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NONLOCAL DIELECTRIC MODEL FOR INTERMOLECULAR
INTERACTIONS AT SECOND AND HIGHER ORDERS

presented by

ANIRBAN MANDAL

has been accepted towards fulfillment
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5/08 KzlProjIAccaPreleIRC/Dateoue.indd

NONLOCAL DIELECTRIC MODEL FOR INTERMOLECULAR INTERACTIONS
AT SECOND AND HIGHER ORDERS

By

Anirban Mandal

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
Chemistry

2009

ABSTRACT

NONLOCAL DIELECTRIC MODEL FOR INTERMOLECULAR INTERACTIONS
AT SECOND AND HIGHER-ORDERS

By
Anirban Manda]
The nonlocal dielectric function of a molecule determines the effective potential at a
certain point due to an applied external potential at a different point, within the molecule.
The effective potential at point r is determined by the nonlocal dielectric firnction

sv(r,r';co) within linear response and by the nonlocal dielectric function

sq (r, r', r"; to, (0') within nonlinear response to the lowest order. The nonlocal dielectric
functions ev(r,r';w) and sq (r, r’,r';co, (0') depend on the charge-density susceptibilities
x(r,r';m) and xq (r,r',r"; a), (0') of the molecule, respectively. This work shows that for a

group of interacting molecules with weak or negligible charge overlap, the nonlocal
dielectric model gives the interaction energies and forces at second and higher-orders, in
agreement with the results from quantum mechanical perturbation theory. The dielectric
model accounts for screening due to electronic charge redistribution in the interacting
molecules; it accounts for polarization and fluctuations in the charge densities that act as
sources of the external potentials. The model applies within the Born-Oppenheimer
approximation.

At second order, both two-body pairwise additive and three-body non-additive
induction energies appear. We prove that the two-body induction energy is determined
from changes in the static Coulomb interactions within each molecule, caused by a

neighboring molecule that acts as the dielectric medium, whereas the three-body

induction energy is determined by the changes in the static Coulomb interactions between
two molecules, due to the presence of a third molecules which acts as the dielectric
medium. Dispersion energy is pairwise additive at second order and results from changes
in the intramolecular exchange-correlation energy caused by the dielectric screening due
to a neighboring molecule. The interaction energies at second order include linear
screening only, while the induction and dispersion forces on nuclei result from both linear
and nonlinear screenings. We show that induction forces result from nonlinear screening
of the potential due to permanent charge distribution of the neighboring molecule and
linear screening of the potential due to induced polarization of the neighbor, while
dispersion forces result from nonlinear screening of the fluctuating charge distribution of
the neighboring molecule and linear screening of the dynamic reaction field from the
neighbor. The linear screening present in the dispersion force is described by transition
susceptibility of the molecule. Moreover, the dispersion force includes effects which
don’t have a dielectric interpretation.

At third and fourth order of molecular interactions, the induction and the
induction-dispersion energies show both linear and nonlinear screening, while the
dispersion energy includes linear screening only. Depending on the nature of interaction,
the induction energy can be classified into different categories, each category showing a
different screening effect. Screening in the dispersion energy can be described either by
the dielectric function of a single molecule, or by the dielectric functions of two or three

molecules.

To my parents

iv

ACKNOWLEDGEMENTS

I would like to express my sincere gratitude to my advisor Professor Kathy Hunt for her
invaluable guidance and enduring patience throughout the course of research at graduate
school. I am deeply indebted to her for her help, support, teaching and encouragement
during my stay at MSU.

I would like to thank my second reader Professor James Harrison for helping me with the
computational methods and for encouraging me to continue research in theoretical
chemistry.

I would also like to thank my committee members Professor Piotr Piecuch and Professor
James McCusker for their great advice and teaching.

I take this opportunity to thank Dr. D. J. Gearhart, Dr. Ruth L. Jacobsen, Dr. Maricris
Lodriguito, Afra Panahi, Mariam Sayadi, Jeff Gour, and Jesse Lutz for their help and
company.

I thank Janet Haun and Zeynep Altinsel for their help and kindness during my hard times.
I am grateful to my friends Anindya and Paramita for their help, support and warm
company.

Finally, I would like to take the immense pleasure to thank my parents, my uncles, my
aunts and my late grandma for their love and support. Without their help, I would not be

able to come this far.

Table of Contents

Chapter 1: Introduction ................................................................................................................... 1

Chapter 2: Nonlocal dielectric functions on the nanoscale: Screening of two-
body induction and dispersion energies at second order ............................................. 35
2.1: Dielectric screening and the induction energy 35
2.1: Dielectric screening and the dispersion energy 38

Chapter 3: Dielectric screening of second-order induction and dispersion

forces on nuclei of interacting molecules ................................................................... 43
3.1: Dielectric screening and the second-order induction forces
on nuclei .................................................................................................................... 43
3.2: Dielectric screening and the second-order dispersion forces
on nuclei .................................................................................................................... 50

Chapter 4: Dielectric screening and the three-body nonadditive interactions

at second order ............................................................................................................ 75
4.1: Dielectric screening of the three-body induction energy at
second order .............................................................................................................. 75
4.2: Dielectric screening of second-order three-body forces on nuclei
of interacting molecules ............................................................................................ 79

Chapter 5: Dielectric screening of three-body and four-body interactions

at third and fourth orders ............................................................................................. 85
5.1: Dielectric screening of the induction energy
at third order .............................................................................................................. 87
5.2: Dielectric screening and the nonadditive
dispersion energy ...................................................................................................... 98
5.2A: Dielectric screening of nonadditive three-body dispersion
energy at third order ............................................................................................... 98
5.23: Dielectric screening of nonadditive four-body dispersion
energy at fourth order ........................................................................................... 105
5.3: Dielectric screening and induction-dispersion energy at
third and fourth orders ........................................................................................... 111
Chapter 6: Summary and conclusions ......................................................................................... 117
Appendices .................................................................................................................................. 134
References ................................................................................................................................... 145

vi

Chapter 1: Introduction

. . . 1 . .
A nonlocal dielectric functlon, 8V(l',l";(1)) characterlzes the screening of an

applied scalar potential (Pex (r';w), due to electronic charge redistribution within a
molecule. Within linear response, the effective potential (peff (run) at a point r within a

molecule is related to an applied potential (Pex (r';o)) of frequency a) by

(Peff = 80 [dr's'v1(r,r';w)<pex(r';w), (1.1)
where 30 is the permittivity of vacuum. Thus, ev(r,r';o)) is defined as the fimction

which acts as the integral kernel to determine the effective potential at point r, within

linear response, when an external potential (Pex (r’;(o) acts at point r'. The dielectric
function av(r,r';(o) depends both on the response point r and on the point r' at which

the external potential acts. In this work we show that a generalized dielectric model of
potential screening holds for two-body interactions (treated here through second order)
and for three- and four-body interactions (treated here through fourth order) of quantum
perturbation theory. For two or more interacting molecules, the source of the external
potential can be either the permanent charge distributions or the charge-density
fluctuations within the molecules. We prove that the interaction energies and the
interaction-induced forces on the nuclei of the molecules are accurately described within
the nonlocal dielectric model, where the interacting molecules behave as the dielectric
medium. In treating intermolecular interactions, we assume that the overlap between the
electronic charge clouds of the interacting molecules is weak or negligible. Once the

interaction energies are known, the interaction-induced forces on a particular nucleus are

obtained as the negative gradient of the interaction energy with respect to the coordinate
of the nucleus. Throughout the derivations, we work within the Born-Oppenheimer
approximation.

The nonlocal dielectric function 8V (r, r';u)) is related to the nonlocal charge-
density susceptibility2 x(r,r';o)) of the molecule. Using quantum perturbation theory,

Jenkins and Hunt1 have proved that the nonlocal dielectric function and the nonlocal

charge-density susceptibility are related by

30 8;] (r, r'; co) = 6(r — r') +(41I80)-1 Idr" r — r" _l x(r", r’; (o) , (1.2)

 

 

within the Bom—Oppenheimer approximation. The nonlocal charge-density susceptibility

in Eq. (1.2) is given by2

mm) = —[<o

 

fi(r)G(<o)fi(r') |0>+ (0

 

l3(r')G(-60) 50) I0) ]- (1 -3)

In Eq. (1.3),

 

0) denotes the ground state of the isolated molecule, {5(r) is the charge-

density operator, G((:)) is the resolvent operator,

G(w)=(1 - 500)(Ho-Eo-hw)'1(1-soo), (1.4)

 

500 denotes the ground-state projection operator |0> <0 , H0 is the unperturbed

Hamiltonian of the molecule, and E0 is the unperturbed ground-state energy. For a
molecule with fixed nuclear positions, x(r,r’;w) determines the change in the electronic
charge density Ap(r,(o) within linear response to a frequency-dependent external

potential ‘Pex (r,o)) , via

Ap(r, w) = Idr’ x(r,r’;0)) ‘Pex (r', (o). (1.5)

The effective potential (peff (r; 0)) within the molecule is given by

(pefi‘ (r, (0) = (pex(r,(1))+(41te0)_1 Idr'Ir—r'|_1Ap(r',o)), (1.6)
without restriction to linear response in calculating Ap(r,o)). Thus, (peff(r,(o) has a
source term containing the sum of the external charge density Pex (r’, (1)) that generates
‘Pex (r',a)) and the change in the molecular charge density Ap(r,o)) induced by
(Pex (r', co) . Hence (peff (r, 0)) satisfies the Poisson equation by

V2 (peff (r, (o) = —[pex (r, (1))+Ap(r, (13)] / 80. (1.7)
The linear response theory suffices to describe the intermolecular interactions at first

order. At second and higher orders, we must include nonlinear response of the molecule

to the applied potentials. Including the lowest-order nonlinear response to (Pex (r', (0) , we

obtain
Ap(r,(o) = Idr' x(r,r';o)) ‘Pex (r', (o) + (1 / 2) £000 dco Idr' dr" C(r,r', r"; (o — (o',o)')

x (Pex (r', (0 — 03') (Pex (r", 05') , (1.8)
where the quadratic charge-density susceptibility C(r, r’, r”; (o — co',w') satisfies the

relation

C(rar'J'm, w') = S 0‘31“"; 60, 60')[<0 50‘) G(wo)fio (I‘")G(0)) 15(1")

+<o
+<0

 

 

0)
°>

o>]. (1.9)

b<r")G*(—w')fi°(r')G*(—wa)fi<r)

 

 

13(r")G*(—w')fi°(r)G(co)fi(r’)

 

 

The operator S(r',r";co,o)') denotes the sum of the terms obtained by permuting the
charge-density operators [1(1") and fi(r") together with their associated frequencies a)

and (o' in the expression that follows S. In Eq. (1.9) and below, 030. = (0+ (0’ , and the

operator fi0(r) is defined by 60(r) =fi0(r)—<O

 

fi(r)|0>. We do not assume that

damping is negligible in general (particularly near resonance), but in cases where
damping is negligible, C(r,r',r';co,w") is symmetric under permutation of all of the
variable pairs:(r,—(oo.), (r',(o), and (r",co"). The method used by Orr and Ward3 to

derive multipole polarizabilities yields Eq. (1.9), when applied to charge-density
operators: cf. Eq. (43b) of ref. 3. Eq. (1.9) is also consistent with the results for multipole
hyperpolarizabilities derived by Bishop, in Eq. (41) of ref. (4). From Eq. (1.8), a

quadratic dielectric fimction can be defined by
so 861031", r”; a), (11’) = (41rso )_1 jdr'" Ir —r"'|_1C(r",r',r",o),w'). (1.10)

Thus, within nonlinear response to the lowest order, the effective potential within a

molecule is obtained as

cpeff(r,w) = so [dr'831(r,r';co)<pex(r',w)
+ (1 / 2) so [:0 dro' Idr' dr' 8211 (r, r', r'; (o — m',(o')

xcpex(r',(o—co')(pex(r",(o'). (1.11)
Within a nonlocal dielectric framework, Eq. (1.11) gives a complete description of the
interaction energies and the interaction-induced forces at first and second orders, and the

interaction energies at third and fourth orders, for three- and four-body systems.

Nonlocal response theory has been applied earlier to study the properties of polar

fluids,5 to find expressions for the equilibrium dielectric constants of fluids consisting of
molecules possessing arbitrary polarizability densities,6 to perform calculations for
solvation of an ion in a cavity,7 to determine the inverse dielectric function of quantum
wells in terms of the random phase approximation,8 and to study the properties of small

conducting particles and thin films in an oscillating longitudinal electric field.9 For

1043 the dielectric function depends only on the

systems with full translational invariance
distance between the response point and the point where the perturbation acts (r - r') ,

while for molecules, the intramolecular dielectric functions depend both on r and r' due

to inhomogeneity within the molecules. Previously, systems with full translational
. . 10-13 . . . . 14.15 . . . .
invariance or spatlal penod1c1ty have been described 1n terms of dielectric

theories, where the dielectric function depends on a single spatial variable.
Dielectric functions depending on a single spatial variable can be described either

by s(r—r') or by its spatial Fourier transform s(k). Dielectric models using the

function s( k,co) have been used to describe quantum many-body problems,'6 to

. . . . . . . 17
characterize the behavror of quantum dots or quantum wrres 1n m1crocav1t1es, metal

clusters and colloidal aggregates,‘8 quantum dot crystals,‘9 localization of hydrated

20,21

electrons, hydration forces,22 dipole-dipole interactions near surfaces,23 and normal

. . . . . . 24 25
mode couplmg 1n semiconductor m1crocav1t1es. Zaremba and Sturn have calculated

s(k, (o) of alkali metals at the level of the random phase approximation, using the soft x-

ray absorption spectra. A cluster expansion method has been used by Felderhof and

Cohen26 to obtain the wave-vector-dependent effective dielectric tensor of a suspension

of spherical inclusions. They showed that in the cluster expansion, the terms
corresponding to the “overlapping” regions of the spherical inclusions lead to the

Clausius-Mossotti formula. Dielectric functions of the form s(k,co) have also been used
in linear response within the density functional theory.27 In liquids, the dielectric

frictionzs’29 and solvation dynamics30'37 depend on the dielectric function s( k,m), and

that affects the fluorescence spectra and rates of charge-transfer reactions38'40 in liquids.

Dielectric functions of the form s( k,co) have been used to study electron transfer,“45

charge-density fluctuations in translationally invariant systems,”48 and polarization

fluctuations in liquids.49
Extension of the dielectric model to the intramolecular domain was suggested in
several early works. Theimer and Paul50 introduced the polarizability density of atoms in

the context of light scattering by monatomic gases. Quantum mechanical calculations of

that function were done for the hydrogen atom. Polarizability density was introduced by
Frisch and McKennaSI as a correction to the local polarization in the study of light
scattering by fluids. Oxtoby and Gelbart52 calculated the pair polarizability anisotropy of

interacting noble gas atoms using the polarizability density instead of point dipoles.

Application of a nonlocal dielectric model instead of a classical continuum dielectric
model to atomic interiors was suggested by Orttung and co-workers.53 Dielectric effects
were used by Levine and Seven54 to calculate the optical polarizabilities and the

photoemission cross sections of nitrogen and acetylene using a time-dependent local-

55-58

density theory. In a few later works, single-point response functions have been used

where one of the two spatial variables r and r’ is being integrated. Response functions

that depend both on r and i" have been used in the study of light scattering by fluids.6’ 36’

”'62 The Nonlocal dielectric model has been applied for theoretical calculations of

surface enhanced Raman scattering (SERS) from metallic nanoshells.63 It has been found

that the nonlocal optical effects can be significant for molecules very close to the shells
and for shells of very small dimensions.
Dielectric models have a wide range of applications in the study of

. . . . . . 64-76
conformational energetics and noncovalent bondlng 1n protems and biomolecules.

For proteins, no universal dielectric constant (or constants) exists because of the
inhomogeneities within the protein interiors. Typically, the dielectric function is
approximated as a constant or as a simple function of distance between a perturbing
charge and a response point. Early calculations on proteins-I7-79 in solutions used an
effective dielectric constant that represents the overall effect of the medium, i.e. water +

protein. However, it was shown80 that this model leads to erroneous results for the
interaction of an ion pair inside the protein.69 The dielectric constant of a protein depends

on the particular property used in its definition.69 For a powder sample of protein, 8 can

be found by applying a weak electric field and using the Clausius-Mossotti equation. This

method has been applied in many physical measurements and the s has been found to be

81,82

quite low (about 2). At a microscopic level, 8 of a protein can be determined from

69,83,84

Coulomb’s law or from The Born formula of self energy.85 Simonson and co-

86 . . . . . . . .
workers have 1nvest1gated the microscoplc mechamsm of charge screemng 1n proteins.

By introducing a fixed, perturbing point charge, they have calculated the Frohlich-
Kirkwood dielectric constants and the generalized susceptibilities of deca-alanine and

cytochrome c. Values of s depend on the particular kind of interaction inside the protein

and can range from s = 2,69 through 45 s S 8 for dipole-charge interactions,70 up to s 2 20

for charge-charge interactions-l"75 Attempts have been made87‘88 to calculate s at the

different sites of protein. However, the solvent reaction field effect was not considered in

any of those attempts. As a result, they all underestimate the value of s. The need for site-

86‘89'93 More recently, Song has

dependent values of s has also been noted in other works.
used the effective polarizabilities of the individual amino acid residues to develop an

approximate site-dependent dielectric function for proteins.94 Spatial variation of the
dielectric function may influence electron or proton transfer, ligand binding,95 molecular

recognition,96 ion transport through channels,97 and conformational dynamics of

biomolecules.

The present work derives a dielectric screening model to treat the two- and three-
body intermolecular interaction energies at second order, three- and four-body interaction
energies at third and fourth orders, and the interaction-induced forces on nuclei at second
order. The screening is nonlocal, since it depends both on r and r’ .

For a pair of interacting molecules, the interaction energy at different orders of

perturbation theory can be obtained by solving the electronic Schrodinger equation

[1316‘ +138 +VAB ll‘VAB) = EAB NAB) , where HOA is the unperturbed Hamiltonian of

molecule A, VAB is the interaction Hamiltonian, E AB is the energy of the A-B pair and

IWAB> is the wavefunction of the interacting system. This particular separation of the

98-100

Hamiltonian is known as the polarization approximation. At first order, the

100.10]

interaction energy for a pair of molecules is purely electrostatic and depends on the

permanent charge densities (and hence on the permanent moments) of the molecules. A

perturbation analysis for the first-order interaction energy AEU) of a pair of molecules A
and B with fixed nuclei and negligible charge overlap yields

—1

AB“) =(41tso)—1 jdrdr' p0A(r)p(1)3(r')|r—r' , (1.12)

 

where p0A (r) and p5 (r’) are the permanent charge distributions in molecule A and B

102.103

respectively. Following Longuet-Higgins, the charge-density operator (3A (r) for

molecule A can be defined as

(3A(r) = Ze6(r—rj)+ZZI6(r—RI), (1.13)
j I

- where the sum over j is for all the electrons assigned to molecule A, with position
operators rj, and the sum over I is for all the nuclei in A with charges ZI and positions RI.
For a pair of molecules with fixed nuclear configurations and negligible charge overlap,
the first-order interaction energy can be expressed using a multipole expansion]04 in
terms of the charge q, permanent dipole no, permanent quadrupole (90, and higher

moments of each molecule

(1)_AB_AB_ A .8
AE —q (p0 “OaSOa (1'3)®0a330a8+'”

_ B A B A B IA

(1):)3 is the potential at the origin RA of molecule A due to the permanent, unperturbed

charge distribution in molecule B. 303a and 3,0118 are the field and the field gradients at

RA respectively, due to the unperturbed charge distribution in B. The Einstein summation

convention over repeated Greek indices has been used in Eq. (1.14). Since the interaction
at first order depends on the unperturbed charge distributions (and hence on the
permanent moments) of the molecules, no screening effect is apparent in the first-order
interaction energy. The first-order force on a particular nucleus I in molecule A can be

obtained from Eq. (1 . 12) using an energy-based approach, employing the relation
171(1) =—vI AB“), (1.15)

where V1 denotes differentiation with respect to the coordinate of nucleus 1, RI.

Electrostatic forces on nuclei can also be derived using the Hellmann-Feynman

105J(X5

theorem and Sternheimer—type shielding tensors.107 The forces on a nucleus in an

atom in presence of an external field have been related to the dipole shielding factors by
Epstein108 and Sambe,109 where the dipole shielding factor 7 has been defined as y = 1 -

o, with o the fraction of the external field actually felt by the nucleus. For diatomic or

polyatomic molecules, the dipole shielding factor is a second-rank tensor represented by

yao, and relates the effective field at the nucleus to the external field by”0

3‘11 = (Sao —yao)So. Calculations on the dipole shielding factors have been done by

lll-l2

several groups. 3 70113 was related to a molecular property in the works of Lazzeretti

10

124,125 109

and Zanasi, Sambe, and Epstein.108 It was shown that Yell} is related to the

derivative of the molecular dipole moment with respect to the nuclear coordinate by
21(6aB—yflfi)=ahg/0RL. (1.16)

Fowler and Buckingham]26 extended the shielding factor to include non-uniform applied
fields and non-linear terms. In terms of their formulation, the net electric field on nucleus

I in molecule A in presence of an external field is

a}, = 3%?” +(5aB ailing +(1/2)(pI 01137 3134736 +...
+(1/3)[(3/2)(R[13—RBA)8W+(3/2)(Rf),—RYA)5ao—(Ra—Ré)éoy

e
”asy]3§y+(“3)§nsye :53 375+“ (1.17)

where we have kept only the terms which depend linearly and quadratically on the fields

and the field gradients. 36 is the external field with gradient 3'6, applied at RA . 7313 is
the Stemheimer shielding tensor, V3107 is the field-gradient shielding tensor, and (p1 (1’57

#1375 are the nonlinear shielding tensors. 39m is the field that acts on nucleus I in

absence of any external perturbation. Using an analogy with the expansion of the induced
dipole moment in powers the external fields and field gradients and the Hellmann-
Feynman theorem, they connected the higher-order nuclear shielding tensors to the
derivatives of molecular multipole moments and polarizabilities with respect to the

nuclear coordinates:

lel

1
wag, = chm/GR , (1.18a)

11

zI [(3/2)R'13 so,Y +(3/2)R§ 50113 41}, 567 +vI ]= 59" MRI}, (1.18b)

(1137 B7
21 s1 - 6A /6R1 (1 18¢)
(11378 _ I375 01 ’ '

a is the dipole polarizability and A is the dipole-quadrupole polarizability. However,
neither in their work nor in the work by Lazzeretti and Zanasi were the derivatives of the
molecular moments and the polarizabilities expressed in terms of molecular response

tensors. Derivatives of the molecular dipole moment and polarizability density with
respect to the nuclear coordinate were evaluated by Hunt,127 and later generalized by

Hunt, Liang, Nimalakirthi, and Harris.128 It was proved127 that the Stemheimer shielding

tensor is related to the nonlocal polarizability density a(r, r') '27’128‘60'62’6’129’130 by

7313 = - Idrdr'TaY(RI,r)aYB(r,r'). (1.19)
Within the framework of the nonlocal polarizability density a(r,r';0), the first-order

forces on nuclei were derived by Hunt and Liang.'31 Specifically they showed that

although the first-order interaction energy depends only on the permanent charge
distributions in the molecules, the first-order force on nucleus I in molecule A depends on
the electronic polarization induced in A by B:

—l

A135“) = zI jdr'pO (r')(R}Jl —r;,) RI —r'

 

 

+21 IdrTao(RI,r)APg\(l)(r). (1.20)
APéA(1)(r) is the first-order polarization in A induced by the field from B and

TaB (RI,r) is the dipole propagator. AP; (1) (r) depends on the nonlocal polarizability

12

density of A. The results in their work were obtained using an equation derived by Hunt
which shows that the infinitesimal motion of a nucleus within a molecule changes the
Coulomb field, and the response of the electronic charge density to that change in the
field is determined by the same nonlocal polarizability density that determines the

induced polarization of a molecule in presence of a static external field. In a later work,

132

Liang and Hunt showed that both the energy-based theory using Eq. (1.15) and the

electrostatic-force theory based on Helhnann—Feynman theorem and the shielding tensors
yield identical result for the first-order forces on the nuclei. In that work, they explicitly
connected the shielding tensors and the nonlocal polarizability densities. The first order

forces on nucleus I due to interaction with B were showed to be

Pg“) = 210303 —Y(IXB)3gB+(l/3)[(3/2)(Rll3 —RBA)80W

+(3/2)(R§, —RYA)5a[3 4R}, 419mm Him 136?” +..., (1.21)

where 3% is the static external field at RA due to the permanent charge distribution in
B, and

V3157 = Idrdr’Ta5(RI,r)a58(r,r')[(3/ 2) (113 430579.

+(3/2)(r1¥-RYA)Sog—(ré—R§‘)8oy]. (1.22)

Thus the force obtained using the shielding-tensor approach was related to the force
derived using the nonlocal polarizability density. However, the polarizability-based
approach was not connected to a dielectric framework.

The intramolecular nonlocal dielectric model was first applied to intermolecular

interactions by Jenkins and Hunt.1 In their work, it was shown that the force on nucleus I

13

in molecule A depends not on the “bare” Coulomb potential due to the unperturbed
charge distribution in B, but on an effective potential. The effective potential on nucleus I

is determined by the nonlocal dielectric function sv(r,r';0) of molecule A due to

redistribution of the electronic charge cloud within A. Using a susceptibility-based

approach, they proved that the first order force 171(1) on nucleus I is

11 I . -1 . B .
F( ) =.2 soa/6r[_[dr sv’A(r,r;O)(pex(r;0)] r=RI’ (1.23)

where the notation [6/ arf(r)] | r = RI means that the derivative is first evaluated with

respect to the coordinate r and then r is set equal to RI .Thus, the first-order force on the

nucleus was connected to the nonlocal dielectric model.
At second order, the interaction energy consists of a sum of induction and

dispersion terms. The induction term AEigc)l is classical and appears from static, linear

response of one molecule to the potential from the unperturbed charge distribution of the

neighboring molecule. The dispersion term AEgzigp is purely quantum mechanical and

depends on correlations between the spontaneous charge-density fluctuations within the

133

interacting molecules. Both induction and dispersion energy are obtained from time-

independent perturbation theory with the second-order correction to the energy given by

AEQ) = —Z Z<OAOB lvABlmAnB><mAnB vABlvoB>/(Em +En —Eo).

 

(1.24)
A B . . - " AB .
m and n are the excrted states in molecules A and B respectively, V 15 the

interaction Hamiltonian, and Em, En, and E0 are the unperturbed energies of excited state

14

and ground state respectively. The sum of the terms from Eq. (1.24) with excited states
confined to either molecule A or B yields the induction energy, while the sum of the
terms with excited states on both molecules gives the dispersion energy.

The induction energy depends on the permanent charge distributions and the

molecular polarizabilities of the interacting molecules. In terms of the permanent

104
moments of the molecules,

B —(1/3)AA sBas -A(1/6)c 3B 3B

2
as? ) d=-(1/2)aa1330a5 33 1111730 33?), 0.1375 30ers eye

B 0A“ A 1A

A s'A (1.25)

SIA IA
_(1/6)CB3075_(1/15)Ea67630030B75+

01878 300113
where 3" is the gradient of the field-gradient, C is the quadrupole polarizability, and E is
the dipole-octopole polarizability.104 Within a nonlocal response model, the induction
energy is described by the nonlocal polarizability density and the static polarizations of
the molecules.l3l The induction energy can also be expressed in terms of nonlocal
charge-density susceptibilities and the static external potentials from the unperturbed
charge distributions of the interacting molecules. We give a detailed description of that in
chapter 2. Since the induction energy depends not only on the permanent moments of the
interacting molecules, but also on the induced polarization in one molecule due to
permanent moments in the other, it is apparent that induction energy can be described in

terms of intramolecular screening. However, the induction energy has not been connected

to a dielectric screening model previously.

15

The induction force at second order can be calculated using Eq. (1.15), where the
induction energy is given by Eq. (1.25) or in terms of the static polarizations and the
nonlocal polarizability densities of the interacting molecules. Within the nonlocal

polarizability density model, the second-order induction force on nucleus I in molecule A

depends on the derivatives of the static nonlocal polarizability density aaAB (r, r';0) and

the field SOAa (r) at B due to permanent moments of A. Unlike the first-order force which

depends on the permanent charge distribution in B and the first-order induced

polarization of A, induction force at second order depends on the first-order induced

polarization in B and the second-order induced polarization in Am:
F5311! = [(1erTaB(RI,r)[APéA(2)(i-)+APIB(1)(i-)], (1.26)

where the second-order induced polarization is determined by the nonlocal

130,134 A
13

afiy(r,r',r';0,0). Induction force on nucleus I can

hyperpolarizability density,

also be determined using Stemheimer-type shielding tensors by132

1(2)_I _I B 11 .B
AFind,a — Z (5018 yaB)SRB+(l/3)Z vaBysRBy

+(1/3)zI [(3/2)(Ré —RBA)80W +(3/2)(R§—RYA)5,,B
—(R(11- Rémm] 11,337 +(1 / 2) ZI¢<£137308133€Y ,

II B ,B
+(1/3)z 5111311530133075 +... (1.27)

16

where 333 and SQRBBY are the reaction field and its gradient from the first-order

polarization of B, induced by the unperturbed charge-distribution in A. Thus, the second-
order induction force on nucleus I depends on linear screening of the reaction field and its
gradients due to the first-order induced polarization of B and nonlinear screening of the
field and its gradients due to the unperturbed charge distribution in B. The two different

approaches used to calculate the induction force are connected to each other by Eqs.

(1.19), (1.22), and the relations132 between the nonlinear shielding tensors ¢€£BY , 5:375

and the nonlocal hyperpolarizability density,

¢<il37 = jdr dr’ dr” Ta5(RI,r) 135137“, r',r") . (1,23a)
51 = Idrdr'dr’T (RI r)|3 (r r' r"
(1137 013 ’ 83¢ ’ ’

x[(3/2)(r; -R$)85n +(3/2)(1‘3 ‘R5A)57<P

—(r;;, —R$)8Y5 ]. (1.28b)
Electrostatic force theory based on the Hellmann-Feynman theorem was also applied by
Nakatsuji and KogaI35 to calculate the forces on nuclei of interacting molecules. In their
work, the force on nucleus A in atom A was expressed in terms of the density matrices of
the atomic orbitals and the forces were categorized into three different classes: 1) the
atomic dipole (AD) force, which originates due to the attraction between nucleus A and
the polarized electron distribution within molecule A, 2) the exchange (EC) force, due to

the attraction between nucleus A and the electron distribution in the region between

atoms A and B, 3) the extended gross charge (EGC) force, caused by the interaction

17

between nucleus A and the gross charge on atom B. At long range, only the atomic dipole
and the extended gross charge forces are significant. At first order the atomic dipole force
and the extended gross charge force correspond to the first-order forces on the nuclei

derived within the nonlocal polarizability density approach, where the AD force depends

on APA(1)(r) and the EGC force depends on of? (r). At second order, the AD force and

the EGC force are related to APAQ) (r) and APB(1)(r) , respectively.

The dispersion energy at second order arises due to the correlation between the

133.2,129,136

charge-density fluctuations in the interacting molecules. The spontaneous

quantum mechanical fluctuations in the charge density in one molecule create a field that
acts on the second molecule and induces a shift in the charge density. The induced shift
in the charge density in the second molecule causes a dynamic reaction field that acts
back on the first molecule and results in a net energy-shift. The induced shift in the
charge density is determined by the dynamic polarizability density a(r,r';r) [or
a(r,r';co) in the frequency-dependent form] or by the dynamic charge-density
susceptibility x(r,r';r) and the charge-density fluctuations are correlated according to the

fluctuation-dissipation theorem.137 Although the charge-density fluctuations and the

reaction field are time-dependent, the dispersion energy is time-independent.138 The

overall dispersion energy for a pair of interacting molecules A and B depends on the

nonlocal polarizability densities (or the charge-density susceptibilities) of the molecules

. . . 129
at imaginary frequencres.

AEQBQ) = —(h/ 271:) go do) Idr...dr" (115‘3 (r, r';io)) T137 (r',r")

l8

xaB5(r", r";ior)T5a(r”', r). (1.29)

The dispersion energy can also be calculated from second-order perturbation theory (cf.
Eq. (1.24)). For example, Longuet-Higgins and Salem103 calculated the dispersion energy

for a pair of interacting molecules at long range, after applying the Unsold
approximation. They expressed the dispersion energy in terms of correlation functions of
the charge-density fluctuations. However, the dynamic natures of the correlation

fimctions were not considered explicitly. The dispersion energy in terms of the
frequency-dependent susceptibilities was first derived by McLachlan.139 In his work the

interaction between two quantum-mechanical systems A and B was considered and the

interaction Hamiltonian was expressed in terms of fluctuations in the physical quantities
x) and y,- , belonging to A and B respectively. The second-order dispersion energy was

given by

= — (h / 2n) 13° déaik(i§)l3ki(i§), (1.30)
where aik (11;) and Bki (i:) are the susceptibilities of A and B respectively and 5 denotes

the frequency. The susceptibilities given in McLachlan’s work depend on the transition
matrix elements of the current and charge-density four vectors, and thus include magnetic
contributions to the van der Waals interaction energy. McLachlan’s work was followed
by the work of Longuet-Higgins,140 where the fluctuations were specifically given in
terms of transition matrix elements of the charge-density operator and the second-order
dispersion energy was showed to be explicitly dependent on the susceptibilities of the

interacting molecules:

19

 

w = —(h/47r2) 1dr) 1dr) Idrf 1de 15° d§ “(qlitu-ifilgigljg (1.31)

a(r1,r2,i§) and 0101' ,rj,i§) are the susceptibilities of the first and the second molecule
respectively, at imaginary frequencies. a(rl,rz,i§) was described as “the mutual
susceptibility at imaginary frequency ifi of two points r] and r2 of the first molecule,”
that measures the response of the electron density at r] due to an applied exponentially-
increasing potential at r2, with time dependence exp(§t). Thus, a(q,r‘2,i§) corresponds

to the charge-density susceptibility x(r,r';i(o) used in later works. The second-order
dispersion energy has been calculated by Langhoff,I41 who has employed the contour-

integration technique of Casimir and PolderI42 to separate the dispersion energy as
integrals over response functions of the interacting systems. Langhoff Fourier-
143

transformed the Coulomb potential to give the dispersion energy. Jacobi and Csanak

followed the same method used by Langhoff to evaluate the dipole-dipole interaction
term in the dispersion energy of two closed-shell atoms. Specifically in their work, the
dispersion energy was expressed in terms of the Born amplitude Xn (q) , which is the

Fourier transform of the transition density Xn (r r). The Born amplitude was then

separated into radial and angular parts and the dipole-dipole interaction term was
evaluated in terms of the frequency-dependent polarizabilities. The dispersion energy can

also be calculated by expanding the Coulomb operator into the interactions between

multipole moments.144 However, this method has a disadvantage that the expansion
diverges asymptotically. KoideI45 has formulated a convergent expansion of the

Coulomb operator into interactions between spherical waves. The charge operators were

20

transformed from configuration space to the wave vector space and the energy
denominators were separated using the contour-integration technique of Casimir and
Polder. The final form of the dispersion energy was obtained in terms of the frequency-
dependent polarizabilities of the interacting atoms in the wave vector space, the spherical

Bessel functions, and the Clebsch-Gordan coefficients. It was shown that in the limit r ——>
0, the dispersion energy goes to zero, while in the limit r -> 00, it yields the well known

multipole expansion series. Derivations and calculations of the dispersion energy in terms

of approximate charge density susceptibilities have been given in Ref. 146 - 152.

153,154

Within the density functional theory (DF T), dispersion energy is contained

within the exchange-correlation energy. Density functional theory has been employed to

calculate van der Waals interaction energies by Langreth and Perdew,155 Harris and

156

Griffin, Gunnarsson and Lundqvist.157 Anderson et al. 158 evaluated the frequency-

dependent polarizabilities for different atoms and ions and the van der Waals C6

constants for several atomic and molecular interactions by using a modified effective

density neff, originally used by Rapcewicz and Ashcroft.159 Several research groupsmo'163

have used the method of a coupling constant 2, that turns on the electron-electron
interactions. Kohn, Meir, and Makarov164 (see also chapter 4) have employed DFT to

calculate van der Waals interaction energy between small and large intersystem distances.

They have approximated the density distribution n(r) by the local density approximation

154

(LDA) or by the generalized gradient approximations (GGA).165 The Coulomb

interaction energy was separated into short and long-range parts, and the long-range

contributions to the interaction energy were expressed using the adiabatic connection

21

formula. Finally, the expression was transformed to the time domain and was applied to
the calculations of the asymptotic van der Waals interaction between H-He and He-He. In

their work, the van der Waals energy was obtained as

 

Evdw = —C6 /R6, (1.32)
where
ll 22
xA (t1)xB (t2)
C =3/n dt dt . 1.33
6 ( >13" 113° 2 ”Hz ( )

x22 is the 2 component of the density response to a perturbation in the z direction. A

seamless van der Waals density functional has been formulated and applied to the

interaction between two self-consistent jellium metal slabs by Dobson and co-workers.166
In their work, the correlation energy (EC) has been determined by the adiabatic
connection fluctuation-dissipation formula (ACFD), which relates EC to the Kubo
density-density response function st , the electron-electron interaction VCoul , and the
Kohn-Sham density-density response function XKS- XKS depends on the average
ground-state electron density. MS is determined from XKS’ the exchange-correlation
kernel fchS , and a modified electron-electron interaction AVCoula by the Dyson-like

screening equation. A related derivation and calculation have been done in later work by
Dobson and Wang!“ For a detailed review of van der Waals studies using conventional
density functionals, we refer the reader to ref. 186. In more recent work, Hunt168 has

derived the electronic energy as a functional of the average electronic charge density and

the average of the gradient of the charge-density fluctuations with respect to an external

22

potential. The functional is nonlocal. However, it is distinct from the other nonlocal

. . 163.164.170-181
functronals berng used.

Jenkins and Hunt182 have derived a model where the correlation between the

polarization fluctuations has been connected to a nonlocal dielectric function

sd (r,r';io)). The nonlocal dielectric function sd (r,r';o)) determines the dielectric
displacement D(r,o)) within a molecule, due to an applied external field E(r’,(o) at r'.
For a translationally invariant system, sd (r, r’; (1)) depends solely on r—r' , and hence
can be represented as the spatial Fourier transform 5d (k; or). For these systems, 3d (k; or)
is connected to the potential screening function sv(k;(o). On the intramolecular scale,

the two dielectric functions are quite different due to the inhomogeneity of the
intramolecular environment.

The dispersion force on nucleus I in molecule A can be calculated by taking the
negative gradient of the dispersion energy with respect to RI , the coordinate of nucleus 1.

Within the nonlocal polarizability density model, the dispersion force on I depends on the

derivative of the dynamic nonlocal polarizability density of A and the derivative of the

correlation of the polarization fluctuations within A, with respect to RI. The derivative of
the frequency-dependent nonlocal polarizability density with respect to RI is related to
128

the frequency-dependent hyperpolarizability. Thus, this component of the dispersion

force results from the interaction between the nucleus and the nonlinear polarization of A

induced by the polarization fluctuations in B. If we denote this component by Ms?) ,

132
then

23

AFAS‘}; = 21 Eodm [drTaB(RI,r)APéA(2)(r,m). (1.34)

Thus, the first component of the dispersion force resembles the first component of the
induction force from Eq. (1.26), with the difference that in the case of the induction force
the polarization in A is induced by the fields from the static polarizations in B, while for
dispersion force the polarization is induced by the fields from polarization fluctuations in

B. This part of the dispersion force also corresponds to the atomic dipole force in the
work by Nakatsuji and Koga,135 although the dispersion forces and dispersion induced
dipole were not considered explicitly in their work.

The second component of the dispersion force depends on the derivative of the
correlation between the polarization fluctuations within A with respect to RI. The

correlation between the polarization fluctuations at r and 1" depends on the imaginary

part of the nonlocal polarizability density by the fluctuation-dissipation theorem137

(1 / 2) (SP5: (1', (1)) 511? (r', to') +59%;A (r', m') 5P5,A (r, 05))

= (h / 27:) 019‘; (r, r'; 0)) 8(a) + 03') coth(hco/2kT) . (1.35)

The infinitesimal shift of nucleus I within molecule A changes the static Coulomb field

that modifies the above correlation. This is similar to the field-induced fluctuation

183,184

correlations studied earlier. Due to the change of the nuclear Coulomb field, the

correlation depends on the imaginary part of the frequency-dependent hyperpolarizability

A!

“BY (1', I", 1’30, 0). Thus the magnitude of the correlation is changed. As pointed

density, [3

out by Liang and Hunt,132 the change in the static external field may also introduce new

24

types of correlations in molecule A. In chapter 3 of this work, we show that the Coulomb
field from nucleus actually brings in new correlation within the molecule. One thing to
note here is that at second order, both the induction and the dispersion force depends on

interaction of nucleus I with the polarization induced in molecule A. Within the context
of dispersion force for interacting atoms, this is known as the Feynman’s “conjecture”.105
Quoting from Feynman’s work on the electrostatic description of forces between
interacting atomslos,

“It is not the interaction of these dipoles that leads to van der Waals’ force, but

rather the attraction of each nucleus for the distorted charge distribution of its own

electrons that gives the attractive 1 / R7 force.”

186 for the

Feynman’s conjecture was first proved by Hirschfelder and Eliason
special case of two hydrogen atoms, both in the Is state. It was also addressed by
Nakatsuji and Koga135 in their work on electrostatic force theory. Hellmann-Feynman

forces on the nuclei of two interacting He atoms have been calculated in later work by

Allen and Tozer.187 The first general proof of Feynman’s conjecture was given by

Hunt.188 In that work it was shown that the dispersion force on nucleus I in molecule A

results from the interaction of I with the dispersion-induced change in the polarization of

A:

APSE) = zI jdrAP$(i-)T¢S(R1,r), (1.36)

where AP? (r) depends on the nonlocal hyperpolarizability density of A and the nonlocal

polarizability density of B, at imaginary frequencies. In the later work by Liang and

25

HuntI32 it was mentioned that the dispersion force on nucleus I depends entirely on the

hyperpolarizability of molecule A; and that it does contain a component that stems
directly from the polarization of B. In chapter 3, we prove that a part of the second
component of the dispersion force actually depends on the linear response of A to the

field from the induced polarization of B, as determined by the transition-

189‘190 of A. Thus, this part of the dispersion force is similar to the second

polarizability
component of the induction force in Eq. (1.26).
For three or more interacting molecules, nonadditivity appears at second and

higher orders. At second order, nonadditivity appears only in the induction energy. At

third and higher orders, the intermolecular interactions consist of induction energy,

253,258,261-264

dispersion energy and induction-dispersion energy . All of the three types of

interactions show nonadditivities. The first calculation of the three-body interaction
energy was done by Axilrod and Tellerm and by Muto,192 who calculated the long-
range triple-dipole energy of three interacting atoms with spherical charge distributions
using third-order perturbation theory. The interaction energy was found to be

1 + 3 cos(91 ) cos(02 ) cos(0 3 )

AEDDD = C9 3 3 3 (1.37)
r r r
12 23 31

 

112,123,131 are the sides and 01,02,03 are the angles of the triangle formed by the

atoms, C9 z (9/16)Vol3 , where V is the atomic ionization potential and a is the

193 to study the

polarizability. The triple-dipole interaction was first applied by Axilrod
preferred lattice structures of the rare gases. Long-range many-body interactions are more

dominant in condensed phases than in molecular clusters.194 The importance of many

26

body effects has been investigated in studies of the thermodynamic properties of

1 97-200

fluids,195’196 calculations of third virial coefficients , and studies of molecular

193,201-205

crystal structures. F orrnisano et al.206 have related the small-k behavior of the

static structure factor S(k) in a noble gas fluid to the two (London)- and three (Axilrod-

Teller triple-dipole)-body potentials. More recently, Jakse and co-workers207 have used a

potential energy function based on the two-body potential of Aziz and Slarnanzog’209 and

the triple-dipole Axilrod-Teller potential to study the structural and thermodynamic

properties of liquid krypton. Evidence for many-body effects has been found in the

193,201,202,204,205

preferred structures and binding energies210 of rare gas crystals. In recent

work, Donchev211 has studied the role of dispersion forces for cubic lattices using a

coupled fluctuated dipole model (CFDM), where the particles have been treated as three-
dimensional harmonic oscillators, coupled by the dipole-dipole potential.

Deviations from pairwise properties have also been noted in studies of structure,

dynamics, light scattering, IR and far-IR absorption spectrazn'220 of van der Waals

216-225 226-235 193,201-205

clusters, liquids, and solids. Computations and experiments on the

spectroscopic properties of van der Waals trimers have specially inspired research in the

field of many-body effects. Studies in this field include rotational spectra of ArzHF,236

6 237 238 9 240 l 242

Al'zDF,23 ArzHCl, ArzDCl, (HCN)3,23 ArzHCN, Ar2COz,24 Ar20CS,

243 243

NezKr, and Ne2Xe2. Far-IR intermolecular vibrations have been observed in

216 217 245

ArzHCl, ArgDCl, and in (1120),.218 Vibrations in the mid-IR in ArnHF,244 DF3,

(HCN)3,,246 and (H20)3247 also show nonadditive effects. The role of nonadditive effects

27

in the properties of water trimer and liquid water has been analyzed by Gregory and
248 . 249 . . . .

Clary, and by L1 et al. Collrsron—rnduced absorption spectra of compressed gases

and liquids show evidence of nonadditive three-body dipoles,250 since the pairwise-

additive dipole is not sufficient to explain the observed intensities of the transitions.

Reddy, Xiang, and Varghese251 detected an absorption between 12 300 and 12 700 cm"1
for compressed H2. This absorption corresponds to the v = 0 —> 1 transition on all three of
the molecules in an H2'"H2’"H2 complex. This particular transition is known as a triple

transition252 and it is forbidden with pairwise additive dipoles.

Early theoretical studies of nonadditive interactions were done using the point-
multipole form. The Axilrod-Teller-Muto triple-dipole formulation was extended by
Stogryn253 to evaluate the three-body dispersion energy of molecules with arbitrary
symmetry. The energy denominator from third-order perturbation theory was separated
using Buckingham’s method of evaluating the second-order dispersion energy and the
dispersion energy was obtained in terms of the polarizability tensors, the mean
polarizabilities of the molecules, and the dipole propagators. The method was applied to

calculate the cohesive energies of molecular crystals. A similar method was used to
calculate the third virial coefficients of C02 and N2, where the spherically symmetric
component of the two-body potential was described by Lennard-Jones (12-6) or (18-6)
potentials. Martin254 derived the three-body dipole moment of three spherical atoms from
the fourth-order perturbed energy with the perturbing Hamiltonian expanded in terms of
the spherical multipole moments of the interacting atoms. For H3, coefficients of the

leading terms were calculated by diagonalizing the unperturbed H-atom matrix with s, p,

28

and d basis sets. The numerical values of these coefficients were then estimated for He3 .

A similar method was followed by Gray and L0255 to evaluate the long range part of the

three-body dipole moment of three interacting atoms and to estimate the density required

to observe the collision induced infi'ared absorption in rare gases experimentally. The

256,257

multipole-moment expansion method has been used by Bruch and co-workers to

study the three-body dipoles of interacting spherical atoms. In later work, Stogryn258 did
a systematic analysis of the third-order perturbation energy for a system of N asymmetric
molecules where the perturbing Hamiltonian was defined in terms of the multipole
moment tensors, as used earlier by Kielich.259 The overall third-order energy was
separated into induction, dispersion and induction-dispersion energies. The induction
energy was further separated into two components: one component (W3) is linear in the
hyperpolarizability and the other one (WA) is bilinear in the polarizabilities. The
dispersion energy was separated into three parts: One involves the polarizabilities of the
interacting molecules at imaginary frequencies (WD), one (WBA) contains the
polarizability of one molecule and the hyperpolarizability of the other, and the last part
(WCD) extends to asymmetric molecules of the result found by Chan and Dalgarno.260

Nonadditivity of the induction energy at second order was derived

explicitly by Piecuch.261 In his work, the Rayleigh-Schrodinger perturbation theory was

used to express the second-order correction to the energy in a system of N interacting

molecules. The pairwise nonadditive induction energy was obtained as

29

A=_ E Z <1Vijlpj><pjlvjk1>‘ (138)
. - E(pj) ’ '

1,1,1?! Pirgj

(kiln)

where Stogryn’s notation258 has been used. i, j, k are the interacting molecules,

 

)and

lp) are respectively the unperturbed ground and excited states, and e (p j) = Epa —Ega .

The interaction Hamiltonian Vij was then expressed in terms of the multipole expansion

using a spherical tensor technique, to obtain

N p I I!
_ li+lj _li_lj—1 _lj—lk —1
A _ —41r. Z Z (—1) Rij Rjk
1,],k.=.1 Ii [j [j [k
(109,1) 1j Lik Li,”-
21,421}- “2 2194.211c “2U ] :1 [1k 3"
2,1, 21k J L116 L119 J J J

l,- +l'j l} +lk Lilg'

X[[[Q1. -D’i(o:1)®Q, .n’k(m‘1)]L. ®a1 .{l'-1'-).D’f(n:1))L, ,
z 1 k k 1k 1 J J _] 1k)

®[Y1,- +1; (Bij)®Yl;- +1k (8 jlt)]1.,.kj 18- (1.39)

Rij is the vector pointing from i to j and (Rijsfiij) are the spherical components of Rij in

the global coordinate system fixed in space. Q11. is the spherical multipole moment of i,

(oi,coj,o)k are the Euler angles describing orientations of local coordinate systems fixed

in molecules i, j, k with respect to the local coordinate system fixed in space. D1 (or) is
the matrix that represents a rotation to in the (2j + 1)-dimensional irreducible

representation of the SO(3) group. aj denotes the irreducible spherical polarizability of

30

atom j and LikaLilg' are tensors which couple the molecular moments and the

polarizabilities. Eq.( 1.39) gives the three-body induction energy at second order, where
the dependence on orientation is simplified as far as possible and the overall energy is
completely separated into the spherical multipole moments and irreducible spherical
polarizabilities of the isolated atoms.

In three successive later works, the same method was used by Piecuch to derive
. . . 262 . . . 263 . . . . . 264
the 1nductron energres, dlspersron energres, and the isotropic interactlon energres

at the first three orders of perturbation theory. In those works, the interaction energies at
first, second and third orders were explicitly derived in terms of the spherical tensor

formalism and the physical significance of the third-order interaction energies fiom
Stogryn’s work258 were explained very clearly. The overall third-order interaction energy
was given by

«as 83:22.rare-Esta:wire

where a is the polarizability tensor and B is the hyperpolarizability tensor. In Eq. (1.40) a
right Q denotes the field from the permanent multipole moment of one molecule and a
left Q denotes the permanent multipole moment of another molecule. The induction
energies were farther separated into two-body, three-body and four-body interactions.
The induction and dispersion energies from Eq. (1.40) correspond to the interaction

energies in Stogryn’s work by, WB 5138);:1522, WA =E822g, WD Z-Eledésp ,

wBA=— _E(3l)3gsp, and WCD a Eggdisl’.

31

A complete description of the three-body interactions must include short range

222-225
0

interactions. These interactions can be included either by ab initio calculations r

by using exchange-perturbation methods such as symmetry-adapted perturbation theory

265-271

(SAP'I) or intermolecular perturbation theory/Moller-Plesset perturbation theory

272-275

(IMPT/MPPT). A symmetry-adapted Rayleigh-Schrodinger perturbation method

was used by Moszynski et al.271 to calculate the nonadditive three-body interaction

energies of van der Waals trimers. The three-body terms from the polarization and

exchange effects were separated and the polarization terms were evaluated using the
linear and the quadratic polarization propagators.276 The three-body polarization terms

were obtained as
- . . (210) (210) . (1,1,1) . .
1nductron. Bind (B (— A,C), Ei n d (A <— B, B <— C), Bind (A (— B,C (— B),

dispersion: E(3.’3)(3,3); and induction-dispersion: E

(2 l 0)
dlsp '

1nd —disp‘ Nonadditive effects have

been calculated by a IMPT/MPPT method developed by Chalasir’rki et al.272'275
Nonadditive dispersion interactions have been described within a reaction-field
. 2.277 . 201 . ,
approach by Linder and Hoemschemyer, and by Langbern. In then works, a
nonlocal response theory has been used to obtain the nonadditive dispersion energies in
terms of frequency-dependent polarizabilities or susceptibilities. Li and Hunt278 have

used a nonlocal polarizability density model to evaluate the three-body polarizations and
three-body forces on nuclei of interacting molecules. The three-body polarizations and
forces were determined from the three-body interaction energies at the third order. The

overall third-order interaction energy was derived as a sum of three-body dispersion

32

energy, three-body induction energy, and three-body induction-dispersion energy. The
induction energy was farther separated into hyperpolarization, static reaction field and
third-body reaction field. It was also shown that the energies obtained using the nonlocal

polarizability density method correspond the third-order interaction energies obtained by

261-264

Stogryn,258 Piecuch, and Moszynski et (11.271 In the present work, we have used the

nonlocal response model used by Li and Hunt to treat the three-body and four-body
interaction energies. However we have worked within a charge-density susceptibility

based model to connect the interaction energies explicitly to a nonlocal dielectric model.

In a later work, Li and Hunt”9 evaluated the nonadditive three-body dipoles of inert gas

trimers and H2--~H2---H2 using the model based on nonlocal polarizability and
hyperpolarizability.

In chapter 2 of this work, we derive the two-body induction and dispersion
energies at second order within the nonlocal dielectric model. These interaction energies
have not been described previously in terms of intramolecular screening. Chapter 3
shows the dielectric screening present in the second-order induction and dispersion
forces. We also derive a new fluctuation-correlation and the physical origin of the terms
present in the second-order dispersion forces. In chapter 4 we extend the nonlocal
dielectric model to derive the three-body induction energy at second order. We prove that
at second order, the nonadditive three-body induction energy results from dielectric
screening, where a particular molecule acts as the nonlocal dielectric medium to screen
the electrostatic interaction between the other two molecules. Finally in chapter 5, we use
the nonlocal dielectric model to derive the nonadditive three-body and four-body

interactions and third and fourth orders. We specifically describe the induction energy at

33

third order and the dispersion and induction-dispersion energies at third and fourth

orders. Chapter 6 includes a brief summary and conclusions.

34

Chapter 2: Nonlocal dielectric functions on the nanoscale: Screening of two-body

induction and dispersion energies at second order

2.1 Dielectric screening and the induction energy

Within quantum perturbation theory for intermolecular interactions, the energy
changes due to static reaction fields determine the induction energy. The permanent
charge distribution of each molecule sets up a field that polarizes the neighboring
molecule; in turn, this produces a reaction field that acts back on the first molecule,

shifting its energy. Thus the induction energy depends on the static, nonlocal

polarizability densities ag‘B (r, r';0) and (12B (r, r';0) of interacting molecules, as shown
in earlier work;131

AEind=—(1/2)jdrdr'aa B(r, r'- ,0)3B a(r)3B OB(r
—(1/2) jdrdr'oaB(r-,r';0)30Aa(r)sA B(r), (2.1.1)

to second order in the intermolecular interaction. In Eq. (2.1.1), 30130 (r) denotes the or
component of the field acting on A due to the unperturbed, static charge distribution
pg (r) of molecule B, and similarly for 38‘“ (r). The Einstein convention of summation

over repeated Greek subscripts is followed in Eq. (2.1.1) and below. The result for

AEind includes higher-multipole polarization, as well as the dipole-induced dipole

interactions, because 01(r,r';03) is defined to allow for the distribution of polarization

within the molecule,

35

o(r,r';to) = <0|f’(r)G((D)f’(r') 0)+(0|r(r')G(-o)r(r)|0). (2.1.2)

 

In Eq. (2.1.2), P(r) is the polarization operator and G((t)) is the resolvent operator
defined in Eq. (1.4). The polarization operator P(r) is related to the charge-density
operator fi(r) by

V-P(r) = —p(r), (2.1.3)
and hence, from Eqs. (1.3), (2.1.2), and (2.1.3),

VV':01(r,r';or) = — x(r,r';o)). (2.1.4)
From Eqs. (1.5) and (2.1.2) -— (2.1.4), a(r,r';c0) functions as an integral kernel, to give

the polarization P(r, (1)) at point r in a molecule by an applied field S(r'mr) acting at r’ .

The field 3’8 (r) due to the unperturbed charge distribution pOA (r) of molecule
- - - A _ ~A _ A __
A rs related to the electrostatrc potentral (00 (r,co — 0) by JO (r)——V(p0 (r,co—0).

Below, we use the notation (06" (r) for the potential (p8 (r,(0 = 0) . Integration by parts in
Eq. (2.1.1) gives the induction energy in terms of the charge-density susceptibilities of A

and B, and the potentials (08‘ (r) and (pg)3 (r) ,
AEind =1/21drdr'x 0-. no 61,30»me

+1/2 jdrdr'x (r,r';0)oOA(r)tp(A(r'). (2.1.5)

With the potentials expressed in terms of the permanent charge densities of the two

molecules,

“1132,3001

 

AEind =1/2(1+5oAB)jdrdr'x (r,r';0)[(4ne0)"1jdr"r—r"

 

36

x[(41tso)_1 jdr'" r'—r"|_1p(l)3(r'")], (2.1.6)

 

where 50 AB interchanges the molecule labels A and B. From Eqs. (1.2) and (2.1.6), with

a change of the labels on the integration variables, it follows that the induction energy is

accurately expressed in the dielectric model, by
, _ /24 ‘1 ddrdnB '1 n, 17—!“1 B I
413m —1 ( nso) (1+sOAB){ j r r r 90(r)[808V,A(r,r .0)1|r rl pom

— Idrdr'p(1)3(r)|r —-r' _1pOB(r')} , (2.1.7)

 

where sv’ A (r, r'; 0) is the static, nonlocal dielectric function of molecule A. The first
term in Eq. (2.1.7) gives the static Coulomb energy of the unperturbed charge distribution

pg (r) of molecule B in presence of molecule A, which acts as a dielectric medium to

screen the interactions within molecule B. The screening is nonlocal, since 8;} A (r, r';0)
9

depends on both r and r'. The second term in Eq. (2.1.7) is the single-molecule, static

Coulomb energy of the permanent charge distribution of molecule B in absence of A. The
operator 50 AB generates the corresponding terms that depend on the static Coulomb
energy of A. Thus the induction energy depends on the difference between the

dielectrically screened and unscreened interactions of the permanent charge distributions

within each molecule.

37

2.2 Dielectric screening and the dispersion energy

The dispersion energy AEd results from spontaneous, quantum-mechanical

fluctuations of the charge density in each of the interacting molecules; these give rise to
fluctuating fields acting on the neighboring molecules, polarizing it, and thus producing a
dynamic reaction field, which acts on the original field source, shifting its energy. To
derive the dispersion energy within the dielectric framework, we start from a standard

expression for AEd , which is obtained both from time-independent perturbation theory

and from reaction-field theory,2‘201

—1

 

AEd = — (h / 27t)_2 (47rso )—2 £0 dco Idr dr' dr' dr" x (r, r'; 1(1)) lr' — r"

—1

X xB(r",r"';iw)lr" — r| (2.2.1)

From Eq. (2.1.4) and integration by parts, Eq. (2.2.1) is equivalent to an expression for

[Ed in terms of the nonlocal polarizability densities of A and B,129
AEd = — (h / 21:) go do) Idr dr'dr' dr'" aaB (r, r';ico) ToY (r', r"
xaB (r" r"'i(0)T (r" r) (2 2 2)
Y5 a 9 6a 9 9 ' '
where the tensor T(r,r') is the dipole propagator,

Tao (r, r') = (47t80)—1Va Vo |r — r'|_1. (2.2.3)

For molecules A and B interacting at long range, Eq. (2.2.2) reduces to the well-known

188
form,

AEd = —(h/21t)TB.Y(R)T5a(R) If doo0AB(ior)oB5(ior), (2.2.4)

38

to leading order in R", where R is the vector from A to B and R = I R | . However, Eqs.

(2.2.1) and (2.2.2) also include the effects of higher-multipole fluctuation correlations
(beyond the dipole); so these equations give accurate results for the dispersion energies of
nonoverlapping molecules, when the molecules can not be adequately approximated as
point-polarizable multipoles - e. g. for large molecules in configurations such that typical
intramolecular distances may exceed the shortest intermolecular distances.

In order to show the connection to charge-density fluctuations at real frequencies
explicitly, we reverse the steps of the derivation by Linder and Rabenold (Ref. 2), Eqs.

(61) - (\67). We assume that the temperature is sufficiently low that the susceptibility

densities xA (r, r';ico) and x3 (r',r"';i(0) change little over an interval of A0) = 27tikT/h
on the imaginary axis; then the integral in Eq. (2.2.1) can be approximated by the discrete

sum,

w!
AEd = — kT (41rso)-2 Z Idr dr’ dr' dr" x (r, r'; 21rinkT/h)
n=0

—1

 

r'—r'

 

x 1130",r"';27rinkT/h)lr""-r1—1 , (2.2-5)

where the prime on the summation indicates that the n = 0 term is multiplied by 1/2.

Equivalently,

—1
r'—r"

AE =(ih/41I)(47t8 )‘2 do) drdr'dr"dr'"xA(r,r';co)
d 0

 

 

xxB(r",r";co)|r"—r|—1coth(h(D/2kT), (2.2.6)

as shown by evaluating the frequency integral in Eq. (2.2.6) in the upper co half-plane,
around a closed contour C that runs along the real 60 axis from o) = - W to co = - s (s > 0),

clockwise around the small semicircle c0 = s exp(i0) from 0 = 7: to 0 = 0, along the real

39

axis from (0 = s to (.0 = W, and then counterclockwise around the large semicircle o) = W

exp(i0) from 0 = 0 to 0 = it. In the limit as W -—> 00, the integral around the large

semicircle vanishes, since both xA (r, r';o)) and xB (r',r"';co) fall off as (0—2 for large (0.

The poles of the susceptibilities are located in the lower complex half-plane, by causality.
Therefore, the only poles within the contour C are those of the hyperbolic cotangent
function, and Eq. (2.2.6) is equivalent to Eq. (2.2.5) by the residue theorem.

The susceptibilities have both real and imaginary parts, denoted by x' and x" ,
x(r,r';o)) = x'(r,r';m)+ix'(r,r’;(o). (2.2.7)
The real part x'(r,r';or) is an even function of frequency, while the imaginary part

x"(r,r';a)) is odd in 0). Since coth(hco/2kT) is odd in 00, Eq. (2.2.6) is also equivalent to

AEd = -(h / 410(4480)‘2 (1 +59A13) 1: do) [or dr' dr" dr" xA" (r, r'; (1))lr' _ .-~ ‘1

 

x XB (r", r"; (0) Ir" - r|_1 coth(hco/2kT) . (2.2.8)

As above, 50 AB perrnutes the labels A and B.

l37,l38,2

By the fluctuation-dissipation theorem, the imaginary part of the charge-

density susceptibility is related to the spectrum of charge-density fluctuations by

— (h / 4n) xA" (r, r'; or) coth(ho)/2kT)

= (1 / 2n) 1:” d(t —t') exp[—io(t-t)]<5pA(r,t)6pA(r',t')> , (2.2.9)
00 +
where
<8pA(r,t)8pA(r',t')> s <5pA(r, t) opA(r', t')+opA(r',t') opA (r, t)>. (2.2.10)
+
Therefore,

40

AEd = (1/8r)(4rtr.0)‘2 (1+ goAB) [:0 do jdr dr'dr'dr" Eda — t')exp[—io(t — t')]

"'1 XB (r", rm; (1)) lrm _ rl‘l

 

x <8pA (r, t) SpA (r', t')> |r' — r”
+

—2 I II M I A A II
=1/4 4 1 d d d d dt—t 5 ,t5 ,t
()(7580) (+60AB)jrrrrE:( )(ptr)p(r)>+

I

x [r —r"‘1xB(r',r";t-t') “‘1. (2.2.11)

 

 

r" — r|
Next, we use the relation between x(r, r';t - t') and 8;} (r, r';t - t’) in the time domain,
‘1 x(r",r';t-t'), (2.2.12)

so 8;,1 (r, r';t — t') = S(r — r') 6(t — t') + (47rso)_l Idr" r — r”

 

 

along with the Born symmetry [xB(r,r';t — t') = x3 (r',r;t — t')] , to obtain
AEd = (1 / 4)(4rtso)‘1 (1 + goAB) Idr dr'dr" [:0 d(t - t’) <opA (r, t) opA (r', t')>
+

—1

I H

><|r —r

 

so 833 (r, r’;t — t')

1‘1 . (2.2.13)

_(1 / 4) (41t80)—1 (1+50 AB) jdrdr'<opA(r,t)opA(r',t')> |r—r
+
The first term shown explicitly in Eq. (2.2.13) gives the Coulomb energy associated with

interactions of the fluctuating charge densities 5pA (r,t) and SpA (r',t’) , in the presence

of molecule B, which acts as a dielectric medium with the nonlocal screening function
333 (r, r",t— t') , introduced by Jenkins and Hunt.1 A charge-density fluctuation at r',t'
sets up a potential at r', t' (in the Coulomb gauge, with retardation neglected). Molecule
B gives a screened potential at r,t , via 83,3 (r, r",t—t') , and the screened potential

affects the energy of a charge-density fluctuation at r,t. The response is integrated over

41

all “time lags” t— t' , but s;1B(r,r',t —t') = 0 if t— t'<0 , so the response is causal. The

second term in Eq. (2.2.13) gives the Coulomb energy of associated with the charge-
density fluctuations in molecule A, in the absence of molecule B. The operator 50 AB
generates the corresponding term, in which molecule A acts as a dielectric medium for
fluctuating charge interactions in B.

The Coulomb energy of interaction between the charge-density fluctuations in
the same molecule equals the intramolecular exchange-correlation energy in density
functional theory (after the self-energy has been removed). With Eq. (2.2.13), this implies
that - for molecules with weak or negligible charge overlap — the dispersion energy is
equal to the screening-induced change in the intramolecular exchange-correlation energy,

summed for the two molecules.

A different dielectric function sd (r, r';i00) at imaginary frequencies183 is directly
related to the correlation of the polarization fluctuations at r and r'. On intramolecular
scale, sd (r, r';60) is distinct from sv(r,r';(o). Thus, sv(r,r';io)) does not relate directly
to the charge density fluctuations. However, Eq. (2.2.13) proves that sv(r,r';t —t') and
hence sv (r, r';(o) is directly related to the screening of the correlations of the

intramolecular charge density fluctuations.
Since the correlations between the permanent charge density and the fluctuating

charge density vanish for each molecule, there is no net Coulomb energy associated with

the interactions between po (r) and 8p(r’, t) . Hence, in the region of negligible overlap,

Eqs. (2.1.7) and (2.2.13) give the energy to second order, in a dielectric framework; and

the results are consistent with quantum perturbation theory.

42

Chapter 3: Dielectric screening of second-order induction and dispersion forces on

nuclei of interacting molecules

In this chapter, we express the second-order induction and dispersion forces on
the nuclei of interacting molecules within the dielectric framework. The force on nucleus

K in molecule A is determined by the negative gradient of the interaction energy of the

molecule with respect to the coordinate RK of nucleus K.
FK =—VK AE, (3.1)

where VK denotes differentiation with respect to RK. In the following sections we use
Eq. (3.1) to derive the induction and dispersion forces on the nuclei. Throughout the
derivations, we use the Bom—Oppenheimer approximation: The forces on the nuclei are
determined as functions of the nuclear coordinates, fixed within individual calculations

but not restricted to the equilibrium configuration.

3.1 Dielectric screening and the second-order induction forces on nuclei

The induction force Filrid on nucleus K in molecule A is given by the derivative

of the induction energy with respect to coordinate RK of nucleus K,

FIE = —VK AEind- From Eq. (2.1.1) and the Born symmetry280 of the polarizability
density 01A (r r" 0) = 01A (r' r'O) we obtain
GB 9 9 GB 9 9 9

FiEd = 1/ 2 Idr dr'aaaAB (r, r';0) / 6RK 3861 (r) 8013“,)

43

+ jdrdr’aaB(r,r';0)630Aa(r)laRK sgBu'). (3.1.1)

The derivative of the polarizability density of molecule A with respect to the nuclear

coordinate depends on the nonlocal hyperpolarizability density B(r,r',r";co,0) of A,127’128

ado/AB (r, r';o)/aR1Y< = zK Idr'Bé‘BSU, r',r";(t), 0) T57 (r", RK), (3.1.2)

where T(r”,RK) is the dipole propagator defined in Eq. (2.2.3) and the

hYPCIPOIarizability density is given byl30,134

Momma)133(r")G<w')tp(n-')

0)

 

 

Baoy(r,r',r';or',(0") = Soy(r’,r",(o',(o")[<0

0)

+ <01 Br (r")G*(—o") 13B) (r")G(—coo )1“), (r')

 

+ <01 Br (r')G*(-o") 1'38 (r)G(w')Po(r’)

 

0).
(3.1.3)
The polarization operator P(r) satisfies Eq. (2.1.3). In Eq. (3.1.3), the operator
Soy (r',r",co',o)') denotes the sum of terms obtained by permuting Po(r’) and P7 (r') ,
together with there associated frequencies 60’ and 60" in the expression that follows the
operator, 006 = 60' + (0" 9 and P8 (r) = Pa (r) — <0l Pa (r) '0) .

The derivative of the field due to molecule A, taken with respect to a nuclear

coordinate in molecule A, is given by

— r r 7—3 I
6361a(r)/6RB<=(41rso) 1jdr(r(,—r(,)r—r 6p8A(r)/6RII3(

 

 

44

r—RK

—3
l , (3.1.4)

+zK(4rrs0)"1 VBK (ra — RaK)

 

and the derivative of the electronic charge density with respect to the position of nucleus

K depends on the charge-density susceptibility of molecule A,

_1 1
r—RKI . (3.1.5)

I

p3_A(r)/aRK — ZK(471:80)

    

         

Eq. (3.1.5) is equivalent (after integration by parts) to the relation derived by Hunt'31
between the nonlocal polarizability density 01(r,r';0) and the change in the electronic

polarization when a nucleus shifts infinitesimally within a molecule.

Thus, from Eqs. (3.1.1), (3.1.2), (3.1.4), and (3.1.5), we obtain

                  

 

 

Filrido" — (1/2)zK Idrdr'dr'B&6(r,r',r";0,0)SB B(r)30BY (r'")Toa(r ,RK)
+ZK(41rso)—2 Idrdr’dr'dr'a -— ”‘3
xx A(r",r"' ,0)vKr —RK|1 30 (r')
r
+ZK(41[8 )‘1jdrdroB(r,r;0)3A'(r)vK( —R )r-RK_3
0 Br 07 'B B)

 

(3.1.6)

K

Next, we show that Find a

in Eq. (3.1.6) is equivalent to the force on nucleus K

calculated with a dielectric screening model and the static - but perturbed — charge

distribution of molecule B.

45

K

First we write Find,“ from Eq. (3.1.6) in terms of APlBa(r,0), the induced

change in the static polarization of molecule B at first order, due to the field from the

permanent charge distribution of A. APlBa (r, 0) is given by
APlBa(r-,0) = 1dr'oaB(r,r';0):sOAB(r').
Then from Eqs. (2.2.3), (3.1.6), (3.1.7) and

V& V3 r"—RK r"-RK ,

[—1

—l
= —v§ vs 1

 

 

we obtain

rigid,“ = _(1/2)zK (4mO)_1 jdrdr'drquAYB(r,r',r";0,0)sBBu)s(Byu')

—l
r"—-RK‘

XVCIf V75

 

— zK (4rtr.0)‘2 [or dr'dr" APlBa(r, 0) VB |r — r" ‘1

 

r"'—RK

-1
xxA(r'. ram 73‘ I

 

—1
-ZK(41t80)-1 jdrAPIBB(r,0)V§ Vo r—RKI .

 

(3.1.7)

(3.1.8)

(3.1.9)

The potential acting on molecule A is the sum of the potential chB (r) due to the

unperturbed charge distribution of B and the potential Act)? (r) due to the change in the

charge density of B, caused by its interaction with A. From Eq. (3.1.9), the force

46

K
Find,01

depends quadratically on (WI)3 (r) , since 353 (r) = —V(p(I)3 (r). The potential Ami3 (r, 0) due
to the first-order change in the polarization of B is given by

r—r'

 

 

A(plB (r, 0) = (4trr.0)‘1 1dr' v2, ’1 APlBa (r', 0), (3.1.10)

since Ap?(r',0) is related to AP1B(r',0) by the same relation that connects the

corresponding operators [Eq. (2.1.3)]. Then from Eqs. (3.1.9), (3.1.10), and repeated use
of the divergence theorem, we obtain

FK

ind,“ = (1 / 2) 2K (4rtr.o)‘1 1dr dr’dr”[Vo V? vs B11116“ r', r";0, 0)]

—1
x (pg (r) (pa3 (r') V§ r" — RK‘

 

—1
—2K (41:.s0)‘1 jdrdr'x (r,r';0)VaK r—RKI Ao1B(r',0)

 

—zK V§[Ao1B(RK,0)]. (3.1.11)

The Born symmetry of xA (r,r';0) with respect to an interchange of its arguments has

been used in deriving Eq. (3.1.11).
From Eqs. (1.9) and (3.1.3), the B-hyperpolarizability density is related to the quadratic

charge-density susceptibility C(r, r’,r";0,0) by
V13 V'y VS B375 (r, r', r") = — C(r,r',r";0,0). (3.1.12)
With this result and a relabeling of the variables of integration, Eq. (3.1.11) becomes

Fill?” = —(1/2)zK (4rtr.0)‘1 jdrdr'dr"gA(r',r",r";0,0)

r n n —1
xtpg(r)(pg(r )6/6ra|r -r| r=RK

47

—ZK(47tso)_1Idr'dr"x (r”,r';0)6/6ra

 

" -1 I
r—rI A(pF(r,0) r RK

—zK V§[AoB(RK,0)]. (3.1.13)

From Eqs. (1.2), (1.10), and (3.1.13), the induction force on nucleus K in molecule A is
given by

K
I:ind,01

= — ZK so 6/6ra[Idr's;91A(r,r';0)A(p?(r',0)] I r : RK
— (1/2)ZKs a/ar [Idr'dr's_l (r r' r” 0 0)
O a qu 9 9 9 9

x oB(r';0)oOB(r';0)] (3.1.14)

 

r=RK'

Eq. (3.1.14) shows that the induction force on a nucleus in molecule A results from

dielectric screening of the potential from molecules B; the first-order change in the
potential A(‘)? (r',0) is screened linearly within A, while the unperturbed potential is

. . K
screened quadratically to give Find, 01'

As shown earlier by Jenkins and Hunt,l linear
screening within A of the unperturbed potential (pg)3 (r';0) from B gives the force derived
from the electrostatic potential energy.

The effective potential (Péfi (r, 0) within A due to B is sum of terms of first and
second order in the A-B interaction, given by

wfiefi(r90) = so Idr’ 83,21 (1', r';0)<p(1)3(r', 0), (3.1.15)

and

48

(pg: eff (r, 0) = so Idr' 8:11, A (r, r'; 0) A(pf3 (r', 0)

+(1/ 2) so Idr' dr" 3211A (r, r', r”; 0, 0) (19(1)3 (r', 0) (03 (r”, 0) , (3.1.16)
respectively. From Eqs. (3.1.14) and (3.1.16),

K __ K A
Find,a——Z a/ar“[‘p2aeff(r’o)]Ir=RK’ (3.1.17)

Hence we conclude that the dielectric screening model gives the second-order induction
forces on each of the nuclei in a pair of interacting molecules, consistent with the results

from perturbation theory.

49

3.2 Dielectric screening and the second order dispersion forces on nuclei

In this section, we derive the dispersion forces on the nuclei of interacting
molecules within the dielectric model. We explicitly show the new field-induced
fluctuation correlations which appear in the dispersion force and we explain the physical

origin of the terms present in the dispersion force using perturbation theory.

The dispersion force Fclf on nucleus K (with charge ZK) in molecule A is

derived from the dispersion energy AEd: F? = -VK AEd. From Eq. (2.2.8) for AEd,

-1

F}? = (h/ 47:) (4r:so )_2 VK £000 d(o Idr dr'dr' dr'x '(r,r’;or) r'—r"

 

 

x XB" (r", r'"; (1)) Ir" — rI_1 coth(ho)/2kT)

—1

 

+ (h / 47:) (41:80 )_2 VK E; d0) Idr dr' dr" dr'" xA" (r, r’; or) r' — r"

 

x xB' (r', r"; 0)) Ir" — rI—1 coth(hco/2kT). (3.2. 1)

Below, the first and the second terms in Eq. (3.2.1) are designated by F31?” and FBIZ) ,

. . . . . . . . 2,137
respectively, for convemence 1n the analysrs. From fluctuatron-d1331patlon theorem,

Fcll((1) = —(1 / 81:) (47:60 )_2 Elder Idr dr'dr" dr‘" VK xA (r,r';(0)Ir — r"'I_1

—1
r'—r" .

x foo d(t — t') exp[—i 60 (t — t') ] <8pB (r", t) 8pB(r'", t')>+

 

 

(3.2.2)

The derivative of the real part of xA (r, r';:o) with respect to RK is connected to the real

part of quadratic charge-density susceptibility via the relation

axA' (r, r'; o) / aRK = -zK (47:.s0)‘1

50

. -3
r1v _ RKI

x Idriv Re CA (r, r',riv;m, O) (RK —riv)

 

(3.2.3)

Eq. (3.2.3) follows from the contraction of Eq. (3.1.2) with Va VB, and integration by

parts. From Eqs. (3.2.2) and (3.2.3), with 8o(r,t) used to denote the potential acting on

A, due to the fluctuations 6pB(r,t) in the charge density of molecule B (and neglecting

retardation effects),

K _ K -1 , . A
Fd(l)—(1/87:)Z (471280) foodo Idrdr dr Reg (r,r,r,o,0)

—3
x(RK —r") r"—RKI

 

x E; d(t — t') exp[ — i a) (t — t') ] <8th (r', t) SoB (r, t')>+ . (32.4)

The quantum mechanical average of S(pB(r',t) vanishes; however, the average of the

product 8oB(r',t) SoB (r,t') is nonvanishing, because the charge-density fluctuations that

give rise to the potentials are correlated.

The Fourier transform of the correlation function <5ch (r',t) SoB (r,t')> in Eq.
+

(3.2.4)is
B r r B n _ r B r B - r . n
<5(p (r ,(o)5(p (r,o) )>+ — foodt Eodt <6q> (r,t)6:p (r,t')>+ exp(ro)t+1o) t').

(3.2.5)

Since <8¢B (r',t) SoB (r,t’)> is a function only of the time interval t— t' , after changing
+

the variables of integration in Eq. (3.2.5) to t— t' and t' , we obtain

51

B I I B II _ _ I B I _ I B I:
<5cp (r,o))6(p (no) )>+ — food“ t') Eodt <6cp (r,t t)5(p (r,t 0)>+
xexp[i(o'(t—t')+i((o'+ 60")t']

= 27: <8<pB(r',o)') SoB (r,t'=0)> 8(or'+(o'). (3.2.6)
+

In Eq. (3.2.4), we express <6qu (r',t) 5(pB(r,t')> as the inverse Fourier transform of its
+

Fourier transform, to obtain

Ffa) = (1 / 87:) (l / 27:)2 ZK (47:so )-1 1:0 do) Idr dr' dr' Re CA (r, r', r"; (o, 0)

-—3
x(RK -r") r"-RKI

 

x fwda—t') exp[-iw(t-t')]

x E; dd [:0 do)" <8th (r', 00') 6(0B(r, 00">+ exp(— i 01' t — i 0)” t') , (3.2.7)

which is identical to

F<Ii((1) = (1 / 87:) (1 / 2702 zK (mm-1
x £060 do) E; d(o' foo do)" Idr dr' dr' Re CA (r, r', r'; (0, O)

-3
x(RK —r") r"—RKI

 

x 1: d(t—t')eXP[-i(w+co') (t-t3leXPl-i(w'+w") t'1

x <6(pB (r', (0') MB (r, (0")>+ (3.2.8)

Next, we evaluate the t — t' integral:

52

Fle) = (1 / 81:)(1/27t) 2K (47:80)_1

x E; do) [:0 do)’ E000 do)" [dr' dr' dr' Re CA (r, r', 1'"; 03,0)

-3
x(RK -—r") r"—RK‘

 

x 8(a) + 03') exp [- i ((1)’ + (0") t' ] <8q>B (r', (0') S(pB (r, (1)")>+ . (3.2.9)

After evaluating the co' integral, use Eq. (3.2.6) for <6<pB (r',-0)) 8(pB(r,o)")> , to obtain
+

F31?” = (1 /81t) zK (mm—1

x E; dco ED do)” Idr dr' dr' Re CA (r, r', r"; o), 0)

—3
x(RK —r") r'—RKI

 

x exp[—i (0)” —w) t' ] <5ch(r', —o)) 5¢B(r, t' =0)> 8(0)" —co) . (3.2.10)
+

Then we evaluate the a)" integral, which gives

F3140) = (1 /81t) zK (47mo)_1

x fwdm Idrdr'dr” Re§A(r,r',r";w,0)

—3
x(RK —r") r'— RKI <6<pB(r',—(o) Sch(r,t'=0)> (3.2.11)

 

+.
The real part of the quadratic charge-density susceptibility, Re CA (r, r', r';a), 0) , has the

permutation symmetry,

53

Re CA (r, r', r'; 0), 0) = Re CA (r', r, r'; —(o, (1)) , (3.2.12)

and it is even in a); so

Re §A(r,r',r';(o,0) = Re §A(r', r, r'; 0), —a)). (3.2.13)
Therefore
FED = (1 / 81:)ZK (mm-1

x fooda) jdrdr'dr'RegA(r",r,r';m.—m)

-3
><(RK —r") r' — RK| <8ch(r',—(o)5(pB(r,t’=O)> . (3.2.14)

+

 

We insert an integration over (0' , using a delta function,

FED = (1 /87t)ZK (mm-1

x foo dco foo dw' Idr dr' dr' Re CA (r", r, r'; (10', —o)) 6((o'-o))

 

—3
x(RK -r") r"-RK| <5ch(r',—w)8(pB(r,t'=O)> . (3.2.15)
+
From Eq. (3.2.6),
2n <6<pB(r',—o)) 6(pB(r,t'=O)> 5(0)’—(o) = <6<pB(r',-(o)6<pB(r,co')> , (3.2.16)
+ +

from which we obtain

FK

d(l) =(1/167t2)ZK(41t80)—1

x foodo) Eodm' Idrdr'dr' Re CA (r',r,r';o)',—(o)

54

 

-3
x (RK — r") r" — RKI <8<pB (r, (0') 5<pB (r', —(o)> , (3.2.17)
+
or equivalently,
FK = (1 /161t2)ZK (mm—1

d(l)

x I: do) [:0 do)’ Idr dr' dr" Re CA (1", r, r'; 03’, (o)

 

x ( VK r' — RK [—3 )<8<pB(r,(o')5(pB(r',m)>+. (3.2.18)
In a form that makes the dielectric screening interpretation clear,
F360) = —(1 / 161t2) ZK 20 V” [:0 do) Eoda)’ Idr dr' dr' 8211’ A (r', r, r'; a), co')
x<5<pB(r,o)) ZSch(r',0)')>Jr r' = RK . (3.2.19)

 

Eq. (3.2.19) shows that the first term in the dispersion force on nucleus K in molecule A
results from the nonlinear dielectric screening of the correlated fluctuations in potential
due to molecule B. This component of the force is analogous to the component of
induction force that results from quadratic screening of the static potential due to
molecule B.

The remaining component of the dispersion force on a nucleus in molecule A is given by

the second term in Eq. (3.2.1), denoted by Fdé2) , with

—1

 

= (h / 4112)(47t80)_2 VK foo do) Idr dr’ dr' dr'" xA" (r, r'; w) r' — r"

 

K
Fd(2)

xxB'(r”,r";a))|r"'—r|—1coth(hw/2kT). (3.2.20)

55

VKXA”

In appendix A, we derive a relation that connects (r, r' ,w) to the 1maginary part

of the quadratic charge-density susceptibility, CA"(r, r', r'; (o, 0) ,

vK xA"(r,r';0~)) = (47:80)-1 Jldrn 2K

     

—1 II
r"—RK| §A(r,r',r',m,0), (3.2.21)

To show the dielectric screening present in F31?” , we take Eq. (3.2.21) and separate the

terms with n = j and n ¢ j. We obtain

     

           

         

         

1211(2): (it/h 2)(h/47t)(47t80)_3 jdrdr'dr'dr"dr“’ 2K r"—RKl
x12 2 < ‘ ">j><j|13(r)|n><n )0>w355(wno—w)
n¢0j¢0,j¢n
+2 2 (0|P(")|J >J<|f>(r")ln><n )0)
n¢0j¢0d¢n

xRe [(193%) —ir- /2—m)‘1]6(mn0 —(o)

J

         

         

         

         

+2 2 <0Ip<r>ln<n> '>j><1‘">0>w3:,a<wno-m>
n¢0j¢0j¢n

+2 2 (OIPWIHX ">110" 30>

n¢0j¢0j¢n

x Re [(033%) -iF ° / 2 — (0)4 ]5(o)n0 — (o)

         

         

J
-2 Z <Olf><r'>ln ><n Ip<r>ljf>< ">0>w3})6<wno+m>
n¢0j¢0,j¢n
-Z Z <0|fi(r'>ln><n ")J><J|fi(r>|0>
n¢0j¢0,j¢n

56

xRe[(m3%) +iFj /2+m)’1]5(mn0 + (1))

           

-ZZ<

n¢0 j¢0,i¢n

-Z Z <0Ip<r>l>< ><jlp<r'>ln><nlé<r>I0>

>j>j<|p<r'n)| ><n n|é<r>|0)w3},6(cono+w)

           

         

n¢0 j¢0,j¢n
x Re [(033%) + iFj /2 + m)—1]8(mn0 + (n)
+ Z < ><n|13(r)- poo (r)| nn)( " ') 0) (0,310 S(wno -m)
n¢0

)0>(Dnlo ammo -co)

+2 (0|13<r>|nn>< |P(r'

n¢0

)nn><

         

         

n) (n | p(r')-p00(r') l n) (n | (S(r) | O>co;110 8((on0 + w)

-Z<“')

n¢0

           

O) (1)510 S(wno + (o)

         

' Z (0113031“><n15(r)-Poo(r)ln)<n “ ")

n¢0

) n)<fl|13(r'>|0>

             

+ Z (015(r>|n><n|13(r' -

     

           

n¢0
x Re [(cono —w)'1] S(mno — co)
+ 2 < A ’) n><n|f>("')-Poo ") n><nlf>0°>|0>
n¢0

>< Re [(00110 —co)"1 6(wno —w)

)n><n “ '> 0>

         

             

- Z <0|fi(r>|n><nlfi(r' -

n¢0

x Re [(cono + (0)4] 5(0),,0 + 01))

57

 

- Z <0|fi(r’)|n><n|13(r")-Poo(r') n><nl13(r>|0>

n¢0
x Re 1(wno +w)"1 S(wno +co)}

. . —1
xIr—r"|‘1xB(r",r",m) r" —r' coth(h(o/2kT). (3.2.22)

 

 

The first eight terms in Eq. (3.2.22) can be described as correlation between the charge-
density fluctuations and the susceptibility fluctuations in molecule A. To show this

correlation explicitly, we define a transition susceptibility of molecule A,
1300', (01;r', (1)2) following first-order transition hyperpolarizability defined by Hanna,

Yuratich, and Cotter in Eq. (2.19) of Ref. (281). In particular, we need only the real part
of the transition susceptibility; assuming for simplicity that the states of A are real, we

have

é<r>lj><jl13(r')l0>Re(coJ-o—iFj/2—m2)"

 

Rexn0(r,co1;r',(o2) =(1/h) Z[(n
j¢0

+ (n|p(r')|j)<j|p(r)|0)Re(coj0 —iI“j /2—m1)'1]. (3.2.23)

From the definition of the transition hyperpolarizability in Eq. (3.2.23), we introduce the

transition susceptibility of molecule A as

onn<wr€0>= Z Homo-">1j><jlé<r>ln>w3$
j¢OJ¢n

+<O|p(r)|j><j|p(r")|n>Re(o)j0-iFj/2—w)'1]. (3.2.24)
Thus, the first eight terms in Eq. (3.2.22) give the correlation between a charge-density

fluctuation at r and a susceptibility fluctuation at r' (and vice versa), within molecule A.

To illustrate the quantum mechanical nature of this fluctuation, we consider the

58

correlation between a susceptibility fluctuation at r with frequency —(0 and a charge-
density fluctuation at r' with frequency (1). In the limit T—+0, the fluctuation-correlation
is given by

(1 / 2)( 6x(r', 0;r. — (0)5962 co) >+

=(1/2)< 6x(r", O;r, —m)5p(r', 0)) + apl (r', (0)5)(1' (r, — co;r”, 0)> . (3.2.25)
Using the facts that
8xl(r,w;r",0) = 8x(r,—o);r",0) , (3.2.26)
and
5pl(r,m) = 5p(r,—co), (3.2.27)
we obtain

(1 / 2)< 5x(r"‘,0;r, -o)) 6p(r', (1)) )+
= (1 / 2) [ mu (1", 0; r, —w) pno (r') 5(wno + w) + 9011 (r') xno (r, w; r", 0) 5(w0n + (0)]
= (1 / 2) “Mn (r', O; r, —co) PnO (r') S(wno + (o) + POn (r') XnO (r, (o; r", O) 5(w-mn0)]

= (l / 2) ”On (r', O; r, —m) PnO (r') S(mno + w) + POn (r') XnO (r, a); r", 0) S(mno — 03)] .

(3.2.28)

From Eq. (3.2.28), we can describe the first eight terms in Fd%2) (which we denote as

Fézm below, for convenience) as
. —1
1&2” = (1 / 4) (mm—3 Eodco jdr dr' dr'dr’" drdr'v zK VK r" — RKI

 

x [( 8x(r",0;r, —(1)) 5p(r', (a) )3 +< 5x(r',0;r', —(1)) 8p(r, (1)) >3 ]

59

. . —l
xlr—r"|—1xB(r",rw;w) er —r'

 

(3.2.29)

 

Eq. (3.2.29) can also be represented as

F(Ii<(2),1 = (1 / 4) (4ne0 )_3 E; dco' f; d0) Idr dr' dr" dr‘" drdrIv

—1
sz VK r'-—RKl 8((o+o)')

 

x[( 5x(r",0;r, (0') 5p(r', 03) >5 +< 8x(r",0;r', (0') 8p(r, (1)) >3 ]

. . —l
xlr —r'"|_1xB(r",rw;o3) rlv —r' (3.2.30)

 

 

Eq. (3.2.30) proves the fact that when a nucleus shifts in the molecule, the change in the
nuclear Coulomb field due to the position shift brings in new correlations within the
molecule. When nucleus K in molecule A shifts infinitesimally, it changes the static

Coulomb field, given by
30.6") .—. (4ne0)‘1 2K Tag (r”,RK)5R[I3< . (3.2.31)

Previously, Liang and Hunt'32 noted that the change in the nuclear Coulomb field may

introduce new types of fluctuation correlations in the molecules, as well as altering the
magnitude of the correlations. Eq. (3.2.29) establishes the fact that the shift in the
position of nucleus K does bring new fluctuation correlations within the molecule,
namely the correlation between charge-density fluctuation and susceptibility fluctuation.
In absence of any external field, the charge-density fluctuations are correlated by the
imaginary part of the charge-density susceptibility, x"(r,r',(o). The nuclear Coulomb

field alters that fluctuation, brings in new fluctuations, and also introduces a new

60

correlation function, §"(r,r',r';co,0). From Eqs. (3.2.21), (3.2.22), and (3.2.30), we can

relate the correlation between the charge-density fluctuation and the susceptibility

fluctuation as

(1 / 2)[< 8x(r',0;r,w') 5p(r',w) >3 +< 5x(r",0;r,co') 5p(r, (0) >2 ]

= (h / 211:) 63:11 0(r,r’,r";co,0) 8((o+ (0') coth(ho)/2kT), (3.2.32)
where 63:21 O(r,r",r";(1),0) means that only the terms with j i n,0 of CA. (r, r', r';tn, 0)

determine the correlation.

Next, we show the dielectric screening present in Fcll((2)1' The spontaneous

charge-density fluctuation in molecule A gives rise to a perturbing potential that acts on

molecule B, shifting the charge density in B and therefore producing a fluctuating

reaction potential 5680, m) that acts on A:

 

 

 

 

8ch (r, (n) = (47:30)_2 Idr.’ dr' dr'" |r — r' —1 x3 (r', r"; w) r" — r"'|—1 5pA (r'", (1)).
(3.2.33)
Using Eq. (3.2.32), we can simplify Eq. (3.2.29) to
—1
Fd((2),1 = (1 / 4) (411280 )—1 foo do) Eodm’ Idr dr" ZK VK r” — RK|
x 6(0) + co')< SxA (r", O; r, (0') S(pB (r, m)>
+
—1
—1 I I a K K n K
+(l/4)(41t80) foodm Eodco Idr dr Z V r —R l
x 5(0) + co')< SXA (r", 0; r', (0') S(pB (r', m)> . (3.2.34)
+

61

The fluctuations in the susceptibility SXA (r”, 0; r, to) correspond to fluctuations in the
dielectric function 68;1A(r”,0;r,co), with the same relation as in Eq. (1.2); thus we

obtain

FdK(2)1 — —(1/4)a/ar' Eldon) Eodco' jdr"

x 50<83 3',"A(r ,O;r, (0') S(pB (r, (o)>

r"_15((o+co')

     

 

 

+r.=RK

—(1/4)a/ar' Eodo) 2K < 8<pB(r',u))> (3.2.35)

 

+ r! = RK
Thus, FcIl((2)1 in the second component of the dispersion force on a nucleus in molecule

A comes from the dielectrically screened dispersion potential, due to the change in the
charge density of B induced by the spontaneous fluctuations in A. In this case, the
average of the dispersion potential from molecule B vanishes; however, the fluctuations
in the dielectric screening function are correlated with the fluctuations that give rise to the
dispersion potential. Hence, the screened field vanishes, but the screening effect does not.

The remaining terms in the second component of the dispersion force can not be
described in the dielectric framework, because they do not stem from the field induced
fluctuation correlations described above. These remaining terms are not related to a

nonlocal response function of molecule A. In the next part we explain the physical origin

of all the terms present in 1736(2) using time-dependent perturbation theory. We use the

fact that a charge-density fluctuation in A at r',t' creates a potential in B at r'v,t' ,

which induces a shift in the charge density in B at r'",t. The induced shift in the charge

62

density in B creates a reaction potential on A at r, t. This reaction potential acts as an
external time-dependent perturbation and perturbs the ground and the excited states of
molecule A. That results in field induced transitions and perturbed transition charge
density in molecule A. Below we show a systematic analysis of these effects and relate
them to the second component of the dispersion force.

The interaction between the charge-density of A, and the reaction potential from

B is given by the Hamiltonian
121(1)“) = jdr p(r) -6(pB(r,t), (3.2.36)

where {n(r) denotes the charge-density operator for molecule A. Using standard time-
dependent perturbation theory, the ground and excited state wave functions of A to first

order in the applied potential are given by

lw1(t)) = In)CXP[-iwn t1

+(1/ih) fwdt' Idr{Z(j|p(r)|n)8(pB(r;t')exp[iwjn t']} |j>exp[-i(oj t] ,
l

(3.2.37)

and

|wr(t)>=l0>eXP[-iwot]

+(1/ih) foodt' Idr{z<j|p(r)|0)5<pB(r;t')exp[icoj0 t']} |j>exp[-i wj t],
1

(3.2.38)

63

respectively. The average value of the charge density at r",t between the perturbed

 

ground and excited states in molecule A is 8p(r",t) = (wf(t)|p(r") wi(t)>+cc , where cc

means the complex conjugate of the expression. From Eqs. (3.2.3 7) and (3.2.38),

6pm» = (Olav)

 

n)exp[i(1)0n t]

+(1/ih) fwdt' jdr{2(o|8(r')
j

 

j><j|13(r)ln)

x 56B(r; t') exp[icojn t']}exp[ iwoj t]

_ (um) foodt' Idr{2(O|p(r)|j)(j|p(r')|n>
j

x 8¢B(r;t')exp[im0j t']}exp[i wjn t]. (3.2.39)

In Eq. (3.2.39), the first term represents the unperturbed charge-density fluctuations

between the ground and the excited states at r’,t. The average of this unperturbed

fluctuation vanishes. The second and the third term are the transition charge densities

between the ground and the excited state, perturbed by the reaction potential. We call
these terms the “first-order transition charge density”, Spgln) (r", t) between the ground

state [0) and the excited state In). Using Eq. (3.2.29) along with a Fourier transform of

the potential to the frequency domain, we obtain

8pgln)(r";t) = (1/2ih) fiat.) foodt' jer(o|,3(r')
l

 

j><j|13(r)ln>

x {8ch (r; (o) exp[i (mjn + (n) t' ] + 863* (r; (1)) exp{ i ((Djn — (o) t' ] } exp[i cooj t]

64

 

—(1/2ih)£:da)foodt'Ier<O|p(r)|j><j|p(r') n)

J

x {5(pB(r; (0) exp[i (woj + 0)) t' ] + S(pB'k (r; (0) exp{ i ((00j — 0)) t’] } exp[ i (Djn t]

= —(1/2h) Eodw jer(Olé(r')|j><le3(r)ln>
J

x{((1)jn +(0)’1exp[i0)t]5(pB(r;(o)

+ ((0 jn - (0)'1 exp [-i o) t] 663* (r; (0)} exp [i wOn t]

+(1/2h) 1:08..) Idrz(o|,s(r)| 001800111)

J

x {((ooj +03)"1 exp[i (0t]8(pB (r; (0)

+ (we) -8)-1 exp 1-i cot] 843*0; m) } expfiwOn 1]
= (-1/2h) fwdmfidco’ jer(0|;3(r')|j)(j|§(r)|n)
j

X{((°jn +(0)'1exp[io)t]6(pB(r;0))

+ ((0jn -0))'1exp[-imt]5(pB*(r;(o)}exp[ i0)’t]8((00n —(1)')

 

+(1/2h)food(0f;d(0' Ier<O|p(r)|j><j| (S(r") n)
j

x{(0)0j+o))'1exp[i(0t ]8(pB(r;0))

+ (cooj — (0)"1 exp [-i (0 t] 8613* (r; (0)} exp[ i (0' t ]6(01)0n — (0') .

(3.2.40)

65

The reaction potential S(pB(r;(0) acting at r in A is due to the shift in the charge density

in B induced by the charge-density fluctuations at r' in A. So

. . . —1
8(pB (r; (0) =(47t80 )-2 Idr'dr" drlv |r — r""|_1 xB (r'",r'v;a)) r'v - r'

 

 

x <n|p(r')|0>8((0-o)no). (3.2.41)
From Eqs. (3.2.40) and (3.2.41), the first order transition charge-density at r',t in

molecule A is given by
1 I! _ I I M .
8.581301): —(1/2h)(41t30) 2 fiodw [:dm jdrdr dr dr"

j><j|p(r)|n>{((0jn +0))_1exp[i((0+0)') t]8((0n0 -0))6((00n —0)')

 

XZ<0lf><r")
j

+((1)jn - m)'1exp [i ((0 — (0') t] 6((0n0 + (0) 8((00n — (0')}

x|r_rIII|—1 XB (rm, riv;m) riV _ r! _ <1] “503)

 

0)

 

 

+ (1/2h) (41:20 )_2 [:0 do) 1:: do)’ Idr dr' dr‘" driv

x Z (0|;3(r) | j) (j| (S(r”) n) { (mm + 0))_1 exp[i ((0 + (0') t] 8(mn0 — (0)8(030n — (0')

J

 

+((1)0j — (0)'lexp [i ( (1) — (0' ) t] 8((0n0 + (o) (5((1)0n — (0' )}

x|r_rM|—l XB (rm, riV;C0) riv _ r! — <n|fi(rl)

 

0) , (3.2.42)

 

 

Eq. (3.2.42) is obtained using the fact that the charge-density operator is not self-adjoint

in the frequency domain. Equivalently

66

8p(1n)(r' ;t)- — —(1/2h) (47:30)—2 Eodm f; dco' Idrdr' dr"’driv

><§jl<

           

><|jp(r)|n n>{((0jn+(0) 1exp[i(()1)+(0')—t]5((0no (0)5((0+(0')

+((0jn - (0)'1exp [i ((0 — (0') t] 8((0n0 + (0) 8(0) - (0')}

(n

xlr_rIII|-1xB(rlI,riv; rlv _rI ) O)

         

     

 

+(1/2h)(4n.e.0)‘2 Eodo) Edd Idrdr'dr’”driv

n>{((00j +(0 )_1exp[i( (0+(0' )t]8((0n0 —c0)5((0+(0')

x2018» 1001 w

j

   

+((00j — (0)'1Exp[i ((0 — (0') t] 8((0n0 + (0) 8((0 - (0')}

(n

rlv _ rI

     

 

) 0). (3.2.43)

         

X lr_rlfl‘_1 XB (rm, riv;

Eq. (3.2.43) shows that although the reaction potential acting on A is time-dependent, the
perturbed first-order charge density in A is time—independent.
From Eq. (3.2.43), we collect the terms with j i n, 0. Then integrating over (0'

and rearranging the terms, we obtain

8p(1n)(r" ;—t) — —[(1/2h)(47r80)_2 l: d(0 jdrdr' dr'dr'v

         

x Z [{(O A "Hill IP(')| “(03-0) +<01P(r)| JXJ l6(r")ln)
j¢0,n

x((0j0 —(0)'1}5((0n0 —(0)

+{< ")J'J'>< |13(r)In 10°35 +<01P(")| >< |P(r')ln>

           

67

+(0|13<r)Ij) (1030") 0(ij + co)'1}5<wno + <01

 

17580"",riv riv —r' p(r') 0)]. (3.2.44)

(11

 

x Ira”!-

 

 

;w)

 

Interchanging r and r' in the terms and adding them to the terms in Eq. (3.244) yield the

terms present in FK . Thus, the first eight terms present in the second component of
d(2),1

the dispersion force arise due to the interaction of the first-order transition charge density

in A with nucleus K. The first order transition charge density at r" is induced by the

reaction potential S(pB (r, t) and is determined by the transition susceptibility of A. From

Eq. (3.2.44), we can write F31?” 1 as

"‘1 (1)
Z 8me (r", t). (3.2.45)

—(1/2)(47:.c.0)“1 [dr'ZK VK IRK —r'
n¢0

 

K _
Fd(2),1 ‘

One thing to note here (and also in the next sections) is that in Eq. (3.2.44), both terms
appear with the same sign, whereas in Eq. (3.2.22) they appear with opposite signs. This
apparent disagreement of sign is due to the fact that in the time-dependent perturbation
formulation, we are considering the charge density fluctuations in the limit of zero
temperature. At a finite temperature, we will have to consider the canonical distribution

of the eigenstates. We will solve discuss this later in this chapter.

Next, we explain the second set of four terms [we denote them by FdEZ) 2]in Eq.

(3.22.2). We use the same formulation as in the previous section, but here we only

consider the terms with j = n in the initial state and j = 0 in the final state (which are

68

considered as secular terms in time-dependent perturbation theory). Integrating over (0' ,

we obtain

500113020 = —(1/2r0(42wo>‘2 1:04P ldrdr'dr"driv [(PIPP'”) 00113009830

 

x { 8(6),,0 - (0) + S(wno + (0)} -(0 P(r') 1‘) <OIP(" )l0> (”I110

 

 

x { 5((0n0 — (0) + S(mno + (0)} ]Ir — r'"I_1 xB(r", riv — (n I p(r')

I

rw —l‘

 

 

0

;m)

 

= _(1/2}‘1)(47:g0)_2 fiodm Idr dr'dr" driv [<0lf3(l’")

 

n)(nl13(r)-Poo(r)ln)

x { 5((0n0 —- (0) + S(wno + (0)} Ir - r"'I—1 1B (r', rIv ; (0)

iv -

xr —r'

 

 

<n|13(r')

 

0) . (3.2.46)

Thus, Eq. (3.2.46) shows that the second set of four terms in Fd((2) are due to the

interaction between the first order transition charge-density and the nucleus, where the

first order transition charge-density is induced by the interaction between the reaction

potential and the difference in the permanent charge densities between excited state In)

and ground state. These four terms can not be explained within the dielectric framework,
because they do arise from the response of molecule A to the reaction potential from B.
In the next part of this section, we explain the last four terms in the second

component of dispersion force. The spontaneous charge-density fluctuation in molecule

A brings it from the ground state to the excited state In). During the time interval t— t' ,

molecule A remains in the excited electronic state In) and that creates a change in the

potential at the nucleus due to the change in the average electronic charge density. When

69

the reaction potential 5pB(r, t) acts back on molecule A, it brings A from exited state
In) to the ground state. The first-order amplitude for this induced transition In) —) I0)

together with the change in the potential at the nucleus determine the last four terms in

F31?” . The change in the potential at the nucleus during the time interval t—t' is given

by

-1
A(pe(RK;t—t') = —(4m;0)‘1 Idr'ZKIRK -r" [<nIp(r')In)—<0Ip(r")I0)].

 

(3.2.47)

From time-dependent perturbation theory, the first order amplitude for the transition

In) —> I0) due to interaction with an external time-dependent Coulomb potential is given

by

c“) (t) = —(1/2h) 1:066 Idrdr'dr'"driv {(0

In)-—>IO) P(rJI n>(w0n + w)"

 

0)(01)()n — (0)-l

 

>< eXP[i(<P0n + 0))t ]<Pex (r, (P) + (0| 130‘) I n>(n I130“)
x exp [i ((00n - (0) t] (sz (r, (0)} . (3.2.48)

Using the reaction potential 8680', (0) in Eq. (3.2.48), we obtain

CIPI->I O>(t) = —(1/2h)(47t80)_2 [lam Idrdr'dr" driv {<0Ip(r)In)<nIp(r')I0)

x ((0011 + (0)'1 exp [i (won + (0) t] 6((0no — (0)

+ (0 I (30') I n) (n I (30") 0)((1)()n — (0)"1 8((0no + (0)}

 

. . —1
X Ir _ I,mI—l XB (rm, l,lv ; (a) r1v _ r'

 

 

70

= (l/2h)(47r80)—2 1:086 Idrdr’dr"driv {<0Ip(r)In)<nIp(r’)I0)((0n0 —6)-1

x 8((0n0 — (0) + (0 I p(r) I n) (n I (30") I 0) (wno + (0)'1 5((0n0 + (0)}

 

 

III—l B III iv iv I—l
xIr—r I x (r ,r ;(o)r —r (3.2.49)
Thus, from Eqs. (3.2.47) and (3.2.49), the last four terms in F3132) are given by
K _ _ e K. _ . (1)
131(2),3 _ (1/2)A(p (R ,t t) Z CIPH O>(t). (3.2.50)

n¢0
These four terms are not connected to the dielectric model, since they do not originate
from the response of A to the perturbing potential.
Eqs. (3.2.45), (3.2.46), and (3.2.50) show the physical significance of the second
component of dispersion force. We have proved the origin of the terms using perturbation
theory. Although the external perturbation is time-dependent, the dispersion force does

not show time-dependent behavior, which is the exact same result obtained using reaction

field theory?"129

Finally in this chapter we discuss the apparent disagreement between the signs of
the terms from perturbation theory and the terms we obtain from the derivative of the
imaginary part of the charge-density susceptibility. The hyperbolic cotangent function in
the dispersion energy appears from the fact that when we consider the charge-density

fluctuations at a finite temperature T, we need to use the canonical distribution of the

138,282

molecular eigenstates. The ratio of the spectrum of the charge-density fluctuations

and the imaginary part of the charge-density susceptibility yields the hyperbolic

cotangent function in the dispersion energy. In the formulation described above, we have

71

used zero-temperature fluctuations. In the limit T -—> 0, coth(h(0/2kT) —> [0((0) -0( - (0)] ,

where 0((0) is the Heaviside step function.278 Thus in the limit T -> 0, Eq. (3.2.22)

simplifies to

 

         

. —1
Fd%2)_ _ (n/hz)(h/4n)(4na0)‘3 Idrdr'dr"dr"drw zK VK r'—RKI
x{ Z Z .. 1') j><jII3(r)In)<n|f)(r')I0)(03I) 8((0n0—(0)
n¢0j¢0,j¢n

+2 2 <o1p<r>1><1p<r~>1n><n11s<r'>1o>

n¢0 j¢0,j¢n

x Re [(003%) — iFj / 2 — 0))_1]8((0n0 — (0)

+2 2 <0|P<r>ln><nl13<r')lj><j‘ ")0

         

)(03I) 8((0n0 — (0)

         

         

           

     

   

       

       

           

n¢0j¢0,j¢n
+2 Z<0|p(r)1n>< ">00” )0)
n¢0j¢0j¢n

x Re[((03I) —iFj /2 -(0)_1]6((0n0 —(0)
-Z Z < 100180111X W>w388<wno+w>
n¢0j¢0,j¢n
-2 2 <0” '>|n><n“ 'JlJ><J'|P(r>lO>
n¢0j¢0,j¢n

xRe[((63I) +il"j /2+(D)_1]5((Dn0 +0))
-2 Z < ‘ ">1><1°113(r')|n><n|13(r)|0>63},S(mno+0)
n¢0j¢0,j¢n

72.

-Z Z <0|P<r’>

n¢0 j¢0,i¢n

 

j)(jI13(r”) n)<nl13(r)l0>

 

xRe[((63I) + i1“ 1- /2 + (0)-1]5((0n0 + (0)

n) (n I p(r)-p00 (r) I n) (n I p(r') I 0) (0510 8((0nO — 0))

+ Z <0113<r">

natO

 

+ Z (011mln><n113<r'>-poo<r'>1n><n1130-") 0104.10 sauna-co)

 

 

 

 

 

 

n¢0
- Z <°|P<r"> n><n1fi<r'>-poo<r)1n>0186101411..56.0+0»
n¢0
- Z <0113<r'>1n><n113<r>-poo(r>1n><n1136-6 06,1058...) +...)
n¢0
+ Z (OIPmIn><nIP(r')-poo(r")|n)(nII3(r') 0)
n¢0
x Re [((DnO ’(Dyll 8(03m) —(0)
+ Z (OIPW) nI<11IP(I"')-po()(r") n)(n|p(r)I0)
n¢0

x Re [(03110 - (6)4] 8(6),,0 -(0)

— Z <0|P(r)|n><nI13(r")-poo(r")

n¢0

 

n><an(r'>|0>

x Re [((0n0 + (6)1] 5(6),,0 +0))

- Z <0|P(r')|n><n|P(r")-Poo(r')|1001130010)

n¢0

x Re [((ono + (0)'1 ] 5((0n0 + (0)

73

—1
[0((0)—0(—(0)]. (3.2.51)

xIr—r"I_1xB(r",riv,w) riv _rI

 

 

Since 0((0) vanishes for (0 < 0, Eq. (3.2.51) yields exactly what we have obtained using

perturbation theory.

Thus, we can write the total dispersion force on nucleus K in molecule A as a

sum of four terms,

K_ K K K K
Fd -Fd(1)+Fd(2),1+Fd(2),2+Fd(2),3. (3.2.52)

Both F31?” and F360” are described within the dielectric framework. FED results from

the nonlinear screening of the fluctuating potentials from B within A, while Fd<(2)1
appear due to linear screening of the fluctuating potential from B, due to the

. . . . . K K . .
susceptlblllty fluctuatlons 1n A. F (1(2) 2 and F d(2) 3 do not have d1electr1c

interpretations, since they are not related to response functions of A and they do not show
any dielectric screening. All of the terms in dispersion force result from the interaction of
the nucleus with an induced shift in the electronic charge density (or change in the

aVerage electronic charge density) of the same molecule. This is known as the Feynman’s
Conjecture about the origin of dispersion forces.105 The first general proof of Feynman’s
Conjecture was given by Hunt,188 where the dispersion force was given as a function of

imaginary frequencies. In this work, we have showed the general proof of Feynman’s

COnjecture in the real frequency domain.

74

Chapter 4: Dielectric screening and the three-body nonadditive interactions at

second order

4.1 Dielectric screening of the three-body induction energy at second order

In this section we show that the nonlocal dielectric model can accurately describe
the three-body induction energy at second order of molecular interaction. In section 2.1,
we have showed that for a pair of interacting molecules, the two-body induction energy
results from changes in the static Coulomb interactions within each molecule, due to the
presence of the second molecule, which acts as the dielectric medium. In the present
chapter, we consider a group of three interacting molecules A-"B-"C with weak or
negligible charge overlap. We prove that At second order, the three-body induction
energy results from the change in the two-body induction energy of a pair of molecules,
due to the presence of a third molecule, which acts as a dielectric medium.

The interaction energy of two test charges is affected by the presence of a
dielectric medium. The shift in the interaction energy of two test charges in presence of a

linear dielectric medium characterized by the potential screening function ev(r, r';0) is

given by

 

AB = Idrdr'dr" p (1‘)P(l'") - Idr dim , (4.1.1)
47t8v(r,r';0) r" —r'I 47:80 Ir-r'I

 

where the first term gives the screened interaction energy due to the presence of the
dielectric medium and the second term gives the direct, unscreened interaction energy of
the two test charges. We prove that Eq. (4.1.1) describes the second-order three-body

induction energy for a group of three interacting molecules, where the permanent charge

75

distributions of two molecules act as the test charges and the third molecule acts as the
dielectric medium with the same nonlocal dielectric function given above.

For a group of interacting molecules AmBmC, the nonadditive three-body
induction energy results as follows: the permanent charge distribution of one molecule
sets up a potential that shifts the charge distribution of the second molecule; in turn, this
produces a potential that acts on the third molecule, creating a non-additive shift in the

energy. Thus the three-body induction energy depends on the static, nonlocal charge-

density susceptibilitiesxA(r,r';0), xB(r,r';0), and xC(r,r';0)of the interacting

molecules:

2,3 I I I
AEInd) = [drdr x (r,t;omgmgm

+ Idrdr'x (r,r';0)<p€(r)<p8(r')
+ Idrdr'x (r,r';0)tpOA(r)(pg(r'). (4.1.2)
Here and below, AEgEn’n) denotes an energy shift in molecule X, of order m in the

interactions among n distinct molecules. In Eq. (4.1.2), (pg)3 (r) denotes the static external

potential acting on A due to the permanent charge distributions in B (and similarly for B
and C). The first term in Eq. (4.1.2) can be interpreted as follows: the static external
potential from C acts at r' in A, creating an induced shift in the charge density at r
within A. The induced shift in the charge density at r then interacts with the static
external potential from B at r, thus producing a net energy shift. Eq. (4.1.2) can be

written for more compactness as

21.8%? = (1+CABC) Idrdr'x (r,r';0)(p(l)3(r)(pg(r'), (4.1.3)

76

where the operator C ABC denotes cyclic permutation of the indices A, B, and C in the

expression that follows.

Expanding the potentials in terms of the permanent charge densities in Eq.
(4.1.3), we obtain
l-l

AEIfiE?) = (1 + C ABC ) (47:80 )_2 Idr dr' dr" dr'" pg (r') r" — r

 

xxA(r,r';0) r' — r"|‘1 pOC(r"'), (4.1.4)

 

where as usual, p; (r) is the permanent charge density at r in molecule X. From Eqs.

(1.1) and (4.1.4), with a change in the labels of the integration variables, it follows that
the second-order three-body induction energy is accurately expressed within the dielectric

model by

2, '- I II - II II I— I
AEInd3) = (1+CABC)(47IEO) 1‘I‘drdr dr pg(r)[eoevl’A(r,r ;0)]Ir —rI 1pOC(r)

— (41:80 )_1 Idr dr' pg (r) Ir - r'I_1 p0C (r')

—(4ue0)‘l Idr dr'p0C(r)Ir — r'I_1 pOA(r')

 

— I I —1 I
—(4ne0) 1[ch-(Ir pOA(r)Ir—r p56), (4.1.5)
where ev,A(r,r’;0) is the static nonlocal dielectric function of A defined in Eq. (1.1).

The first term in Eq. (4.1.5) gives the screened interaction energy due to the Coulomb
interaction between the permanent charge distributions of two molecules, in presence of a

third molecule, which acts as the dielectric medium. For example, the first term in Eq.
(4.1.5) gives the static Coulomb energy of the interaction of p30) and p8 (r') in

presence of molecule A, which acts as the dielectric medium to screen the interaction

77

between the unperturbed charge distributions of B and C. The screening is nonlocal,
since 8;1A (r,r";0) depends both on r and r" . The sum of the last three terms in Eq. (4.1.5)

gives the static Coulomb energy of the unscreened interactions between the unperturbed
charge distributions of the molecular pairs. Thus, the three-body induction energy at
second order depends on the difference between the dielectrically screened and
unscreened interactions between the unperturbed charge distributions in two molecules.
The results are in accordance with Eq. (4.1.1) for the interaction between test charges in
presence of a dielectric medium. A fundamental difference between the two-body
induction energy and the three-body induction energy at second order is that the two-
body induction energy results from screening of intramolecular interactions, while the

three-body induction energy results from screening of intermolecular interactions.

78

4.2 Dielectric screening of second-order three-body induction forces on nuclei of
interacting molecules
In this section, we prove that the dielectric screening model can also describe the
three-body induction forces at second order on nuclei of interacting molecules. Following
Eq. (3.1), the second-order three-body force on nucleus K in molecule A is given by

(2 3)

K

K(2,3) _
Find _ (4.2.1)

where as usual, VK means derivative with respect to the coordinates of nucleus K.
Following an expression derived earlier by Li and Hunt, we can write the three-body

induction energy at second order as

735%?) =—(1+CABC)_Idrdr'aA 130" r'- ,0)30a (nag 0‘30- (4.2.2)

where 3013a (r) denotes the 01 component of the field acting on A due to the unperturbed,

static charge distribution p0B (r) of molecule B, and similarly for 3 0CB(r'.) In Eq. (4. 2. 2)

a:B(r,r';0) is the static nonlocal polarizability density of A, defined in Eq. (2.1.2).

Expanding Eq. (4.2.2) and using Eq. (4.2.1), we obtain

K(2, 3)_
Find

—'Idrdr 66f:,l3(r,'r,0)/aRK:s(m(r)3C(r0(1rB
+(1+goBC) Idrdr'aa Bu, r ;0)630A0L (manK 3CB(r',) (4.2.3)

where the operator PBC perrnutes the labels B and C in the expression that follows.
Using the relation between the derivative of the nonlocal polarizability density with

respect to RK and the nonlocal hyperpolarizability density from Eq. (3.1.2), the derivative

79

of the field 30Aa(r) with respect to RK from Eq. (3.1.4), and the relation between the
derivative of the permanent electronic charge density pgA (r) with respect to RK and the

nonlocal charge-density susceptibility xA (r, r';0) from Eq. (3.1.5), we obtain

 

 

K 2, I II I II I II
131151.63) = zK Idrdr dr BgySO’J'J ;0,0)3(I)3B(r)38y(r)T5a(r ,RK)
K “'2 I II III B I, II-3
+(1+pBC)Z (47t80) Idrdr dr dr afiy(r,r,0)(rB-rI§)Ir—r
A I K K‘1 C
xx (r,r”';0)VCl r"-—R I 3070")
K —1 I B I, C I
+(l+pBC)Z (47:80) Idrdr (167(r,r,0)307(r)
K K K_3

 

The first two terms in Eq. (4.2.4) yield the force on nucleus K due to its interaction with
the second-order three-body polarization of A. To show it explicitly, we note that the [3

component of [the field at r in A due to the permanent charge density in B is related to the
potential at r by 38350) = —VB (p530) = —VB (41E80)-1 I‘dr"'Ir—r"'I"1 p530") and the

charge density at r‘" in B is related to the polarization by Eq. (2.1.3). Thus, the first term

in Eq. (4.2.4) gives

zK I dr dr' dr" (31% 5 (r, r',r";0, 0) 3533 (r) 33 7 (r')T5a(r", RK)
= ZK Idr dr' dr' dr'" dr1v [33:5 (r, r', r"; 0, 0) [ TBS (r, r") P610") ]

x [Tm(r',riv)P§n(ri")]1‘5a(r',RK). (4.2.5)

80

Using the relation between the charge-density susceptibility and the nonlocal
polarizability density from Eq. (2.1.4), the second term in Eq. (4.2.4) can be written as

(1+(oBc)zK(4ne0)-2 Idrdr'dr”dr"a .—3

                  

xxA(I-, r'; 0)VK"

 

K
__ K I II III iv A III II, II B I.
— (1+goBC)Z Idrdr dr dr dr [011150 ,r ,0)T5I3(r ,r)afiy(r,r,0)
xrygu', r'v)P08 (r'v)]Tan(r" ,RK.) (4.2.6)

Thus from Eq. (4.2.4), (4.2.5), and (4.2.6), the sum of the first two terms [denoted by

Filrfd,a(1)] are given by
513522) = zK IdrTaI3(RK,r)P:‘((12B3), . (4.2.7)

in accordance with Eq. (58) in ref. (278) derived by Li and Hunt. The last term in Eq.

(4. 2. 4) [denoted by FiIrEd (10)] yields
5113812) = (1+5/JBC)ZK IdrTaB(RK, r) PB(1)(r) (4.2.8)

In the next part of this section, we connect the three-body induction force on

nucleus K to the dielectric model. First, we write the induction force from Eq. (4.2.4) in
terms of APllill (r, 0) , the induced change in the static polarization of molecule B (at first
order), due to the permanent charge distribution of C (and similarly for C), given by

APE0L (r) = [dr' “(100’ r'; 0) SOCB (r') . (4.2.9)

Using Eqs. (2.2.3), (4.2.4), (4.2.9) and (3.1.8), we obtain

81

FK(2, 3)=

mda —ZK(47£80)-1 Idrdr'dr"B[’;Y8(r,r',r';0,0)3 BB(')30CI (r)

xVaKVS

 

r'-RKI

-[1+C(B—>C)][ zK(4ne0) 2 Idrdr"dr"'APll,3

            

x xA (r', r"; 0) VaK

 

r"—RKI

+ZK(47t80) 1IdrAPB B(r)VKVB

 

r— RKI—l ]. (4.210)

The operator C ( B —> C ) in Eq. (4.2.10) means replacing the molecular label B by C in
the expression that follows. The potential acting on molecule A is the sum of the potential
due to the unperturbed charge distributions of B and C, and the potential due to the shift
in the charge density of B, induced by the potential due to the unperturbed charge
distribution in C (and similarly the potential due to the shift in the charge density of C,
caused by the potential from B). Thus from Eq. (4.2.10), the three-body induction force

on nucleus K depends quadratically on the potentials from the permanent charge

distributions in B and C and linearly on the potential, A0)? (r) , due to the first-order shift

in the charge distribution in B, ApI3 (r’) (and similarly for C). Using A(pI3 (r) from Eq.

(3.1.10) in Eq. (4.2. 10) and by repeated use of divergence theorem, we obtain

FK(2,3)

in d,a = ZK (47mg).1 Idr dr' dr"[VI3 V7 V3 BIPKY 6 (r, r',r"; 0, 0)]

636056-68

 

r”—RKI

82

—1
—[1+C(B—+C)][ZK (41(80)—1 Idrdr'x (r,r';0)VK r—RKI

 

xA¢P(r',0)
+sz§[A(plB(RK,0)]], (4.2.11)

where we have used the Born symmetry of the charge-density susceptibility xA (r,r';0)

with respect to an interchange of its arguments.

Next, we use the relation between the nonlocal hyperpolarizability density
[3&5 (r, r', r"; 0,0) and the quadratic charge-density susceptibility C(r,r',r';0,0) from Eq.

(3.1 . 12) and a relabeling of the integration variables, to obtain

FK(2,3) =

ind, a — ZK (47t80 )-1 Idr dr' dr” CA (r, r', r'; O, O)

x (pg)3 (r) (0% (r') (3 / are, Ir" — rI—1

 

r=RK

—[1+C(B—>C)][ZK (mm—1 Idr'dr"x (r',r;O)

 

r—RKI A(pB(r',0)

 

—zK VaK[A(pI3(RK,0)]] (4.2.12)

Using Eqs. (1.1), (1.10), and (4.2.12), the second-order three-body induction force on

nucleus K in molecule A is given by

1K(2,3) z
1nd,a

v,A K

—[1+C(B—>C)]ZK 806/6ra[_Idr'8_ (r,r';0)A(pI3(r',0)]
r

83

_ ZK 80 ('3 / (3ra[ Idr' dr' 8:1,1A (r, r', r”; 0, 0)]

xeguzoxpgozon RK.
, _

(4.2.13)

Equation (30) shows that the second-order three-body induction force on a nucleus in

molecule A results from screening of the potentials from neighboring molecules B and C;

the first-order potential due to the induced shift in the charge density in B (or in C) is

screened linearly within A , while the unperturbed potentials from B and C are screened
quadratically.

At second order, the eflective three-body potential within molecule A is given by

(p:f§2,3) (r, 0) = [1 + C ( B —) C ) ] 80 Idr' 8;,1A (r, r'; 0) A(pI3 (r', 0)

+ 80 Idr'dr" a; 1 (r, r', r";0, 0) (0(1)3 (r’, 0) (pg (r', 0) . (4.2.14)

Thus, using Eqs. (4.2.13) and (4.2.14), the second-order three-body induction force on

nucleus K is

K(2,3) _
Find . —

A(2,3)

—zK a/ar[quff (r,0)] (4.2.15)

 

r = RK .
Eqs. (4.2.13) and (4.2.15) prove the fact that the three-body induction force on nucleus K

in molecule A at second order can be exactly described by the dielectric screening model.

84

Chapter 5: Dielectric screening of three-body and four-body interactions at third

and fourth orders

In this chapter, we derive the three-body and four-body intermolecular interaction
energies at third and fourth order within the dielectric model. We show that the results are
in agreement with the results from quantum perturbation theory. In chapter 4, we have
showed that at second order, the three-body induction energy results from dielectric
screening of the Coulomb interactions between the permanent charge densities of two
molecules and the screening arises due to the presence of a third molecule which acts as
the dielectric medium. In the present chapter, we prove that at third order the induction
energy results from either intermolecular or intramolecular screening, depending on the
type of interaction. Moreover, nonlinearity appears in the induction energy at third order,
resulting quadratic response and nonlinear screening. At second order, the interaction
energies show linear screening only.

Nonadditivity in dispersion energy first appears at third order. In chapter 2, we
have proved that the second-order dispersion energy results due to the dielectric
screening of the intramolecular exchange-correlation energy. We show that at third and
fourth orders, the dispersion energy still appears due to screening-induced change in the
intramolecular exchange-correlation energy. However, at third and fourth order, the
dielectric medium consists of two and three molecules respectively and that brings many-
body effects in the screening function. We derive the many-body dielectric functions
from the many-body susceptibilities of the interacting molecules and we describe the

screening of the dispersion energy at third and fourth orders in terms of these dielectric

85

functions. We also prove that the dispersion energy at third and fourth orders results from
screening of the dispersion energy at second and third orders respectively, and that
screening appears due to the presence of a third or a fourth molecule which acts as the
dielectric medium.

The third category of interaction that appears at third and fourth orders is the
induction-dispersion. It results form the perturbation of the dispersion energy by a static
external field. The external field perturbs the response function of the molecules and
brings in new type of fluctuation correlations. Dispersion energy shows linear screening
only, but the perturbation by an external field produces nonlinear response, and hence
nonlinear screening in the induction-dispersion energy. At third order, induction-
dispersion energy includes nonlinear screening only. At fourth order, both linear and
nonlinear screenings appear.

We prove that the screenings present in the interaction energies at third and
fourth order are described by the nonlocal dielectric functions introduced in chapters 2

and 3.

86

5.1 Dielectric screening of the induction energy at third order

In this section, we show that at third order, the three- and four-body nonadditive
induction energies are described within the nonlocal dielectric model. We work within
the third order of perturbation theory and express the nonadditive induction energies in
terms of the static nonlocal charge-density susceptibilities of the interacting molecules.

Then we relate them to the nonlocal dielectric functions 8V(r,r';0) and 8q(r,r’, r'; 0,0)

introduced earlier, in order to show the dielectric screening.

Depending upon the type of interaction (and hence the molecular excitation
pattern), both linear and nonlinear responses contribute to the induction energy at third
order. In presence of a dielectric medium, the interaction energy of two test charges is
screened. The shift in the interaction energy caused by a nonlocal dielectric medium with
linear screening is described by Eq. (4.1.1) in the last chapter, and we have proved that
Eq. (4.1.1) accurately describes the dielectric screening in the induction energy at second
order. In the present section, we prove that the same equation still applies for the third-
order induction energy. However, at second order the interacting test charges correspond
only to the unperturbed, permanent charge distributions of the molecules. At third order,

p(r) in Eq. (4.1.1) can be either the permanent charge distribution of a molecule, or the

induced shift in the charge density of one molecule cause by the permanent charge
density of another molecule. Depending on whether the two interacting charge
distributions are permanent or induced shifts caused by an applied potential, the
interaction can be categorized as a particular many-body type (i.e. three-body, four-body

etc.).

87

Nonlinear screening appears at the third-order induction energy. In presence of a
nonlinear dielectric medium characterized by the quadratic dielectric firnction

sq (r, r', r'; 0, 0) , the interaction energy of three charge distributions is given by

AB = (41(80)—2 Idrdr' dr' dr‘" drivp(r)[41t8q (r, 1", l‘";0I 0) l—1

x r" — l’Iv

r'-r"'I_1 p(r'") p(riv). (5.1.1)

 

 

 

At third order, the charge distributions in Eq. (5.1.1) correspond to the permanent charge
densities of the interacting molecules. Unlike the screening caused by the linear response
of the dielectric medium, nonlinear screening does not stem from the screening of a
lower-order interaction. This is because, at different orders of perturbation theory the
interactions arising purely due to the nonlinear response are characterized by response
functions of different orders and they can not be interrelated to each other.

From intermolecular perturbation theory, the third-order energy for a cluster of
molecules AmBmCmD of arbitrary symmetry, interacting at long range is given by

<o1v1m><m18° n><n1v1o>

(Em — E0)(En — E0)

 

AE(3) = Z Z

m¢0n¢0

, (5.1.2)

 

where Im) and In) are the excited states of the molecules. Vis the interaction
Hamiltonian and for a pair of molecule A and B,

vAB = (47:80)_1 Idr dr' 6A6) 1330") r—r'l‘l. (5.13)

 

88

0

In Eq. (5.1.2), the operator \7 = V—(OIVIO). Following Li and Hunt, we separate the

third-order induction energy AESZi into the hyperpolarization energy AElr3y)p , the static

(3)

reaction potential energy AEsrp , and the third-body reaction potential energy AE(3)

tbp'
Hyperpolarization energy results from Eq. (5.1.2), with the excited states Im),
In) confined to one molecule and m, n ;15 0. For molecule A, the three-body and four-

body hyperpolarization energies at third order are given by

AEQJS’” = (1 / 2) (1 + SOBC) Idr dr' dr' CA (r, r', r'; 0, 0) ([353 (r) ((3%): (r') ((363 (r")
+(1/2) (1+SOBD) Idr dr' dr"(A (r. r'.r";o.0) ((1,361 086') 480")
+ (1 / 2) (1 + (ecu) [dr dr' dr' CA (r, r'Ir"; 0, 0) (pg 0‘) 98 (r') (19(1)) (r"),
(5.1.4)
and
ABSJSA) = Idr dr' dr' CA (r, r’, r'; 0, 0) (p(r)3 (r) (p((): (r') (p5) (r") , (5.1.5)

respectively. In Eqs. (5.1.4) and (5.1.5), CA (r,r',r"g0,0) is the quadratic charge-density
susceptibility of molecule A. The operator @BC perrnutes the labels B and C in the
expression that follows. The hyperpolarization energy of A in Eqs. (5.1.4) and (5.1.5) can
be interpreted as due to reaction field effects, where the potentials from the permanent
charge distributions of two neighboring molecules create a nonlinear shift in the charge

density at r in A. The induced shift in the charge density interacts with the potential due

to the permanent charge distribution of the third molecule, thus resulting an overall

89

energy shift of A. To connect the hyperpolarization energy to the dielectric model, we

expand the potentials in Eqs. (5.1.4) and (5.1.5), and then use Eq. (1 .10), to obtain

AEQSS’” = (1/2) (1+goBC)(41t30)_2 Idrdr'dr'dr"drwp0 B(r)

rII _ rrv Pg (rlv)

 

 

x [ 80 86.1.4. (r, r', r'; 0, 0)] Ir' — r"'I_1 pg (r'")
—2 I II III ivp B I II
+(1/2)(1+pBD)(4n80) Idrdr dr dr dr Oq(r)[808 1A(r, r ,r ,,0 0)]

rv p5) (rrv)

 

x r'—r"'I_1p(I))(r") r"—r

 

 

+(1/2) (1 +50CD)(41t80)—2 [drdr'dr'dr' dr'v pC(r) [e0 sq 1A(,r r', r"; 0, 0)]

o -1 o
I "—1 M N
r —r I p80 r —rIv p80”),

 

   

 

(5.1.6)

and

A 4 — I l M . - I N
AEhyg’ ) = (41w0) 2 Idrdr dr dr drwpg(r)[80 eq’1A(r,r,r ;0,0)]

0 _1 o
I "—1 M "
xIr —r I p0C(r )r —rIv p80”).

 

 

(5.1.7)

If we define the two-body and the three-body effective potentials at r due to nonlinear
screening by (p(2 )(r) and (p(3 )(r) respectively, then

A(3,3)_

2,AC
AEhyp ( )(r )

—(1/2)IC(B—+C)1]drp0(r)<p

+(1/2)[C(B—+D)]Idr130(09f,2 ADM)

90

+(1/2)[C(C——>D)] Idrp(l)3(r)(pgi~AD)(r), (5.1.8)
and
AEQJSA) = Idr pg(r)cpg%fACD)(r). (5.1.9)

The effective potentials in Eqs. (5.18) and (5.1.9) are given by

(pngfAC)(r) = Idr'dr'[80 8211, A (r, r',r";0,0)](p8j (r’) (08 (r") , (5.1.10)
and
(péifACD) (r) = Idr'dr'[80 8211,14 (r, r', r";0, 0)](pg: (r') (pg (r'). (5.1.11)

The operator C (B —+ C) in Eq. (5.1.8) replaces the labels B by C in the expression that
follows it.

Eqs. (5.1.6) — (5.1.9) prove that the hyperpolarization energy at third order is
accurately described within the dielectric framework, where one molecule acts as the
nonlinear dielectric medium to screen the interaction between the permanent charge
distributions of other two or three molecules. The results are consistent with Eq. (5.1.1).
The net three-body and four-body hyperpolarization energy at third is obtained by

(3,3)
hyp

(3,4)

summing the AE and AEhyp for A, B, C, and D.

Static reaction-potential effects correspond to the dynamic effects in dispersion
interaction, with the difference that the reaction-potential is produced in response to the
permanent charge-density in this case, rather than the charge-density fluctuation. Static
reaction potential energy results from linear screening and is obtained from Eq. (5.1.2)

with m ;E 0 in one molecule and n It 0 in the other molecule. For molecule A,

91

A(3, BC) A(3, BD) A(3, CD)

AE(3 3)— —AES

srp, A + AES

+AES

= (1 / 2) (471180)—3 [ Idr dr' dr' dr‘" driv drv pA ' —1 xB (r', I"; 0)

             

 

         

 

             

 

         

 

             

 

 

—l
xr"—r"'I 1)(C("',r r —rv p0A(rv)
+ Idr dr'dr' dr'" driv drv pA 1 x8 (r', r';0)
II 1 XD 1v v"1 A v
xr — r’"I (r’”, r r —r pO (r )
+Idrdr'dr"dr"'driv drv pA — '_1xC(r',r";0)
1 XD v_1 A v
xr"— r"'I (r, r —r pO (r ), (5.1.12)

        

 

where each term accounts for two different polarization routes. For example, AEngE)

accounts for the polarization routes A—IB—>C—>A and A—IC—>B—>A. Summing

AE,(3 3)
AE,srp A

C(3, 3) D(3, 3)

from Eq. (5. 1.12) with AEB(3’ 3), ABS and AES gives the total

A13(3 3) A(3, BC)

static reaction-potential energyAE at third order. AES can be viewed as the

induction energy of molecule B, in the presence of the unscreened external potential fi'om
molecule A and the screening potential from the shift in the charge density induced in C
by A. Alternatively, it can be interpreted as the unscreened interaction between the first
order shifts in the charge densities of B and C, caused by the permanent charge

distribution in A. To show the dielectric screening present in the static reaction potential

energy, we take AEB(3’BC) from Eq. (5.1.12), the second-order two-body induction
SFP

92

energy of B from Eq. (2.1.5), and the relation between the nonlocal dielectric function

and the charge-density susceptibility from Eq. (1.2), to obtain

 

 

 

 

AEQQ’BC) = (1/ 2) (4Tt80 )—2 Idrdr'dr" dr'" driv pOA(r)Ir—r'_1xB(r',r”;0)
'1 II iv iv III _1 A III
><[80 8v,C(r ,r )] r —r pO (r)
A A
—(1/2) [drdr'x (r,r';0)<p (000 (r')
0
= (1 / 2) (47t80)_2 Idrdr'dr" dr’" driv pOA(r)Ir—r'_1xB(r',r';0)
'1 II iv iv III —1 A III
x[eo 8v,C(r ,r )] r —r pO (r )

 

 

- (1 / 2) (47!:80 )_2 Idr dr' dr' dr'" x (r, r'; 0)

II‘] II
xIr-r p0A(r )

 

 

r'—r"I‘1p€‘(r'). (5.1.13)
The first term in Eq. (5.1.13) gives the screened induction energy of B due to its
interaction with A, in presence of C which acts as the dielectric medium. The second
term is the unscreened second-order induction energy of B. Eq. (5.1.13) proves that the

static reaction potential energy at third order results due to the difference between

dielectrically screened and unscreened second-order induction energy of a molecule.

Note that ($9134) = 0. Four-body terms in the static reaction potential first appear at the

fourth order of perturbation theory.
Third-body potential energy has terms with polarization routes that begin and end
at different molecules. At third-order, the third body potential energy shows both three-

body and four-body effects and is obtained from Eq. (5.1.2) with the same excitation

93

pattern as in the case of static reaction potential energy. Thus, it includes linear screening
only. First we consider the third-body potential energy corresponding to the polarization

routes A—>B—>C—>B and B—->C—>B—IA.

 

 

 

 

AEIBSIABC = (1/ 2) (47:80 )"3 Idr dr' dr' dr'" driv drv pg (r) Ir — r' _1
. . —1
X xC (rI’ rII; O) rII __ r"II—1 XB (r", rlv ; O) rlv _ rv p8 (rV) .
(5.1.14)
AEI13)§)ABC can be described as the interaction between the permanent charge density of

B and the first-order shift in the charge density in B induced by A, in presence of C
which acts as the dielectric medium. The first-order shift in the charge density of B

induced by the potential due to the permanent charge distribution in A is given by

 

ApIB(r) =(41rgo)-1 Idr'x (r,r';0)Ir—r’—lp(‘:‘(r'). (5.1.15)
From Eqs. (5.1.14) and (5.1.15),
3,3 — I n III I "1
AEfbijBC = (1/2)(41t80) 2 Idrdr dr dr pg(r)Ir—rI
xxC(I-',r'; 0) r" —r"'|‘1 Ap1B(r"') . (5.1.16)

 

Energy shift of B due to direct intramolecular interaction between its permanent charge

density and the first-order induced shift in the charge density is

‘lAplB(r'). (5.1.17)

 

AEB = (1/2)(4m.~0)‘1 Idrdr'p(l)3(r)Ir—r'
From Eqs. (5.1.16), (5.1.17), and (1.2), along with a change of the integration variables,

we obtain

94

-1 '
APIB (r)

 

r"—r'

£8133,ch = (1 / 2) (mm—1 Idr dr' dr' pg(r)[eoe;}c (r, r";0)]

 

—(1 / 2) (411.90)“1 Idrdr'p(l)3(r)Ir—r’|_1Ap?(r'). (5.1.18)

The first term in Eq. (5.1.18) is the intramolecular interaction between the permanent
charge density of B at r and the first-order induced shift in the charge density at r' , in
presence of molecule C which acts as a dielectric medium to screen the interaction. Thus,
within the dielectric model the three-body terms present in the third-body potential
energy depend on the difference between the dielectrically screened and the unscreened
Coulomb interactions between the permanent charge density and the first-order shift in
the charge density within a molecule. This result can be compared to the second-order
two-body induction energy described in chapter 2, where we showed that the two-body
induction energy at second-order depends on the difference between the screened and the
unscreened Coulomb interactions between the permanent charge densities within a
molecule. Thus, the three-body terms in the third-body potential energy show the similar
screening effect, but at the next order.

Finally in this section, we derive the four-body effects in the third-body potential
energy within the dielectric framework. Interactions present in the four-body terms in the
third-body potential energy at third order are purely intermolecular. For example, the

interaction energy associated with the polarization route B—>D—>C—>A is given by

“5%:th = (1 / 4) (41:30 )_3 Idr dr' dr” dr'" driv drv pOA (r) Ir — r' -1 xc (r', r”; 0)

 

. . -1
r"—I-"'I"1)(12(r"',r";0)r'V—rv 656V). (5.1.19)

X

 

 

 

95

Eq. (5.1.19) can be viewed as the induction energy of C, in the direct potential from the
permanent charge density in A and the screening potential from the induced shift in the
charge density in D, caused by the potential from B. Alternatively, it can be described as
the energy due to the intermolecular interaction between the first-order shift in the charge

density of C induced by A, and the first-order shift in the charge density in D induced by

B. Note that this energy corresponds to the term E82323 introduced by Piecuch,262 with

i It j It k. To connect AESS‘IBDC to the dielectric model, we take Eq. (5.1.19) along with

all the polarization routes with C and D in the excited states, the second-order three-body
induction energy of C from Eq. (4.1.2), and the relation between the nonlocal dielectric

function and the charge-density susceptibility from Eq. (1 .2), to obtain

’1 XC (r',r'";0)

 

4 - I N M . II
131383;),th = (47:80) 2 Idr dr dr dr drIv pfj‘(r)Ir —r

. . —l
><[80 e;{o(r".r“’ ;0)1 r'v —r' p80)

 

 

— (41:20 )72 Idr dr’dr"dr"’ p0 (r) Ir — r" ”1 xc (r', r'";0)

 

xIr" -— t-'|'1 of? (r'). (5.1.20)

In Eq. (5.1.20), the first term is the induction energy of C due its interaction with A and
B, in presence of D which acts as the dielectric medium to screen the interaction. The
second term is the unscreened three-body induction energy of C at second order. Thus,
the four-body effects in the third-body potential energy are obtained as the difference
between the screened and the unscreened second-order three-body induction energy. The

second-order three body induction energy itself results from screening of the Classical

96

electrostatic interactions between two molecules. Hence, the four-body terms in the third-
body potential energy are described as the screening effects present in the same type of

interaction, but at the next higher order.

 

 

97

5.2 Dielectric screening and the nonadditive dispersion energy

In this section, we derive the nonadditive dispersion energy within the dielectric
model. We focus on the dispersion energy of a particular molecule and relate the change
in the correlation between the intramolecular charge density fluctuations of that molecule
with the nonlocal dielectric functions of other molecules. In section 5.2 A, we show the
screening present in the three-body dispersion energy. In section 5.2 B, we derive the
nonadditive four-body dispersion energy and show the screening present in the four-body
dispersion energy of a particular molecule. The direct effects of overlap damping are
included in the expression of the dispersion energy, but not modifications due to

exchange or orbital distortion.

5.2A Dielectric screening of nonadditive three-body dispersion energy at third

order
Previously, Li and Hunt278 have developed a theory for the nonadditive three-

body dispersion energy, based on the correlations in the fluctuating polarization of
interacting molecules A, B, and C. The three-body dispersion energy derived in their

work is given by
AEE13’3) = —-r1 Eodm Idr dr'dr" dr'" driv drv Tr[T(rv,riv)0lC (riv,r";i(0)
II
III II B II I, - I A V, '
xT(r ,r )01 (r ,r,1(0)T(r,r)01 (r,r ,1(0)]. (5.2A.1)
In Eq. (5.2A.1), a(r,r';i(t)) denotes the nonlocal polarizability density generalized to

imaginary frequencies, and T(r,r') is the dipole propagator defined in Eq. (2.2.3). Tr

means the trace of the expression that follows. The result in Eq. (5.2A.1) is derived after

98

 

summing the three-body dispersion energies of A, B, and C, and using the Kramers-
Kronig relation between the real and the imaginary parts of the nonlocal polarizability
density.

In the present section we use a susceptibility based approach that we used in
section 2.2 to Show the screening present in the intramolecular charge density
fluctuations due to the two-body dispersion interaction. Following the same approach that
we used in section 2.2, we show that the average energy shifi of molecule A at third order
due to the correlation between the charge density fluctuations at points r and r’, in

presence of molecules B and C is given by

A _ I I M . " I
AEd (3’3) = —(h/47t)(47t80) 3(1+@BC)E:Od0) J'drdr dr dr drIv drvxA (r,r;oo)

-1 —1

IV

f 0

r —r —1xB(r",r";w) r'"—r xC(riv,rv;co) rv —r

X

 

 

 

 

 

 

x coth(h(o / 2kT) , (5.2A.2)

where xA"(r,r';w) is the imaginary part of the charge-density susceptibility of A,
defined in chapter 2. Charge-density fluctuations in A at time t' induces a shift in the
charge density in B at time t" which eventually induces a shift in the charge density in C
at time t, thus creating a reaction potential on A at time t. Using the fluctuation-

dissipation theorem from Eq. (2.2.9), we obtain
A 3,3 — I II III ' I
AEd( )= (1/4)(41t80) 3(1+goBC)[dz-dr dr dr dr" dr" fwda—t)

v—l
r—r

x flaw—t')<5pA(r,t)SpA(r',t')>

 

 

+

99

 

. . -1
xxC(rV,rlv;t --t") r” —r"' xB(r",r”;t"-t') r"-—r' _1 . (5.2A.3)

 

 

 

 

From Eq. (5.2A.3), the two-body dispersion energy of A from Eq. (2.2.11), and the
relation between the nonlocal dielectric function and the charge—density susceptibility in

the time domain from Eq. (2.2.12),

AE§‘(3’3) = (1 / 4) mam-2 (1 + gOBC) jdr dr' dr" dr'" driv J: d(t - t') j: d(t" — t')

iv _1
r—r

 

 

x<6pA (r,t)éipA (r’,t')> 30 8;1C (riv,r"';t—t")
+ 9

X xB (rm, rII; tII _ tr) ru _ rIl—l

 

_(1/4)(4u.~20)‘2 [1+C(B —) C)] Idrdr'dr'dr’" fde—t')

>< <59A(r. t)59A(r’, t')> 1" - r" ‘1 xB(r'Ir"';t - t') r'" - fl"1 ,
+

 

 

 

(5.2A.4)
where the operator C (B —> C) replaces the label B by C in the expression that follows.
The first term in Eq. (5.2A.4) gives the screened two-body dispersion energy of A due to

interaction with B (or C), in presence of C (or B), which acts as the dielectric medium
with the nonlocal dielectric screening function a;lc (riv,r'";t—t"). A charge density

fluctuation in A at r',t’ sets up a potential in B at r',t' (in the Coulomb gauge, with
retardation neglected) which induces a shift in the charge density in B at r',t’. The

reaction potential in A at r,t due to the shifi in the charge density in B is screened by the

presence of C via its nonlocal dielectric function e;1C(rw,r"';t—t”), and the screened

100

 

potential affects the two-body dispersion energy of A. The second term gives the second-
order two-body dispersion energy of A in presence of molecules B and C. Thus, the
three-body dispersion energy at third-order depends on the difference between the
dielectrically screened and the unscreened two-body dispersion energy at second order.

In an alternate way, we can describe the three-body dispersion energy of A in
terms of an effective, two-body susceptibility of B and C. Following Kohn, Meir, and

Makarov,I64 if the long-range interaction between B and C acts as a small perturbation,

this two—body susceptibility is given by

 

xBC(r,r';co) = 9» Jdn drz xB (r,r1;co) 4 xC(r2,r';co), (5.2A.5)

n80 ln - 12|
where it is a coupling constant that “turns on” the long-range interaction between B and
C. If B and C are non-interacting, the overall susceptibility is given by

XBC = xB(r,r1;w) + xC(r2,r';w). (5.2A.6)

283

Previously, Li and Hunt have showed that for a pair of interacting centrosymmetric

linear molecules A and B, the overall polarizability in presence of an external field Se is

given by

A B

lim :56 —> o ——a(““ + ”(1)
63E

_ A B ind
— aaB +aaB “Mali , (5.2A.7)

where (19B and (12B are the polarizabilities of the isolated molecules and Aug}? is the

collision-induced electronic polarizability of the pair. Using a self-consistent solution of a
set of equations that relate the induced dipole moment to the local field, they showed that,

at first order

101

Aug}? = (1 + goAB )aég, TYMR) (1533 . (5.2A.8)

In Eq. (5.2A.8), (13b is the dipole polarizability of molecule A, R is the vector from an

origin at the center of symmetry of molecule A to the origin at the center of molecule B,
TaB (R) is the dipole propagator given by, TaB (R) = VaVB (R_1) , and 50 AB permutes

the labels A and B in the expression that follows. The collision-induced electronic
polarizability defined in Eq. (5.2A.8) determines the first-order dipole-induced-dipole.
Here we use the same method to derive the two-body susceptibility for a pair of

interacting molecules A and B, in presence of a fluctuating external potential (Pext (r; (0).

Within linear response, the shifts in charge densities of A and B are related to the applied

potential by the equations
ApA(r;co) = J'dr'x (r,r';w)(pé},p(r’;(o). (5.2A.9)
ApB(r;(o) = jdr'x (r, r';o))(pgpp(r'; 0)). (5.2A.10)
The applied potential at A is related to the external potential (Pext (r; m) and to the
potential due to the shift in the charge density of B, ApB (r; 0)) by

ApB(r';m)

A I
r;o) = r;o) + dr
(Papp( ) (Pext( ) I 4 O|r_rI

 

= <Pext(r;C0)+ [dr'dr' 4 x3 (r',r";co)(pext(r";o)). (5.2A.11)

7:80 Ir — r'l
The self-consistent solution of Eqs. (5.2A.9) — (5.2A.1 1) yields

APA(r;w)+ApB(r;w)

102

 

= Idr'[xA(r,r';m)+xB(r.r';w)]<pext(r':w)

 

+(1+50AB) Idr'dr'dr"xA(r,r";CD) 41w

1 III I I
II IIIIXB(|' Ir;0))(Pext(r;(D)-

 

(5.2A.12)
From Eq. (5.2A.12), the two-body part of the susceptibility of molecules A and B, within

linear response is given by

 

AXAB (r, r';0)) = (1+ (GAB) Idr" dr'"x (r, r";o)) 4 x3 (r",r';(o) .

 

7:80 r' — r'"|
(5.2A.13)

From Eqs. (5.2A.3) and (5.2A.13) the two-body susceptibility in the time domain is
AXBC (r, r';t — t') = (1+ SOAB) Idr' dr'” £000 d(t” — t') x3 (r, r"; t" — t')

1

,, xC(r”’,r';t—t"). (5.2A.14)
r —l'

 

"I

 

X
47:80
Using the two-body susceptibility from Eq. (5.2A.14) and a change in the integration

variables, the three-body dispersion energy of A from Eq. (5.2A.3) is written as

A5303) = (1 / 4) (mm—2 jdr dr'dr"dr”' food (t — t')< 5pA(r, t) 5pA (r', t') >+

xIr—rn‘"1xBC(rfi,rM;t_tI) _1.

 

 

r'" —r| (5.2A.15)

Finally, using Eqs. (5.2A.15), (2.2.11), and (2.2.12), we obtain

AEdA(3’3) = (1 / 4) (41:80)_1 jdr dr' dr' food (t — t')< 5pA(r, t) 8pA(r', t') >+

x rI_rII—1

 

 

so 8;",ch (r", r; t — t')

103

 

1‘1 . (5.2A.16)

_(1/4)(47t80)—1Idrdr'<6pA(r,t)5pA(r',t')> |r—r
+

The first term in Eq. (5.2A.16) gives the Coulomb energy associated with interactions

between the fluctuating charge densities SpA (r, t) and SpA (r',t') in presence of the pair
of molecules B and C, which together act as the dielectric medium with the nonlocal

screening function .9;ch (r’,r;t—t'). The second term is the Coulomb energy of direct

intramolecular interactions of the charge-density fluctuations in molecule A, in absence
of B and C. Thus, Eq. (5.2A.16) proves that the three-body dispersion energy of A results
from the difference between dielectrically screened and unscreened interactions between
its intramolecular charge-density fluctuations, where the dielectric screening is

characterized by a two-body screening function, defined in Eq. (5.2A.14).

104

 

5.2B Dielectric screening of nonadditive four-body dispersion energy at fourth

order

In this section, we describe the nonadditive four-body dispersion energy at fourth
order Within the nonlocal dielectric model. We consider the dispersion energy of
molecule A in a cluster of interacting molecules A---B---C°-°D, and derive the dispersion
energy using the same susceptibility base approach that we used in chapter 2 and in the
previous section. In appendix B, we derive a new equation which gives the four-body
dispersion energy of the A---B°-~C---D cluster at fourth order. We show that

A 4,4 3h " II—
AEd( ) = —-;(41t80)_4 Soda) Idr...drv“ xA r le

(r',r";ic0)

 

         

rIII _ l,lv iv r;v V]

X

xC(r

 

       

 

 

V“ —r - (5.23.1)

 

 

XXD (rVi,rVii;im) r

In terms of real frequencies, the four-body dispersion energy of A is given by

 

 

             

AEdA(4’4)= —(h/41t)(41c30)4(1+5oBCD) Idr...drVii [:0de x ",(r r' 00,)
. —l
X |r’ _ rII "' rIII _rIV x C(rivr V ;(D)
x rv —rVi — xD (rVi,rVii;co) rVii —r— coth (hail 2kT).

 

 

 

 

(5.23.2)

In Eq. (5.2B.2), the operator (QBCD permutes the labels B, C, and D. Using the

fluctuation-dissipation theorem from Eq. (2.2.9),

105

AEdA(44)" _(1/4)(4n30) —4(.1+5oBCD)Idr .drVii Eodfl- 1W)J:Od(tt

.—1
rvii

xfood(t'— t'A)<5p (r, t)5pA (r', t'r-)>

 

 

+

Vll’r V

XC (rv’riv;tIII _ tII)

 

TV] —I‘

       

xx D(r

iv _1
r —r"'

II

r —r’|_1.

 

>< xB(r’”.r";t" -t')

(5.23.3)

 

 

From Eqs. (5.28.3), (2.2.12), and the three-body dispersion energy of A from Eq.

 

 

 

 

 

(5.2A.3), we show that
4 I ”I N
A932?4 )— —(1/4)(47t80) 3..(1+5oBCD)jdr .drv'food(t—t)J:od(t —t)
xfo d(t"—t') 8 A(r t)8 A(r' t') r—rVi _1
00 P 9 P 9 +
X8 -1 Vi V, _ III C V iv, III_ II
08v,D(r ,r ,t t))( (r ,r ,t t)
. —1 B _1
X rIV _ rIII x (rm, rII; tn _ tI) rII _ rIl
A(3,3)
_AEB CD . (5.2B.4)

In Eq. (5.2B.4), the first term gives the screened three-body dispersion energy of A in

presence of molecule D (or B or C) which acts as the dielectric medium with the nonlocal
screening function 5v,D (rVi ,rvgt — t’"). The second term gives the unscreened three-

body dispersion energy of A at third order in the molecular cluster A-"B-"C-"D. Thus,

from Eq. (5.2B.4), the four-body dispersion energy of A depends on the difference

106

 

between its dielectrically screened and unscreened three-body dispersion energies at third
order.

Finally in this section we show that the four-body dispersion interaction can also
be described as screening to the direct intramolecular interactions between fluctuating
charge densities, where the other three molecules provide the dielectric screening. To
show this screening effect, we derive a three-body susceptibility in terms of the nonlocal
charge density susceptibilities of three interacting molecules. Previously, Champagne, Li,
and Hunt284 have showed that for a cluster of non-overlapping, isotropic species A, B,
and C, interacting at long range, the nonadditive three-body polarizability at second order
is

280151263) = (1A (13 (1C 3 ABC Ta5(RA,RB)TB5(RA,RC)

+ (1 / 3) s ABCCA aBaC Tay5(RA,RB), (5.23.5)

where the species centers are located at RA, RB, and RC respectively; SABC denotes the

sum over all permutations of the labels A, B, and C in the expression that follows it; and

the propagators T aB...(o (r, r') of arbitrary rank are given by

Tag...m(r,r') = vavB mvm lr—r'I—l. In Eq. (5.23.5), (1A is the dipole polarizability of

the isotropic species A, and the C tensor determines the quadrupole induced by a uniform
field gradient, within linear response. Here we derive the three-body susceptibility of
three molecules A, B, and C, interacting at long range, in presence of a fluctuating
external potential. Following Eq. (5.2A.9) — (5.2A.10), the shifts in the charge densities

of A, B, and C, within linear response is given by

107

APA (r; 0)) + ApB (r; (o) + ApC (r; 0))
= Id!“ X (1’, I"; (0) (PaApp (r'; (o) + Idr' x (r, r'; w) (ngp (r'; (o)

+ Jdr'x (r, r';(o) (pgpp (r';o)) . (5.2B.6)

The applied potential at A depends on the external potential and on the potentials due to

the shifts in the charge densities of B and C. Thus,

ApB(r';co> +1 r, ApC(r';w)

A . _ . I
(papp(r,(o) — cpext(r,(o)+ Idr 41:80lr—r’|

, . (5.23.7)
41:80 lr—r

 

The charge density shift of B, in turn, depends on the charge density shift of C (and vice

versa), which finally yields

I II 1 I II 0
(paApp(r;w) = (pext (r; (o)+ Idr dr —, xB(r ,r ;m) (Pext(r ;co)
41:80 lr—r |

 

 

+ Idr' dr' ——————,- xC (r', r'; (o) (Pext (r"; 0))
47:80 |r — r
' 1
+ 1+ dr'...drIv — B r',r";w
xxC (r",riv;o)) (Pext (riv;o))+... (5.2B.8)

Solution of Eqs. (5.2B.6) and (5.2B.8) gives

ApA(r;co> + 413%: co) + 4pc (r; co)
= [dr' x (r,r'; 0)) (Pext (r'; m) + [dr' x (r. r'; 03) ‘Pext (r'; w) + [dr’ x (r, r'; 0)) <9ext(r'; 0))
+ Idr' AXAB (r, r'; (o) ‘Pext (r’; (o) + Idr' 13ch (r, r'; (o) (Pext (r'; (0)

+ [dr' AxCA (r, r'; (o) (Pext (r'; 00)

108

 

+ Idr' AXABC (r, r'; w) (Pext (r'; (o) + (5.28.9)

In Eq. (5.28.9), AXAB (r, r';to) denotes the two-body part of the susceptibility defined in

Eq. (5.2A.13), and AxABC (r, r';(o) is the three-body part of the susceptibility, given by

 

 

 

ABC(r,r';00) = SABC Idr'...drv xA(r,r";(o) 1' X B(r"'r iv ;(n)
41:80 r r'"|
>< 1. XCO'V , r';t») , (5.23.10)
47:80 rIv —rv

 

 

In time domain, the three-body susceptibility is
ABC I, _ I _ II V III _ II II _ I A II, _ III
(r,r,t t)-SABC Idr...dr food“ t )fwda t)x (r,r ,t t )

1 BIIIivIIIII 1 ivr,v

,,,x (r ,r ;t —t) . x C(r ;"'t -t').
lV_rV

 

 

               

 

 

411280 r

(5.28.11)
Using the definition of the three-body susceptibility in the time domain from Eq.
(5 .28.] 1) and a change in the integration variables, the four-body dispersion energy of A

in Eq. (5.28.3) can be written as

AE§(4’4) = (1/ 4) (47320 )-2 Idr dr' dr' dr'" E d (t — t')< 8pA (r, t) 8pA (r', t') >+

—1 XBCD

I

xlr_rII II III

—1
—r|.

 

 

(5.23.12)

 

   

   

 

Thus, from Eqs. (5.28.12), (2.2.11), and (2.2.12), we obtain

A3§(4I4) = (1/4)(41I280)_1 jdr dr’dr' fiod (t — t')< 8pA(r, t)8pA(r',t') >+

x rI_rII—1

 

 

80 83,130) (r", r;t — t')

109

‘1 (5.23.13)

 

—(1/4)(47t80)—1 Idrdr'<5pA(r,t)8pA(r',t')> |r-r'
+

Eq. (5.28.13) proves that the four-body dispersion energy at fourth order results from the
screening of the intramolecular exchange-correlation energy, where the dielectric
screening depends on molecules 8, C, and D which together act as the dielectric medium.

Thus, the many-body effects are contained within the nonlocal dielectric function.

110

5.3 Dielectric screening and induction-dispersion energy at third and fourth
orders
In this section, we prove that the dielectric screening model also describes the
simultaneous induction-dispersion energy of a cluster of molecules. Simultaneous
induction-dispersion effects appear, because the permanent charge-density of one

molecule acts as the source of a static external potential (00 that perturbs the two-body or

three-body dispersion interaction of two or three other molecules respectively: Each of
the two or three molecules in the cluster is hyperpolarized by the simultaneous action of
the static external potential and the fluctuating potentials from its partners. The static
external potential also alters the correlations of the spontaneous, quantum mechanical
fluctuations in the charge densities of the other interacting molecules. Within the
dielectric model, the induction-dispersion interaction can be interpreted as the
perturbation of the dielectric medium by the external potential. This perturbation brings

in nonlinear screenings into the dielectric medium, which are of secondary importance in

183,184

the case of pure dispersion interaction. Previously, Hunt and Bohr developed a

theory for the dispersion dipole of an A---B pair, based on the change in dispersion energy

due to a uniform, static external field‘ Se. Li and Hunt278 applied the same analysis to the

dispersion energy of the A---B pair in presence of the static external field SOC due to the

permanent charge distribution of molecule C, after allowing for the nonuniformity of the
field. In the present work, we relate the three-body and four-body induction-dispersion
interactions to the dielectric model. We focus on the energy shift of a particular molecule

due to the correlation between its intramolecular charge density fluctuations in presence

111

of the static external potential. We use the charge density susceptibility based reaction
potential approach that we used in the previous section, to describe the induction-
dispersion energy of the interacting molecules, where the overlap between the charge
distributions of the molecules is assumed to be weak or negligible.

In presence of the static external potential (pOC due to the permanent charge

distribution in molecule C, the two-body dispersion energy of molecule A is given by

A3303) = (1 / 4) (47t80)_2 jdr dr'dr' [:on — t')< 8pA(r,t)5pA(r',t') >+

C

—1 II III I "‘1
XB(r 9r 9(p0 9t-t) '

 

(5.3.1)

r"—r|

 

xlrI_rII

In Eq. (5.3.1), xB (r',r’”;(pg,t -t’) denotes the nonlocal charge density susceptibility of

molecule 8 in time domain in presence of the static external potential (pg. If the external

potential is significantly small, x((p,t — t') can be expanded in Taylor series,

x(tp0C,o)) = x(t —t')+:;(t —t',0)chC +..., (5.3.2)
where x(t — t') and C(t — t',0) denote the linear and the quadratic charge-density

susceptibilities respectively, in the absence of the perturbing potential. Substituting Eq.

(5.3.2) in Eq. (5.3.1), we obtain

AES‘Q’Z) = (1 / 4)(47t80)—2 jdr dr'dr" [:Oda — t')< 6pA(r, t)8pA(r', t') >+

rI_rn—1xB(rl,rn;t_tI) “'1

X

 

 

 

r'"-r|

—2 iv I A A I I
1/4 4 d...d d - 6 , 6 I
+< )(mo) Ir r I: (t t)< p (rt) p (r t)>+

112

 

 

r"‘1((r',r, " r";—t t,0)|r"—r|_1<p0C(riv). (5.3.3)

The first term in Eq. (5.3.3) is the unperturbed two-body dispersion energy of A. The

second term gives the three-body induction-dispersion energy of A at third order.
Expanding the potential (pf)3 (riv) in Eq. (5.3.3),

A33? 3) — —(1/4)(47tao)_3 jdr...dr" Eod(t—t')<5pA(r,t)5pA(r',t')>+

. -1
xr'— _1§B (r",r " ,riv;—t t',"0)|r— r| 1rW—rv pg(rv).

 

 

 

 

(5.3.4)
The energy in Eq. (5.3.4) can be interpreted as the hyperpolarization energy of 8 caused
by the fluctuating potentials from A and the static potential from C. The nonlinear shift in
the charge density of 8 caused by the potentials due to the charge density fluctuations in
A interacts with the static external potential from C, thus giving an overall energy shifl.
Using Eq. (5.3.4) and the nonlinear dielectric function in the time domain, the three-body

induction-dispersion energy of A is given by

AB“: 3) — _(1/4)(4n.«~.0)’2 [dr...driv Ed(t—t')<59A(rat)59A(r',tI)>
+

XI”—

 

 

"‘1[eosq1,r,B(r'r'" r'v,t— t',"'—0)]|r r|1pg(ri").

(5.3.5)
Thus from Eq. (5.3.5), the induction-dispersion energy of A is described within the
dielectric model as the energy shift due to the nonlinear interaction between the charge-
density fluctuations in A and the permanent charge distribution in C, in presence of B

which acts as the nonlinear dielectric medium. This interaction is analogous to the

113

hyperpolarization energy described in section 5.1, with the difference that the
hyperpolarization energy appears due to the nonlinear interaction between permanent
charge distributions, while the induction-dispersion energy is due to the nonlinear
interaction between permanent charge distribution and charge-density fluctuations.
Following the same line of argument, we can relate the four-body induction-
dispersion energy of molecule A with the dielectric model. Here we consider the three-
body dispersion energy of A in presence of 8 and C, perturbed by the external potential
due to the permanent charge distribution of D. Expanding the charge density
susceptibilities of 8 and C in Taylor series with respect to the external potential and

keeping only the lowest order terms, the four-body induction-dispersion energy of A is

 

 

 

 

givenby
A44 — ' I II I
AEi+(d’ )=(1/4)(41:80) 3(1+goBC)Idr...drv' f: d(t-t) f: d(t —t)
A A —1 3 iv ‘1
x<5p (r,t)8p (r',t')> r'—r" x (r',r'";t"—t') r'—r
+
C iv v vi v _1 D vi

xC (r ,r ,r ;t—t",0)r —r (90 (r ). (5.3.6)

 

 

Eq. (5.3.6) can be interpreted as the hyperpolarization energy of C (or 8) caused by the
fluctuating potential from A, potential due to the shift in the charge density of B (or C)
caused by the fluctuating potential from A, and the static external potential due to the
unperturbed charge density in D. Using Eq. (5.36) and the relation between the quadratic

charge-density susceptibility and the nonlinear dielectric function, we obtain

A(4,4)

AEi+d = (1/4)(47re0)_3(1+gOBC) Idr...drVi Eda—t') [luv-t')

114

. -l
rI _ rII _1xB(rII,rIII;tII _ tr) rIII _ rlv

 

 

x< 6pA (r, t) SpA (r', t')>+

 

 

x[803;,1C(r'v,rv,rw;t—t',0)]rv—r p80"). (5.3.7)

 

 

Eq. (5.3.7) proves that the four-body induction-dispersion energy of A arises due to the
interaction between the charge density fluctuations at r and the reaction potential caused
by the shift in the charge density of B (or C), induced by the potential from the charge
density fluctuations in A atr' , in presence of C (or B) which acts as a nonlinear dielectric
medium, perturbed by the potential from the permanent charge density in D.

This interaction energy can also be explained as screening to the three-body
induction-dispersion energy of A defined in Eq. (5.3.5). To show this screening effect, we
consider the cluster of molecules A, B, and C, in presence of the external potential from
the permanent charge density of D, where the dispersion energy arises due to the charge

density fluctuations in A. The overall induction-dispersion energy of A in this case is

. 3,3 3,3
given by the sum of the three-body terms ABE A-))B _) A) (—D and ABE A-—))C __) A) (—D ,

and the four body terms from Eq. (5.3.7). Using the relation between the nonlocal
dielectric function and the charge-density susceptibility from Eq. (2.2.11), we can show
that

A(4,4)

Alai+d = (1/4)(41ra0)_2(1+5oBC)Idr...drv Eda—t’) Eod(t'—t')

X< 5pA(r’1)5pA(rI’tI) > |rI_rII _1[80 8B1 (r", I.III;tII_tI)]
+

 

. —I
p33)

 

 

XISO 8;,1C(r",r'v,rv;t-t",0)]

115

— (1/4)(41t30)_2[1+C(B —> C) jdr...dri" [:odO—t')

 

I 0‘1 —1 II III iv, I
r —r [aoeq’B(r ,r ,r ,t—t,0)]

 

x< 5pA (r, t) 8pA(r', t') >+

xIr" 414,580“). (5.3.8)

In Eq. (5.3.8), the first term gives the screened three-body induction-dispersion energy of
A. Here molecule 8 (or C) acts as the dielectric medium to screen the induction-
dispersion interaction, and the screening is linear. The second term gives the unscreened
three-body induction-dispersion energy of A. Thus, the induction-dispersion energy at
fourth order depends on the difference between the screened and the unscreened three-

body induction-dispersion energy.

116

Chapter 6: Summary and conclusions

The present work proves that the intermolecular interactions at second, third and
fourth orders are accurately derived in terms of the nonlocal dielectric model, where the
overlap between the interacting molecules is weak or negligible. Within linear response,
the nonlocal dielectric function av (r, r';(o) determines the effective potential at r , when
an external frequency—dependent potential (p(r',(o) acts at r'. A separate dielectric
function ad(r,r';o)) relates the dielectric displacement D(r,o)) to the external field
E(r',w). For translationally invariant systems, the isotropic average of ad(r,r';m)
reduces to 3V(r,r';o)). Within the intramolecular environment, ad(r,r';m) and
ev(r,r';(o) are different. The nonlocal dielectric function av(r,r';w) is related to the
nonlocal charge-density susceptibility x(r,r';(o) by Eq. (1.2). x(r,r';w) determines the
induced shift in the charge density 6p(r,co) at point r in the molecule due to an applied
potential (p(r',o)) at r'. Molecular properties which are related to the charge-density

129,188

susceptibility are nonlocal polarizability density, infi'ared intensities,127 the

Stemheimer electric field shielding tensor, 127 charge reorganization terms in vibrational

force constants,286 and the softness kernel of density fimctional theory.287’288

Dielectric response of translationally invariant systems or systems with spatial
periodicity can be described by a dielectric function which depends only on the distance
between the response point r and the point r' where the external perturbation acts. For

these systems a convenient choice is to use the dielectric function 8( k,o)) , which is the

spatial Fourier transform of e(r—r',o)). Dielectric functions of the form e(k,oo) have

117

been used in order to study quantum many-body problems, properties of quantum dots,

solvation dynamics and polarization fluctuations in liquids, and electron transfer.
Dielectric models have been applied to study interactions within proteins and

biomolecules. Models have been developed to probe the dielectric environment inside

protein molecules and these models have been used to interpret different experimental

observations such as determination of pKa shifis of inserted amino acid residues,289
dynamic shifts of the fluorescence, for the markers placed at various sites of protein,290
measurements of Stark effect on absorption bands of different chromophores,“ and

determination of the apparent basicities of the different charge states of protein”:2

Inhomogeneities inside the protein molecules make it impossible to define a universal
dielectric constant (or constants) for proteins. The choice of the value of protein’s
dielectric constant depends on the particular property or interaction to be studied and the
model used to study those properties. To give a complete description of the dielectric
nature within the protein environment, it is necessary to develop a model in terms of the
site-dependent dielectric constants.

Extension of the dielectric model to describe interactions within the
intramolecular framework was suggested in several works. Early works on light
scattering by fluids and collisional polarizability anisotropy of interacting noble gas
atoms used a polarizability density instead of the point dipole approach. In later works on
light scattering by fluids, response functions were used that depend both on r and r'.
Importance of nonlocal response has been noted in recent works on surface enhanced

Raman scattering by metal nanoshells.

118

Intermolecular interactions at first order are purely electrostatic in nature. Within
quantum perturbation theory, first-order intermolecular interactions are obtained as
Coulomb interactions between the unperturbed charge distributions or polarizations of the
molecules. When the molecules are far apart, the electrostatic interaction energy is
obtained as a sum of the interaction energy of permanent multipole moments of the
molecules, given by Eq. (1.14). For a pair of interacting molecules, the first-order force

on nucleus I in molecule A is calculated by taking the negative gradient of the first-order

interaction energy with respect to the coordinate RI of nucleus 1. Thus, it depends on the

derivative of the permanent charge distribution of A with respect to RI. When using a

multipole expansion, the first-order force is given in terms of the derivatives of the
permanent multipole moments of A. First-order forces on nuclei can also be calculated
using the electrostatic Hellmann-Feynman theorem and the Stemheimer-type shielding
tensors, where the force on nucleus I is obtained as sum of the interactions between the
charge on nucleus I and effective fields and field gradients at I due to molecule 8, given
by Eq. (1.21). The effective field at I due to molecule 8 depends on the field from 8 due
to its permanent moments and the nuclear shielding tensors of I. At first order, the
effective field and the field gradient originate due to linear screening of the external field
and the field gradient and are determined by the linear shielding tensors of nucleus 1.

These linear shielding tensors are related to the derivatives of permanent multipole
moments of molecule A with respect to RI [Eqs. (1.16), (1.18b)]. Physically, the

shielding appears due to the electronic screening of the external field. Within the nonlocal
polarizability density model the electronic screening is shown by the nonlocal

polarizability density a(r,r';0) , where the first-order force depends on the fields at I from

119

the unperturbed charge distribution in B and the first-order induced shift in the

polarization of A [Eq. (1 .20)] which is determined by a(r,r’;0) of A and the field due to
unperturbed charge distribution in B. The Stemheimer—type shielding tensor 7:13 is

connected to the nonlocal polarizability density “YB (r, r') by Eq. (1.19). The first-order

force on nucleus 1 in molecule A was first derived within the nonlocal dielectric model by
Jenkins and Hunt. A susceptibility-based approach was used to express the first-order
force on I in terms of the static nonlocal charge-density susceptibility of A and the
potential from the unperturbed charge-distribution in 8. Using the relation between the
charge-density susceptibility and the nonlocal dielectric function from Eq. (1.2), the first-
order force was expressed [Eq. (1.23)] as interaction between nucleus I and the external
potential from B in presence of the intramolecular dielectric medium A which is

characterized by the nonlocal dielectric function av, A (r, r'; 0).

In chapter 2 of this work, we have proved that the induction and dispersion
energies at second order are derived within the nonlocal dielectric model. Using quantum
perturbation theory, the second-order induction energy is obtained from Eq. (1.24) with
the excited states confined to either molecule A or molecule 8. At second order, the
induction energy depends on the static fields due to the permanent charge distributions of
the interacting molecules and the responses of the molecules to those static fields.
Induction energy is given within the nonlocal polarizability model by Eq. (2.1.1). When
the molecules are far apart, the induction energy can be written in terms of the permanent

moments and the multipole polarizabilities of the interacting molecules [Eq. (1.25)].

120

In section 2.1 of chapter 2, we derived the second-order induction energy in
terms of the static charge-density susceptibilities and the potentials due to the permanent
charge distributions of the interacting molecules [Eq. (2.15)]. In order to derive Eq.
(2.15), we expressed the fields in terms of the potentials and used the relationship
between the charge-density susceptibility and the nonlocal polarizability density from Eq.

(2.14). Then using the relation between the charge-density susceptibility and the nonlocal

dielectric function ev(r,r';0) in Eq. ( 1.1) we have proved that the induction energy at

second order results from the difference between the dielectrically screened and the
unscreened Coulomb energies due to the permanent charge distributions within a
molecule, where the second molecule acts as the nonlocal dielectric medium. The result
is given in Eq. (2.1.7), where the first and the second term give the screened and the
unscreened Coulomb interactions respectively, within a molecule. Thus, we conclude that
the two-body induction energy at second order is derived within the nonlocal dielectric
model as the difference between the dielectrically screened and the unscreened
intramolecular interactions between the unperturbed charge densities of the molecules.

In section 2.2, we have derived the second-order dispersion energy within the
dielectric framework. Dispersion energy results from the correlation between the charge-
density fluctuations or polarization fluctuations within a molecule. Using a reaction field
method, the dispersion energy is derived as an integral over frequency, where the
integrand is factored into the nonlocal polarizability densities of the molecules at
imaginary frequencies. This result is given in Eq. (1.29). Dispersion energy is also
obtained from the second-order perturbation theory using Eq. (1.24) with excitations

confined to both molecules. Second-order perturbation theory has been applied to

121

calculate the dispersion energy in several works. Within the density functional theory,
dispersion energy is obtained as the exchange-correlation energy.

In the present work in section 2.2, we have expressed the dispersion energy in
Eq. (2.2.1) as an integral over frequencies with the integrand factored into the charge-
density susceptibilities of molecule A and B at imaginary frequencies. Then we have used
a contour-integration technique and the symmetries of the real and the imaginary parts of
the susceptibility to write the dispersion energy in terms of the charge-density
susceptibilities of the molecules at real frequencies [Eq. (2.2.8)]. Using the fluctuation-
dissipation theorem from Eq. (2.2.9), we have expressed the dispersion energy in the time
domain. Finally, from Eq. (2.2.9) and the relation between the charge-density
susceptibility and the nonlocal dielectric function in the time domain from Eq. (2.2.12),
we have derived the dispersion energy as the difference between the screened and the
unscreened Coulomb interactions between the charge-density fluctuations within the
molecules. The final result is given in Eq. (2.2.13). The first term in Eq. (2.2.13) is the
screened interaction between the charge-density fluctuations in A in presence of B, which
acts as the dielectric medium (and similarly for B). The second term gives the unscreened
interactions between the charge-density fluctuations within the molecules. Thus, we have
proved that the second-order dispersion energy results from the screening of the
intramolecular charge-density fluctuations.

Induction and dispersion forces on nuclei at second order are calculated by taking
the negative gradients of the second-order induction and dispersion energies with respect
to the nuclear coordinates. At second order, the induction force on nucleus I in molecule

A depends on the first-order and the second-order induced polarizations in A, as given in

122

Eq. (1 .26). The second-order polarization in A results from the nonlinear response of A to
the static fields from B and is determined by the nonlocal hyperpolarizability density

A
Bafir

(r, r',r';0, 0) of A. The first-order polarization of A in Eq. (1.26) is induced by the
first-order polarization of 8 caused by the. unperturbed polarization in A. Thus, the
second order induction force on I is related both to the linear and the nonlinear response
of A. It is important to note here that the induction force on nucleus I does not stem from
the interaction of I with the polarization of B, but from the interaction of I with the
polarizations induced in A. The induction force can also be described in terms of the
nuclear shielding tensors by Eq. (1.27). The terms which depend linearly on the reaction
field from 8 and its gradients, correspond to the first term of Eq. (1.26) [i.e. the first-
order induced polarization in A]. Terms depending quadratically on the reaction field and
its gradients are related to the second term of Eq. ( 1.26) [nonlinear polarization of A].
The nonlinear shielding tensors depend on the derivatives of the molecular polarizability
densities with respect to the nuclear coordinates [Eqs. (1.18a), (1.18c)] and hence, on the
nonlocal hyperpolarizability densities [Eq. (1.28a) — (1.28b)]. Eqs. (1.19), (1.22), (1.28a)
and (1.28b) connect the electrostatic and the second-order induction forces calculated
within the nonlocal polarizability density model to the forces calculated applying the
electrostatic Helhnann-Feynman theorem.

In section 3.1 of chapter 3, we have derived the second-order induction force on
nucleus K in molecule A in terms of the nonlocal dielectric model. We began the
derivation with the induction energy expressed in terms of the static polarizabilities of the

molecules and the fields from the unperturbed charge distributions in the molecules.

Within this approach, the second-order induction force on nucleus K depends on the

123

 

 

derivatives of the nonlocal polarizability density (1A (r, r';0) of A and the field 365‘ (r) at
B, with respect to RK. The derivative of aA(r,r';0) with respect to RK depends on
BA (r, r';0) , the nonlocal hyperpolarizability density susceptibility of A [Eq. (3.1.2)]. The
field 380) in molecule 8 depends both on the electronic and the nuclear charge

densities of A and the derivative of the electronic charge density with respect to RK is

related to the nonlocal charge-density susceptibility xA(r,r';0) of A., given by Eq.

(3.1.5). Thus, we have expressed the induction force on nucleus K in terms of the
nonlocal charge-density susceptibility and the nonlocal hyperpolarizability density of A.
Next, we have written the fields in terms of the potentials, used the potential from the
first—order shift in the polarization of B from Eq. (3.1.10), and used the relation between
the nonlocal hyperpolarizability density and the quadratic charge-density susceptibility

C(r,r’,r';0,0) from Eq. (3.1.12), to obtain the second-order induction force on K within

the susceptibility based approach in Eq. (3.1.13). Eq. (3.1.13) shows that the induction
force on K depends on the nonlinear screening of the potentials due to the unperturbed
charge distribution in 8 and linear screening of the potential from the first-order shift in
the charge distribution of B. This result is consistent with the induction force obtained
previously using the nonlocal polarizability density model and the nuclear shielding
tensors. In order to show the nonlinear screening in the induction force, we have used the
nonlinear dielectric function given in Eq. (1.10). Finally using the effective potentials in
A due to linear and nonlinear screenings we have expressed the induction force on K by

Eq. (3.1.17). Eq. (3.1.17) proves that the second—order induction force on nucleus K

124

 

depends on the intramolecular screening of the external potential acting on A. This result
is similar to the first-order force derived by Jenkins and Hunt. The difference is that the
first-order force includes linear screening only, while the second-order induction force
results both from linear and nonlinear screenings within molecule A.

The dispersion force on nucleus K can be calculated the same way, by evaluating
the negative gradient of the second-order dispersion energy with respect to RK. Within

the real frequency domain, the dispersion force on K in molecule A contains two

different terms: one includes the derivative of the frequency-dependent polarizability

a(r,r’;(o) of A with respect to RK, and the second one contains the derivative of the

correlation between the polarization fluctuations within A with respect to RK. The

derivative of (1A (r,r';(o) with respect to RK depends on the nonlocal hyperpolarizability
density BA (r,r',r";o),0) of A. From the fluctuation-dissipation theorem, the correlation
between the polarization fluctuations is related to the imaginary part of the nonlocal

polarizability density, 01A" (r,r';co). Thus the derivative of the correlation is given by the

derivative of 01A" (r,r';o)) with respect to RK, which is related to the imaginary part of

the nonlocal hyperpolarizability density BA" (r,r',r";co,0). The first component of the

dispersion force resembles the first component of the induction force, with the difference
that in the case of later, the external fields are time dependent and the nonlocal

hyperpolarizability density depends on frequency. The second is quite different, since it

depends on the imaginary part of BA (r,r',r";(o,0) and shows no linear screening like in

the case of the induction force. In earlier work, it was concluded that this part of

125

 

 

dispersion force might depend on the polarization of 8. Moreover, since the correlation
between the polarization fluctuations is affected by the change in the nuclear Coulomb
field, it was noted that this field might bring new correlations and could even change the
magnitude of the correlation function (field induced fluctuation correlations). However,
those new type of correlations were not derived explicitly.

In section 3.2, we have derived the second-order dispersion force within the
nonlocal dielectric model. The dispersion energy was written within the frequency
domain, where it depends on the real part of the charge-density susceptibility of one

molecule and the imaginary part of the charge-density susceptibility of the other [Eq.

(3.2.1)]. Using Eq. (3.2.1), the first part of the dispersion force (F3609 on nucleus K has

been derived in terms of the real part of the quadratic charge-density susceptibility of A
and the correlation of the charge-density fluctuations in B. This part of the dispersion
force appears due to the nonlinear screening of the fluctuating potentials from molecule
8. Thus it is similar to the first part of the induction force. We have showed the nonlinear
screening present in the dispersion force using the frequency-dependent nonlinear

dielectric function of A [Eq. (3.2.19).

The second component of the dispersion force (RED) depends on the derivative

of xAfl(r,r';(o) with respect to RK, and hence on CA"(r,r',r';m,0). We have expressed
PEG) explicitly by expanding CA"(r,r',r';m,0) in terms of the charge-density matrix

elements of the unperturbed eigenstates and the unperturbed Bohr frequencies. From that,
we have separated the terms with j = n and j 36 n. The terms with j ¢ n have been written

in terms of the transition susceptibility of A. Then we have showed that this part of the

126

second order dispersion force [designated by FK ] actually results from the
d(2),1

correlation between the charge-density fluctuations and the susceptibility fluctuations
within molecule A. This result has been given in Eq. (3.2.30), which proves the fact that
when nucleus K shifts infinitesimally within molecule A, the change in the Coulomb
from K modifies the correlation of the charge-density fluctuations in A and actually
introduces new type of correlation, namely, the correlation between charge-density
fluctuations and susceptibility fluctuations. Finally, the dielectric screening present in

Ffile has been given by Eq. (3.2.3 5), which shows that F312” results from screening

of the fluctuating potential from 8 within A and the screening depends on the fluctuation

of the nonlocal dielectric function 8v, A(r", O; r, 0)). Terms with j = n in Fcll((2) were
separated into two sets. One set of terms where r" is directly connected to RK (denoted

by Fci((2) 2) and the other set where either r or r' is directly connected to RK (denoted

by F(11((2) 3 ). We have used time-dependent perturbation theory to explain the physical

significance of all the terms present in FCIl((2),1’ Fcll<(2),2’ and FC11<(2)I3' The external

potential in the perturbation Hamiltonian is the reaction potential from molecule 8. Using

the first-order perturbed wave functions for the initial and final states of molecule A, we

have showed that 61(0),] and Fcll((2),2 result from the interaction between the transition

charge density of A with nucleus K [ Eqs. (3.2.45) and (3.2.46)]. The difference between

and F<i<(2),2 is that FC11((2),1 is determined by the transition-susceptibility of A,

K
Fci(2),1

127

while F31?» 2 appears due to the transition charge density induced by the interaction of

the reaction potential from B with the difference between the permanent charge densities

of A in excited state n and in the ground state. Thus, Fcll((2) 2 does not have a dielectric

interpretation. [4111(0) 3 results from the fact that when molecule A remains in excited

state 11 during the time interval t— t' , the potential at nucleus K due to the average
electronic charge density changes. The transition of molecule A from state 11 back to the

ground state is induced by the reaction potential from B. The first-order amplitude for

this transition, along with the change in the potential at nucleus K, yield Fcll((2) 3. This

result is given by Eq. (3.2.50). Like Fri<(2) 2, FcIl((2) 3 can not be connected to the

dielectric model.

Nonadditivities appear in the intermolecular interactions at second and higher
orders. At second order, only the induction energy is nonadditive. At third and higher
orders, all kinds of interactions show nonadditivity. For three or more interacting
molecules, if one of the molecules has a permanent charge density, that brings a new type
of interaction, known as the induction-dispersion. This interaction first appears at third
order. Importance of nonadditivity has been noted in several experimental and theoretical
works on thermodynamic properties of fluids, structures, dynamics, and spectroscopic
properties of van der Waals complexes and collision induced transitions in compressed
gases. Several theoretical works on nonadditivity have employed the multipole expansion
technique to evaluate the many-body interaction energies, dipoles and higher multipoles

at second and third orders. Short range effects are incorporated in the many-body

128

interactions by using methods that include exchange (examples are SAPT, IMPT/MPPT).
Nonadditive interactions are also described within the reaction field method.

In chapter 4 of the present work, we have proved that the nonlocal dielectric
model describes the nonadditive three-body induction at second order. In section 4.1, we
have showed that the three-body induction energy at second order results from the
dielectric screening of the first-order Coulomb interaction between two molecules. The
result is given in Eq. (4.1.5). The screening appears due to the presence of a third
molecule, which acts as the dielectric medium. Thus, the second-order three-body
induction energy depends on the intermolecular screening of the Coulomb interaction
between the permanent charge distributions of two interacting molecules. This result is
different from the second-order two-body induction energy, which depends on the
intramolecular screening of the Coulomb interactions within two molecules. In section
4.2, we have derived the second-order three-body induction force on nucleus K. We have
followed the same method that we used to derive the second-order induction force in
section 3.1. The second-order three-body dispersion force on nucleus K has been derived
in terms of the effective three-body potential in molecule A, which depends on nonlinear
screening of the potentials from B and C due to their unperturbed charge distributions and
linear screening of the potential from the first-order shift in the charge distribution of 8
induced by C (and vice versa). The results are given in Eqs. (4.2.14) and (4.2.15).

In chapter 5, we have derived the nonadditive three-body and four-body
interaction energies at third and fourth order. We have considered a cluster of four
molecules A-"BmC-"D and specifically focused on the three and four-body induction

energies at third order and three and four body dispersion and induction-dispersion

129

 

 

energies at third and fourth orders respectively. The induction energy has been described
in section 5.1. The induction energy at third order has been separated into three
categories: hyperpolarization, static reaction potential, and third-body potential.
Hyperpolarization energy results from nonlinear response of one molecule to the
potentials from the permanent charge distributions in the other molecules. At third order,
nonadditive part of hyperpolarization energy contains both three-body and four-body
terms, given by Eqs. (5.1.4) and (5.1.5) respectively, where we have considered the
hyperpolarization of molecule A. Within the dielectric model, hyperpolarization energy is
described by the nonlinear screening of the interactions between the charge distributions

and the screening is determined by the dielectric function sq (r, r’, r'; 0, 0) [Eqs. (5.1.6) —

(5.1.7)].
Both the static reaction potential and the third-body potential effects result from
linear screening and the screening present in them are characterized by the dielectric

function av(r,r',r";0,0). In static reaction potential terms, the polarization starts and

ends at the same molecule. Thus they are purely three-body effects at third order. In Eq.
(5 .1.13), we have showed that the static reaction potential results from the dielectric
screening of the two-body induction energy at second order and is given by the difference
between the screened and the unscreened two-body induction energies. Third-body
potential effect includes terms that start and end at different molecules. Thus at third
order, third-body potential contains both three-body and four-body terms. In Eq. (5.1.18)
we have proved that the three-body terms in the third-body potential energy results from
screening of the intramolecular interactions between the permanent charge distributions

and the first-order shift in the charge distributions within a molecule. Eq. (5.1.20) shows

130

that the four-body terms in the third-body potential energy depend on the difference
between the screened and the unscreened second-order three-body induction energies,
where the fourth molecule acts as the dielectric medium. Hence we conclude that the
three-body terms in the third-body potential energy result from intramolecular screening,
while the four—body terms result from intermolecular screening.

We have derived the three-body and four-body dispersion energies in section 5.2
within the nonlocal dielectric model. We have used the susceptibility based reaction field
approach that we used in section 2.2 and in section 3.2. In section 5.2A we specifically
considered the three-body dispersion energy of A in presence of B and C. The three-body
dispersion energy at second order has been obtained as the difference between the
screened and the unscreened two-body dispersion energy of A at second order, where the
third molecule (either 8 or C) acts as the nonlocal dielectric medium. We have expressed
the screening in Eq. (5.2A.4) where the nonlocal dielectric function and the charge-
density fluctuations are given in the time domain. We have also showed that the three-
body dispersion energy of A results from the screening of the interaction between the
intramolecular charge density fluctuations in A, where both 8 and C act as a combined
dielectric medium. To derive this result, we have used the two-body susceptibility used
by Kohn, Meir and Makarov (Eq. 5.2A.5) and the two-body collision-induced electronic
polarizability (Eq. 5.2A.8) introduced by Li and Hunt. Finally, we have derived the
screening in the three-body dispersion energy in Eq. (5.2A.16) in terms of the two-body
nonlocal dielectric function in the time domain. Eq. (5.2A.16) proves that the three-body
dispersion energy of A depends on the screening of the intramolecular interactions of the

charge-density fluctuations in A, due to the presence of B and C which together act as a

131

single dielectric medium. Following the same approach in section 5.28, we have showed
that the four-body dispersion energy of A at fourth order results from screening of its
three-body dispersion energy, due to the presence of a fourth molecule (8 or C or D).
This result has been given in Eq. (5.2B.4). We have also proved that the four-body
dispersion energy can be obtained as the difference between the screened and the
unscreened interactions of the charge-density fluctuations within A, due to the presence
of B, C, and D which together act as a three-body dielectric medium. In order to derive
that result, we have used the three-body collision-induced polarizability introduced by
Champagne, Li, and Hunt [Eq. (5.28.5)]. The final result is given in Eq. (5.28.13), where
the screening has been described in terms of a three-body nonlocal dielectric function.
Finally in section 5.3, the induction-dispersion energies at third and fourth orders
have been derived in terms of the dielectric model. The induction-dispersion energy
results from the perturbation of one molecule in the dispersion cluster due to static
external potential from another molecule. The molecules are hyperpolarized by the
simultaneous action of the fluctuating potential of one molecule and the static external
potential from the other molecule. Thus, the induction-dispersion energy results from
nonlinear response, and hence from nonlinear screening. We have derived the dielectric
screening in terms of the nonlinear dielectric function in the time domain. Eq. (5.3.5)
proves that the induction-dispersion energy at third order results from nonlinear
interaction of the charge-density fluctuations in A and the permanent charge density in C,
due to the presence of B which acts as the nonlinear dielectric medium. Next, we have
derived the four-body induction-dispersion energy within the dielectric model in Eq.

(5.3.7), where we have proved that the four-body induction-dispersion energy of A

132

results from nonlinear interaction of the intramolecular charge-density fluctuations in A,
first-order induced shift in the charge distribution in 8 and the permanent charge
distribution in D, in presence of C that acts as the nonlinear dielectric medium. We have
also showed that the four-body induction-dispersion energy depends on the difference
between the screened and unscreened three-body induction energy, where the screening is
linear [Eq. (5.38)].

Our work derives a generalized dielectric model to treat the intermolecular
interactions at second, third and fourth orders. The nonlocal dielectric functions

av(r,r';o)) and sq(r,r',r';w,w') provide complete description of the intermolecular

screenings and the effective potentials are accurately described by Eq. (1.11) within the
Bom-Oppenheimer approximation. As a first step of our firture work, we would like to

compute the numerical values of the nonlocal dielectric functions for diatomic molecules.

Numerical value of the frequency-dependent charge-density susceptibility of H2 has been
computed at the CISD level in ref. 285. To calculate av(r,r';o)) we will have to use Eq.

(1.2), where we will have to evaluate the nucleus-electron attraction type integrals over
the atomic orbitals. We would also like to compute the nonlocal dielectric functions for
larger molecules. Practical application of the dielectric model to larger molecules can be
done by coarse graining the model, i.e. averaging over regions around each nuclear
center, or over functional groups. Another possible future work will be to include short-

range interactions within the model to account for the overlapping molecules.

133

APPENDICES

134

Appendix A. Derivation of the relation between C”(r,r',r";co,0) and the derivative of
X"(r,r';m) with respect to RK

To derive VK x"(r,r';(o) , we first differentiate with respect to a real quantity (the
nuclear coordinate) and then take the imaginary part of the result. We use the inverse

radiative lifetime Fn for state In) to approximate the effects of damping near resonance.
This approach neglects branched or successive decays of the excited states.

VK x(r,r';o)) = —(1/I‘1)VK Z [(0 p(r)|n><n|p(r')|0> ((ono —iFn /2—o))_—1

n¢0

 

+(0|(3(r') n><n|p(r)|0>(mn0+iFn /2+co)‘1]. (A.1)

 

The derivatives of the states are given by

V§|g> =—(1/h) Z lj)(le<11< H|0>m}i)
j¢0

I K‘1 . .. I -1
r—R I Z|J><j|p(r)|0>wj0 (A.2)
j¢0

= —(4nsoh)‘1 [dr'zK Vii

 

and similarly

I K71 . , -1
r—R ’ 2 |m><m|p(r)n)comn, (A.3)

m¢n

VaK |n> = -(47t80h)-1 [dr' ZK VaK

 

 

where damping is neglected for static perturbations. Also

v§ (tong +11“n /2i(o)_1 = -(1/h)(wn0 +irn /2i(o)—2 [<n|V§ H|n>—(0|VaK H|o)]

—1
= -(oon0 + iFn / 2 i 00-2 (47530”).1 idr' ZK V‘If r, —RKi

 

X (“I50") - Poo (r") | n) I (A-4)

135

where we have used the Hellmann—Feynman theorem. In Eq. (A.4) we have neglected the
dependence of the inverse radiative lifetime on the nuclear coordinates, treating I], as a

property of the entire adiabatic electronic state. The off-diagonal elements of the
derivatives of the charge-density operators themselves vanish, because the only explicit

dependence of p(r) on the nuclear coordinates comes from the nuclear contribution to the

charge density, and states I j) are orthogonal Bom-Oppenheimer electronic states. From

Eqs. (A.2) — (A.4), the derivative in Eq. (A.1) becomes

VK x(r,r;—o))— (1/h)2(411:30)_1 Idr' zK

     

H-RKI

XIXZI

           

><Ip<r>lnn><I13<--'>I0>w,-‘o1 (wno—irn/z—wr‘

 

     

   

     

   

 

     

     

         

         

       

           

         

n¢0j¢0

I42 2(0 p(r)|k)(k )n)(n|,3(r')|o)m;;(mn0—irn/2—m)"1
n¢0k¢n

XI 2 z <0Ip(r)In><n ~ ) k)(k|,s(r')|o)m;}1 (mnO—iFn/Z—(o)_1
n¢0k¢n

x12 233413313 ')r->013 ~>o)w;g<wno-irn/2—wr1
n¢0j¢0

x12 Z<0 ><Ip<r>l nn>< nIé<r>I0>wgJ<wno+irnam)“
n¢0j¢0

x12 Z<0 '><>Ikk >n><nlé<ril0>wgg<wno+irnmm)"
n¢0k¢n

x12 Z< ” '>n><n )k><klé(r>I0>w}3<wno+iFn/2+co)“
n¢0k¢n

136

XIX 20>

n¢0j¢0

'n)l><n|i‘1(r)lj><j“ ")

0) 0):;01 (cono+il‘n / 2 +00)—1

         

       

X Z <0lf><r>|n><nlfi(r'>|0><nlf>(r")—poo(r")In><cono —iI‘n /2-co)‘2

n¢0

X Z <0|13(r'>|n><n|13(r>|0>(n|f>(r">-poo(r")ln>(wn0+irn / 2+w>'2].

n¢0
(A.5)

Next, we separate the terms with k = 0, in the summation over k i n; this gives

vK x(r,r';w) = (1/15)2 (41:2.0)‘1 Idr'ZK

     

—1
r"—RKI

><I Z Z <0I13<r">I1><1I1’1<n->In><nli><r'>l031,701 «one —irn /2-w>“
n¢0j¢0

><I Z Z (0113(r>|n><n ‘

n¢0j¢0

XIZZX

n¢0j¢0

XIX Z< ‘ '>

n¢0j¢0

XIX Z<0lfi<r>lk>< ‘ '>
n¢0 k¢0
k¢n

') .00 A ")

0) (03.01 (wno — 11“,, / 2 — (.5)—1

         

         

           

">1'0>

)nn)(n|3(r)|o)m‘j’01(mn0+irn /2—m)’1

         

           

n><nIp(r)Ij><iji(r”)IO>m3701(cono urn/2+1»)-1

n) (n I p(r') I 0) (31:111 (tong — iFn / 2 — co)—1

           

           

n)<nII3(r')I0)(1)6r1l (wno — iFn / 2 — 03)—1

XI 2 <0|13(r>|0>< ‘ ">

n¢0

x[ z z (0|(3(r)In><n * ~) >013, (cone—in. /2—m>‘l

n¢0 k¢0
k¢n

k>< * )0

           

         

137

x [ Z <0Ifi(r)In><nIfi(r") 0) (0|{5(r')|o) (1)511 (mno —1rrl /2—co)"1

n¢0

><I Z Z (0|13(r')
n¢0 k¢0
k¢n

 

 

1904150")

 

 

n) <nIp(r') monk-11(0)” +il‘n /2 +00)-1

x1 2 <0|13(r')

n¢0

XI 2 Z (0030')
n¢0 k¢0
k¢n

 

0> <0l13<r">

 

n) (n I p(r) I 0) (1)611 (con0+ iFn / 2 + (of—1

s(r') k)(k|13(r)|o)m;11 (wno +11“n /2 +111)—1

 

n) (n

 

 

x[ Z <0Ip(r')In)<nIf>(r”) 13(1')I0>0)6r11((0n0+irn/2+03)_1

n¢0

 

0) (0

 

x I Z (013(I)|n)(n|,s(I-) axiom) —1rn /2-o))_2

n¢0

 

 

0)<nl13(r")-poo(r")

>< [ Z (0 l 66') n) (n l 130') I 0) (n l 130'") - 9000'") l 0) (01110 + irn / 2 + co)—2 ]

n¢0

 

(A.6)

1th

We interchange the labels n and k in the 7th and 1 terms in Eq. (A6) and combine the

terms, to obtain

—1
vK x(r,r';co) = (1/11)2 (471.90)‘1 Idr'zK vK r"—RKI

 

—1

><I Z Z (000’)

n¢0j¢0

 

 

j)<jl13(r)—poo(r)ln>(nlf1(r')

 

 

X Z Z (0|13(r>|n>(n|f>(r'>-Poo(r'> j>0|i3(r")

—1
o)m.‘1(mn0 —1rn /2—(o)
jO
n¢0j¢0

138

—1
)nn>< Ip(r)I0>co coj()1(0°n0+irn/7—‘*'(0)

X Z Z (0|P(r">| >10 |P(r'

             

   

             

         

             

n¢0j¢0
XX Z<0|P(r'n><>ln|P(r>-Poo(r>|1'1'><|P(r" 0>mj-01(wn0+irn/2+w)—
n¢0j¢0
X Z 20) ><k|P<"” >PP>< '>0>(<Pno-i1“n/2-011>‘1
n¢0k¢0

X Z Z <0|P(r'>|k>0< ‘

n¢O k¢0

> n><nlé<r>l0><wno+irn >2+wr1

             

       

x((un0 —iFn /2 —m)‘1 ]. (A.7)
We assume that the inverse radiative lifetimes l"n and Fk are much smaller than the

transition frequencies mnk, (an- Using the identity

 

lim 1, =PI—l—)$i715(y), (A.8)
y__)()xi1y x

we take the imaginary part of Eq. (A7) and the relabel the summation indices to obtain

—1
VK)("'(r,r';o))=(11;/h)2(47t.s())_1 Idr'ZK r'—RKI

     

“XXI

n¢0j¢0

         

)j)0li‘1(r)- 1100(r>|n >01

           

) 0)co;01 S(mno — (0)

+ Z Z <i>IP<r->In><nIi><r'>—poo<r'>I1><1I13<r">I0>w;01 5011110 —w>
n¢0j¢0

- Z 2 <0|P(r">l >10 |P<r'-

n¢0j¢0

)nn>< Ip(r)I0>o)j018(con0+o1)

             

139

 

- Z Z <01P(r'>|n><n|P(r>-poo<r>|j>0|1><r">

—1
0) ij S(mno — 0))
n¢0j¢0

 

n><P|P<I">

 

+ Z Z(0|P(r)lj)<jl13(r")-poo(r") 0>5(mn0—m)

n¢0j¢0

xRe[(mj0—irj/2—m)‘1]

 

+ Z Z <01P<r>ln><nIP(r">-poo<r') j><j|é(r')|0>8<wno -3)

n¢0j¢0

xRe[(ij—irj/2—m)‘1]

 

‘ Z Z <0|P("'>IJ'>0'IP("'>~Poo(r"> n)<n|13(r)|0>5(wno+m)

 

 

n¢0j¢0
xRe[(o)jO+iFj/2+o))_1]
- Z Z<0|13(r') n)(nl13(r")-Poo(r") j) (il13(r)l0)5(wno+w)
n¢0j¢0
xRe[((ojO+iFj/2+o))_l]}. (A9)

A comparison of Eq. (A9) and 8;"(r,r’,r";o),0) yields

—1
VK x"(r,r';(o) = (471:80 )_'1 Idr" ZK VK r" — RKI (;"(r,r',r";(o,0) . (A.10)

 

140

Appendix B: Derivation of the four-body irreducible dispersion energy for the
cluster of four interacting molecules

In this section we derive the four-body irreducible dispersion energy of a cluster
of interacting molecules A-"B-"C-"D using the reaction field approach. We derive the
dispersion energy as an integral over frequencies, where the integrand is factored into the

charge-density susceptibilities of the interacting molecules at imaginary frequencies.

The charge density fluctuation SpA (r';to) in molecule A at r' produces a

fluctuating potential that propagates through the surrounding molecules inducing
transient changes in their charge densities. The potential which acts back on molecule A

at r and time t is the reaction potential, and it’s given by

r _ rv11

—1 .. .
XD (rvn , l,vr ; (o)

5(pA (r;m) = (41180)—4 sBCD Idr'...drv“

 

 

_1 M

—1 6pA (r'; (o) .

 

. . —-1
XC (rv , l,rv ; co) r1v _ r XB (rm, I,II; co) Ir. _ rI

 

 

 

 

(8.1)
The operator SBCD denotes the sum of the terms obtained by permuting the labels 8, C,

and D in the expression that follows the operator. The average energy shift of molecule A

due to the interaction between the fluctuating potential S(pA (r;t) and the charge density
. A . .
fluctuation 8p (r,t) rs
ABA = (1 / 2) Idr< 5pA(r,t)6(pA (r,t) >. (8.2)

From Eqs. (8.1) and (8.2) along with the Fourier transforms of the charge-density

fluctuation and the potential, we obtain

141

I'Vil

         

 

ABA = (1/2)(47t80)_4 SBCD End") Eda) Idr" drVii<A

vii rvi v _

”I

1
xB(r".r';w>

 

rw —l‘

     

1 .
xC(rv,r'v,

 

 

xx I)(r ;1'01)v —-r

I I—1

xr—r

 

 

6pA(r',0))> exp[—i(0)+0)')t. (8.3)
+

The charge density fluctuations 8pA(r,0)') and 8pA(r',0)) are correlated by the

fluctuation-dissipation theorem 1 37

(1/ 2)< SpA (r, 0)') SpA (r', 0)) + 8pA (r', 0)) SpA (r,0)')>

= _ (h / 2n) XA'(r, r'; 0)) 500+ 0)')coth (710) / 2kT) . (3.4)

Substituting (8.4) in (8.3) and integrating over 0)’ , we obtain

 

 

 

 

     

 

.. .. —1
ABA =—(1/2)(h/27t)(47c80)—4 53CD Eodw Idr...drV“r—rV“ x D(rV“, rV';m)
. —1 c . 1 3
erI—rv X (rv,r1v, r1v_rIII X (r..,rII; 0))
II I—l A" I .
xr —rI x (r,r,(o)coth(h(o/2kT). (8.5)

 

In the limit T —> 0, the hyperbolic cotangent function simplifies to [0(01) - 0(-0))], where

0(0)) is the Heaviside step function (cf. chapter 3). Thus, in the limit T —+ 0,

 

 

 

 

 

     

.. ..—1
ABA = —(1 / 2) (h / 21:) (41:80 )—4 sBCD Re fiodco Idr...drv" r—rv"
vii vi v v"1 C v iv 1v — 8
xx [’0 ;w)r '—r x (r .r . r -r" x (r",r';w)
><Ir"—r'1)(A (r', r; 0)), (8.6)

 

142

 

 

 

 

where Re denotes the real part of the expression that follows. Using the Kramers-Kronig

relation between the real and the imaginary parts of the charge-density susceptibility,

x'(r,r';0)) = (UK) P L: dx x'(r,r';x) (x — 0))_1 (8.7)

(where P denotes the Cauchy principal value of the integral) we can write the charge-
density susceptibility as

x(r,r';0)) = x'(r,r';co)+ix'(r,r';(o)
_ II I, _ “-1 - _ II I.
— (l/It)P E000 dx x (r,r ,x) (x 0)) +1 [:Ddx 5(x 0))x (r,r ,x)

= lim (1/112) dxx"(r,r';0))(x—0)—i§)_1. (8.8)
~§—>O

In deriving Eq. (8.8), we have used the identity given in Eq. (A.8). From Eq. (8.8) and

the fact that x”(r,r';0)) is an odd firnction of frequency, we obtain

x(r,r';0)) = lim (2/ It) dxxx"(r,r';x)[x2 —(0)+iE,)2 ]—1 . (8.9)
§—>0

We use Eq. (8.9) in Eq. (8.6), take the sum AEA + AEB + ABC + AED in the limit I; —>
0, and then use the symmetry of the charge-density susceptibility with respect to

interchange of its arguments, to obtain

AE(4 4): —(3h/21t 4)(4neo)‘4 1:011 [Zodx Eody Eodz Ian. .dl'v“

{[uxyiu2 —22)“ (x2 —22)‘1 (y2 —22)“1+1xyz(x2 -u2)‘l (y2 _u2)—1(22 —u2)‘11

+1uyz<u2-x2)‘1 02 —x2)'1 (22 —x2)‘11+1uxz<u2 —y2)“ (x2 —y2)“<22 —y2)“1}

143

vii—1D» vii v1 vi v‘ C» v iv iv ‘1
xr—r x (r ,r ;z)r —r x (r ,r ;y)r —r"'

 

 

 

 

 

 

>< xB" (r'", r”; X) 11"’-r’|—1 xA' (r', r; n) . (B. 10)

 

The frequency integral over u, x, y, and 2 can be separated into independent quadratures

using the identity
[min2 —22)‘1 (x2 -22)“ (y2 —22)“1+Ixyz<x2 -u2)"1 (y2 _u2)-1(22 _u2)_1 1

+1uyz<u2 —x2)‘1 02 —x2)“(z2 —x2)‘11+1uxz(u2 —y2)“ (x2 -y2)'1 (22 —y2)“1

= (2/1t)£0d0)uxyz[u2 +012]_l[x2 +(I521‘1[y2 +021‘1[22 +0214. (3.11)
Using Eq. (8.11) and in Eq. (8.10) and then using the relation
2 2 —1 II I, _ '.'
dxx(x +0) ) x (r,r ,x) — (rt/2)x(r,r,10)) (8.12)

which holds in the limit of infinitesimal damping, we obtain

—1

 

 

 

 

 

 

AEE14’4) = _ 2?. (41:30)”4 £0 (10) Idr...drVii r — rVii xD (rVii , rVi ; i0)) rVi — rv
7t
x XC (rv , riv ; i0)) riv — r'" xB (r", r";i0))
x r'—r'I—1xA(r',r;i0)). (8.13)

 

Eq. (8.13) gives the irreducible four-body dispersion energy at fourth order for the

cluster of interacting molecules A---B---C---D.

144

 

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