowdin’s formalism, proof of the existence of T hinges on proofs of operator
invertibility. Wilson and Sharma [6,7] have analysed the invertibility
requirements for one form of the T operator. By relating T to an ”outer
projection" of the Hamiltonian, Lowdin has recently shown that the operator
T remains regular as needed for the construction of perturbed eigenfunctions
[4]; he has also applied the partitioning method to derive rational
perturbation approximations [4].

An alternative derivation of Lowdin’s perturbation expansions is
presented in this chapter, based on Kato’s theory. This approach does not
involve manipulation of inverse operators, although it does require proper
manipulation of resolvents and consideration of their domains. In section
2.2 we review the aspects of Lawdin’s work necessary for comparisons [3,5],
with brief reference to other work in this field [8]. In section 2.3 the
calculation of the perturbed eigenvectors and eigenfunctions is treated from
the viewpoint of complex variable and Hilbert space theories [9-13].
Expansions are developed in section 2.3 for perturbed Hamiltonians with

arbitrary dependence on a perturbation parameter x. The Hamiltonians

considered by Lowdin (in Refs. 4, p. 79 and 5) can be understood as one-
parameter dependent expansions in which only the first-order correction term
is present. The results and conclusions from section 2.3 are discussed in
section 2.4, while section 2.5 contains a digression on the interrelation

among several perturbative treatments used in quantum mechanics.

2.2. ~ ON LOWDIN’S STUDIES IN PERTURBATION THEORY
The Hamiltonian H of the system under consideration is expressed in
terms of the Hamiltonian H0 for the unperturbed system and the correction
(1) Hm

term H as H = H0 + . The wave operator U relates the perturbed

eigenfunction w to the unperturbed eigenfunction $0:

w = uwo. (1)

When the operator t is constructed from the reduced resolvent T and the

perturbation term H(l) as

t = H(1) + H(1)TH(1) = H(1)U , (2)

where

U = 1 + mm (3)
the eigenvalue x associated with dimay be written as

x = x0 + <t>0 , (4)
where the brackets with subscript zero denote an expectation value in the
state vb.

It is useful to express T in terms of the projection operator P that
projects out the eigenfunction $0 of Ho from an arbitrary function. P
projects onto a one-dimensional subspace of 356, the separable Hilbert space
taken as the set of functions that are square-integrable and complex-valued
on the configuration space of the Hamiltonian operator H. The projector
onto the orthogonal complement with respect to the range of P is (l-P).

The reduced resolvent T in Eq. 3 is the operator
T(€) = [c — (1-r)u]’1 (l-P) (5)

evaluated at €=x (in general e is a complex variable). T(€) may be

expressed equivarently in terms of the resolvent R i (6—H)-1 of the outer
projection H = (l-P)H(l-P) of the Hamiltonian H:

T<c> = (1—p) (e—fi>’1 (1—P) . (6)

In general we may assume 3(x—H)—1 only when x¢o(H), where 0(H) is the
discrete or point spectrum of H, even though x may be a point of constancy
of E(x) or a point of continuity of E(x), where E(x) stands for a resolution
of the identity [14]. In deriving conventional perturbation theory
formulae, we may take T(€) with 6 in the spectrum of H, provided that € is
not in the spectrum of H; in particular, we may set €=x, because <w|¢b>x0
ensures that T(€) remains regular at x [4].

Series for t and U [5] may be derived by using the unperturbed reduced
resolvent To(€), defined as T0(€) = [€-(l-P)Ho]-1 (l-P), with appropriate
choice of € to obtain Brillouin-Wigner, Rayleigh-Schrodinger or
"intermediary" perturbation expansions [4].

With the reduced resolvent S = [xo-(l-P)H0]-1(l-P), if |<SV'>| < l,
where

v’ = n<1> - <wolt|wb> = H‘l) — <t> (7)

o O
the following expansion is obtained for t:
t 2 H(l) + H(1)SH(1)+H(1)SV’SH(1) + H(1)SV’SV’SH(1) + . . . . (8)

Hence

V’=H(l)-<H(1)>o - <H(1)SH(1)>0 - <H(1)SH(1)SH(1)>0 +

1)>OSH(1)>0 + <H(1)S<H(1)SH(1)>OSH(1)>0 + . . . , (9)

(l)

+ (H S<H<

and so

t = [3(1)] + [n(l)sn(1)] + [H(1)S(H(1)-<H(3) > )SH(1)] +

[H(1)S(H(l) - <H(1)>O)S(H(l) - <H(1)>O)SH(1) —

- <H(1) SH(1) >OSSH(1)] + . - - 3 t1 + t2 + t3 + . - - (10)

u = 1 + [sn(1)] + [s(n(1) - <n(l)>o)sa(1)] +

+ [s(H(1) - <H(1)>0)S(H(l) - <H(l)>o)SH(1) (11)

_ (DU) 2(1) ...=
(H SH >08 H ] + — 1 + 111 + 112 + 03 +

where the ith term in square brackets has been identified with ti in
expression (10) or with Ui in (11). The expansions for t and U given in
(10) and (11) let us calculate corrections to $0 and x0 consistently with

the choice of H.

A related perturbation formalism has been developed by Speisman [8],

using the operator

T =—élEEl . (12)
o >~o
We should emphasize, though, the essentially symbolic character of this
expression in his work. This character becomes evident when, after taking
x0 as an isolated eigenvalue of Ho, Speisman in fact equates (12) to the
reduced resolvent, i.§., I’(x - x0)-1 dE(x), where the prime in the
integration means that it is performed in the whole range of x except within

a properly defined neighbourhood of x0.

2.3. - THEORETICAL ANALYSIS

This section begins with a brief review of the aspects of Kate’s
perturbation theory that are central in this analysis. Rigorous treatments
of the perturbation methods, regular or asymptotic expansions and
convergence properties may be found in Kato’s original papers [9-12,15].
The derivation of the perturbation series is carried out and the connection
with Lowdin’s work [4,5] is established.

Following Kato, let us consider HK a self-adjoint or hypermaximal, but

not necessarily bounded operator such that

W

Hx = I xdEK(x) (13)

Q

where the system of projections EK (x) is the resolution of the identity
corresponding to HK and the sub-index x stands for the dependence on a

parameter xefli. For £€A(HK), the resolvent RK(R) is defined by

Rx(“) = (“K‘“)—1 = I <x-1)'1 dEx(*) . <14)

m
and it is a bounded analytic function of 1. [\(H‘) is the resolvent set of
Rx, 1.3., the set of real numbers not belonging to the spectrum of Ex, and
the non—real complex numbers.

When x0 is an isolated eigenvalue (in general with finite multiplicity

m) of H within a properly chosen closed curve F belonging to)A(Ho), we

0

shall write

Rx(n) = (x0 - n)’1 so + S.(“) . (15)

9

10

where

l

SK(R) = I, (x—i)“ dEK(x)

(I stands for an integration except at the point x=x0, [12]).

(15)

SK(1) given by (16) is the reduced resolvent with respect to x0, and

E is the projection operator onto the space associated with x

0
be emphasized that SK(R) has a singularity at 1=x0.
(1)

Let x“ be a regular perturbation and
E =—:l— (j) R (i)di

x Zni x O

r
Since
_ a n n (1) n

R (Q) - R (i) Z {-1) x [H R (1)] .

x 0 “:0 0

we obtain

where

(n) _ _ n—1_1__ (1) .n.
A - ( 1) 2.1 gir Roma Roma

with H(1) appearing n times in the integrand.

(n)

We take from Kato’s papers the expression for A

k

1 card (L) k1 (1)5k2 ..9. 1) n+1

1‘") = (-1)"‘ z (s H
i=1

. H‘

where card(L) = number of elements in L, and

S )i

It should

0.

(17)

(18)

(19)

(20)

(21)

ll

n+1
L : { (k1,...,kn+l) : jg] kJ ; n . kJ 2 0, V5 : 1 g j s n+1} (22)
s0 = -E s = s ( ) (23
‘ o ' ‘ ‘0 >‘0 )
(An explicit derivation is given in Appendix A.)

In the more general case

9)
ax = no + xn(1) + x2u(~' + - - - , (24)
we obtain
card (I) card (J) k (v ) k k (v ) k
A(") = - z (—1)p z (s 1H 1 s 2 - - - 5 PH P s p+1)i‘j ,
i=1 j=1 (25)
where card(I) = number of elements in
P v
I = {(v1,-",vp)353vj=n A VJZI: JZISJSP}

and card(J) = number of elements in
2+1

J = {(k1,...,kp+1)3§3kj=p A kJ20,Vj:l$j$p+l}

It has been proven [10] that there exists a unitary operator UK such

that
a) U0 = l,
b) UK E0 is regular wherever EK is regular,
and c) E = U E0 U—l .
K K K

(n)

Let us now consider explicitly the terms A , for n = l and 2

12

(1) _ _ (1) _ (1)
A - EOH S SH E0
A(2) = E0H(1)SH(1)S + SH(1)EOH(1)S + SH<1)SH(1)E0 -
_ , (1) (1) 2_ (1) 2 (1) __ 2 (1) (1) _ (2) _
ECHO EOH S EOH S H E0 5 H EOH E0 EOH S
_ (2)
SH E0 . (25)
If we define
(1) - _ (1)
al ; SH E0
19> _ sows
(2)- (1) (1) _ 2(1) (1) _ (2)
a1 ; SH SH E0 S H EOH E0 SH Eo
aéz) 5 E0H(1)SH(1)S - E0H(1)E0H(1)S2 - E0H(2)S (27)
then from (26) and (27)
(1) _ (1) (1)
A — a1 + 82
A(2) 2 aiz) + aéz) + SH(1)EOH(1)S - E0H(1)SZH(1)E0 . (28)
Substitution of (28) into (19) yields
Eszo+ x[a§11 aé1)] + x2[a§2)+ aé2)] +...+x2[SH(1)E0H(1)S-

13

9
+...+ {5(x“)Tr-—O} (29)

where the braces in (29) include all those terms whose order in x is equal
or greater than 2 and whose trace vanishes, the first of which is

(1) (1) _ (1) 2 (1)
SH EOH S EOH S H E0.

Let us denote by EK the operator obtained by removing from E all
K

those terms whose trace vanishes. We may take

EX 2 exl + e"2 (30)
where
_ (1) 2 (2)
exl - E0 + x81 + x 81 +
(1) 2 2) (3”

,(
exZ ‘32 * “ d2

Rearrangement of the terms in the series (19) is justified by the fact
that the expansions (19) and (29) are absolutely convergent when Ix] is less

than a bound that must be determined in each particular case [12]. Also

ex2w0 = 0 and Exwo = ex (32)

Using for the unitary operator the same notation for removal of terms

with vanishing trace, we write

. _ - - -1 _ - - -1
Ex - UxEon - (UKE0)(E0UK ) . (33)

The following relationship is trivially fulfilled when

14

co co
AiltZa. andB-I1+Zb.:
. 1 . 1
1:1 1:1
Q as
A - B 1 A + B + Z Z aib. -l (34)
1:1 3:1 J

provided that A and B are absolutely convergent.

Since

6 1 + KU(1) + x2fi(2) + .

. (35)

E B E + E B '1 + [ z 2 x(1+J)U(1)EOU(J) ] — E0 (36)
i=1 3:1

as may easily be deduced by using (34).

Comparing (36) and (30), we identify

e = U E = E + xa(1) t x a + . . . (37)
x 1 1

x1 0 0

The expansion for UK may be written from (37) and (27) as

“ 1 - xSH(1) + x2[SH(1)SH(1) — 52H(1)EOH(1) — sn‘2)] + .

C
II

1 + [‘xS + x2(SH(1)S - SZH(1)E°) +...]H(1)+

+ [x2(—S)+...]H(2)+ . .

= 1 + f1(x)H(1) + {2(x)n(2) + . . . = 1 + E: f (x)H(n) , (38)
n=l n

where fn(x) stands for an expansion in powers of x, the first term being of
(1), H(2)

order n. It contains the operators H ,

According to (32),

15

(39)

If the traceless terms in EK had not been omitted, a different U
K

would have been obtained, and it would yield (see (39)) a wave function that

would satisfy the normalization criterion (O lwx) = 1, rather than <¢b|$x) =
K

1 .
When only one perturbative term is present in the Hamiltonian, U;
becomes
E : 1 + f1(x)H(1) (40)
K
and
E = E E w = 6 w = w + f (x)H(1)¢ (41)
x x 0 0 x 0 0 1 0
where
f1(x) = "KS + x2(SH(1)S — $2H(1)Eo) + . (42)
Let
T a f1(x=1) = —S+SH(1)S-SZH(1)E0+... = s’+sn(1)s-szu(1)so+ ... (43)
where S’ 5 -S. Now
w = (1 + TH(1))EO = (1 + w(1) + w(2) + ... ) E0 (44)
where w(1) = S’H(1)E

0

16

.I l f)
w(2) : su‘l’sn‘1)20 ~ s“H(1)E0HmE0 . (45)

Formulae (45) are exactly the same as those given by Lowdin, as may be
seen in a straightforward way, once (37) has been slightly transformed by

using

2 (1) (1) u 2 (1) (1) - 2 (1) (1) (46)
s H EOH E0 - s H EOH EOEO — s H <H >0E0 ’

(1)<H(1)>0 for SZH(1)E0H(1) in Ex.

Let us now compare our results with the expression for the energy

and by substituting 32H

given in ref. [5]. If x = 1,

>’
ll

x0 + <¢0|H(1)W|¢0> = x0 + <¢0|H(1)(1 + TH(1))I¢(0)>

(1) (1) (1)
x0 + <¢0|n |¢0> + <¢0|H TH |¢0>

x0 + [<¢0|H(1)|¢0>] + [<¢0|n(1)s’n(1)|wo>] +

[<¢0|n(1)sn(1)sn(1)|¢o> — <¢b|H(1)82H(1)E H(1)|¢0>] + .

0

x0 + (t1)0 + (t2)o + (t3)0 + . . . (47)

where <ti>0 has been identified with the ith term in square brackets.

According to (24) and (38), we may write in general

t = E an(n)[1 + in fm(x) H(m)] (48)
n=l m=l

and if, in particular, we take only n = l and x = 1

l7

(1) (1)[ (1) (l)

t = H(1)[l + fl(x=1)H 1 = H 1 + TH 1 : H u (49)

The last expression above is identical to the expression given by
Lowdin for the reaction operator t.
Kato [12] found the expansion of wx in a slightly different way. He

considered the adiabatic transformation Wx that allows us to obtain on from

¢

 

 

0 9

w = W W0 (50)

K K
where W is taken as U -E and

x x 0

dWK dEK

= - W
dx dx x (51)

The formulae to be substituted in an expansion of WK at x = 0 are
found by calculating the qth derivative of WK with respect to x at x = 0 and

by comparing with (19),

a q
W = Z -}— [ £1— 1:0q'Kq : E0 + KA(1)EO +
x q=0 q. dxq x x
+ K2{ E0 + % (4(1))230} + . . . (52)
where

 

[‘dE" ] “(1)
dx x=0

18

d2
E <2) (53)

 

, have been used.

The perturbed wave function is obtained from (50).

- _ (1) 2 (1) (1) _ , (1) 2 (1)
4x - 40 KSH 4b + x {SH SH *0 \H >05 R 4b

1 1 2 2
- 5 IISH( )woll 4:0 -— SH( >40} + . . . (54)

Derivations of the expressions equivalent to A(n) for the energy are

analogous (9:. [9,12]).

2.4. - DISCUSSION

Several authors ([16] and references therein) have formulated
diagrammatic representations of the Rayleigh-Schrodinger perturbation
theory. The use of the rules from such techniques is a convenient way to
write to any order the correction to both the energy and the wave function,
without requiring an explicit calculation when the expressions are needed.

A(“) and its

Similarly we point out that the sum rules in expressions for
equivalent correction terms to x0 are not particularly complicated. Once
these expressions have been written, it is a straightforward matter to

(n)

obtain correction terms directly either by introducing A in the
expansion of WK to obtain the wave function or by calculating several
traces to obtain the eigenvalue.

The method has the advantage that we may keep consistency between the
order of the correction in our expansions for xx and w“ and the order in x
kept in the expansion for Hx, since we have formulae that are general

enough for both A(n)

and its equivalent form for xx; i.g., the formulae are
not restricted to Hamiltonians like HK = H0 + xH(1), but Hx may be taken to
the desired order in x.

Another significant aspect within this context is that w; is obtained
from $0 as the result of an adiabatic transformation, WK or Ux'EO' The
operator that carries out this transformation acting on an arbitrary

function yields an eigenfunction of H“. The unitary operator, UK or Ux,
represents how the function describing the state of the system evolves as
the effects of the perturbation change it.

The relationship with the variational principle has been discussed by

Kato in his original papers. The degenerate case is also treated in the

19

20

references given to Kato’s work.

Following Kate’s original work, the usual series in perturbation
theory have been deduced, with the same normalization criterion as in
Lowdin’s work [3-5]. The series have been written in terms of a reduced
resolvent, for which no problem about singularities arises. This work
establishes the consistency between results obtained from Ldein’s method
and expressions developed strictly within the framework of complex—variable
perturbation methods applicable for Hilbert spaces.

A different method from that used by Kato [12] has been given to
derive an expansion for the wave operator. It allows us to visualize
different ways to obtain the perturbed wave function and its different

normalization conditions.

2.5 - RELATIONSHIPS AMONG PERTURBATION METHODS.

This section should be regarded as a complement to the main contents of
this chapter. We include it here because we have seldom encountered
references to Kato’s work in the literature on perturbation theory; yet
Kato’s rigorous treatment of perturbation theory provides a reliable and
powerful algebraic language to deal with perturbative problems. The
diversity of perturbative treatments in textbooks and research papers and
their apparent independence and specialized character make it difficult to
develop a unified conception of several areas of nuclear, atomic and
molecular quantum mechanics because of an apparent lack of common
principles. We therefore present in this section the underlying
fundamentals, which interrelate different mathematical formalisms through a
common background.

It is customary to regard interactions as corrections to an
unperturbed Hamiltonian, H0. The total Hamiltonian HK is decomposed into
Ho and the perturbation xH(1) associated with the interaction. The limit x
= 0 correspondents to the unperturbed system whereas x = 1 corresponds to
the real system. In the study of atoms, for instance, H0 may include the

H(1) would

kinetic energy, nuclear attraction and central potential terms;
include the Coulomb repulsion minus the central potential terms, and
perhaps magnetic interactions or interactions with an external field. The
whole Hamiltonian is thus split into two terms formulated according to the
independent-particle model.

Kato’s formalism, as formulated in the references given in this

chapter, yields expansions commonly called Ragleigh-Schrbdinger expansions

for the perturbed eigenvalues and eigenfunctions. Besides mathematical

21

IN)
(‘3

rigour, Kato’s treatment has the strength of dealing with projection
operators, EK or E0, which are uniquely determined, whereas eigenvectors
lack this property. Some of the results of this chapter have been
obtained because of this feature, g;g;, different normalization criteria
imposable on the wave function are easily visualized in terms of the
operator that yields the perturbed state vector when it acts upon the
unperturbed wave function. The exact energy, xx, does not appear explicity
in the expansions. This fact renders the expansions for the exact
eigenvalues and eigenfunctions very useful for calculations. Size
consistency is another interesting feature of the expansion for the exact
energy given by Eq. (47). Brueckner [17] considered the Rayleigh-
Schrodinger expansion and showed that the terms having a non-linear
(aphysical) dependence on the number of particles of the system cancel with
each other. The linked—cluster diagram theorem was later proven by time-
dependent [18] and time-independent [19] methods. This theorem states that
the aphysical terms cancel through all orders of diagrammatic perturbation
theory, only linked diagrams appear in the series for xx.

Although it is not conventional in the theory of operators on the
Hilbert space, we could consider the inverse (xx - Ho)- instead of the
resolvents of either HO or H“, as taken previously. We would thus generate
expansions analogous to those given in previous sections of this chapter,
but they would have xx instead of xb in the resolvents. The perturbation
expansions with explicit dependence on the perturbed eigenvalue xx are
known as the Brillouin-Wigner expansions. The analogue to Eq. (47) so
constructed is the Brillouin-Wigner expansion for the energy, and that of

Eq. (49) is designed the transition matrix or T-matrix [20-22] in

23

scattering theory, whereas its expanded form is known as the Born (or
Newmann) series [20~22]. "Reaction matrix" or "K—matrix" replaces the term
"transition matrix" in studies on nuclear matter [23]. Eq. (47) with x0 in
the resolvents replaced by xx may provide a justification for the
terminology "effective interaction", as t is sometimes called.

The sum of the zeroth- and first-order correction energies is the
expectation value of HK in the state represented by the model function ¢b
in both the Brillouin-Wigner and the Rayleigh"Schrodinger perturbation
expansions (see Eq. (47)). When H0 is the central-field Hamiltonian and

Hm

includes at least the noncentral electrostatic interaction, the sum x0

+ («no I 11(1)

I¢O> is called the Hartree—Fock energy because this is the
quantity minimized in a Hartree—Fock procedure [24-27]. The remaining part
of the energy is the correlation energy. Methods such as CI, MCSCF,
electron—pair theories,..., [28] are employed to obtain quantitative
information on the terms in Eq. (47) beyond the first two. The terms in
Eq. (47) of second order and beyond represent true many-body effects, for
the Hartree-Fock approximation only takes into account the effects on each
electron of the average field of the remaining electrons and nuclei. Thus,
the electrons move independently of each other and the instantaneous motion
of the electrons or the correlation between them in their mutual Coulomb
field is not taken into account as such [29].

Size consistency is a desirable property of any method used in
computations in chemistry because it is the differences between two
quantities what are often most interesting. In contrast to the Rayleigh-

Schrodinger perturbation expansion however, the Brillouin-Wigner expansion

for the energy is not size consistent. Nevertheless, we know [9] that the

24

eigenvalue xx of the perturbed Hamiltonian is regular in x and power—series
expandable with a non-vanishing convergence radius. The eigenvalue is an
analytic function of x and xx * x0 as x + 0. Therefore, it is legitimate
to think of an expansion about xx = x0 of the resolvents in the Brillouin-
Wigner series. Such an expansion allows us to recast the Brillouin-Wigner
series in the Rayleigh-Schrodinger form. Furthermore, Brandow proved
[31,32] that this procedure allows for cancellation of all the unlinked
terms in the Brillouin-Wigner series, as should be expected because the
Rayleigh-Schrodinger series is size consistent.

Feshbach’s operator [34,35] in nuclear physics is an effective
operator that yields the exact energy when operating on a model function.
Lowdin’s [3,5] treatment of the partitioning technique may be considered a
development of Feshbach’s previous work.

Finally, the resolvent operator (14) may be written as
_ —-l_ . -1_ +
R.(“) - (HK-R) - — (c - Hx+1n) = — c (a) (55)

with t, n e]R,and £5£+in. The zeroth-order Green’s function operator or

propagator G; (C) is obtained from 6+ (t) by replacing Hx by H0. Now

(a-H +1.) = (E‘Hx+in) + “(1) (56)

0
from which it is a simple matter to obtain the Dyson equation

6+(z) = 63(c) + GS<£>H(1)G+(:) (57)
closely related to the wave-operator relationship (40). The limit of £+ xx
and n + 0 yields a distribution which establishes the connection between
the Green's function-opeator notation of the resolvent and the singular (or

Sochozki’s) generalized functions [36] of use in scattering theory.

CHAPTER 3.

3.1. INTRODUCTION

Interactions between colliding molecules in gases or liquids cause
shifts in the charge distributions of the collision partners. These shifts
result in differences between the net dipoles of colliding pairs (or
clusters) and the vector sums of the dipoles of the molecular constituents,
if unperturbed. Collision-induced changes in dipole moments are manifested
in the dielectric and spectroscopic properties of bulk samples. For
example, infrared and far infrared absorption processes that are single-
molecule forbidden may be observed in compressed gases and liquids as a
consequence of transient, collision-induced dipoles. Such pressure-induced
far IR absorption has been observed experimentally in inert-gas mixtures
[37-39], H2 [38], N2 [38,40-42], 02 [38,40,41], CH4 [43], and SF6 [44] and
forbidden near IR spectra have been studied for the diatomics, triatomics
such as CS2 [45], and other polyatomics. Useful information on dynamics in
dense gases and liquids can be obtained by analyzing the lineshapes for
single-molecule forbidden spectra (and for collision-induced contributions
to allowed spectra), if the collision-induced dipoles are known as functions
of intermolecular separation and relative orientation. In analyses to date,
collision—induced dipoles for small molecule pairs have often been
approximated as sums of classical multipolar contributions, with short-range
anisotropic overlap corrections represented by parametrized exponential
functions. At this stage, direct calculations of overlap effects on pair
dipoles are needed in spectroscopic applications. It is also of interest to
compare the calculated dipoles of van der Waals complexes in their

equilibrium configurations with the dipoles determined experimentally by

25

26

molecular beam electric resonance studies of the Stark effect on rotational
transition frequencies characteristic of the complex [46,47]. Calculations
or measurements of collision—induced dipoles provide information on
molecular interactions complementary to that obtained from potential energy
surfaces, and may indicate the relative importance of classical
electrostatic interactions, charge transfer, and short-range overlap and
exchange effects [48].

For molecules interacting at long range, only classical-multipole
polarization (cf. [48]) and dispersion effects [49] contribute to the
collision—induced change in dipole moment. The net dipole for well
separated molecules can therefore be determined if values of the single-
molecule multipole moments, polarizabilities, and nonlinear response tensors
are known [50—55]. For molecules interacting at short range, definitive
results can only be obtained by ab initio calculation [56-62]. Calculations
including correlation effects have been performed for the pair dipoles of
He...H [57,59,60], He...Ar, He...H2, and H2...H2 [57], while calculations
restricted to self-consistent field level are available for the dipoles of
the inert~gas heterodiatoms Ne...Ar, Ne...Kr, and Ar...Kr [56,58], and for
the Ne...HF dipole [48]. Quite large basis sets are usually needed in pair
property calculations [61], with the consequence that the computational
requirements are substantial. This prompts interest in approximations that
yield good results in the region where overlap is small, but nonnegligible.
Since numerical precision is most difficult to attain in ab initio
calculations on molecules at intermediate and long range, approximations
applicable near the van der Waals minimum may be used to join the known

long-range forms of the interaction—induced dipole to ab initio results at

27

short range. Additionally, quantum mechanical approximations for collision~
induced properties provide information on the effects of long-range
electrostatic interactions, overlap, exchange, hyperpolarization, and
dispersion, which may prove useful in selecting basis sets for subsequent
ab initio work.

To evaluate proposed approximations for pair properties, it is
necessary to compare the results with the accurate ab initio results
available for properties of small molecular pairs. Comparisons of ab initio
and approximate collision-induced polarizabilities of H...H in the triplet
state [62] and He...He [61] have been used to test electrostatic overlap
models [63], exchange perturbation methods [62,64—66], polarizability
density models [67~69], and exchange antisymmetrization approximations
[70,71].

In this chapter, we report a calculation of the collision-induced
dipole moment of He...H as a test of label—free exchange perturbation theory
[72—75] at lowest order. The label—free exchange perturbation method
constitutes a direct Rayleigh-Schrddinger perturbation theory with fully
antisymmetrized zeroth-order wavefunctions. By construction in terms of
projection operators, the unperturbed Hamiltonian and the perturbation term
are individually invariant with respect to exchange of electrons between the
interacting molecules. The label-free exchange perturbation formalism is
reviewed briefly in section 3.2, for application in Computing collision-
induced dipoles. In section 3.3 the label-free theory is related to other
exchange perturbation approximations, and the collision-induced dipole is
shown to separate into polarization and exchange contributions. It is also

shown that results for the pair dipole at zeroth-order in the label-free

28

theory are identical to the results of an exchange-antisymmetrization
approximation developed by Lacey and Byers Brown [70].

We have calculated the He...H dipole moment in two ways: first, by
evaluating the expectation value of the dipole operator with the zeroth-
order pair wavefunction; and second, by computing the energy for He...H in
the presence of a uniform applied electric field and then differentiating
with respect to the field to obtain the dipole. Also, we have carried out
the calculations at two levels of approximation for the single-atom
wavefunctions. In the first, ls Slater orbitals are used on each center;
and in the second, an extended Gaussian basis is used at each center.
Evaluation of the dipole expectation value in the Slater basis has been
reported previously by Buckingham [49] and by Mahanty and Majumdar [76]; we
obtain identical results in this case and new results from the other three
calculations. The methods of calculation and the selection of basis sets
are described in section 3.4. Results are presented in section 3.5. We
find that closest agreement with accurate ab initio results [57] is obtained
from the Gaussian-basis calculations of the dipole as an energy derivative.
Errors in the overlap dipole are typically 20-30% at this level of
approximation. Significantly, the errors in this approximation appear to be
smaller than the discrepancies between the two reported ab initio

calculations of the He...H dipole [57,60].

3.2 EXCHANGE PERTURBATION THEORY IN LABEL-FREE FORM

For molecules interacting at short range, both polarization and
exchange effects contribute significantly to the interaction energy and to
interaction-induced changes in electric properties. The label—free exchange
perturbation theory developed by Jansen [72-75] treats these effects within
a direct Rayleigh-Schrodinger formalism. Projection operators are used to
define an unperturbed Hamiltonian and a perturbation term that are
separately invariant with respect to electron permutation. Also, within
this formalism the expectation value of any dynamical variable for a cluster
of interacting molecules can be separated into additive contributions from
each of the molecules in the cluster.

The Hamiltonian H for interacting molecules A and B with a total of N

electrons is

2
. 7 Zb 1

V .
( -z-—§-—Z-——)+2—~- (1)
2 a raj b '"bj k<j rkj

:1:
H
H M2

j l
where Z8 is the charge of nucleus a in molecule A, Zb is the charge of
nucleus b in B, j and k are electron indices and rq‘j is the distance from
electron j to nucleus q. Atomic units are used throughout this work, and
the nuclear positions are assumed to be fixed.

To carry out a Rayleigh-Schrbdinger perturbation expansion with a
fully antisymmetrized pair wavefunction as the unperturbed wavefunction, an
appropriate partitioning of the pair Hamiltonian into an unperturbed
operator and a perturbation term must be found--but the electron labels (j

l to N) are not assigned exclusively to molecule A or to molecule B, and the

antisymmetrized wavefunction is not an eigenfunction of the single-molecule

29

30

Hamiltonian corresponding to any fixed electron assignment. Within the
label free exchange perturbation formalism, this problem is resolved by

splitting H into the unperturbed Hamiltonian

“ p " P (0) <0)
: ‘: "7
Ho .3: ”01 E: [HA1 ”'31 ]Ai’ ‘4
1-1 1—1
and the perturbation V due to molecular interactions,
A P
V 3 OZ Vi/Ki. (3)
i=1
In these equations Hg?) is the Hamiltonian for the isolated molecule A,
given that A has been assigned the ith possible set of NA electrons selected
from the total of N (and similarly for Hég); Vi represents the interaction-

energy operator for the ith assignment of electrons to molecules A and B.

The number of ways of assigning NA electrons to molecule A and NB to

+ N , is

molecule B, from a total of NA B

9
_ (NA+ NB).

P- v 0
NA' NB.

(4)

The operator/li projects out the simple product term for the ith electron
assignment, when applied to an antisymmetrized pair wavefunction. The
zeroth-order wavefunction 00 is constructed by antisymmetrizing the product

of the ground-state wavefunctions ¢A and $8 for isolated A and B molecules:

u p41:

1% (8)

Q0 = f 2.0P.Pi[¢A¢B] = f
1 1 1

31

where f is a normalization constant and 2 OP Pi is the intersystem
i i

antisymmetrizer. By definition the projection operator/i,i acting on QC
yields f¢i'
]f a uniform, static electric field E is applied to the A B pair, the

A

full Hamiltonian HF is

fiF = fi (5 = 0) + 8’ (g) (6)

where H (E20) is given by Eq. 1 and

A P
I! (E) - ’13 . E _ ..F . )2 Bi Ai - -E .i§1(gi + Bi) A1 (7)

The dipole operator Bi is the same for all electron assignments i, but the

dipole operators 2? and B? for A and B depend upon the electron assignment.
At self-consistent field (SCF) level, the wave function for molecules

A and B is given to first order in an applied field E and zeroth order in

the molecular interaction by

N
- -1/2 “(M -1/2 ~(1) _ l *3/2 ~(0)
Qb - C1 Q + E [ c1 151 91 2 gzcl Q ] - (3)
The constants c1 and 92 are
c1 = <$(°)| $(O)> (9)
c = 2 ; <$(°)|$(1)> (10)
2a i=1 in '

5(0)

is an antisymmetrized product of unperturbed orbitals centered on A and

B, and the tilde superscript indicates that the function has not been

32

~(1)

normalized. “in is the antisymmetrized product with the unperturbed

orbital i replaced by the first order correction to orbital i in an

applied field Fu in the u direction. The vector égl) has components
"(1) "(1) ~(1)
(pix ' ’iy ’ and ’12 ‘

To lowest order in the interaction, the total energy E of the A—B pair

is

- - <0) <0) -1 ~(0) P ~(0)
E-<@0|H|¢o>—EA +13 +c <<I> [xvi/tile >+

B 1 i=1
—1 ~(0) ,. N ~(1) —2 ~(0> “ ~(0) -
+E-{2c1 <<I> IH(£=0)IJEIQ>‘j >-<,;2c1 <<I> IH(£=0)|§ >-g } (11)
where
§= awhile, jLI350)»;1 (12)
1:

Thus, to lowest order, the energy is the sum of three terms: first, the

(0) (0)
A I EB

the absence of the applied electric field; second, a field-independent term

total energy E for molecules A and B at infinite separation and in
equal to the A-B interaction energy to lowest order if exact wavefunctions
wk and “h are used to construct 0b [77,78]; and third, a term linear in the
applied field.

From the energy expression, the dipole moment may be obtained by

differentiating with respect to the applied field:

2%
B aF

- 19 «p Inn; > - -[<fl |H|~p > +
‘ a§ o 0 i=0 ’ aE 0

N

{=0

33

+

.811 gal:
(QOIaEI@O> + <¢0|H| 8E )]E = 0

A N
Q - 2c‘1<¢(°)|n(520)| z

(1)) + c_2c
1 ~
1—l

§i 1 <5(°)|§(§=0)|$(°)>. (13)

2

In Eq. (13) g is the dipole moment calculated directly as the expectation
value of the dipole operator with the zeroth~order, zero-field wavefunction,
as in Eq. 12. The remaining terms in Eq. (13) represent the non-Hellmann-
Feynman contribution, which would vanish if Q0 were exact.

In the label-free perturbation formalism, the operators R0 and 7 are
not Hermitian individually, although the sum of the operators is Hermitian.
As a result, the pair energy at first order in the interaction formally
includes the correction term <¢0|H0|01>, which depends on the first-order
change Q1 in the pair wavefunction due to the perturbation V. Since this
term is expected to be substantially smaller than the standard first-order
energy shift (Q0|V|@b> included in Eq. 11, the correction <¢b|H0|@1> is
usually grouped with the leading second—order terms [72,73].

To second order, the interaction energy may be computed from Q1 in a
sum—over-states form [72], with contributions from the continuum generally
nonnegligible [79]. Alternatively Q1 may be approximated by finding the

stationary vector of the functional (cf. [80])

J[q,] = <m|H0—E0|\p> + <q;0|v —E1|q:> + (\IIlV-EIIKIJ0> (14)

subject to <Q|Q0> = 0. If SJ vanishes with an arbitrary variation 6% of Q
away from the stationary vector 31, then the expansion coefficients ak for
$1 satisfy

34

 

l
<¢k|vnp0> + 2 31:30 kaa[<4’k'"o'%> + <¢Q|H0|¢k>1
8k: Ei— E ('5)
0 k

in the basis @k of orthogonalized, antisymmetrized product states for A and
B at infinite separation. In the approximation that the sum on the right-
hand side of Eq. 16 is negligible, $1 = @1. An equivalent approximation has
been made by Jansen in a direct expansion of 01 in the basis ¢k [72].

If $0 and @1 have been determined, then the sum of the unperturbed
energy E0, the full first-order energy E1 (including the correction term

<¢0|H0|¢1>), and an approximate second~order energy E for the A-B pair in

2

an external field can be obtained from

E0 4 E1 + E2 : <mO|H (g = 0) — e - g | ¢b> + <mllvl¢0> . (16)

The interaction~induced dipole may then be computed from the derivative of
Eq. 16 with respect to E, as in Eq. 13.

In this work, the calculation of exchange-overlap contributions to the
He...H pair dipole is based on Eqs. 11 and 12, which represent the lowest-
order effects. The exchange~overlap terms fall off exponentially with
increasing R. In contrast, the second-order dispersion contribution to the
He...H dipole (present in B derived from Eq. 16) varies as R".7 at long
range. This contribution originates in the correlations between the
fluctuating charge moments on the H and He atoms, and it dominates the
exchange-overlap contribution for sufficiently large R. Accordingly, in
calculating the He...H dipole for internuclear distances R > 4.0 a.u., we

have added the leading dispersion dipole to the lowest~order exchange-

35

overlap dipole from Eq. 11 or 12. Higher-order polarization and exchange
effects are neglected at this level of approximation for the pair dipole;
overlap damping of the dispersion contribution and the effects of He—
electron correlation on the He-H correlation and exchange energies are also

neglected.

3.3. COMPARISON WITH OTHER EXCHANGE PERTURBATION METHODS

In this section the label-free exchange perturbation theory of
interaction-induced properties is related to other exchange perturbation
approaches. We show that the results from the label-free theory can be
separated into polarization and exchange contributions [2,81-83]. We also
show that the zeroth—order expression for the pair dipole in the label-free
theory is identical to the exchange-antisymmetrization approximation used by
Lacey and Byers Brown in calculations of inert—gas heterodiatom dipoles
[70].
3.3.1 Exchange perturbation theory: Polarization and exchange
contributions to interaction energies and pair properties
In one standard form of exchange-perturbation theory [81] the nth

approximation to the wavefunction On and the nth approximation to the

interaction energy En of an AeB pair are obtained iteratively from

@n ¢o l R0 (En " V1) @n—l (17)

and

E
n

ll

<¢0|v1|¢n_1> , (18)

where ¢ is the simple product of the ground—state wavefunctions ¢ and w ,
0 A

V1 is the perturbation term for the assignment of the first NA electrons to

molecule A and the remainder to B, and R0 is the reduced resolvent

|¢k><¢kl

kxo Ek ’ Eo

R = Z (19)

0

‘Phe function ¢k is the kth simple—product excited state of A and B at

infinite separation and Ek is the corresponding energy. The intermediate

36

37

normalization <¢OIQ> = 1 has been imposed. Symmetry forcing is accomplished
by choosing the normalized antisymmetrized product of ¢A and wB as the

zeroth-order approximation @0 [2,65,66]. At first order the A-B interaction

pol

energy is the sum of a polarization term E1 and an exchange term EeXCh:

l

<¢0|v1|r¢0> — <¢0|v1|¢0><¢olrl¢o>
1 + <¢0|P¢0>

_ pol exch _
E1 ~ E1 + E1 ~ <¢0|V1|¢0> +

 

<¢0|V1|¢0> + <¢0IV1|P¢0>

= , . (20)
1 + <¢0|P¢0>

 

where P is the intersystem antisymmetrizer (see Appendix B).
In the label-free exchange perturbation theory, the interaction energy
at first order is

p
E1 : «11021-21 ViAil‘I’o'i . (21)

This expression for E1 may be simplified by splitting the full
antisymmetrizer in @0 into terms that involve electron permutations within
A, within B, and between A and B. In Appendix B, the permutation invariance

of V is used to obtain

E-l[<¢w|vlww\+/¢¢|§:v1\|flfli: PAB{x }>l(2°)
1 ' s A B 1 A 3’ ‘ A 3 i=2 i i 'A BJ;1 OJ 5 1"‘XN ‘

The expressions for wA and $3 have been restricted to the self~consistent
field level, and the set {x1...xN} contains the orbitals occupied in the

ground states of isolated A and B molecules. The operators P23 acting on
{x1...xN} perform intersystem permutations only. In Eq. 22,.AA is the

antisymmetrizer for electrons assigned to molecule A,.AB is the

38

antisymmetrizer for B, 01 is the parity of the ith permutation among the
total of 2, and

s ; l + <¢0|P¢0> . (23)
Thus the interaction energy at first order is given as the sum of the
polarization contribution <¢0|V1|¢0> and an exchange correction. Each of
the operators/l.i in the second term of Eq. 22 projects out the single term
from the ket for which the assignment of electrons to A and B is consistent
with the form Vi for the perturbation. In the product ¢A¢B the first NA
electrons are assigned to A and the remainder to B. Relabelling of dummy
indices suffices to show that Eqs. 20 and 22 are identical. At first order,
equivalent results are also obtained from variants of exchange—perturbation
theory with stronger symmetry forcing [2,84-87].

The one-particle density matrix [88~90] can also be separated into two
terms, one without electron exchange between A and B and a second with one
or more interchange contributions. Consequently the expectation value of
any single—particle operator decomposes into polarization contributions and
exchange-overlap contributions [2]. McWeeny and Sutcliffe [89] and
Magnasco, Musso, and McWeeny [91] have performed the separation explicitly
for the one-electron part of the Hamiltonian for two interacting molecules.
The analysis for the dipole moment operator is analogous.

3.3.2 Density-matrix perturbation theory

Lacey and Byers Brown have evaluated the dipole moments of several

diatoms by first finding the density matrix associated with the normalized,

—l/2 5(0)

antisymmetrized product c1 of the isolated atom wavefunctions [70].

The spinless electron density p(0)(r) in the absence of an applied field is

39

obtained from

0 -1 ~
P( )(El) = Nc1 I ds1 I dxz...dx ¢(O

N )(51’52"°§N)$(0)*(§1’§2’“§N) (24)

(0)

where o and c1 are defined as in section 3.2, and E, s, and x denote
spatial variables, spin variables, and collective space-spin variables,

respectively. The exchange—overlap dipole of the atom pair AB is then

r + Z r (25)

B 2’ - IE p(0)(:) d3£ 't Z A B~B.

A

At lowest order in the label-free perturbation theory, the expectation
value of the dipole is identical to the Lacey-Byers Brown result. This

equivalence can be shown explicitly starting from

P
g=<~plzgiAinJ> (26)
121

with the pair wavefunction Q from Eq. 8. The operator Bi is the same for

each of the electron assignments i and

A. =1. (27)

LEM”

i

Thus

t
H
/\
'6'
(C
H
'6‘
V

= - I 5 9(a) d E + Z 5 + Z 5 (28)

In terms of the applied field E

p(1;) = 9(0)“) + g-e‘llr; + 003) (29)

40

(0)

The electron density p (r) for the AB pair in the absence of an applied
field (determined from Q) is identical to p(0)(r) from Eq. 24, so the field-
independent dipole is equal to B from Eq. 25.

It should be noted that this approach yields only the term i in Eq.
13. At this level of approximation the Hellmann-Feynman theorem is not
satisfied, and the pair dipole moment obtained by differentiating the energy
differs from E.

The lowest—order correction to the pair electron density in an applied

field satisfies

N
(1) - -1 ~(0) ~
8 (r1) — NcLE1 Ids1 Idxz...de [ é (51...xN) 9i (x1...xN)
*
+ §§1)(x ...x 5(0) (x ...xN) ]

_ —2 ~(0) ~
Ngzc1 Ids1 dez...de Q (x1...§ ) Q (§1°°'§ ) . (30)

N

Substitution of this result for 3(1)(r) into Eqs. 28 and 29 gives an
approximation for the exchange—overlap contribution to the pair

polarizability.

3.4. COMPUTATIONAL METHODS.
In this section the methods used to approximate the He...H
wavefunction and to calculate the pair dipole are described. In each of the
calculations we have employed a wavefunction of the form given in Eq. 8 for

the He-H pair in an applied field in the z direction

3
. ~(0) _ 1 , <0) <0 —<0
w1th Q —Jr§- E opxtxfl <1) Xfle 22) xhe 23)} (32)

prior to normalization. P runs over all the elements of the symmetric group
83 and oP = :1 depending on the parity of P. The functions 51 are defined
by

$§1> = x§1><1)w‘°)<2.3) — x§1)<2) w(°)(1.3) + xé1)(3) ¢(°)(1.2) <33.a)

(1‘

5%]i: x§°)<1) ¢[1)<2.3) — x;°)<2) w§1)(1.3) + x§°)<3) w, ’(1.2) (33 b)

~<1)_ <0) <1) _ (0) <1) <0) (1)
i3 - xH (1) wz (2,3) xfl (2) $2 (1.3) + x” (3) t2 (1.2) (33.c)

<0) . -__1_ <0 — (o _ — (o (o
a» (1..) -- J2- { xfle 21) xHe 22) xfle (1) 4,6 32) } <34.a)
(1)- _1 <1) ~(0), _ —(0) (1)

w, - J2 a,“ (1.) xfle (2) 4,8 <1) xHe (2)} <34.b)
<1)- _1 <0) —<1) _ —<1)' (0)

$2 — fl {xfle (1)xHe (2) xfle (1)><H,3 (2)} (34.62)

In each of the terms above, x(i) represents x(§i)u(i) and 2(i) represents

x(§i)B(i), where u and B denote spin states. xéo) is the lowest—energy

(1)

atomic orbital for an isolated atom q and xq

(0)
X9

is the first-order correction

to for a single q atom in an applied field in the z direction.

41

42

Within the label-free exchange perturbation theory, the 2 component of
the He...H pair dipole can be expressed as a sum of H—atom and He-atom
. H He
: Z + 0 ‘~
dipoles (”2) (p2) (pz > (35)

To lowest order in the atomic interaction and to first order in the applied

field F ,
2
H ~(0) ~(0) 3 ~(0) ~(1)
(“2) r N2 (F‘z ) {<o I“ bl > + F2 2 (é I“ %|
23:1
l 0
+Fz: <<1>(.)Ip'z‘<'1'>>l()1 (36)
1:1
and
3
< He > - N2 (Fz ) {<¢(0)| “Ola (0)> + F2 2 <¢(°)|$ %|$(1)> }
1:1
3 ~(1) “He ~(0)
+ F, 1: < <1. I.) |<1> >1 (37)
z ._ j z
le
2 _ —l -l . .
where N (F2) ~ c1 (1 * cmqzc1 ) and C02 is the z component of go given by

Eq. 10 with N=3. If the origin of the coordinate system coincides with the

H nucleus and the He nucleus is located at R on the +2 axis,

“H

“z 2 ‘2 ziAi (38)
1
and
11:8 = — [ (22+ 23- 2R)/\1+ (21 + 23- 2R)/\2+ (21+ 22— 2R)/\3 ]. (39)

The result for (pz> is then

43

00
s
HHe oo oo 2 _ oo 01 10
(“2)“1_Soo 2 [22 Zane Msnne1+F 1_Soo 2 [ RSHHe(SHHe+SHHe)+

HHe HeH

oo 01 +810 )_Z;g

+zHeH( HeH HeH

10 00 10 00 2
+2 zHHeSHeH zHeHe(2-SHHe )
800
10 00 HHe 00 00 01 10 00
+ZHeHSHHe+ 00 2 [2% flSHHe](SHeH+SHeH )SHeH } (40)
l- -S
HHe
where
t. (t t t
S a;: (x 5 ix £5; and z ab: (x ; izlx (Si ; t,s = 0 or 1; a,b = H or He .

The field-independent term yields the approximate zero-field pair dipole
(and the term linear in F2 approximates the exchange—overlap contribution to
the 22 component of the pair polarizability tensor.

It has been suggested [92] that calculating the energy of the He-H
pair in an applied field F2 and then differentiating with respect to F2 in
order to find the dipole (Eq. 13) should yield more accurate results than a
direct calculation as an expectation value (from the field—independent term
of Eq. 28). In general, though, no a pgigri choice between the two methods
is possible, and we have performed calculations to test each of the methods.
The explicit expression obtained from Eq. 11 for the He-H pair energy (to
first order in the atomic interaction and first order in the applied field
F2) is given in Appendix C.

We have obtained wave functions for the isolated atoms in an external

field at two levels of approximation. The A.0.’s x are constructed with

44

Slater bases in the first approximation and with Gaussian bases in the

(0)

second. In the Slater basis, X“ is the H~atom ls orbital, and the first

(1)

order correction to the hydrogenic wave function XH is found
analytically[93]. xél) can be expressed in terms of modified 2pz and 3pz

Slater orbitals:

iél) =-;% ’ (1+§ )r cos a (41)

In the first approximation for He [76], we use the optimal ls Slater-

(0 ) (0 )

orbital for‘xHe and the unperturbed wave function ¢H He satisfies

W(O)

wHe "7? : expi-{(r1+r2) ][u(1)B(2)-B(1)u(2)] (42)

with § 2 27/16. The first order correction to the He wave function is

approximated variationally by minimizing

_ 3 3 l (l) . (l) _ (l) ¢(1) l _ l _ l 27 2 (l)2

I—Jd rld r2[4 Ylee lehe + 4 Y2¢He V2¢H + (2r12 r1 r2+ 2(16) )whe
l

+<z +22”. if? 12.)] (43>

<1) _ _,H<1) (0)

using a trial wave function of the form wHe wHe

. (1) _
with H — (21+22)Fz'

(The subscripts l and 2 in the functional I refer to the electrons in the He
atom.) The value obtained for the variational parameter is t = —0.72231,

and

45

32.2: = ...;3

In the calculations at this level of approximation, the matrix elements
appearing in Eqs. C.l-C.ll for the pair energy and in Eq. 40 for the pair
dipole were evaluated by use of standard formulas for integrals involving
Slater orbitals [94-102] (see Appendix D for a hybrid integral involving the
modified Slater orbital 3pz).

In the second approximation SCF calculations were performed for He and
H using uncontracted bases of ten 3 Gaussian functions to determine x(o).
The Gaussian orbital exponents [103] and the calculated coefficients for
each of the functions are given in the Table. The calculated ground state
energies are -2.8616692 a.u. for the He atom and -0.49999862 a.u. for the H
atom. As outlined below, xél) was determined from a series of SCF
calculations on the H atom, with a bare positive electric charge placed on
the z axis at a distance ranging from 45 a.u. to 60 a.u. from the H nucleus
[104]; xéi) was determined in the same way. An uncontracted Gaussian basis
set of six p functions was used for each atom, together with a single
contracted function x(o) formed from the ten 8 functions with contraction
coefficients from the Table. The orbital exponents of the p Gaussian
functions for hydrogen [105] and helium [62] are also listed in this Table.
The p function coefficients obtained directly from the SCF calculation
reflect linear and nonlinear polarization of the atoms by the field E and

(1)

field gradients E’, E"... of the point charge. To fix x , it is necessary
to determine the contribution to each coefficient that is first order in the

electric field Fz. This contribution was obtained by fitting each

contraction coefficient ci to the expansion

46

C

. = C(1)F + C€2) F3 + c (3) F F’ (45)
1 1 z

1 z i z 22'

(l)

The resulting Ci values are listed as the p contraction coefficients in
the Table .

As a check on the quality of the Gaussian bases, we have calculated
the atomic polarizabilities of H and He. For the H atom the Gaussian basis
yields a = 4.4989 a.u., while the exact value is a = 9/2. For He, in the
Gaussian basis a = 1.3087 a.u. More accurate values for the He
polarizability from near Hartree—Fock calculations differ by less than 2%
from this result [61,106,107].

The integrals needed in the Gaussian basis calculations were performed
using the program SOINTS, written by R. Pitzer (Ohio State University) and

maintained by the Argonne National Laboratory Theoretical Chemistry Group.

Function

10

ll

12

13

14

15

16

Type

47

TABLE 1

Orbital exponents and contraction coefficients
of the Gaussian basis (lOs 6p) on each atom

 

Exponent H Coefficient

(Normalized)
1170.498 7.37776139.10'
173.5822 5.83581828-10‘
38.65163 3.18288336-10"
10.60720 1.38031549-10”
3.379649 4.89937724-10‘
1.202518 1.42487999-10’
0.463925 3.12524622-10'
0.190537 4.13000752-10‘
0.0812406 2.02671433-10’
0.0285649 7.74758325-10"
3.009711 —1.5993250-10‘
0.710128 —1.4338116-10‘
0.227763 —8.7766919-10’

0.0812406 -1.4154398

0.0356520 —1.3241078-10"
0.0154420 —3.0605474-1o‘

5

4

3

2

2

l

l

l

l

3

2

l

1

l

2

 

2

2

1

l

l

l

2

3

He
Exponent Coefficient
(Normalized)
3293.694 9.59977811-10
488.8941 7.61241309-10
108.772 4.11477457-10
30.1799 1.72174980-10’
9.789653 5.70398703-10’
3.522610 1.49210258-10”
1.35436 2.82269836-10“
0.5561 3.59695130-10’
0.2409 2.51503899-10’
0.10795 5.17599605-10'
6.6 -2.4617864-10‘
2.1957216 —1.9220719-10’
0.5693178 —1.4916053-10'
0.4223072 -2.4362999-10"
0.2007022 -3.4883846-10'
0.0799030 —1.0301889-10’

5

4

3

2

l

3

l

1

3.5. - RESULTS AND DISCUSSION

The He...H dipole obtained by differentiating the interaction energy
with respect to an applied field (Eq. 13) shows a roughly exponential
dependence on internuclear separation in the range 4 a.u. S R $ 8 a.u. This
range extends beyond the van der Waals minimum near 7 a.u. [60]. In the
Gaussian basis, the dipole from Eq. 13 reaches a local maximum at very short
range (R ~ 0.6 a.u.), and vanishes as R approaches zero, as expected. At
short range, where the dipole is determined primarily by overlap and
exchange effects, the polarity is He+H—, consistent with the rule that the
larger atom is negative [49]. At long range, the pair dipole results
entirely from dispersion effects, and its sign changes to He-H+.
Asymptotically [49,53,54]

p(R) = D7R—7 + DQR—9 + . . . (46)

As noted in section 3.3, we have approximated the pair dipole by adding the
leading dispersion term (with D7 = 120 a.u. [53,54]) to the exchange-overlap
dipole calculated from Eq. 13 or from Eq. 12 (i.e., by direct calculation of
; alone). In the Gaussian-basis calculations using Eq. 13, the dipole
changes sign at R ~ 8.7 a.u.

Ab initio calculations of the He...H dipole including correlation
effects have been reported by Bender and Davidson for the single
internuclear distance R = 3.0 a.u. [59], by Ulrich, Ford and Browne for the
R range from 0.5 to 20.0 a.u. [60]; and by Meyer for R between 5.0 and 11.0
a.u. [57]. At intermediate and long range, there are substantial
differences between the available ab initio results. The dipole calculated
by Ulrich gt gl. changes sign at R ~ 7 a.u., while the dipole obtained by

Meyer does not change sign until R reaches ~ 8.5 a.u. Between 9.5 and 11.0

48

49

a.u., Meyer’s results approach the known long-range form of the dipole
closely. In contrast, the dipole calculated by Ulrich gt gt. exceeds the
leading dispersion dipole by a factor of ~ 6 in this R range. For He-H
distances between 6.0 a.u. and 11.0 a.u., Meyer’s results are used as the
basis for assessing our approximations, and the results of Ulrich gt gt. are
used at shorter range.

Our results for the He...H dipole are plotted for comparison with the
gb initio dipole in Figs. 1 and 2. The long-range form is also shown in
Fig. 1. Our approximation exhibits the same qualitative features as the gb
initio dipole. Fig. 1 shows that the results obtained by differentiating
the interaction energy with respect to the applied field to find the overlap
dipole (Eq. 13) and then adding the asymptotic dispersion correction agree
quite closely with Meyer’s results in the range from 4 a.u. to 11 a.u. In
fact, the discrepancies between Meyer’s results and this approximation are
smaller than the discrepancies between the two sets of gt tgitig results
[57,60]. Relative to Meyer’s values, our closest approximation is typically
in error by 20~30% over the range from 4 a.u. to 11 a.u. The remaining
differences result from neglect of the higher-order exchange, dispersion,
and orbital distortion effects, overlap damping of dispersion (gt. [108—
111]), and the effects of intra—atomic correlation in He on the He-H

exchange and dispersion dipole. Some cancellation of error occurs, since

the omitted effects are not all of the same sign. The error is g 5% at R

4.0 a.u., ~20% at R = 7.0 a.u. (the van der Waals minimum), and g 5% at R
11.0 a.u. It is large (roughly a factor of 2) at R ; 8.0 a.u., but near
this point large relative error can result from a slight displacement of the

zero of p(R) between the accurate and approximate analyses.

50

Computing the expectation value of the dipole with the zeroth-order
antisymmetrized wavefunction (Eq. 12) gives a smaller exchange—overlap
dipole than that found by differentiating the energy with respect to an
applied field (Eq. 13). The latter approach yields better results. It is
interesting that the gb initio value of the pair dipole lies between the
Gaussianabasis values obtained from Eq. 12 and those obtained from Eq. 13
for exchange~over1ap effects, (with the dispersion correction added in each
case), when R<9.5a.u.

As R decreases from 5.0 to 1.0 a.u., the dipole moment increases
rapidly. Low-order perturbation theory breaks down at short range, and the
exchange—overlap contribution to the dipole is overestimated by Eq. 13. The
level of agreement between the results of Eq. 13, evaluated either with the
Slater basis or with the Gaussian basis, and the results of Ulrich gt gl.
for R < 4 a.u. is surprisingly high, as Fig. 2 shows. This agreement must
be fortuitous. The local maximum in p(R) occurs at larger R (near 1.0 a.u.)
in the g9 tgtttg calculations than in the approximate work, and the value of
p(R) from Eq. 13 is smaller at the maximum. In the Slater-basis
calculations with Eq. 13, we did not find a local maximum in p(R) over the
range of R values studied. At short range, the dipole computed from Eq. 12
is substantially smaller than the gg initio dipole. On the scale of Fig. 2,
results from Eq. 12 are essentially unaffected by the choice of the Slater
or Gaussian basis.

The dipole obtained by differentiating the energy with respect to an
applied field (Eq. 13) differs from the expectation value of the dipole

moment (Eq. 12) by the correction term Apz [112]

51

at!
—-|

Au 2 “ <¢ I
z E=0 0 322 o E=0 (39)

~

This term is nonvanishing because the Hellmann-Feynman theorem is not
satisfied at lowest order in label-free exchange perturbation theory. The
ratio of the correction term Apz to the total dipole calculated from Eq. 13
is plotted in Fig. 3 for both the Gaussian basis and for the Slater basis.
Comparison of the dipoles calculated with Eqs. 12 and 13 provides one
indicator of the uncertainty in the results. Calculations of Apz have been
used previously to indicate uncertainties in the correlation contributions
to molecular dipoles, in cases where the wavefunction at self-consistent
field level approaches the HartreeeFock limit. In this work, Apz is
definitely smaller for the Slater basis (see Fig. 3), but the results from
the Gaussian basis and Eq. 13 agree more closely with Meyer’s work.

At lowest order in the labe1~free exchange perturbation theory, for
the He...H dipole we have found better agreement with accurate gb initio
results when the dipole is determined by differentiating the pair energy
with respect to an applied field than when the dipole is computed directly
as an expectation value. The dipole calculated as an energy derivative is
significantly larger than the dipole calculated from the expectation value
(by factors as large as 4 for Gaussian—basis calculations in the range 4.0
a.u. < R < 7.0 a.u.). In this context, it is interesting to compare recent
self-consistent field (SCF) results [56] for inert-gas heterodiatom dipoles
with estimates based on the exchange-antisymmetrization approximation
developed by Lacey and Byers Brown, since the exchange-antisymmetrization
approximation is equivalent to a direct calculation of the dipole

expectation value with the zeroth-order wavefunction. The SCF results

52

exceed the exchange—antisymmetrization estimates by ~ 60% for Ne...Ar for
internuclear distances between 4.0 and 6.0 a.u., and by ~ 50% for Ne...Hr
(with smaller differences for other pairs). Although the discrepancies are
significantly smaller than for He...H, they are of the same sign. Improved
agreement with SCF results might be obtained from exchange—perturbation
calculations that employ energy derivatives. Results for the He...H dipole
suggest that the label-free exchange perturbation method can provide a
useful approximation for collision-induced properties in the region of

charge overlap.

Fig. 1.

53

 

 

 

 

 

 

 

 

Collision-induced dipole of He...H as a function of the
internuclear separation R in the range from 4.0 to 11.0 a.u.
Curve A shows the ab initio results obtained by Meyer, and Curve
B the ab initio results of Ulrich gt gt. Curves C and D have
been obtained by adding the long-range dispersion dipole to the
exchange-overlap dipole computed form the energy derivative (Eq.
13). Curve C shows the dipole in the Gaussian basis, and D the
dipole in the Slater basis. Curves E and F have been obtained
similarly, but the exchange-overlap dipole has been computed as
an expectation value from Eq. 12; Curve B shows results from the
Gaussian basis, F from the Slater basis. Curvg G is a plot on
the leading term of the dispersion dipole, D R . The ordinate
is scaled logarithmically and the dipole p(R3 is given in a.u.

Fig. 2.

ab initio results of Ulrich gt gt.

54

 

 

 

 

3.0 1 . r
1
'1
\
’ x. 1 B
/ ‘-
\. \
2.0 7 ./ \i. "
- .1 '5
i C t
: \
' 9"}.
1 "3}.
g A .5" \'\._
IO _. j 0" \. \.\ ..
i ; \Kgbx
i 95.x.
\‘g--.,\.,
I 6 \§&::“\
0 I 1 ‘~~t-~“¢‘:fi
O l 2 3

Collision-induced dipole of He...H as a function of internuclear
separation R in the range from 0 to 4.0 a.u. Curve A shows the
Curve B and C show the
lowest-order exchange-overlap dipole, Obtained from Eq. 13, B in
Slater basis and C in Gaussian basis. Curve D shows the
exchange-overlap dipole computed as an expectation value with
the zeroth-order wavefunction; results from the Gaussian and
Slater bases superimpose on this scale. The dipole p(R) is
given in a.u.

55

A P2 4’2
0.9
0.8 r

0.7 - \

 

 

0.5 - \
0.4 - \

 

 

0.1—
0 24 6 8|O|2|4|6R

 

Fig. 3. Ratio of the correction term by to the exchange-overlap dipole
for He...H, plotted as a functign of the internuclear separation
R (in a.u.). The exchange-overlap dipole has been computed as
the derivative of the first-order interaction energy with
respect to an applied field. Curve A shows the results from the
Slater-basis calculation, and Curve B the results from the

Gaussian basis.

CHAPTER 4.

4.1. INTRODUCTION

Absorption in the IR region of the spectrum by non-polar fluids is a
collision-induced phenomenon [115], and a number of molecular properties
may be determined from the spectra [116]. Collision-induced IR spectra
have been studied for atoms and non-polar molecules [37-38, 40-41, 117—
119]. Furthermore, cooperative many-particle effects in polar fluids
affect the absorption line shape in this region of the spectrum.
Experiments have been performed [120] in order to ascertain the importance
of the collision-induced absorption relative to the absorption associated
with the permanent dipole.

Both the intermolecular potential and the interaction-induced dipole
are needed [121] as functions of the intermolecular separation and
relative orientation in the calculation of the absorption line shape.
Pairwise additivity of the potential is usually assumed [122] in the
simulation of collision dynamics, but higher order or nonadditive effects
[81,123-124] play an important rgle in many physical situations. For
example, the inclusion of three-body forces is known [125-126] to bring
results of molecular dynamics simulations into closer agreement with
experimental data. Molecular dynamics simulations with many-body
potential functions have been suggested [127]. Some calculations [73,128]
have been reported with long-range triple dipole potentials [129-130].
When the intermolecular interaction is expanded in a Taylor series, the
first nonadditive part of the third-order perturbation energy of three
nonoverlapping molecules is the triple-dipole potential. Nonpolar species

can have a triple-dipole potential; if we consider three indentical rare

56

57

gas atoms a, b and c, an instantaneous dipole on atom a polarizes b, whose
dipolar field perturbs c, and the dipole on c interacts with a. The long-
range triple-dipole energy may be written as a sum-over-states expression
[129] or as an integral over imaginary frequencies [130]. Bounds to the
three-body long-range interaction coefficients have been obtained [131]
for a number of atoms. It was established [132c] in the Drude model
approximation that the triple-dipole energy term is a good approximation
to the long-range nonadditive energy for the rare gas crystals, for the
many-body terms that involve more than three bodies in the dipole
interaction approximately cancel the remaining three-body multiple
interaction terms. Nevertheless, the addition of higher order
contributions than the triple-dipole term was later claimed [133] to be
necessary for Xe. A combination of two-body Lennard-Jones and three-body
triple-dipole and experimental exchange-overlap potentials has been used
[134] in the study of atomic cluster growth. The relevance of nonadditive
interactions has prompted the generalization [135] of the correction to
the basis-set-superposition error for the calculation of many-body effects
[136-138]. Also, nonadditive effects in the potential have been studied
for hydrogen [129,139-140] and rare gases [73,135,137,141-153].

Barker gt g1. [154-158] have claimed with the support of experimental
data that overlap-dependent manyebody interactions in rare gases must be
irrelevant. However, the fact that the addition of long-range three-body
interactions to the pair potential brings calculated properties of rare
gases into agreement with experimental data does not in itself provide a
high degree of physical insight [159] into the nature of the interactions.

While the three-body exchange terms are considerably smaller than the two-

58

body [132d] at distances corresponding to the minimum of the van der Waals
potentials, the three-body exchange contribution is significant [137,160-
161] when it is compared with the triple-dipole energy [129]. Although an
overlap-dependent modification of the long-range forces [1323,162-163] has
been used to explain the stability of the different rare gas solid
structures, three-body exchange energy had earlier been considered crucial
by Jansen gt gt. [164-171] in explaining the polymorphism of the rare gas
and another isoelectronic solids. In spite of points made [172] in reply
to criticisms [173], it has been suggested later [132a-b, 174] that Jansen
gt _t. overestimated the nonadditive exchange contributions. The
unavailability of reliable information on many body forces has prompted
their accurate quantum mechanical computation [156,174], consistently to
some specified order of magnitude [174]. Despite Barker’s remarks [157—
158] and the interpretive challenge posed [158] by the agreement of
experimental data and results of calculations that omit three-body
exchange effects, it is believed [127, 153, 174-177] that overlap~
dependent three-body forces cannot be fully disregarded and that they
alone or in combination with other many-body effects may be the cause of
discrepancies between experimental data and calculations of the dynamic
structure factor and other properties [127,178]. Furthermore, it does not
seem totally convincing to argue that overlap-dependent contributions are
made irrelevant by the small statistical weight of regions where overlap
is important, given that these contributions may be very large in those
regions (orders of magnitude larger than in the statistically favored
regions).

The moment induced in a group of three atoms or molecules introduces

59

three-body effects in the ternary absorption coefficient [115,179]. This
coefficient has been calculated [180] from experimental data for C02.
Pair additivity in the dipole moment is usually assumed in dynamics
calculations [39,181], but it has been suggested that many-body effects
associated with the irreducible triplet dipole moment must be included in
order for the calculated collision-induced absorption to reproduce the
experimental line shapes [39,182]. Although binary collisions are the
most important [183] at sufficiently low densities, the irreducible
triplet dipole moment must make the leading contribution to absorption by
monatomic unicomponent gases. Buckingham stated [184] that a set of three
spherically symmetrical charge distributions should be polar due to
induced moments, hence capable of absorbing electromagnetic radiation in
the IR region of the spectrum. A model has been proposed [185] to
calculate the spectrum due to ternary collisions in pure rare gases.
Studies on the triplet dipole moment [186] have yielded the long range
contribution as a power series of the internuclear distances. The
coefficients of the leading terms have been calculated [187-188] for three
hydrogen and three helium atoms, and the dispersion triplet dipole moment
has been evaluated [189] for the Drude model. A variational method has
been used [190] in order to explore the dipole moment of a hypothetical
model neutral system with one electron in the field of three identical
nuclei with a fractional charge each. The effects of applied pressure
[191] and many-body effects [181,192-195] on interaction-induced
depolarized light scattering [196] have also been discussed and
calculations have been performed [197] to determine the triplet

polarizability of helium.

60

A system of three hydrogen atoms in its quartet state is a good
candidate to study three-body effects on the dipole moment with no
interference from two-body contributions, and, in general to model
interaction-related properties or potentials for three closed shell atoms.
Nonadditivity in the potential energy has been studied [140,198-201] for
spin-polarized H3, and the potential energy curves [199,201-204] and the
Axilrod—Teller triple-dipole term [139,205] have been calculated at
different levels of approximation. Hecht [206] predicted that gas or
liquid completely spin polarized hydrogen would be endowed with superfluid
behavior, and that it would recombine at densities where the three-body
collisions became important unless an external magnetic field precluded
such recombination. Monte Carlo calculations of bulk properties showed
[207-208] that the system would exist as a gas (18,28) or liquid (3H), and
liquid-to-gas and solid-to-gas phase transitions were studied [209-211]
subsequently. Measurements on completely polarized atomic hydrogen at low
temperatures have been reported [212-218], transport properties [219-220]
have been determined, and the quantum Boltzmann equation [221] has been
established for the gas in a regime where only binary collisions are
important. Three-particle recombination has been considered [222] in the
kinetics studies of the polarized gas, and experimental evidence for such
a process has been reported [223-225]. In order to achieve Bose-Einstein
condensation, magnetic confinement of the gas is under investigation
[226].

Label—free exchange perturbation theory [74-75,]68,l72,227] has been
applied to the study of interaction energies [73,168,171-172],

interaction-induced molecular properties [76,228], and hyperfine structre

61

spectral line shifts [75]. A reformulation of the label-free exchange
perturbation theory has been proposed [229] since model calculations for a
non-interacting system [230] suggested that the original formulation [227]
could not be satisfactory in practice. Further calculations [75,228] have
shown that this method can be useful in approximating interaction-induced
properties when overlap is non-negligible. We derive in Section 4.2 the
expression for the dipole moment of H3 in its quartet spin state.

Although the Hellmann-Feynman theorem is not satisfied at this level of
approximation, for this system it can be shown that the dipole moment
calculated as an energy derivative will be more accurate than the dipole
calculated as an expectation value (section 4.2). It has been established
[228] that the Lacey and Byers Brown method [70], which gives the lowest-
order expectation value, accounts for only a small part of the exchange-
overlap pair dipole moment. We conclude on the grounds of the treatment
given in Section 4.2 that the Lacey-Byers Brown method provides no
information at all about the dipole moment of certain triplets of
identical atoms. Furthermore, the Gaussian model yields in both its
original [164,166,231] and modified [232] forms an electron density which
renders the exchange-antisymmetrization dipole moment [70] equal to zero
for any cluster of identical atoms. The second-order dispersion triplet
dipole moment has already been calculated [187-188]; the question then
arises whether the total dipole can be approximated simply by adding the
first-order exchange contribution. The issue of approximating interaction
energies as the sum of second-order dispersion and first-order exchange
contribution has been addressed by Margenau [233]. The calculations that

he performed with a rather simple wave function showed that the

62

approximation was not acceptable for hydrogen, but it was almost correct
for helium and it should be legitimate for heavier structures. Second-
order nonadditive overlap-dependent contributions to the energy have been
studied [166,170] in the Gaussian effective electron model, but it has
already been established [132a,l74] that this approximation overestimates
the second-order energies. The results of more accurate calculations
[234] on hydrogen have shown that the first-order exchange energy between
two ls atomic orbitals accounts for ~90% of the difference between the
energies of the lowest singlet and triplet states. Second-order
interaction energies involving exchange are considerably smaller than the
first-order exchange contribution in the He...He interactions [235].
Higher—order exchange energies are expected to be a small percentage of
the first-order exchange energy unless a strong bond is formed [132d].
Furthermore, it has been claimed that it is legitimate to calculate
induction and first-order exchange energies independently, and this
procedure has been used in the computation of interatomic interactions
[236]; the results seem to support the validity of the approximation
[236]. The same approximation for the second-order exchange contribution
to the triplet dipole moment is adopted here. The results of our
calculations discussed in Section 4.3 show that the exchange-overlap
contribution to the triplet dipole moment cannot be disregarded in

comparison with the long-range dispersion contribution.

4.2. DIPOLE MOMENT CALCULATIONS.

In this section, the dipole moment for a system of three interacting
hydrogen atoms in the lowest quartet state is obtained in the label-free
exchange perturbation formalism.

The Hamiltonian, H, for the system in a uniform static electric field
is

F {j (1)

A A
H = H(~=Q) + H(§) = HO+ V — ~
H(§) is associated with the interaction of the external electric field E
and the hydrogen triplet. H(§=Q) is the sum of the Hamiltonian operators
for the three non-interacting hydrogen atoms plus a term, 3, which
contains all the electrostatic interatomic electron- nucleus attractions,
and nucleus-nucleus as well as electron-electron repulsions.

Each term in the Hamiltonian may be written in a form which is
invariant with respect to assignment of the electrons to the interacting
atoms:

A p

1:].

A A A A
where 0 represents one of the operators HO, V or B in (1), and

" p
O :2 z OpiAi - . (3)

i=1 p
The subscript p=a, b, c labels each hydrogen nucleus in the triplet

system. The number of ways NP electrons from a total of N = §.NP may be

assigned to three different centers is

63

p _ (Na+Nb+NC)!

' o
N82Nb3NC.

so in this case, p = 6.

 

(4)

When applied to an antisymmetrized triplet wave function 9.111
projects out the simple product term for the ith electron assignment. If
the zeroth order save function, Qb’ is constructed as an antisymmetrized

product of the ground-state save functions @a’ $5 and $6 for the isolated

atoms:
P [ 1 P .
‘I’o‘ ff: °p. Pi ‘I’a‘pb‘l’c ' 5:24“ ’ (5)
1- 1 1-1 p
where f is a normalization constant and Z 0 Pi
i=1 i

is the intersystem antisymmetrizer. Thus, [1i is not self-adjoint and

Ai‘l’o = “’1‘

The triplet wave function to first order in the applied field E and

zeroth order in the interatomic interaction is given by

3

_ -1/2 (0) , -1/2 (1)_ l -3/2 (0)
0b- c1 Q + 5 [Cl 151 g i 2 gzcl é ] (5)
The constants c1 and 92 are
c1 = <<1>(°)| <I>(°)> ('7)
and
c = 2 ‘32: <¢(°)| em> - (8)
2a 121 in

¢(0) is an antisymmetrized product of unperturbed orbitals x‘o) centered

p
(1)

in is the antisymmetrized product with the unperturbed

on p = a,b,c and é
orbital i replaced by the first-order correction to orbital i in an

applied field Pa in the a direction. The vector 3:1) has components

65

(l) (1) §(1)

Qix , Q1 , éiz . To lowest order in the interaction, the triplet energy
E is [228]
= <QO|H|¢0 > - 3};(0)=c‘1'1 <<p(°)|z v. 1A1 |<1>(0)> +
i=1
A _
+F -'1'1<<1>{2c<¢(°)|H(F=-0)|£ g<1>>——czc12<.<°>.u(§=0)..<°>>_,} , (9)
j: lJ
where
6
g = <¢‘°)| z piAi]§(0)>Cll (10)

i=1

With possible basis-set-dependent effects [81] confined to second and
higher orders [228], the energy is given by the sum of three terms: the
total energy of the three non-interacting atoms 3Eé0); the zero-field
triplet interaction energy to the lowest order, and a term linear in the

electric field.

The dipole moment may be obtained as a field derivative of the energy

(9):

(C
II

a“ a1:

A o
311%] = -81”. IHI~P0>+<FOZ glqo>+<eo ~>}F=0 =
~:0 ...
(11)

w

= B - 2c;1 <<I>(°)|§(F=0)| 24315 + Cl 2c2<<1>(°)m(§=-0)|q>(°)>
" " i=1 Q1

For the three H—atom problem, if the term 7 were removed from the
Hamiltonian H in (1) at §=0, an exact eigenfunction could then be
calculated. Moreover, even if an external electric field were turned on,
an exact wave function could still be obtained to any desired order in the
electric field. The presence of 7 in the Hamiltonian introduces an error
8&0 in the zeroth—order wave function when 2:0. Therefore, when an
electric field is applied the error has the form 8Q0(§)=60b(1+g(§)) where

g(§) is some function of the electric field. The error in the dipole

66

moment calculated as

A
- _ _ .81!
is first order in GO . Generally, the error in the field derivative of

0
the energy may be first or second order in 6&0' When 6wb(§) is of the

form considered above though, (aE/aE)F=0 is second order in SQb [237].
Hence, the dipole moment as given by F5. (11) is in this case more
accurate than the expectation value of the corresponding operator, as in
(12).

The AO’s x in the wave function 9 are constructed with Slater bases;

xéo) is the H—atom ls orbital and the first—order correction to the

hydrogenic wave function xél) is found analytically [93] and written in
terms of modified p—Slater orbitals. We find that 9259’ according to the
properties of overlap integrals with Slater orbitals on identical centers.
The exchange-overlap contribution [70] to the dipole moment given by
Eq. (12) may be analysed as follows. Let us consider a system of p
identical atoms with a total of N electrons. Let us assign the set

_ i ’
P1 - {xl, . . "XN/p} of A0 s to the first atom, the set

"U
N
H
Fla,
2‘
'3‘
z
w

to the second atom, and so forth. The set

,nx } is assigned to the pjib atom. The atomic

P: a
p {Km-m”
P

orbitals of the non-interacting atoms satisfy

i

i i
1.8 n 6 ...-S o ; , ’ : 1, so. , o 13
J J) PP J P P P ( )

i _
<(x )Pl(xj)P.> - (8

Thus, there are p2 (5 x3 )-dimensional blocks in the overlap matrix S.

67

All the diagonal blocks are (3 x 2 )-unit matrices. Consequently, we

can decompose S as the sum of the unit matrix A and a matrix T which
contains p (g x g )-diagona1 blocks whose elements are identically equal

to zero. The inverse of the overlap matrix may be expanded in a power
series (below, the implicit summation convention applies to the contracted
covariant and contravariant indices, but Eq. 14c does not involve

summation over i values):

S-1=(A+T)-1=A‘T+T2-T3+. . . (14a)
-li__i ik_ik9.. . .

(S ) J— T 3+ T kT J T k? RT J , 1 t J (14b)
-li_ ik_ik 9.

(S ) i— 1 + T kT i T k1 l T i (14c)
k=k_k k n_

1 1- T n T an1* T menT 1 ... (14d)

All the contracted indices are distinct from i (diagonal and off-diagonal

matrix elements) and from 3 (off-diagonal matrix elements). If

p(1.1’) = lx5(1)>(S—1)ij<xd(1’)l (15)

is the first-order reduced density matrix, the urcomponent of the

el

electronic contribution to the dipole moment Ba [70] is given [228] by

;°1 = - I dv1 p(1,1') qa = - (s’1)ij<x?lqulxi> -(S’1)1J.(qu)ji . (16)

a 1’91
Hence

-e1__ 1 i k k 1 i _ _ i i _ i k 1 J

”a ’ [5 i+T RT 1 I 1T 1] ((111) i [T a” kaj T kT 1T j](qu) i .(17)
Now

i - i _ i p i _ i
(qu) i- (x Iqulxi> - (x Iqu lxi> + (x lxi>RuP- 0 +RuPS i , (18)

68

where q: is referred to the p-nucleus and Rap is the coordinate of the p-
nucleus in a given reference frame. Then

Si.(q )i.= 2n 2 51.: 3 ER
1 a 1 P up iePP 1 p P

up . (19)
The product of a component of the dipole moment and an atomic orbital in
Eq. (17) is expressed in terms of modified AO’s, and the spherical
harmonics are transformed to the forms referred to canonical axes by means
of the corresponding rotation matrices. The use of orthogonality

relationships and specialization to s-type atomic orbitals let us write

(see Appendix E)

-el _ :N
”a - E ( p) Rap (20)

The nuclear a—component is

N

- nuc N l
p = Z " R -’ dv p(1,1’)‘-’ Z“ R ; (21)
0. pP upN [1,.4 1 pp up

so

— _ -e1 -nuc _

”u - pa + ”a - 0 . Va (22)

Therefore, the exchange-overlap contribution [70] to the dipole moment
given by Eq. (12) vanishes when a cluster of hydrogen atoms is described
by an approximate wave function such as @b° Furthermore, this result
applies to any spatial arrangement of any number of atoms whose electrons
are described by zeroth order s—orbitals, or by an effective 5 Gaussian
function [164,166,231-232,242]. Thus, the dipole moment is given by
-1 (0)“ 3 <1)
B = -201 <Q IH ( 20)] Q. > (23)
~ ~ ._ ~1
1-l

Further elementary consideration on the properties of integrals involving

Slater orbitals on identical atoms allow for additional simplification in

69

the expression of 2* namely, Eq. (23) becomes

B 3‘2cil

P103

¢.(1)> (24)

A
“MW, “1
l

i

where 6’ is the operator 6 after removal of the terms which involve
nucleus-nucleus repulsions. The constant c1 is given by

C1 ; 6[:1“Sib-Sic—slz)c+28absacsbc ] ’ (25)
where Spp, is the overlap integral between two 1s hydrogen orbitals on
p,p’ = a, b, or c. The explicit formula for pa is given in Appendix F.
Modified Slater-type orbitals for xél) have been taken in the directions 2
and x for a = z, x, respectively. The total dipole moment has been
computed as the sum of the overlap-exhcange (Eq. (24)) and the long-range
dispersion [187*188] contributions. Overlap damping, orbital distortion
and higher—order effects are neglected. The calculations have been
carried out for several values of H and Q (or XC for the right triangle
configuration) between 3.0 a.u. and 10.0 a.u.; R, Q and KC are defined in
Fig. 1 in the linear and equilateral, isosceles and right triangular
configurations of the three hydrogen atoms a, b, c. This range of
internuclear separations includes the van der Waals minimum of 32: H2
[243-244] and it covers the region where the pair distribution function is
appreciably different from 0 and 1 [208].

All the integrals over Gaussian expansions [245] of the Slater-type
orbitals which appear in the expression for the triplet dipole moment have
been computed with the program ARGOS [246-248] maintained by R. Shepard

(Argonne National Laboratory).

- {I

‘9

4.3. RESULTS AND DISCUSSION.

We have proven that the Lacey and Byers Brown [70] approximation
yields zero for the triplet dipole moment when the calculation is
performed for identical, spherically symmetric, neutral charge
distributions described by s-type functions. This result generalizes to
any number of such systems in any spatial configuration.

Overlap exchange (Eq. (11)) and dispersion [118] triplet dipole moment
calculations have been performed for the quartet H3 system in its right-
triangle (e=w/2 rads) and linear (a=n rads) spatial configurations, with 3
a.u. $8 $10 a.u. and 3 a.u. $XC $10 a.u. (right triangle) and 3 a.u. $0
$10 a.u. (linear). The results for the triplet dipole moment in a.u. are
plotted in Figs. 2-7.

The X-dispersion dipole moment points from the pair to the single atom
in most of the region XC > R (Fig. 2) whereas the reverse trend is
observed for the Z-component in most of the region XC (B (Fig. 3). The
behaviour of the X-overlap-exchange dipole moment (Fig. 4) is similar to
that of the X-dispersion dipole moment (Fig. 2), but the Z-overlap-
exchange dipole moment (Fig. 5) is directed form the pair to the single
atom in most of the region R) XC. The overlap-exchange dipole moment
(Fig. 6) for the linear configuration is directed from the single atom to
the pair whilst the dispersion contribution (Fig. 7) points in the
opposite direction, t.g., from the pair to the single atom. The order of
magnitude of both contributions is comparable for all the configurations
of Figs. 2—7. We do not present results for the isosceles-triangle
configuration, but a similar trend is observed for its dispersion and

overlap-exchange contributions. Therefore, the tiplet dipole moment is

70

71

substantially affected by overlap-exchange effects in addition to the
dispersion contribution. An analogous result holds for the corresponding
contributions to the energy [174].

The controversy over the actual rgle that the overlap-exchange
contribution plays in the many-body interactions has been noted in the
introduction. We have found that short-range effects in a molecular
property are very important when they are compared with the long-range
contribution. This theoretical result may be checked against experimental
data by computing the dipole moment of a unicomponent triplet, as in
Section 4.2. The collision-induced absorption spectrum may thus be
calculated [182] and the importance of dispersion and overlap-exchange
contributions properly analysed. Further research along this line is
likely to shed some light on the relative weight of short- and long-range
contributions to the many—body effects on properties of systems for which
the leading terms are due to triplet and higher-order associations. The
behavior of the total dipole moment (taken as the sum of the different
contributions) may permit inferences [249] about the form of the triplet

dipole moment time-correlation function.

72

 

 

 

 

 

 

Figure 1. Spatial configuration of three identical nuclei
(a, b, c). a=w, "/2, n/3 for the linear, right-triangle

and isosceles-triangle configurations.

73

 

._'a

X
o
mhwmuoocoo

 

 

 

345678910
R

A \ A

Figure 2. H3 dispersion dipole moment.

X component. Right-triangle configuration.

74

 

 

‘\ i’°""
b_ ’2' ls". .l.
1 ‘ .10'3

 

(N-bwmxlCDKDO

 

 

345678910
R

Figure 3. H3 dispersion dipole moment.

Z component. Right-triangle configuration.

75

1O

 

 

\‘ -3 ‘
,, \C‘1O \

‘ \

-2 1
“x40 ‘\
\ . .

345678910
R _

 

 

wbwmuooco

 

Figure 4. H3 overlap-exchange dipole moment.

X component. Right-triangle configuration.

76

1O

 

 

 

 

(N .2: (II 15) ‘4 (I) “0

K10“2
34 56 7 8910
R

 

Figure 5. H3 overlap-exchange dipole moment.

Z component. Right-triangle configuration.

77

 

1O

 

 

 

345678910
R

 

(Al £> ()1 (3) ~J ‘00 “D

 

Figure 6. H3 overlap-exchange dipole

moment. Linear configuration.

10

CM .35 (11 15) ‘4 (I) (I)

78

 

 

 

 

 

 

345678910
R

Figure 7. H3 dispersion dipole

moment. Linear configuration.

CHAPTER 5.

5.1. - INTRODUCTION.

The interaction between two non-overlapping spherical neutral charge
distributions is dominated by the London dispersion term [250-251]. When
the charge distributions are so far away from each other that retardation
effects must be taken into account, the interaction is still attractive,
but its functional form in terms of the internuclear separation is
slightly different. The use of the quantum theory of radiation [252]
yields an attractive interaction inversely proportional to the seventh
power of the separation. The retarded dispersion forces between
macroscopic bodies may be calculated within the framework of classical
electrodynamics [253], and the van der Waals forces between individual
atoms and molecules are obtained as a special case [254-255]. Other
treatments involving quatum electrodynamics have also been developed [256—
257] to account for retardation effects. The study of retarded dispersion
forces covers the interaction between molecules [258], effects in
dielectrics at finite temperatures [259], three-body dispersion forces
[130] and the interaction between an atom and a surface [260]. The
results of such theoretical studies have in common that the retarded and
unretarded dispersion forces between macroscopic and microscopic bodies
are given in terms of properties of the individual interacting species,
315;, in terms of susceptibilities. Therefore, dispersion forces may be
viewed from a unified standpoint gig susceptibility theory [261]. The
relationship among electromagnetic field fluctuations, susceptibilities,
zero-point energy and long-range forces is discussed by Boyer [262].

Whether retarded (t.g., proportional to “-7) or not (i.e., proportional to

 

79

80

8.6), dispersion forces undergo a power-law divergence as the separation
between the interacting systems goes to zero. Effects due to overlap and
exchange should prevent this collapse by damping the power law which gives
the dispersion energy. Second-order perturbation theory yields the
dispersion contribution when the perturbation in the Hamiltonian is
associated with the intersystem interaction. The matrix elements which
involve the perturbation must be properly handled in order to include the
overlap effects. The multipole series (which indeed is not intrinsic
[88,263] to London’s theory) cannot be used to expand the perturbation,
for the series expansion require that the charge distributions do not
overlap. Matrix elements of operators such as er - rBI-l, where ru is
the position of a particle of system a = A, B have to be evaluated. Their
evaluation without use of a multipole expansion is usually carried out by
separating the coordinate rA and r8 first.

Several techniques (all relying on use of the convolution theorem)
allow for such a separation. The Fourier transform of the potential is
used by Koide [264]. His method has been developed and used to calculate
damping function for Hez, Bez, HeH [265], and (32:)H2 [266], and to
calculate interaction energy curves as well as dispersion damping
interaction functions for He2 [267], Ar2 [268], Xe2 [269], (32:)Li2[270]
and He...H2 and Ar...HCl [271]. An exchange correction term has been
incorporated into the damped dispersion energy for (32:)Li2, (32f)LiNa,
(32:)Li; and (22f)NaAr [272]. Other gt tgtttg computations of dispersion
energies and damping functions have been performed [273] for Ar2 and L12.
A universal, empirical damping function has been proposed and applied

[274] to the calculation of the van der Waals potentials of (323H2, Arz,

81

(BZDNaK and Lng.

Nonlocal polarizability densities [275] and Coulomb interactions in k-
space have also been used to derive expressions for damped dispersion
energies [276].

In other work, the Fourier transform of the transition-density matrix
in the dispersion energy has been used [277] to provide a dispersion force
between closed-shell atoms finite at all distances. This method has been

extended [278] to represent the transition amplitudes in terms of Slater-

 

type ortitals. Damping effects for the first-order Coulomb molecule-

molecule interactions have been calculated on the basis of a two-centre or

bipolar expansion fo er — rBI-1 [279], and the damped second-order

Coulomb energy has been given in terms of response functions [280]. i;

The full Coulomb interacting potential is also used in a reaction-
field approach. A generalized form factor is related in this approach to
the charge-density susceptibility with which we may calculate the second—
order interaction energy. This method [281] is applied to the study of
the He2 [281-282] and Ne2 [283] potentials.

Damping in the dispersion energy has been treated empirically in
electron-gas calculations, applied to the potential energy of H2...He,
H2...Ne, H2...Ar [284] and N2...N2 [285]. Other procedures [286] are
based on fitting expressions involving exponential damping to reliable
results or experimental data.

The dispersion contribution to the dipole moment [53-54,287-290]; has
a power-law divergence (OCH-7) at zero internuclear separation.
Expressions for the damped dispersion dipole moment have been given [291—

292] in terms of properties of the isolated systems. Nonlocal

 

82

polarizability densities [275] and the full Coulomb interaction have been
used to calculate damped molecular properties such as dipoles, quadrupoles
and polarizabilities [293-294] within a reaction-field theory. The damped
dispersion-pair dipole is given [293] in terms of the Fourier transforms
of the nonlocal frequency—dependent polarizability density of one system
and the first hyperpolarizability density of the other system. We present
in Section 2 of this chapter a general method to transform the nth-order
charge-susceptibility density as obtained from generalized susceptibility
theory [295] into a form which can easily be compared with the fully
contracted nth—order nonlocal hyperpolarizability. We derive an
expression for the nonlocal polarizability and hyperpolarizability densities
in Section 5.3 and we give in Section 5.4 the expression for the damped

dispersion pair dipole moment in terms of reduced matrix elements.

5.2.-GENERALIZED FUNCTIONS AND CHARGE-SUSCEPTIBILITY DENSITIES

The expression

1 ln6(w) , (1)

 

lim .[ = PV

{*0 u_1C
where PV stands for "principal value" and 8 is Dirac’s distribution, is
used without proof in most work involving charge-susceptibility densities.
Here we present a general method based on the theory of generalized
functions [36,296] to carry out the limit involved in the charge-
susceptibility density as obtained from perturbation theory. The method
is equally applicable to susceptibility tensors, for the nature of the
matrix elements in the numerators is irrelevant in this treatment. The
method is general because it may be used with susceptibilities of any
order - it is not restricted to the first order, to which Eq. (1)
corresponds. This feature makes the method of potential use in the study
of nonlinear phenomena, both in the nonresonant and resonant regions.

The third-order nonlocal charge susceptibility density is [292]

 

 

 

X(2)(£”a£’, E; w’, u) = [1 + P(£’,w’;£, (0)] 11!!! 75—2
£40
“40
. p0k(£”),pkn(£’).pnotg)
'Z Z{ (w +u’+ u-it in) ' (w +u- in) +
k n 0k 0n
90k(5’) , Pkn (5”),pn0 (5)
* («bk-out) - (no; O-in) "
p (5’), (5), (5")
+ 0k, . Pkn POn’ . _ } . (2)
(”OR'” +1£)°(ubn-uru +1: + 1n)

where u and u’ are the frequencies of the field due to an external source,

83

84

”bk is the difference of the energies (in units of 8) of the ground and
excited (k) unperturbed states, p(£) is the charge-density operator and
the primes in the sums stand for k # 0 and n x 0. The variables t and n
are real and P(£’,w’;£,w) denotes the permutation of E, with E and of (u,
- it) with (u-in) simultaneously.

The matrix elements in the numerators of Eq. (2) are not relevant in

the analysis that we perform. Let us consider the representative term

 

lim 1
(*0 [(ubk’ u’ +i€)'(wbn+ w“ in)]
n40

Define uOk-m’ 5 x, won+u 5 y

Let l/(x.y) be a singular generalized function ( the terms
distribution and generalized function are considered synonymous in this
analysis).

Let C3(fl?2) be the space of test functions ¢ = ¢(x,y) required to be
infinitely differentiable and to vanish identically outside some finite
interval; ttgt, {¢} are a class of infinitely differentiable functions
with bounded support.

The test functions ¢ = ¢(x,y) may also be elements of 5( [17(2), the set

of infinitely differentiable functions such that

k +k
p a 1 2
sup llzll I—k k ¢(X.Y)| < .. . 5 5 (my) .
(5) ac 1 ay 2

1:92: the set of functions which belong to C” and (together with their
derivatives) vanish faster than any power when Igleta, although they do

not have bounded support. Also, there must be constants

K for all g and every k1, k2, p = 0, l, ..., such that

k1.k2.p

[a kl+k2
k

llgllp- 1 k2 ¢(X.y) l <x
ay

kikzp

We have O:(fl?2) CIj([R2). In this case, l/(x.y) is a tempered
singular distribution [296]. We now find an equivalent expression for the

singular distribution

 

SE _1_ : 11m+ 1 ,
xy {*0 (x:i£)'(ytin) (3)
p40

where o = 1 l, j = 1,2. For every o 6 C3 ([RZ), we have

¢<x.y) = o . V(x.y)s€[a.-a] x [b.—b1c Rz

<Sl¢ > 41”: .[bb (m it) (ragin)
n*0

(x-a lit) (Y 02in)
:11“ ...-gr“) 2 2 [¢(X,0)-¢(X,0)+¢(0,Y)-¢(0,Y) +
X2+€ 2) (Y +n )

¢(x.y) dxdy =

 

 

+¢(0.0)+¢(X.Y)-¢(0.Qfldxdy - (4)

The integrations in Eq. (4) are performed as follows

 

(X-o it) (Y‘Oz i n)
O 1
11m _ _ ¢(x,0) -—- dxdy =

¢(x.0)
=::: -02 21 arctg (b/n) [Ea 87;:zigz)

n40
-_ - £1.21. -_ . , sigma).-
ozlfljfa x dx - 021" [Eb 8(y)dy PV ~a x

dx]=

86

-olazuzfibfiadxdys(x,y).¢(x,y) . (5)

Analogously,

. (x-a i£)'(Y“o in)
11m a l 2 _
:40 [with “0’” 2 2 2 2 d“ d” ‘
n*0

 

(X +€ )‘(Y +n )

= - oliquadx 6(x) W [13315]?!) dy - °1°2"2EaJ1—)b¢(x’Y) 6(X.y)dxdy -(6)

(x-oli€)'(Y‘ozin)

(X2+£2)'(y2+n2)

 

1m ¢(o,0)[_a _b dxdy :

= ¢(0.0) lim “-0121 arctg (a/£))'(-0221 arctg (b/n))] =
{40

- n2 «0.0) . (7)

(x-oli{)'(Y—ozin)
2)

 

(114; fiafib [ ¢<x.y> +¢<0.0)—¢<x.0)-¢<0.y)dedy =

“40
- ng.x) . a 95 . 12.. a.212.92 , gx
' PVJEAJEb xy dXdy1+ ¢(0’0) Pv -a x PvIEb y PV -a X dx PvJEb Y

(X2+€2)‘(Y2+n

-Fvfia-‘1“; -FvJ‘:bM%‘ll dy = Pvffibflfifl dxdy , (8)

For FvJEaQ—f = o in Eq. (8).

Equations (5), (6), (7) and (8) allow us to write Eq. (4) as

_ _ . a ggx,y) _ a ggx,y)
<S|¢> - oziwjgbdy 6(y) PV -a x dx al"l-adXS(x) PVJEb y dy -

87

- zalazuzjfaijdxdys(x.y) ¢(x.y> +

 

2 a a x

+0102» [_aijs(x,y)¢(x,y) dxdy + PVI_8JEb$L;;Xl dxdy . (9)

Therefore,
. 1 _ _1 _ 2 _. 1

11m (x+a i£)'(y+a in) - PV xy alazn 6(x,y) 1n[018(x)PVy +
{*0 l 2
n*0

+0 5(y)Fvl ] (10)

2 x ’

where 6(x,y) is a two-dimensional Dirac distribution
The representative result given by Eq. (10) is used to perform the

limits in Eq. (2) to obtain

x(2)(£”o£’,£;w’.m) = [1+P(£,ou,;£.u] ‘5-2 '

. . 90k(§”),pkn(§’),pn0(§)
E W <~..+~'+~>-<~..+~>

 

+ Pou‘i')-"kn(5”)-Pno‘5) +
("Gk-w, ) . (”Onfi")

 

 

(r’) (r), (5”)
+ POk ~ 'Pkn ~ FDn }

(ubk‘u’)'(wbn-u-u’)

’"2{ °11°21P0k(5")'Pkn(5')'Pno(5)'““bk+“'+°'”bn+”)

+012°2290k(5'"'Pkn‘L"’)"’no(5)'““bk"""“bn*“’)

88

Em}

' o
+"13"23"0k(E ) Pkn (5)

 

+ “{Pm‘i >‘Fk.<£ >°Pno<£><-°u$<~ok+w +w>PV.,On..,, -

‘0216(wbn+w) PV

 

_ ’. H. _ _’
P0k(5) Pkn(5 ) Pno(5)( °125(“0k ” ) PV 0 +m

___l__.
-0228(w0n+u) PV ,)

l

 

_P0k(5 ) 'pkn(5)-p0n(§; )(‘0135(00k“m ) PV LOT-3;, -
*0 6(0 ‘u-u’) PV 1 ) }] (ll)
23 On “bk-u, '

where 012 and 021 are equivalent to al and 02 in Eq. (10) and l = 1, 2, 3

h term in the sum within brackets in Eq.(2).

refers to the it
The procedure that we have followed here may be applied to a term with
n factors in the denominator, t.g., to the (n+l)-order nonlocal
susceptibility density. The analogue of Eq. (10) in such a case would
involve the principal value of the inverse of the product of n variables,
plus a combination of n, (n-l), (n-2),..., l-dimensional Dirac
distributions each multiplied by the principal value of the inverse of the

product of 0, l, 2,..., (n-l) variables (respectively) that are not

contained in the corresponding Dirac distribution.

89

We obtain the expression for the linear (or second order) charge
susceptibility density given by Linder and Rabenold [295] when either Eq.
(10) or Eq. (11) is particularized to the case n = 1. Furthermore, the
limits in the nth -order charge susceptibility tensor [297] and the nth
order conductivity tensor [298] are easily obtained when we take £=n in

Eq. (10), t.g., when we carry out the transition to purely real

frequencies by taking a common imaginary component of all frequencies

[297-298] to zero.

5.3.—NONLOCAL POLARIZABILITY AND HYPEHPOLARIZABILITY DENSITY
CALCULATION

Response tensors may be given as sum-over-all-states [293,299-300] or
as restricted sum-over-states expressions [293,300-303]. The expressions
given as sums over all states have apparent divergences (secular
divergences) which should be removable [300] because their origin is a
number of redundant terms coming from a phase factor in the wave function.
This phase factor is expanded in powers of the field. Here, we first
derive and expression for the first hyperpolarizability by removing the
secular divergences from a sum-over-all-states formula [300]. Secondly,
we calculate the reduced hyperpolarizability and polarizability which we
shall need in section 5.4 for the calculation fo the damped dispersion
dipole. The derivation is restricted to systems whose ground-state dipole
moment is zero.

The first nonlocal hyperpolarizability BuBY(5,§’,5”;iu,-iu,0) is

[304]

[F _(rnofltr awn“In [Pg 9* >1 0
(u m‘+iw)(mho+iu)

 

(r,r’, r’ ’;iu,- 11.1.0): 132 Z[l+c.c. ]{—
m,n

8,3,

[P u(r)]0m[PB (r’ )] manY(E”)]n0

(u w'+iu)qho

 

 

+

[P(r’)] [Pam )1 [P(r”)l
+ fl~ 0m mm I.” n0 } ’ (12)

(me-iu) Un0

 

where Pu(§) is the u—component of the polarization and c.c. indicates that
one must take the complex conjugate of the expression following it. We

now split the unrestricted sum in Eq. (12) and perform the integration

90

91

over the g”-variable to obtain the reduced hyperpolarizability

 

 

 

 

 

 

 

A ’.o _o )
8187(5’!’ ,10, 1:.) .
.[P (1')] [p] [P (r’)]
A 9.- _- _ ‘2 u~ W1kfl“ n0
Bwr(r,[ ,1... 1...) — F. [l+c.c.]{m’n (”no+i“’)(“’no+i°) (13.1)
+ Z.[Pu(§)]om[PB(g’)lmnlurlno (13.2)
m,n (“ho+lw)“ho
.[P(1;’)] [P (5)] [ J
+ z B %‘9 E. )4" ”I “0 (13.3)
m,n ”hm 1” ”ho
ll] (uw+iu)im .
n iw(uno+iw) °
n 1:.) (duo
+_2:[PB(£’)]oo[Pu(§)]on[ur]no } (13 7)
n uh0(-iu) , O

0
Where 2: stands for the sum over m and n with the restriction
m,n
m s 0 and n t 0. In the integration to obtain Eq. (13) we have used

I dE"[PY(E”)loo = [urloo = o (14)

where pr is the recomponent of the dipole moment [293], t.g., the reduced
A _

hyperpolarizability BnBY(£,§’;iu,-iu) is specialized to a system without a

ground-state dipole. After partial-fraction decomposition of the terms

(13.4) and (13.5) above and some rearrangement in the sum [l+c.c.],

92

cancellation occurs and we obtain the reduced hyperpolarizability as given

in Eq. (15)

. [P (r)]0m[uY]m[PB(§’)]no

(5.5:;iu.-iw>=2l1+° C 1 {5 (“Midway“)

8&81
[P n(r)10m[PB (r’)]mn [u ran

(“ho+1°)”no

 

[PB(5’)JOmIPu(;)]mn[urlno

(”mo-i“) mn0

 

[Pu(E)]om[UY1molpfl (5’)]00
”mo(”ho+1”)

 

1 £3g(£)]00[PB(E’)JQE[”Y]m0 } O (15)

 

— Z w (u -io)
m m0 m0
Integration over the 5— and E’-variables in Eq. (15) yields the (fully
reduced) first hyperpolarizability in complete agreement with the

expression given by Buckingham [303]. This first hyperpolarizability,

BuBY(i°’-iu)’ gives the static nonlinear dipole in the presence of the
electric fields Hawk-‘1‘"t and Eé.°)ei°t. The static nonlocal

hyperpolarizability density obtained from Eq. (15) by taking w = 0 agrees
in full with the expression given by Hunt [293] in formula (2.37).

Now the first-order correction to the wave function for a system
perturbed by a uniform, static electric field in the u-direction is given

by

93

(fllu '0) a [P lmo
(l) ’ u _ an
N )=£——N’>=ZI
a m ‘“mo m m m0

olm) . (15)

Reshuffling of indices in Eq. (15), the use of Eq. (16) and Fourier

transformation allow us to write the hyperpolarizability in k-space as

M(I)IP (k)|n><n|P (k’)|0>
7K wno +11»)

A

88711

 

(15,15’;iu,-iu)- [1+C. c.

         

(1) a
“(4!“ IPLQS )Im)<m|PB(k) I0)
‘meO-iw)

 

+2
111

.[PB(5)]0m[uu]mn[PY(k’)]0

 

 

25:” (uno‘tiw) (umo'tiu) 1'12
n 0n0(wh0 +1”)

fin0(”ho Si”)

 

“1} . (17)

where c.c.’ denotes the Fourier transform of the terms associated with
c.c. in Eq. (15).

We shall need later the doubly contracted hyperpolarizability in kf
space, gig., kek;§BYu(k,k’;iu,-iu), where the sum over repeated subscripts
is implied. The divergence of the polarization is proportional to the
charge density, and its Fourier transformation satisfies the condition

5 2(1) = -i.p(g). (18)

The use of Eq. (18) in Eq. (17) allows us to calculate the doubly

contracted hyperpolarizability

<¢;1’|p(5>|m><m|p(g*)|o>
fl(uh0‘iw)

 

A 9
kBk Yem(g,g :1w,’1w) = [l+c.c] {-5

.<wél)lp<5’)Im><mlp(g)10>

 

fl (who-iu)

.<0|P(E)|m><m|uu|n><nlp(h')|0>

51(who+iw)(who+iu)

 

n,m

<0|P(§)I0><0|uu|m><mlp(k')|0>

+iw)

 

+ 1:2 E":

m ”mo(“’mo

<0|p(5’)|0><0|uu|m><mlp(g)l0>
+ _. 1}. (19)
ull0(ulllo 1”)

The multipole moments Q:(k) are defined [264] by the equation

 

m _ 4n 1/2 §21+123 .

o,(k)i. 53Zu‘5iii) Y:(°a'¢e) , . JR(kru) (20)
a 2 1.

. . th .

J1 is the Q spherical

Bessel function, and (ra,eu,¢a) is the position vector of the particle a.

where Zn is the electric charge of particle d,

Then using the Rayleigh expansion of the plane wave exp(ig.£a)in the

Fourier transform of the charge density p(k), we may write

.. 9. , a "
pm: 2 X "(34)? W141 Y,”(e.¢)* 020:) . (21)

~ [=0 m=-k

 

 

With 5 = (k, 90 ¢)'

Let the function «I»: (k,u) be defined by

1 l

u + u
no

 

q}; (k,u) a if 2’ |n><n|Q:(k)|0> . (22)
n

95

We rewrite Eq. (19) with the aid of Eqns. (21) and (22) and we

specialize it to spherically symmetric systems. Thus,

a R 1’
(ng’iiwfiu) = [l+c.c.] Z Z Z GULF) .

t A
k k B
Y BY“ l’lizo m:_n m!:_1)

B
[’Y'i'ke'.¢'>*¢E<e.¢>*<¢1mnIn?“"i:<“"iw>>5m~.m+m’ ‘M’fl

* ’ a y * m, ’ m —’
"YJ;(9,¢) YE,(9 9¢ ) < ¢1m19|01’(k )|Ql(k’ 1m)>6m”’m’+m6£,i’t1

’ , , t m _. m’ , .
‘Y';(99¢) YIE’(9,¢ ) <¢l(ka 1U)|Uu|§1!(k ’1w)>69_’1’:16m,m”+m,]

at H -”
4" m +1y3(e.¢)*Y'{' (e’.¢')<0|vu|9'i' ("”iw)><°'°3(")'°>

+— :10 [(-—1>

I},

+YT”(9.¢)*Y3(6’.¢’)*<0|v¢'°lf’,(k”i“’)><o'Qg(k’)Im] } ‘23)

where

1 : 1+9~’

: o 1/2
(21)!(2£ )2 [(21+l)(21 +1)]

 

C(k,1’) 5

In Eq. (23) 5i j is Kronecker’s unit tensor, and the electric field has

been taken in the direction YT . We adopt the convention 3" E ~m

Eq. (23) and within this context.

9! in
The doubly contracted hyperpolarizability given by Eq. (23) is in a
suitable analytic form for computations, since the matrix elements can be
calculated [266].
The doubly contracted Fourier transform of the nonlocal polarizability

density is [276,293] the susceptibility x(§,-5’;iu) ; with the same

conventions and definitions as made for the hyperpolarizability

L

x(g,-g';iu) e 2 z: C(L)v:(e,¢)£:(e’,¢')*<olo¥(k)*|¢$(k’,ie)+

L=0 M=-L

+ §:(k’ ,‘iu))

and

L22L 2

= _. 2L 4?!
C(L) ’ ( 1) [ (2L)! ] 2L+1

 

 

(25)

(26)

5.4.- DAMPED DISPERSION DIPOLE MOMENT CALCULATION

The u~component of the dispersion dipole on A in the A...B pair is

given in terms of properties of molecules A and B by [293]

A _ s ” _ _ ,.. _. B _ ,,.
(”disp)u- (2")7 IodeBABm( L": I}, 9109 10)-u6€( 11:15, ’1”).T86(1‘S).

 

.T (k’) dg dg' , (27)

-—kk . (28)

Let g be the vector from a center in A to a center in B and refer the

position of all the particles in grspace to the origin of 3. Then

2 REE
A ‘4“! g . - B S '
(”disp)u= ‘ (2")7 E®I83m(l£!]£ :1U9-10)k2ka2 (_1)X (E's—B, i1“)

.e-i(k+k’)'g dk dE" (29)

To analyze Eq. (29), the plane—wave exp(-ig.g) is written again as a

Rayleigh expansion

'E'B _ a x .x p p *.
e - Z 2' 1 4"Yx(e’¢)'yx(°R’¢R) Jx(kR) , (30)

x=0 p=rx

where g = (R’eR’¢R)' Let us define

97

98

C(1.l’.L) 5 C(l.l’)-C(L) (31)

with C(l,1’) and C(L) defined by Eqns. (24) and (26), respectively.

The next step is to perform the angular integration in Eq. (29), where
the B-susceptibility and the doubly-contracted A—hyperpolarizability have
been written as in Eqns. (25) and (23), respectively. To do so, we make
extensive use of the properties of spherical harmonics and Clebsch—Gordan

coefficients, the Wigner-Eckart theorem, and the relationship

2 (abaB Ie€>(ed€ 6|cr><bd86l f¢> =
8.6.6

= [(2e+l)(2f+1)]1/2 W(abcd;ef)<afu¢lcr>, (32)

where <aba8le€> denotes a Clebsch—Gordan coefficient and W(abcd;ef) is the
Racah coefficient, closely related to the 6-j symbol. we obtain [305] the

damped dispersion dipole moment for a spherically symmetric system

A _' g_ I“ I“, , ” .x+x’. . , , .
(”disp)z' 1‘ "3 od‘I’Od-k dk xzx’=01 J)‘(kR)J)" (k R )

. 112w) sz'nnl/z

4" (xx 00|10> -

<LL00IOO><OIIQL(k)IIQL(k’,iu)+§L(k’,-iu)>[l+c.c’.]-
L=0

1/2
[. 15%— 11:.» cm mum.)- [<0l|00(k)||0><0llulld>1(k’.iu)>

+ <0||c)0(k')||o><<1>l(k,i..,)llulI0>]esL’x

+ E C(1,1’,L).A(L,n’,x’)[(2£+1)(2x’+1)]1/2 W(11’xL;nx’)
1,1’=0

.{h(X.L.1)[<¢lllQn(k)Iléh,(k’aiw)>+<¢llIQ£,(k’)II§£(k,-iu))]3%r +

+—1gLLL5%- <§1(k,-im)llull§ .(k’.iw) }]» (33)
(22+1)

where

A(a,b,c) H)” [QWEQMD 11/2 [a b c]

4w 0 o o ’ (34)

the double-bar matrix elements are the reduced matrix elements generated

A

A
by the use of the Wigner-Rckart theorem, and we have taken R =
Notice that the Clebsch-Gordan coefficients involved in Eq. (33) imply
the existence of the factors 6

£,£’:l and 62.x’il . Therefore, we may

rewrite Eq. (33) as

(u‘ )- E “L Ex 5 (kmjdk’j (k mj'd... cw, k’ i... 9., >. 1.) (35)
d‘sP z 1,L=0 x=l1-L| *1
(l+x+L= 2)

where G(k,k’;iu;£,x,L) in Eq. (35) is defined by comparing this equation

100

and Eq. (33).

An analysis analogous to Koide’s for the damped dispersion energy

[264] holds for Eq. (35) (or Eq. (33)). Consequently, we may say that

(pA. ) vanishes as R goes to O and as R goes to infinity. Computations
disp z

of (”disp)z and the dipole moment overlap-damping functions may be

performed with the analytic expression given by Eq. (33).

APPENDIX A.

A derivation of Eq. (21) is carried out in this Appendix. Let us

rewrite Eq. (20)

 

(n) _ _ “’1 _L (1) on.

A _ ( 1) Zni ér 90(9)H . . . 30(9)d9 (9.1)
R (9) - E + s + (9— )s2 + (9— >253 + (A 2)
0 - xo-l >~O x0 ’ ' ' '
30(9) 2 s + (9—90) 52 + (9—90)zs3+ . . . (9.3)
S E 80(xo) (A.4)

From (A.l) and (A.2), taking into account definitions (A.3) and

 

 

(9.4),
A‘“) - (-1)“"1 _1- § I E0 + s (9) J “(1) '“' [E 0 +
- Zni F xO-l 0 ' ° ' xo-R
+so(9)] d9 . (9.5)

In particular

E
(1) 1 (1) 1 o
A = . a (9)9 a (9)49 = . [ _ +
2w1§r0 o 2111i. N)“

+ 80(1)] d1:

E
(1) __Q_
+ 50(9)]H [ *0-1

10]

102

1.
-— 80(1)H(1——Qd£ +
xo-D. xo- —Qd9. 2"1¢r))‘0-9‘

___. -.__ (1) _1_ (1)
+ . 4%. H 30(9)d9 + Zni §; 50 (9)H 30(9)d9

= -SH(1)E - E H<l>

0 0 S (A.6)

E

(2)- _:l (1) (1) _ _:l 0
- 2.1 (fir 90(9):; 90(9)!) 90(9)d9 27:91 [W1 +

 

E

<1) 0 (1)
[ t— 9

+ So(l) ] H + 50(1) 1H 2 + S 0(1) 1 d1

0

z _-_1§ _E_0 H (hi9 H(1)119)
2191 90—9 xo-D. xo-D.

1"

d9.-

E

E
__Q (1) (1)__Q _
li.>O-9 H so”)H 90-9 d“

E
(1) 0 (1)
- . S (9.)H H
@1‘ 0 xo-R. xo-D.

E
(1) (1) 0

- . S (9.)H S (1)11 d9. -
g. 0 0 )0-

E E
- __l__ __9_ (1) __Q_
2111 § xo-B. H xo-D.

H(l)so(9.)d1—
r

E0

-§%§xo—o—9.-

F

H(1) SH(£)H(1)S (9)d9—

103

E

(1) 0
xO-R

H(1)So(£)d1 -

_1_
’ 2ni 4i.so(n)"

(l) (1)

SO(Q)H So(l)dfl =

_1_
- 2ni éi'so(9)a

_ (1) 2 (1) _ 2 (1) (1)
30H 3 a 30 s H 80H 30 +

(1) (1) _ (1) (1) 2
+ SH SH E0 EOH EOH s +

+ EOH(1)SH(1)S + SH(1)E0H(1)S (A.7)

where Cauchy’s integral theorem has been used in both (A.6) and (A.7). It

should also be noted that 50(1) fulfills

_QE _ . 9+1

The generalization to obtain (21) is now straightforward.

APPENDIX B

Expectation values of operators in label-free form

In this Appendix, a decomposition of the antisymmetrizer for an N-
electron system is used to evaluate the expectation value of a permutation-
invariant operator 9 in the state 0. The function Q is assumed to be a

normalized, fully antisymmetrized product of N molecular orbitals on system

A

A and Ni on system B, with N=NA+NB.

The N-electron antisymmetrizer.fl is related to an idempotent

projection operator 0 by

,4 "1%: zapszN‘efl (3.1)

P

N’ and GP =

11, depending upon the parity of the permutation P. The operator 0 commutes

where the sum runs over the permutations in the symmetric group S

with any operator 9 that is invariant with respect to each of the
permutations. In particular

[0.11] = 0 (3.2)
where H is the full N-electron Hamiltonian for the "supermolecule" AB; and
if the unperturbed Hamiltonian H0 and the perturbation V due to A-B
interaction are written in label-free form (Eqs. 2 and 30f chapter 3.2),

[0,190] = [ON] = 0 ‘ (8.3)

The operator 0 may be decomposed into terms that perform the

antisymmetrization within systems A and B and terms that perform the
antisymmetrization between systems. Lagrange’s theorem applied to the

symmetric group S and its subgroup SN ® SN ensures that 0 may be split in

N
this way [113,114]:

104

105

N !N ! R
1 A 3 AB
= __, Z a P = —, O 0 (ll + Z o.P. ) (8.4)
(NA+NB)‘ P P (NA+NB). A B i=1 1 1

In Eq. 8.4, Ci is the antisymmetrizing projection operator for molecule A
with NA electrons, defined by analogy tattlfor the full N-electron system,
and.cg_is the corresponding operator for molecule B. n.denotes the identity
operator. PiAB exchanges one or more particle labels assigned to A with
labels assigned to B; it does not involve permutations within the set of

electrons assigned to A or to B. The number 1 of operators PiAB is

n
9 = Z (NA) ("3) . (3.5)
5:1 J .5

Since the upper limit n on the number of electrons exchanged is min (NA’NB)’

N!
9 o
NA'NE'

l = - l. (B.6)

It is convenient to define P by

9
P = 2 0.3.“ . (3.7)

. 1 1

i=1

Then the expectation value of Q in the state 0 is

«FIRM» = N! <6{xl...xN} |n| 00%...pr / s (3.8)

where the xi’s are orbitals located on A or B and

106

N! (UD‘I"'XN} |O{x1...xN}>

(MA!)2 (NB!)2
8 = N! < 5A 03 (1+3) {x1"'xN} |OAOB (1+3) alum”) >

 

NA! NB! < 0A OB {x1...xN} '09 GB (1+3) {xlmxfl} >

< ¢A¢B l (1 + P) ¢A¢B > . (8.9)

In Eq. 8.9 wk is the normalized antisymmetrized product wavefunction for
system A (and *3 for B). An analysis similar to that for 5 shows

_ l
<q9|nlxp> - s <¢A¢B|n|flAflB (1+3) alum”) > . (3.10)

Substituting the form 2 QiAi for 0 yields

_ 1
<~9|n|w> - S [<¢A¢B|nllwAwB> +

p 9
93
+ (wAwBIifzniAilflAflB {:1 0ij {x1'"xn}>]’ (3.11)

where/l1 is the projection operator that assigns the first NA electrons to
molecule A and the remainder to B. Each of the operators Ai in the second
matrix element projects out a single P‘jAB term from the ket. Eq. 22 in

section 3.3.1 is obtained by identifying n with V.

Appendix C

Pair energy of He...H

The first-order pair energy E for the He...H system

_ (0) (0) _ _-
E - EH + EHe + V(Fz-0) + F2 {T1 + T2 + T3} ”ze (C.l)

from Eq. 11, with

-1 —1 -1
{-2<xu'rneIXa>+2<xnlxae><xae'raeixn>‘2<xnelru lxue>

anus
mh—a

V(Fz:0) : +

+<xn'xne>[<xuelr§1'xn>+<xa|xue><xue'rfilwe”

-1 -1
-<xnxuelrlzlxaexu>+ 2<xnxaeir12 Max":

—<xHexHe'r-lélxflxfle)<xflixfle>} ’ (C.2)

T = .2. (M)

1 . <xn|xae><3§°’<xneIX§>-<xue"In”; >>’
T2 =5 - <xue'hue'mexs'xn;[<3;'%.>+<><§|xne>1-<m"meWexalflma

- <41'hn.|><§.><>3|xne>-<xnxae|riélxuexaexxililxug

-1 1 —1 1
- <xn><ue|r12|xnexue><xn|xue> - <xuxnelrlz|xaexae><xa|xne>

+ 3&3)<xu|xae>[<xu.IX§> + “9'31...” ] 9 “W
__2_ _ -1 1 _ —1 1 -1 1
T3 - s 2<XB'rHelxh> 2<x3e"n IxBe) + 2<xhxhelr12|xhxfle>

+ 2<xnxne|riilxuée> ’ 9% <xuelxu><<xnebé> + Whig)

107

108

-1 1 -1 1
+ 2<XHeIrHe|xll><xlllxfle> + 2<xfle|rfle'xfl)<xfl|xfle>

+

<xa'rfillxue><xae|x;> + <xu'rél'xiiexxne'xm

<xue'rfillxue><xu|xue><<xuelx;> + “uni.”

+

+

—1 1 2 -1 1 -1 1
<3).er Ixne><xa|xne> - <xnexu|r12|xuxae> - <xnexa|r12|xnxue>

<xn|xue><<sexaelriéul>3e> + <xaexue|riélxn><§e>>

- Quasar}:Ime><><n'><§e>+m‘z=°> “me”? + <xrl(e'xu>><xuelxu>]’

(0.5)
and
_ < I >
p2 = ffl_§fl§_ 2<xH|z|xhe> - 2R<xh|xhe) J . (C.6)
The notation
_ v2 -1
hq — - E - quq (C.7)
- 2
s - l - (Xfllxfle> (C.8)
_ 3
<qu3|xr> - [d rlxq(1)g(1)xr(1) (c.9)

<qu§|riélxmxu> = I d3r1d3r2x§(l)xv(2) r1; xh(1)xu(2) (C.10)

_ 3
(WV _ Id rlxq(l))%(l) (0.11)

109

is used in Eqs. C.1-C.6. Also, the internuclear distance is R, r12 is the

interelectronic distance, and rq is the distance from an electron to nucleus

q; g is any one—particle operator and q, v, m, u = H or He. The functions
(1) (0) . . 1

xh and Xq from the main text are abbrev1ated as xq and xq,

respectively. Zq is the atomic number of nucleus q.

APPENDIX D

. . -1 H .
The hybrid integral (lsflelsflelr12|gpzlsfle> needed for the calculations
in Chapter 13 is not one of the 79 hybrid integrals in terms of auxiliary
functions compiled in Ref. 67. If we adopt the same conventions and
symbology as in Ref. 67, the calculation proceeds in the following way.

. . . . = H =
Define the charge d1str1butions Ra - lsflelsHe and nab - lsHe 3pz where a -

He and b E H. Q is a E —type distribution according to the standard

ab

two-centre charge distributions given in Table III of Ref. 67.

Furthermore,
fl = k w(£.n)exp(-m£)exp(-Bn) (D.1)
4“ _1_______ 22
E152 ‘n 2)w(£.n) - 2’ Z on fit n J: 4Jr§a (C‘n)(1‘{n)({ “n ) (D-Z)
n= 0 j: —0
w(€an) = 5%;F§3~ (c-n)(1-:n) (3.3)
k = 3 2(:1:)3/2(:: pz)”2 (v.4)
(§)2(z—n)n = §;§r§5 (11:)3/2(:? 2) 1/2 ; 8XP(‘P£-PTn)(€ -n)2 (1— (n) (n. 5)

and M = 0 = q. The expression (0.5) expands by one the number of terms
given in Table III of Ref. 67.

The one-centre distribution is ”He = [ISHe], and

l-exp(-Sa(z-n))[1+% ;a(€-n)l = g “+3)U“ae (v.6)

Therefore, u0 = 1, 111 = 1/2

110

111

_ (5‘3) 1
3 Is 3/2 z-1/2 4 .. 1 File
[ISHellsHe3pz]: (:H 8) (:HP ) pH [0 dcj;ldn[1—e 9E3
“933: ((+3 ) kle P“*"“’4rl—~ (z- “)2 (1— an):
_ 1 ls 3/2 3pz —1/2 4 .. 1 _ _
- HJHH (:He) (:H ) P3 jldsj_1dn{<c n)2(1 an)
-; (z—)
-e He n [(z-n)2(1—cn) —
-%; Hét+“)(€-n)2(1‘€n)l} e-P(£+1")=
4 3p 1
_2___ 1s 3/.2 z)—1/2 _ 001 _4__1010
_ 43—6 ((He) (:H [002° (150“ (2“) “€2,2(P3e ,pH)]- .5?) 102 (3,7)
H _ M/ H-l/z _ X _ *
[lsnellsneapz] "' HHe {H0 (9H8 99H) H0(PHe!PH) pH1(PHe!PH)} (D08)
From (0.7) and (0.8),
4 010 (3.9)

JB—CUZ =Hu
Thus, the number of monopole integrals given in Table VI of Ref. 67

may be increased by adding the term

 

 

[Q a I Q ab ] H 0 H 1 cf
H 010 010
[1sHe | lsHe3pz ] c02 ’ c12 4/ J30

APPENDIX E.

We outline here the proof that leads to Eq. (20) from Eq. (17) in
chapter 4. We give first the expressions for the spherical-tensor dipole
moment, the product of two spherical harmonics on one center, the product
of a component of the dipole moment and a Slater-type orbital, and the
transformation of spherical harmonics under rotations. We then write the
u-component of the dipole moment from which Eq. (20) is derived. We refer
to standard books [238-241] as the source of the mathematical
relationships used in this appendix.

8.1. Spherical-tensor dipole moment.

r0 = 2 (El)
rt1 = ; (2) _1/2(x:iy) (E?)
or = ( 43) /2rY1m(e,¢) , with m = o, 1 1. (33)

E.2. Product of two spherical harmonics on the same centre.

(9. ¢) and Y (e,¢).

9‘1‘“1 12m 2

The product is given by the following expansion in terms of the Clebsch-

Let the spherical harmonics be Y

Gordan coefficients C(1,£’,R”;m,m’,m”)

(29.1 +1) (2112 +1)
4n(2£+1)

(3.9). Y (e, 9) = 2 m[ 11/2C(11121;000) .

 

Y1 m

1 1 92m

2
. C(91129;m1m2m) YHIll (e,¢). (E4)

When we take A = 1, the properties of the Clebsch-Gordan coefficients let

1

us write

112

113

3(212+l) 1/2

Y (9.¢).Y (e.¢) = Z [———-—————— ] c<11 1;ooo>.
1m1 lzmz 1:112:11 4w(2£+l) 2
. C (llzl;m1,m2,m1+ m2) Yfi,m + m2(6.¢) (E5)
1
where 2 stands for the sum over at most two possible values

1=|1211|

for 1, gig., i + 1 and [£2 — 1|.

2

E.3. Product of a component of the dipole moment and a Slater-type

orbital.

Let the normalized complex Slater-type orbital on centre a be

1
(n.1.m>a= (zza)"*’[<2n)sl‘l/Zrn’le"a’ Ylm(ea’¢a) . (E6)
Then 1
- fl! 1/2 n+- , ‘1/2 n -{ r _
rnl.(n.R2.m2)a — ( 3) (2(8) 2[(2n)-] r e a Ylm1(ea.¢h)Y£2méea.¢h) -

_ 2 [2124-1 11/2

1=Il211I _Ei:l— C(1121;000).C(112£;m1,m2,m1+m2) .

[2n+2)(§p+ll 1/2

2{a (n+l,9.,m1+m2)a , (E7)

where Eqns. (E3, ES, ES) have been used and (n+1, k, m1+m2)a’ is a

modified Slater-type orbital on a defined by

3

_1 ..
(n+1,£,ml+m2); = (2:8)“+5[(2n+2)s] /2rne ‘a’Yl,ml+m2(ea,¢h). (as)

0.4. Transformation of spherical harmonics under rotations.
Let (X’, Y’, 2’) be the canonical reference frame. 0 denotes the
rotation (passive interpretation) carrying the system (X, Y, 2) into

coincidence with (X’,Y’,Z’). The transformation law is

114

t o _ X
YXLJ(6 ,¢ ) — E Dmu(fl).Ym(e,¢),

(89)

where 0;” (Q) is the rotation matrix [241] defined by the convention given

by Rose [238] and Messiah [239].

E.5. u—component of the electronic contribution to the dipole moment.

The use of 8.1 through 8.4 let us write Eq. (17) in chapter 4 as

e1__'ik_ik9. i_
”a“ [611+TkTi T117931] (‘11) i

_ _ i i _ i k 1 j _
[ T 3* T kaJ T k? llT 3 ] (r ) i _

k k 1

=3; (_12 )-212 z (1711*
P up p21 up l€Pp k

i k l

P . . .
+ z z 2 [T1.- T113".- 14".“
p=1 iEPp jGPp J J
(i#J)

1 J

p . . .
- z z z z [411 .- T113“.- le-rk
p=1 iEPP jGPp 1=|1i21| 9
(1:3)

A

[ 21i+1
21 + 1

(2ni+2)(2ni+l)J1/z

2(i

[ ((1131319) [(Biifiiw +

p . . .
+ Z Z Z [T1 .41ka .+T1kwlen .] Ru
p=1 iePp j¢PP 3' J J P

J J J. . _
. <(n 1 m )|(nillimi )>

- T1 w T .) +
1
T1.] -

9.
T J.]

1/2
] . C(l 111:000)C(1Ril;m1,mi, m1+ mi) .

J _ 1- _
x: Z 11“., .(n1)n} ml).
’ ' J J m- m.
1111 mJ’ m m 1 1

(E.10.l.l)

(E.10.1.2)

(E.10.2.l)

(E.10.2.2)

(E.lO.2.3)

(8.10.2.4)

115

p . . .
— Z 2 z [7T1.+T1ka.-T1kwknTn.] -
p21 iePp j¢Pp J J J

21.+1 1/2

 

 

Z 2., z - ‘ C(11.i;000)
m; mJ n=|£itll[ 2Q+1 ] 1
L(2ni+2)(2ni+1)]1/2
. C(liil;m1,mi,ml+mi) 2:1 .

J J J — - It” -1 “i '1
. < (n R m ’)l(n.1 m.)’> . D ., .(Q )'0 , (fl ), (8.10.2.5)
1 1 mJ mJ mimi '

where n.5 n.+1 ; m.E m +m. , and m.’ E m +m 3
1 1 1 l 1 1 l 1

The terms in (8.10.2.1) come from the sum (8.10.1.1). The sums
(8.10.2.2) and(8.10.2.3) include the terms in (8.10.1.2) with the orbitals
i and j on the same nucleus. The sums (8.10.2.4) and(8.10.2.5) include
the terms in (8.10.1.2) with orbitals i and j on different nuclei. 8g.
(87) has been used in (8.10.2.3) and (8.10.2.5). The inverse of Eq. (89)
has been used in (8.10.2.4) and (8.10.2.5).

Two spherical harmonics in the canonical reference frame on i and j

satisfy
(Y IY . .> = 6 j . (811)
“imi iJmJ ”1”
Furthermore,
<Y1.m.'Y . .)+<Ynjm.|Y1.m.> = 0 , (812)
1 1 1 m1 1 1 1

when 1i = IlJtll. The use of 8qns. (811), (812) and some orthogonality

116

relationships in Eq. (810) and its specialization to s—orbitals (i.e., 1i

= 0, Vi) yield
P p . . . .
11:" = E z (-Ru ) - z Ru 2 z [T1.TJi—T1k-rk.TJi] +
P p=1 P p=1 Pie? j¢P J
p . . . .
+ Z .Z X l?.,_p-['1".'rJ.-T1 7k.TJ.] =
p=1 1€Pp j¢Pp J l R J 1
N P
- p Z (-Rup) , q. e. d. (813)

9:1

APPENDIX F

The dipole moment for the H3 system in its lowest quartet state is given
by Eq. (24) in Section 4.2. The explicit expression for the transition

element in Eq. (24) is

3
(0) A, (1) - 1 2 _ 2
<¢ |v ligl Qiu >-6{<xalrfi Ixua >(SbCS 8C 2) <Xg|rc 1|xua)(8bc Sab) +

+<xcrb1|xuc>(S2-SZL) +

—1 1 -1 1
+ (Sab~SaCSbC)((xb|rc 'xfia>+<xalrc Ixub)) +
+ (58 ac -sabsbc )(<xC|rb'1Ixia>+<xa|rgllxic>) +
+ (Sbc—Sacsab)(<xb|ra l|xhc >+<Xt'ra 1lxub» +
+5 (< 1 -1| 1 >—< | ‘1; 1 >+< l '1 | 1 > —
ab xcxb r12 “taxi xbxc r12 x'uaxc xaxb r12 Xucxs
—2< Ir-ll 1 >—< lr-ll 1 >) +
J‘c’Sa 12 xuc’ia xcxb 12 Xaxuc
-+< |r_1| 1 >-< I ’ll 1 >) +
JTBS: 12 Xc’ina cha r12 xcxub
+s (< '1 >-2< '1 1 >+< "1 1 >+
ch xbxa'rlzb‘aaxc m'rm'xuaxc xcxlrm'xuaa.
+ < Ir-ll 1 >-< | ’1| 1 >+< | ’II 1 >—< | -1| 1 >) +
xbxa 12 xaxhc Xaxs r12 xaxuc chs r12 xaxab xsxc r12 Xaxub
+s (< | ’II 1 >—< | ”1| 1 >+< | ’1| 1 > -
ac xbxb r12 xhaxb “ext r12 Xuaxb xbxa r12 xficxb
- < Ir-ll 1 >+< |r‘1| 1 >—2< Ir—ll 1 >+
xaxb 12 xucxs “bxc 12 Xaxub thb 12 thhb
+ magma.»

117

118

The notation

<quglxir> =I33F1X§(1)8(1)Xfi: (1) (F2)

-1 _ 3 3 —1
(qurlrlzlxnxu> _ Id rld r2x§(l)xv(1)r12xh(l)xb(2) (23)

is used in Eq. (81). r12 is the interelectronic distance; g is the
electron-nucleus distance; q, r, v, m, and u label the different nuclei a,

1

b, and c. xéo) and x(1) from the main text are abbreviated as xh and xh.

The remaining symbols retain the same meaning as in Section 4.2.

[ 1].

[ 2].

[10].
[11].
[12].

[13].

[14].

[15].
[16].
[17].
[18].

[19].

BIBLIOGRAPHY.

P. Arrighini, "Intermolecular forces and their evaluation by
Perturbation theory". Springer Verlag. Berlin, 1981.

B. Jeziorski and W. Kolos in "Molecular Interactions". H. Ratajczak
and W. J. Orville-Thomas eds., vol.3 pp.1 gt seq. John Wiley & Sons,
1982.

. P. 0. stdin, J. Math. Phys. 3, 1171 (1962)

. P. O. Lowdin, Int J. Quantum Chem. 31, 69 (1982). See also: P. O.

Lowdin, in ”Perturbation theory and its applications in Quantum
Mechanics" Ed. by C. H. Wilcox, pp. 255 et §_g. Publication n9 16 of
the Mathematics Research Center. United States Army. The University of
Wisconsin. J. Wiley and Sons, Inc. , New York, 1966. P. 0. Lawdin, J.
Math. Phys. 6, 1341 (1965). P. 0. Lowdin, Phys. Rev. 139A, 357 (1965).
P. 0. Lowdin, J. Chem. Phys. 43, $175 (1965). P. Q Lowdin, Int. J.
Quantum Chem. 38, 137 (1968); 58, 685 (1971).

. P. O. L5wdin, J. Math. Phys. 3, 969 (1962)

. R. G. Wilson and C. S. Sharma, Int. J. Quantum Chem. 33, 927 (1981)
[ 7].
. G. Speisman, Phys. Rev. 193, 1180 (1957)

R. G. Wilson and C. S. Sharma, Int. J. Quantum Chem. 33, 183 (1982)

. T. Kato, Progr. Theoret. Phys. 3, 514 (1949)

T. Kato, Progr. Theoret. Phys. 3, 95 (1950)
T. Kato, Progr. Theoret. Phys. 3, 207 (1950)
T. Kato, J. Fac. Sci. Univ. Tokyo Sect.I 3, 145 (1951)

F. Riesz and B. 82. Nagy, "Legons d’ Analyse Fonctionnelle”. Akadémiai
Kiado. Budapest, 1968.

M. H. Stone, Amer. Math. Soc. (Colloquium Publications) 13, 184
(1932) '

T. Kato, Trans. Amer. Math. Soc. 39, 195 (1951)

W. R. Salzman, J. Chem. Phys. 43, 3035 (1968)

K. A. Brueckner, Phys. Rev. 199, 36 (1955).

J. Goldstone, Proc. Roy. Soc. (London) 5333, 267 (1957).

N. H. Rugenholtz, Physica 33, 481 (1957).

119

[20].

[21].

[22].
[23].
[24].
[25].
[26].
[27].
[28].

[29].

[30].

[31].

[32].

[33].

120

N. F. Mott and H. S. W. Massey, "The Theory of Atomic Collisions".
Clarendon Press. Oxford, 1965.

R. G. Newton, "Scattering Theory of Waves and Particles" McGraw Hill
Book Co. New York, 1966.

M. L. Goldberger and K. M. Watson, "Collision Theory". New York, 1964.
K. A. Brueckner and C. A. Levinson, Phys. Rev. 33, 1344 (1955).

v. Fock, z. Physik g1, 126 (1930).

L.

. C. Slater, Phys. Rev. 33, 210 (1930).
C. C. J. Roothaan, Rev. Mod. Phys. 33, 69 (1951).
G. C. Hall, Proc. Roy. Soc. (London) 3333, 541 (1951).

A. Szabo and N. S. Ostlund, ”Modern Quantum Chemistry: Introduction to
Advanced Electronic Structure Theory". MacMillan Publishing Co. New
York, 1982.

At this point we briefly analyse the nature of long-range interactions
between molecules. If the instantaneous motion of the electrons is
intercorrelated, we may expect that the charge distributions of two
interacting atoms far away from each other will relax in such a way
that the electron-correlation effects will yield minimum energy. The
relaxation of the electron charge distribution is sometimes referred to
as an interaction between instantaneous virtual electric moments on
each interacting system. Although the time average of such moments is
zero, the average of their effect on the other system is not. The
effect of the virtual electric moments is to yeild a charge
distribution on each centre polarized towards the other. Hence, each
nucleus undergoes an attraction towards the centroid of its own
electron charge distribution, and a net attractive force, 1.9., van
der Waals force, exists between the nuclei [30,31]. One way of
incorporating the effect of the virtual electric moments in the
efpression for wx (8g. (54)) is through a multipolar expansion of

H , even though such an expansion is not intrinsically associated
with the theory of long-range interactions. We shall refer to this
point in chapter 5.

R. P. Feynman, Phys. Rev. 33, 340 (1939).

R. F. W. Bader in "The Force Concept in Chemistry". Edited by B. M.
Deb. Van Nostrand Reinhold Co. New York, 1981.

B. H. Brandow in Proceedings of the International School of Physics
"Enrico Fermi". Course 36. Edited by C. Bloch. Academic Press.
London, 1966.

B. H. Brandow, Rev. Mod. Phys. 33, 771 (1967).

[34].
[35].

[36].

[37]
[38]
[39]
[40]
[41}
[42]

[43]

[44]
[45]
[46]

[47]

[43]
[49]

[50]
[51]
[52]
[53]
[54]

121

H. Feshbach, Ann. Phys. N. Y. 3, 357 (1958).
H. Feshbach, Ann. Rev. Nucl. Sci. 3, 49 (1958).

V. S. Vladimirov, "Generalized Functions in Mathematical Physics".
Mir Publishers. Moscow, 1979.

Z.J. Kiss and R.L. Welsh, Phys. Rev. Letters 3, 166 (1959)
0.8. Bosomworth and H.P. Gush, Can. J. Phys. 33, 729,751 (1965)
G. Birnbaum and R.D. Mountain, J. Chem. Phys. 31, 2347 (1984).

R. Heastie and D. H. Martin, Can. J. Phys. 30, 122 (1962).

~

8.8. Cohen and G. Birnbaum, J. Chem. Phys. 33, 2443 (1977).
J.D. Poll and J.L. Hunt, Can. J. Phys. 33, 1448 (1981).

G. Birnbaum, L. Frommhold, L. Nencini, and H. Sutter, Chem. Phys.
Letters 133, 292 (1983).

G. Birnbaum and H. Sutter, Mol. Phys. 13, 21 (1981).
P.A. Madden and T.I. Cox, Mol. Phys. 53, 287 (1981)

W. Klemperer, Ber. Bunsenges. Phys. Chem. 33, 128 (1974) and

refs. therein; J. Mol. Struct. 33, 161 (1980). Principles, design and
properties of molecular beams as well as the photochemistry of some
van der Waals molecules are discussed by 0.8. Levy in "Quantum
Dynamics of Molecules". Edited by R.G. Woolley. NATO-ASI Series B,
Physics, Plenum Press. New York, 1980.

J. M. Steed, T.A. Dixon, and W. Klemperer, J. Chem. Phys. 33,
4095 (1979). See also feature article by R.8. Miller, J. Phys. Chem.
33, 3301 (1986).

P.W. Fowler and A. D. Buckingham, Molec. Phys. 33, 1349 (1983).
A. D. Buckingham, Propriétés Optiques et Acoustiques des Fluides
Comprimés et Actions Intermoléculaires (Centre National de la
Recherche Scientifique, Paris, 1959) p. 57.

L. Galatry and T. Gharbi, Chem. Phys. Letters 33, 427 (1980).
K.L.C. Hunt, Chem. Phys. Letters 33, 336 (1980).

M. Giraud and L. Galatry, Chem. Phys. Letters 33, 18 (1980).

W. Byers Brown and D. M. Wishnant, Mol. Phys. 35, 1385 (1973).

~

0. M. Whisnant and W. Byers Brown, Mol. Phys. 36, 1105 (1973).

[55]

[56]

[57]

[58]
[59]

[50]

[51]
[62]
[63]
[64]
[65]

[56]

[67]
[68]
[69]
[70]
[71]

[72]

[73]

[74]

[75]

[76]

122

D.P. Craig and T.Thirunamachandran, Chem. Phys. Letters 33, 14
(1981).

G. Birnbaum, M. Krauss and L. Fromhold, J. Chem. Phys. 33,
2669 (1984).

W. Meyer, in Proceedings of the NATO Advanced Research Workshop on
Phenomena Induced by Intermolecular Interactions, G. Birnbaum, ed.
(Plenum, New York, 1985).

R.L. Matcha and R.K. Nesbet, Phys. Rev. 160, 33 (1967).

C.F. Bender and 8.R. Davidson, J. Phys. Chem. 33, 2675 (1966).

B.T. Ulrich, L. Ford, and J.C. Browne, J. Chem. Phys. 33,
2906 (1972) .

P.D. Dacre, M01. Phys. 33, 17 (1982).

R.L.C. Hunt and A.D. Buckingham, J. Chan. Phys. 33, 2832 (1980)
R.L. Clarke, P.A. Madden, and A.d. Buckingham, Mol. Phys. 33,
301 (1978).

P.R. Hilton and D.W. Oxtoby, J. Chem. Phys. 33, 1824 (1981).
J.N. Murrell and G. Shaw, J. Chem. Phys. 33, 1768 (1967).

A.T. Amos and J.I. Musher, Chem. Phys. Letters 1, 149 (1967); J.I.
Musher and A.T. Amos, Phys. Rev. 133, 31 (1967).

D.W. Oxtoby, J. Chem. Phys. 33, 1184 (1978).

D.W. Oxtoby, J. Chem. Phys. 33, 5171 (1980).

R.L.C. Hunt, J. Chem. Phys. 33, 393 (1984).

A.J. Lacey and W. Byers Brown, Mol. Phys. 33, 1013 (1974).
D.W. Oxtoby and W.M. Gelbert, Molec. Phys. 33, 535 (1975).

L. Jansen, Phys. Rev. 162, 33 (1967); L. Jansen and 8. Lowardi,
Chem. Phys. Letters, 1, 417 (1967).

8.8. Polymeropoulos, J. Brickmann, L. Jansen and R. Block,
Phys. Rev. A 33, 1593 (1984).

A.B. Ritchie, Phys. Rev. 131, 125 (1968); J. Chan. Phys. 49
2167 (1968).

A.A. Rumyantsev, Zh. 8ksp. Teor. Fiz. 33, 926 (1973).[Sov.
Phys.-J8TP 33, 459 (1974)].

J. Mahanty and C.K. Majumdar, Phys. Rev. A 33 2334 (1982).

[77]

[78]

[79]

[80]

[31]

[82]

[83]

[84]
[85]
[86]
[87]
[88]
[89]

[90]

[911

[92]
[93]
[94]

[95]

[96]

[97]

123

H. Margenau and P. Rosen, J. Chem. Phys. 31, 394 (1953)

N. Rosen, Phys. Rev. 33, 255 (1931). (Corrections in Phys.
Rev. 33, 2099 (1931)).

J.O. Hirschfelder, Ann. Rev. Phys. Chem. 33, l (1983).

S.T. Epstein, "The Variation Method in Quantum Chemistry",
Academic Press, 1974.

P.R. Certain and L.W. Bruch, in "MTP International Review of Science",
Phys. Chem. Series 1, volume 1: Theoretical Chemistry, W. Byers
Brown, ed. (Butterworths, London, 1972).

U. Landman and R. Pauncz, Chem. Phys. Letters 3, 489 (1971).

J. 0. Hirschfelder and P.R. Certain, Int. J. Quantum Chem. S
3, 125 (1968).

H. Eisenschitz and F. London, Z. Physik 33, 491 (1930).
A. van der Avoird, J. Chem. Phys. 32, 3649 (1967).

J.D. Hirschfelder, Chem. Phys. Letters 1, 325 (1967).
J.D. Hirschfelder, Chem. Phys. Letters 1, 363 (1967).

R. McWeeny, Proc. Roy. Soc. London A 333, 242 (1959).

R. McWeeny, and B.T. Sutcliffe, Proc. Roy. Soc. London A 273,
133 (1963).

P.D. Dacre and R. McWeeny, Proc. Roy. Soc. London 1 313, 435
(1970).

V. Magnasco, G.F. Musso, and 8” McWeeny, J. Chem. Phys. 33,
4617 (1967).

R.D. Amos, Chem. Phys. Letters 33, 89 (1982).
P.S. Epsetin, Phys. Rev. 33, 695 (1926).

R.S. Mfilliken, C.A. Rieke, D. Orloff, and H. Orloff, J. Chem.
Phys. 13, 1248 (1949); 9:. also Ref. 95.

A. Lofthus, Mol. Phys. 3, 105 (1962).
M. Hotani, A. Amamiya, 8. Ishiguro, and T. Himura, ”Table of Molecular
Integrals”. (Maruzen, Tokyo, 1955).

M.P. Barnett and C.A. Coulson, Phil. Trans. 1 333, 221 (1951).

[98]

[99]
[100]
[101]

[102]

[103]

[104]
[105]
[106]
[107]
[108]

[109]

[110]

[111]

[112]

[113]

[114]

[115]

[116]

[117]

[118]

124
M.P. Barnett, in "Methods in Computational Physics" vol. 2, B.
Alder, S. Fernbach, and M. Rotenberg, eds. (Academic Press, New
York, 1963).
C.C.J. Roothaan, J. Chem. Phys. 13, 1445 (1951).
K. Rfidenberg, J. Chem. Phys. 13 1459 (1951).
C.C.J. Roothaan and Rfidenberg, J. Chem. Phys 33, 765 (1954).

K. Ruedenberg, C.C.J. Roothaan, and W. Jaunzemis, J. Chem.
Phys. 33, 201 (1956).

D.B. Knowless, J.N. Murrell, and J.P. Braga, Chem. Phys. Letters
113, 40 (1984).

A.D. McLean and M. Yoshimine, J. Chem. Phys. 33, 3682 (1967).
S. Huzinaga, J. Chem. Phys. 33, 1293 (1965).
J.L. Dodds, R. McWeeny, and A.J. Sadlej, Mol. Phys. 33, 1779 (1977).
8.0. Roos and A.J. Sadlej, Chem. Phys. 33, 43 (1985).
A. Koide, J. Phys. B 3, 3173 (1976).

M. Krauss, D.B. Neumann, and W.J. Stevens, Chem. Phys. Letters, 11,
500 (1980).

A. Koide, W.J. Meath, and A.R. Allnatt, Chem. Phys. 33, 105 (1981).

P. Malinowski, A.C. Tanner, K.F. Lee, and B. Linder, Chem. Phys. 33,
423 (1981).

G.H.F. Diercksen, B.O. Roos, and A.J. Sadlej, Chem. Phys. 33, 29
(1981).

C.D.H. Chisholm, "Group Theroetical Techniques in Quantum
33emistry", Academic Press, New York (1976).

J.C.C.M. van Duijneveldt-van de Rijdt and F.B. van Duijneveldt,
Chem. Phys. Letters. 13, 425 (1972).

J. van Kranendonk, Physics 23, 825 (1957).
G. Birnbaum, Proceedings of the International School of Physics
”Enrico Fermi". Course LXXV. Edited by J. van

Kranendonk. North Holland Publishing Co.; London, 1980.

I. R. Dagg, L. A. A. Read and W. Smith, Can. J. Phys. 33, 1431
(1982).

U. Buontempo, S. Cunsolo, P. Dore and L. Nencini, Can. J. Phys.

125

[119] For a list of refs. on experiments carried out with liquids, see

[120]
[121]

[122]

[123]

[124]

[125]

[126]

[127]

[128]
[129]
[130]

[131]

[132]

[133]

[134]

[135]
[136]
[137]

[138]

B. Catlow and J. Yarwood, J. M01. Struct. (Theochem) 133, 415

(1986).

A. I. Baise, J. Chem. Soc. Faraday Trans. II 33, 1907 (1972).
S. Weiss, Can. J. Phys. 33, 584 (1976).

F. Sullivan, R. 0. Mountain and J. O’Connell, J. Comput.
138 (1985).

Phys. 31,

A. D. Buckingham and B. D. Utting, Ann. Rev. Phys. Chem.
(1970).

g1, 287

H. Margenau and N. R. Kestner, "Theory of Intermolecular Forces".

Pergamon Press; Oxford, 1969.

J. A. Barker, R. A. Fisher and R. 0. Watts, M01. Phys. 31, 657
(1971).

M. Wojcik and 8. Clementi, J. Chem. Phys. 33, 5970 (1986).

J. J. Salacuse, W. Schommers and P. A. 8gelstaff, Phys. Rev. A 33,
1516 (1986).

C. Hoheisel, Phys. Rev. A 33, 1998 (1981).
B. M. Axilrod and 8. Teller, J. Chem. Phys. 11, 299 (1943).
A. D. McLachlan, Mol. Phys. 3, 423 (1963).

J. M. Standard and P. R. Certain, J. Chem. Phys. 33, 3002
(1985), and refs. therein.

a) K. F. Niebel and J. A. Venables; b) J. A. Barker; c) R. J. Bell
and I. J. Zucker; d) J. N. Murrell, in Rare Gas Solids. Edited
by M. L. Klein and J. A. Venables. (Academic Press. London,
1976). Vol. I.

J. A. Barker, M. L. Klein and M. V. Bobetic, IBM J. Res. Develop.
33, 222 (1976).

I. L. Garzén and 8. Blaisten-Barojas, Chem. Phys. Lett. 133, 84
(1985).

B. H. Wells and S. Wilson, Chem. Phys. Lett. 131, 429 (1983).

B. H. Wells and S. Wilson, M01. Phys. 33, 1295 (1983).

. Wells and S. Wilson, Mol. Phys. 33, 199 (1985).

. Wells and S. Wilson, Mol. Phys. 32, 21 (1986).

[139]
[140]
[141]
[142]

[143]

[144]

[145]

[146]
[147]

[148]

[149]

[150]

[151]

[152]

[153]
[154]
[155]

[156]

[157]
[158]
[159]
[160]

[161]

126

R. J. Bell, J. Phys. B Atom. Molec. Phys. 3, L63 (1970).
W. Kolos and A. Leé, Chem. Phys. Lett. 13, 167 (1972).
P. Rosen, J. Chem. Phys. 31, 1007 (1953).

A. Shostak, J. Chem. Phys. 33, 1808 (1955).

8. W. Schmid, J. Schwager, Y. C. Tang and R. C. Herndon, Physica
31, 1143 (1965).

D. R. Williams, L. J. Schaad and J. N. Murrell, J. Chem. Phys. 32,
4916 (1967).

R. F. W. Bader, 0. A. Novaro and V. Beltréh-Lépez, Chem. Phys.

Lett. 3, 568 (1971).
0. A. Novaro and V. Beltrén-Lépez, J. Chem. Phys. 33, 815 (1972).
L. W. Bruch and I. J. McGee, J. Chem. Phys. 33, 409 (1973).

L. Gonzélez-Tovany, V. Beltrén-Lépez and 0. Novaro, J. Chem. Phys.

g9, 1694 (1974).
A. J. c. Varandas, M01. Phys. 1g, 817 (1983).

M. Bulski and G. Chalasifiski, Theoret. Chim. Acta (Berl.) 33, 199
(1980).

M. Bulski, Chem. Phys. Lett. 33, 361 (1981).

J. P. Daudey, O. Novaro and M. Berrondo, Chem. Phys. Lett. 33, 26
(1979).

M. Bulski and G. Chafasifiski, Chem. Phys. Lett. 33, 450 (1982).
J. A. Barker and M. L. Klein, Chem. Phys. Lett. 11, 501 (1971).
J. A. Barker and M. L. Klein, Phys. Rev. 3 Z, 4707 (1973).

J. A. Barker, R. 0. Watts, J. K. Lee, T. P. Schafer and Y. T. Lee,
J. Chem. Phys. 31, 3081 (1974).

J. A. Barker, Mol. Phys. 33, 755 (1986).
J. A. Barker, Phys. Rev. Lett. 33, 230 (1986).
B. J. Alder, Ann. Rev. Phys. Chem. 33, 325 (1973).

. McGinnies and L. Jansen, Phys. Rev. 131, 1301 (1956).

. O’Shea and W. J. Meath, Mol. Phys. 33, 1431 (1974).

[162]

[163]
[154]
[165]
[166]
[167]
[168]
[169]
[170]
[171]
[172]
[173]
[174]
[175]

[176]

[177]

[178]

[179]

[180]

[181]

[182]
[183]

127

K. F. Niebel and J. A. Venables, Proc.
365 (1974).

Roy. Soc. (London) 5 333,

S. Carter and J. N. Murrell, Mol. Phys. 33, 1269 (1976).

L. Jansen, Phys. Rev. 133, A1292 (1964).

Rev.

E. Lombardi and L. Jansen, Phys. 133, A1011 (1964).

L. Jansen, Adv. Quantum Chem. 3, 119 (1965).
L. Jansen and E. Lombardi, Disc. Faraday Soc. n9 49, 78 (1965).

Lett.

L. Jansen and E. Lombardi, Chem. Phys. 1, 417 (1967).-a-

G. G. Chell, J. Phys. C (Proc. Phys. Soc.) 1, 266 (1968).

E. Lombardi and L. Jansen, Phys. Rev. 133, 822 (1968).

E. Lombardi, L. Jansen and R. Ritter, Phys Rev. 133, 1150 (1969).

L. Jansen and E. Lombardi, Chem. Phys. Lett. 1, 33 (1967).

C. E. Swenberg, Phys. Lett. 341, 163 (1967).

W. J. Meath and R. A. Aziz, Mol. Phys. 33, 225 (1984).

J. A. Barker and M. V. Bobetic, J. Chem. Phys. 33, 6306 (1983).

R. A. Aziz in Inert Gases. Edited by M. L. Klein. Springer Series
in Chemical Physics 34. (Springer Verlag. Berlin, 1984).

R. A. Aziz, J. Chem. Phys. 33, 5787 (1980).

P. A. Egelstaff, W. Glaser, D. Litchinsky, E. Schneider and
J. B. Suck, Phys. Rev. 4 33, 1106 (1983).

J. van Kranendonk, in Propriétés Optiques et Acoustiques des
Fluides Comprimés et Actions Intermoleculaires. CNRS.
Paris, 1959.

G. Birnbaum, W. Ho and A. Rosenberg, J. Chem. Phys.
(1971).

§§, 1039

D. Frenkel, in Proceedings of the International School of Physics
"Enrico Fermi". Course LXXV. Edited by J. van Kranendonk. North
Holland Publishing Co.; London, 1980.

A. Raczyfiski and G. Staszewska, Mol. Phys. 33, 919 (1986).
R. L. Welsh, in MTP International Review of Science. Spectroscopy.

Physical Chemistry. Series One. Vol. 3.Edited by D. A. Ramsay.
Butterworths. London, 1972.

[184]

[185]
[186]
[187]

[188]

[189]
[190]

[191]

[192]
[193]
[194]
[195]

[196]

[197]

[198]

[199]

[200]

[201]

[202]

[203]

[204]

I 205]

128

A. D. Buckingham in Propriétés Optiques et Acoustiques des Fluides
Comprimés et Actions Intermoléculaires. CNRS. Paris, 1959.

S. Weiss, Chem. Phys. Lett. 13, 41 (1973).
C. G. Gray and B. W. N. Lo, Chem. Phys. Lett. 33, 55 (1974).
P. H. Martin, Mol. Phys. 33, 129 (1974).

L. W. Bruch, C. T. Corcoran and F. Weinhold, Mol. Phys. 33,
1205 (1978).

L. W. Bruch and T. Osawa, Mol. Phys. 43, 491 (1980).
G. Adam and J. Katriel, Chem. Phys. Lett. 31, 451 (1973).

M. Udagawa, K. Nakamura, M. Murakami and K. Ohbayashi, Phys.
Rev. 3 34, 1563 (1986).

U. Balucani and R. Vallauri, M01. Phys. 33, 1099, 1115 (1979).
P. A. Madden, Mol. Phys. 33, 365 (1978).
W. M. Gelbart, Adv. Chem. Phys. 33, 1 (1974).

B. J. Alder, J. J. Weis and H. L. Strauss, Phys. Rev.~A Z, 281
(1973).

A. D. Buckingham and M. J. Stephen, Trans. Faraday Soc.
(1957).

9.5.3. 884
J. J. Perez, J. H. R. Clarke and A. Rinchliffe, Chem Phys. Lett.
134, 583 (1984).

H. Margenau and J. Stamper, Adv. Quantum Chem. 3, 129 (1967).

R. Block, R. Roel and G. Ter Maten, Chem Phys. Lett. 11, 425
(1971).

S. F. O’Shea and W. J. Meath, Mol. Phys. 31, 515 (1976).

J. N. Murrell, A. J. C. Varandas and M. F. Guest, Mol. Phys. 31,
1129 (1976). ’

A. V. Zaitsevskii, A. V. Nemukhin and N. F. Stepanov, M01. Phys.
41, 377 (1980).

K. Kaufmann, M. Jungen and H. J. Werner, J. Phys. Chem. 33,
3806 (1983) .

A. C. Roach and P. J. Kuntz, J. Chem. Phys. 34, 822 (1986).

T. Koga, J. Chem. Phys. 34, 1636 (1986), and refs. therein.

[206]

[207]

[208]

[209]

[210]

[211]

[212]

[213]

[214]

[215]

[216]

[217]

[218]

[219]
[220]
[221]

[222]

[223]

[224]

129

C. E. Hecht, Physica 33, 1159 (1959).

J. V. Dugan Jr., and R. D. Etters, J. Chem. Phys. 33, 6171
(1973).

R. D. Etters, J. V. Dugan Jr., and R. W. Palmer, J. Chem.
Phys. 33, 313 (1975).

M. D. Miller, L. H. Nosanow and L. J. Parish, Phys. Rev. Lett.
33, 581 (1975).

W. C. Stwalley, Phys. Rev. Lett. 33, 1628 (1976).

W. C. Stwalley and L. H. Nosanow, Phys. Rev. Lett. 33, 910
(1976).

Crampton, T. J. Greytak, D. Kleppner, W. D. Phillips,
Smith and A. Weinrib, Phys. Rev. Lett. 43, 1039 (1979).

>.u:

S.
D.
W. N. Hardy, A. J. Berlinsky and L. A. Whitehead, Phys. Rev.
Lett. 43, 1042 (1979).

Hardy, M. Morrow, R. Jochemsen, B. W. Statt, P. R. Kubik,
Marsolais and A. J. Berlinsky, Phys. Rev. Lett. 43, 453
0).

5‘?

W.
R.
(19

no

I. F. Silvera and J. T. M. Walraven, Phys. Rev. Lett. 44, 164
(1980).

R. W. Cline, D. A. Smith, T. J. Greytak and D. Kleppner, Phys.
Rev. Lett. 43, 2117 (1980).

R. W. Cline, T. J. Greytak and D. Kleppner, Phys. Rev. Lett. 43,
1195 (1981).

J. T. M. Walraven, I. F. Silvera and A. P. M. Mattbey, Phys.
Rev. Lett. 43, 449 (1980).

C. Lhuillier and F. Laloe, J. Physique 43, 197 (1982).

C. Lhuillier and F. Laloe, J. Physique 43, 225 (1982).

J. P. Bouchaud and C. Lhuillier, J. Physique 43,1101 (1985).
Yu. Kagan, I. A. Vartanyants and G. V. Shlyapnikov, Zh.
3333i)Teor. Fiz. 81, 1113 (1981). (Sov. Phys. JETP 34, 590

R. Sprik, J. T. M. Walraven and I. F. Silvera, Phys. Rev.
Lett. 31, 479, 942 (1983).

H. F. Hess, D. A. Bell, G. P. Kochanski, R. W. Cline, D.
Kleppner and T. J. Greytak, Phys. Rev. Lett. 31, 483 (1983).

[225]

[226]

[227]

[228]

[229]
[230]
[231]
[232]
[233]

[234]

[235]

[236]

[237]

[238]

[239]

[240]

[241]

[242]

[243]
[244]

[245]

130

H. F. Hess, D. A. Bell, G. P. Kochanski, D. Kleppner and T. J.
Greytak, Phys. Rev. Lett. 33, 1520 (1984).

R. V. E. Lovelace, C. Mehanian, T. J. Tommila and D. M. Lee,
Nature 318, 33 (1985).

L. Jansen, Phys. Rev. 162, 33 (1967).

J. Juan6s i Timoneda and K. L. C. Hunt, J. Chem. Phys. 34, 3954
(1986).

F:

Byers Brown, Chem Phys. Lett. 3, 105 (1968).
S. T. Epstein and R. E. Johnson, Chem. Phys. Lett. 1, 602 (1968).
L. Jansen, Phys. Rev. 133, 1798 (1962).

W. Kolos and A. Les, Int. J. Quantum Chem. 3, 1101 (1972).
H. Margenau, Phys. Rev. 33, 1000 (1939).

P. R. Certain, J. O. Hirschfelder, W. KoIos and L. Wolniewicz,
J. Chem. Phys. 43, 24 (1968).

K. Szalewicz and B. Jeziorski, Mol. Phys. 33, 191 (1979).

E. E. Nikitin and 8. Ya. Umanskii. Theory of Slow Atomic
Collisions. Springer Series in Chemical Physics 33.
(Springer Verlag. New York, 1984), and refs. therein.

8. T. Epstein, in The Force Concept in Chemistry, edited by
B. M. Deb (Van Nostrand Reinhold, New York, 1981), Chap. 1.

M. E. Rose, Elementary Theory of Angular Momentum. J. Wiley
& Sons. New York, 1957.

A. Messiah, Quantum Mechanics. North-Holland Publishing Co.;
New York, 1963.

M. Weissbluth, Atoms and Molecules. Academic Press; New York,
1980.

C. G. Gray and K. E. Gubbins, Theory of Molecular Fluids.
Clarendon Press; Oxford, 1984.
E. Lombardi, C. Almirante and L. Jansen, Chem. Phys. Lett.
133, 1 (1984).

W. Kolos and L. Wolniewicz, J. Chem. Phys. 43, 1426 (1967).

W. KoIos and L. Wolniewicz, J. Chem. Phys. 43, 2429 (1965).

R. F. Stewart, J. Chem. Phys. 33, 431 (1970).

[269]

[270]
[271]

[272]

[273]

[274]

[275]

[276]
[277]

[278]

[279]

[280]

[281]

[282]

[283]

[284]
[285]
[286]

[287]

[288]

[289]

131

M. Krauss, W. J. Stevens and D. B. Neumann, Chem. Phys.

Lett. 31, 500 (1980).

M. E. Rosenkrantz, 186§ACS Meeting 1983, Phys. Abstract 87.
M. E. Rosenkrantz, 187Q ACS Meeting 1984, Phys. Abstract 56.
M. E. Rosenkrantz, R. M. Regan and D. D. Konowalow, J. Phys.
Chem. 33, 2804 (1985).

P. J. Knowles and W. J. Meath, Chem. Phys. Lett. 134,

164 (1986).

K. T. Tang and J. P. Toennies, J. Chem. Phys. 33, 1496 (1977);
3Q, 3726 (1984).

W. J. A. Maaskant
(1964).

and L. J. Oosterhoff, Mol. Phys. 3, 319

x. L. c. Hunt, J. Chem. Phys. zg, 6149 (1983).

N. Jacobi and Gy. Csanak, Chem. Phys. Lett. 33, 367 (1975).

M. Battezzati and V. Magnasco, J. CHsm. Phys. 33, 2924
(1977).

A. Koide, T. R. Proctor, A. R. A11natt and W. J. Meath,
M01. Phys. 33, 491 (1986).

P. W. Langhoff, Chem. Phys. Lett. 33, 33 (1973).

B. Linder, K. F. Lee, P. Malinowski and A. C. Tanner, Chem.
Phys. 33, 353 (1980).

P. Malinowski, A. c. Tanner, K. F. Lee and B. Linder, Chem.
Phys. 33, 423 (1981).

S. Wozniak, K. F. Raof and B. Linder, J. Chem. Phys. 33, 220
(1983).

R. LeSar, J. Phys. Chem. 33, 4272 (1984).
R. LeSar and M. S. Shaw, J. Chem. Phys. 34, 5479 (1986).

J. Hepburn, G. Scoles and R. Penco, Chem. Phys. Lett. 33,
451 (1975), and references therein.

W. Byers Brown and D. M. Whisnant. Chem. Phys. Lett. 3,
329 (1970).

S. F. Abdulnur, Chem. Phys. Lett. 33, 355 (1973).

K. L. C. Hunt, Chem. Phys. Lett. 23, 336 (1980).

[290]
[291]
[292]
[293]
[294]
[295]

[296]

[297]

[298]

[299]

[300]

[301]

[302]

[303]

[304]

[305]

132

K. L. C. Hunt and J. E. Bohr, J. Chem. Phys. 33, 5198 (1985).
M. Giraud and L. Galatry, Chem. Phys. Lett. 33, 18 (1980).

B. Linder and R. A. Kromhout, J. Chem. Phys. 34, 2753 (1986).
K. L. C. Hunt, J. Chem. Phys. 33, 393 (1984).

K. L. C. Hunt and J. E. Bohr, J. Chem. Phys. 34, 6141 (1986).
B. Linder and D. A. Rabenold, Adv. Quantum Chem. 3, 203 (1972).

R. D. Richtmyer, "Principles of Advanced Mathematical
Physics", vol. I.(Springer Verlag. New York, 1978).

J. Ducuing, in Proceedings of the International School of
Physics "Enrico Fermi". Course XLII. Edited by R. J.
Glauber. Academic Press. New York, 1969.

P. N. Butcher and T. P. McLean, Proc. Phys. Soc. 31, 219
(1963).

S. Kielich, Acta Phys. Polon. 33, 875 (1966)

D. C. Hanna, M. A. Yuratich and D. Cotter, "Nonlinear Optics of
Free Atoms and Molecules". (Springer Verlag. Berlin,1979).

N. Bloembergen, "Nonlinear Optics". (W. A. Benjamin Inc.
Reading (MA), 1965).

B. J. Orr and J. F. Ward, Mol. Phys. 33, 513 (1971).

A. D. Buckingham, in ”Nonlinear Behaviour of Molecules,
Atoms and Ions in Electric, Magnetic or Electromagnetic
Fields". Edited by L. Neel. (Proceedings of the 3lst
International Meeting of the Societé de Chimie Physique,
1978). (Elsevier Scientific Publishing Co. Amsterdam, 1979).

We focus our attention on the susceptibility at nonresonant
frequencies. Therefore, we drop all the terms in Eq. (11)
which involve Dirac distributions since we consider in

this section the susceptibility and hyperpolarizability
densities at purely imaginary frequencies.

K. L. C. Hunt and J. Juanés i Timoneda, to be published.