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dissertation entitled
LINEAR RESPONSE FUNCTIONS OF AN INTERACTING
FERMI GAS AT T = 0

presented by

Raj asinghe Nimalakirthi

has been accepted towards fulfillment
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LINEAR RESPONSE FUNCTIONS OF AN INTERACTING FERMI GAS

AT T=O

By

Rajasinghe Nimalakirthi

A DISSERTATION

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

DOCTOR OF PHILOSOPHY

Department of Chemistry

1993

ABSTRACT
LINEAR RESPONSE FUNCTIONS OF AN INTERACTING FERMI GAS
ATT=O
By

R. Nimalakirthi

The response of an interacting Fermi gas in its ground state to weak external
electromagnetic fields is analyzed in this thesis. The system’s response to arbitrary scalar
and vector potentials has been Studied within the random phase approximation.

The electrical response is characterized via the nonlocal polarizability density
denoted by a(r, r’; to), which gives the polarization P(r, (0) induced at point r in a system
by a perturbing electric field E(r’, to) acting at the point r’, within linear response. A
homogeneous electron gas at zero temperature is selected as a well characterized system,
for the purpose of determining the nonlocal polarizability density and thus gaining
information about the nature and functional form of a(r, r’; to). The longitudinal
component (in It space) of the nonlocal polarizability density a(r, r’; to) is connected to the
dielectric function £(k,m), and this connection is used to obtain results at two levels of
approximation. Results from the Thomas-Fermi ('I'F) form and the random phase
approximation (RPA) for 8(k,tr)) are compared. At TF level, the nonlocal polarizability
density is evaluated analytically, while within the RPA asymptotic analytical results are

obtained. The RPA results are qualitatively distinct from the TF results, which diverge as

Ir - r’ I approaches zero. Within the RPA, there are two long-range components in
OLO‘, r’; O): the first is a monotonically decreasing component that arises from the net
charge screening in the electron gas, and varies as I r - r’ I’3. The second is an oscillatory

component with terms of order Ir — r’ I’“ (n 2 3). The latter is associated with Friedel
oscillations in the electron density as found in Langer and Vosoko’s study of the screening
of an impurity charge. The results indicate the possibility of long-range, intramolecular
terms in the nonlocal polarizability densities of individual molecules.

For molecular systems, it is shown that the change in nonlocal polarizability
density due to an infinitesimal shift in nuclear position is determined by the
hyperpolarizability density. The same hyperpolarizability density describes the
electronic charge distribution’s nonlinear response to external fields.

A method is provided to obtain the asymptotic form of the dynamic charge density
susceptibility, as a function of space and time variables, for a homogeneous electron gas
treated within the RPA. It is shown that the calculation reduces to a single quadrature over
frequency.

Explicit expressions for current density susceptibilities as a function of transferred
momentum and frequency are obtained within the RPA in a gauge in which the scalar

potential is zero.

TO MY PARENTS

iv

ACKNOWLEDGMENTS

My sincere thanks go to Professor Katharine C. Hunt for her guidance,
encouragement and support throughout this Study. I would also like to thank to my
committee members Professor R. I. Cukier, Professor S. D. Mahanti and Professor P. M.

Duxbury for their valuable suggestions.

TABLE OF CONTENTS

LIST OF FIGURES

Chapter

1.

2.

5”

INTRODUCTION

THE GENERAL THEORY OF LINEAR RESPONSE

a. Quantum Mechanical Response Theory

b. Double-Time Green Functions and the Time Correlation Functions

COLLECTIVE OSCILLATIONS AND THE RANDOM PHASE
APPROXIMATION

a. Equivalence of Self-Consistent Field Approach and Collective
Approach based on RPA

THE LINEAR RESPONSE TO SCALAR AND VECTOR
POTENTIALS IN THE RPA

CHARGE DENSITY SUSCEPTIBILITY

a. Relation between the Nonlocal Polarizability Density and
the Dielectric Constant

b. Asymptotic results for the Nonlocal Polarizability Density
in the RPA

c. Fourier transform of Dynamic Charge Density Susceptibility
CURRENT DENSITY SUSCEPTIBILITY
a. Transverse Current Density Susceptibility

b. Longitudinal Current Density Susceptibility

viii

Page

11

15

19

23

33

34

38

49
50

52

7. CHANGES IN ELECTRONIC POLARIZABILI'TY DENSI'TIES DUE
TO SHIFTS IN NUCLEAR POSITION

8. SUMMARY AND CONCLUSIONS
APPENDIX A
APPENDIX B

REFERENCES

54

61

63

68

69

LIST OF FIGURES

Figure 1. Asymptotic form of the nonlocal polarizability density in the
random phase approximation, from Eq. (23). The figure shows
“RP A a (21:)3 3/1: kF'3 103 aLzz(r, r’; 0) as a function of
kFx a kF (r - r’)x and sz a kF (r - r’)z, with rs = 4. The range of

plotted am, A values is [-2.0, 2.0]. Friedel oscillations are evident
in the plot.

Figure A. 1. Contour in the complex k plane, used for evaluation of the nonlocal
polarizability density in the RPA. Branch cuts are shown as striped
lines; the symbol x marks the branch cut origins at the points
i2kpiin, apole at k=0, and apole neari kTF'

viii

1. INTRODUCTION

Within linear response theory [1, 2], the properties of a perturbed system are
expressed in terms of its equilibrium properties. References 1 and 2 treat the response
of a system to an external force by a simple perturbation method, assuming that the
external perturbation can be expressed as an additional term in the Hamiltonian and
that it does not drive the system far from equilibrium. Kubo [2] has shown that the
complex susceptibility and complex conductivity, and hence the electric polarization
and induced currents, can be rigorously expressed in terms of equilibrium time
correlation functions of the associated dynamical variables. For example, it was
shown by Kubo that the conductivity tensor for a given frequency of the applied field
can be rigorously expressed in terms of electric current components fluctuating
spontaneously in the equilibrium state. There has been considerable interest in
evaluating transport coefficients given by Kubo’s formulas.

One of the important tools of quantum field theory is the Green function, which
is convenient for the study of the properties of interacting quantized fields. These
tools turn out to be useful also in statistical mechanics, in cases where one can sum
the same type of perturbation theory diagrams. The application of Green functions is
fruitful in the quantum theory of fields when combined with spectral representations.
Spectral representations for the time-correlation functions and for retarded Green
functions were first established and used in statistical mechanics in the theory of
fluctuations and in the statistical mechanics of irreversible processes, beginning

with a paper by Callen and Welton [1]. Retarded and advanced Green functions, and

their simplest applications to the theory of irreversible processes are discussed at
greater length in Zubarev’s paper [4]. They are very convenient for application in
statistical mechanics as they can be analytically continued in the complex plane.
Linear response functions are most simply expressed in terms of retarded double
time Green functions.

Because of the long range of the Coulomb force, the interactions in a collection
of electrons involve many particles simultaneously. It is well known that an electron
gas of high density can undergo organized oscillations resembling sound waves.
These oscillations, the so—called “plasma oscillations” represent the effect of the
long-range correlations of electron positions brought about by Coulomb interactions.
A description in terms of these organized oscillations therefore provides a natural
way of heating the long-range electron interactions, and leads to greater insight into
the dynamical behavior of the electron gas. We can distinguish between two kinds of
response of the electrons to the field. In one of these, the phase difference between
the particle response and the field producing it is independent of the position of the
particle. This is the response which contributes to the organized behavior of the
system. In the other the phase difference between the field and the response depends
on the position of the particle. Because of the general random location of the particles,
this second response tends to average out to zero when we consider a large number
of electrons, and we can neglect the contribution arising from this. This procedure is
the “random phase approximation” (RPA). The collective description is similar to a

complete perturbation theory treatment in that the perturbation is applied to the

collective particle motion.

The behavior of the electrons in a dense electron gas can be analyzed in terms
of their density fluctuations. These density fluctuations may be split into two
components. One component is associated with the organized oscillations of the
system as a whole. The other is associated with the random thermal motion of the
individual electrons and shows no collective behavior. This split up of the density
fluctuations corresponds to an effective separation of the Coulomb interaction into
long-range and short-range parts; the separation occurs at roughly the Debye length.
Bohm and Pines [5] used the above split-up of the density fluctuations to study the
collective response of the electron gas to the field of an individual charged particle

moving with a specified velocity v0. When Va is less than the mean thermal speed of

the gas, they found that the collective response is just such as to screen out the field
of the specified particle within a distance of order of the Debye length.

The self consistent field (SCF) method, in which a many-electron system is
described by a time-dependent interaction of a single electron with a self-consistent
electromagnetic field, has been shown to be equivalent to the dielectric approach of
Noziéres and Pines [9] to the many electron problem, in work by Ehrenreich and
Cohen [10]. Ehrenreich and Cohen’s paper establishes the relationship among the
equation of motion approach, the quantum kinetic equations, the calculation of the
dielectric function, and Landau’s Fermi liquid theory. The simplicity and ease of
interpretation of the SCF method commends it for problems such as the calculation of
the dielectric constant and the response of the system to a general external

perturbation. By studying the system’s response to an arbitrary scalar potential and

a vector potential we can evaluate the induced particle density and current density
respectively. By using the facts that the transverse vector potential excites only a
transverse current and is not screened by the longitudinal Coulomb interaction we
separate the vector equation into its transverse and longitudinal parts. By using the
equation of continuity, we obtain a result for the longitudinal current self-consistently.

A knowledge of the RPA dielectric constant at arbitrary frequency enables one
to calculate a number of interesting properties of the electron gas. It also predicts

correctly a number of properties of the electron gas such as the plasmon frequency. In the

RPA, since Im §RPA(q,co) vanishes for the plasmon modes, there will be no damping

of the plasmons. The plasmon frequency is determined by the dispersion relation

Re §RPA(q,o)) = 0 [11]. At small values of q, Im §RPA(q,to) is proportional to to at
small values of to. The proportionality to to is an important feature of Im §RPA(q,to).

The long-wavelength plasmon will now be damped. The linear dependence of

Im gRPA(q,(o) on to must also occur for the exact dielectric function. The physical

property under consideration is the rate at which electron—hole pairs are made in the
electron gas [12]. A hole is a state which has an electron removed from the filled Fermi

sea. An initial electron of momentum p and energy SP is excited by a perturbation

with (q,(o). Thus it is excited to a new state with momentum p+ q and energy

£P+q = ep+ to. The electron can only be scattered into states which are previously

unoccupied, so 8 must be above the occupied Fermi sea. Thus the basic process

P“!

takes an electron from below to above the Fermi level. Thus the excitation processes

make electron-hole pairs. By simple arguments, it can be shown that the rate of
making electron-hole pairs is proportional to a), at small values of to. At small values
of a) we can expect damping to be small.

The calculation of the induced screening charge density (about an impurity) in
the RPA has been carried out by Langer and Vosko [13]. In contrast to the Thomas-Fermi
calculation, the screening density at the origin is finite in the RPA. In the RPA, the
screening density does not go to zero exponentially, but rather oscillates at large values of
r. The second feature had appeared in earlier calculations of Friedel [15]. These

oscillations come about as a result of a logarithmic singularity in §RPA(q,0). At q = 2P1:

its first derivative becomes infinite. The physical origin of the singularity is not difficult to

trace. When q < 2pF, one may excite an electron from one part of the Fermi surface to the
another; the electron hole excitation spectrum begins at zero energy. When q > 2pF, one

must instead supply energy to excite an electron; the corresponding excitation spectrum
begins at some finite energy. Friedel oscillations are an exact microscopic property of
normal electron liquids; they may be considered as a direct reflection of the sharpness
of the Fermi surface [11, 16].

This thesis focuses on susceptibilities of the electron gas and molecular systems,
specifically the charge-density susceptibility, the current-density susceptibility, the
nonlocal polarizability density, and the hyperpolarizability density.

The nonlocal polarizability density is a linear response tensor that gives the
polarization P(r, or) induced at a point r in a molecule by a perturbing electric field

E(r’, to) that acts at the point r’ [18-23]: The nonlocal polarizability density was

introduced by Maaskant and Oosterhoff in a study of optical rotation in condensed media
[18]. In their work, o(r, r’; or) is expressed as a difference of two components that are
individually divergent as (u —-) O, and each transition matrix element in the expression
involves an operator specified only as an infinite series [8]. For response to fields that are
derivable from a scalar potential, a(r, r’; to) can be recast in a computationally tractable
form, via a connection [22] between its spatial Fourier transform and that of the charge-
density susceptibility a(r, r’; to) [24-29]. The function a(r, r’; 0)) determines the change
in charge density at point r in response to a perturbing scalar potential acting at r’, within
linear response. This connection gives the longitudinal component of the polarizability
density aL(r, r’; (o) [22].

In chapter 2, we briefly review the general theory of linear response. Collective
oscillations and the RPA are discussed in chapter 3. Chapter 4 contains a discussion of the
linear response to scalar and vector potentials acting as perturbations. There we obtain the
transverse and longitudinal current density susceptibilities within the RPA in gauge-
nvariant form. It is shown that the longitudinal current density is screened within the
RPA. The new results of this thesis are contained in chapters 5, 6, and 7.

In chapter 5 we give a generalization of the Langer and Vosko result by evaluating
the charge density susceptibility and the nonlocal polarizability density, for a scalar
potential of arbitrary spatial variation. We establish a connection between the static,
longitudinal component of the nonlocal polarizability density in position space and the
dielectric function E(k, 0), and then use the connection to obtain results at two levels of
approximation to 80‘, 0): we compare the results from the Thomas-Fermi (TF) [11,12] and

the RPA approximation. Within the TF approximation, we obtain analytical results,

while within the RPA, we obtain asymptotic analytical results.

In chapter 5 we also describe the extension of the approach used in previous
sections to evaluate the dynamic charge-density susceptibility. We develop a method to
evaluate the asymptotic form of dynamic charge density susceptibility within the RPA.

In chapter 6 we study the response to a field of arbitrary polarization. It is
most convenient to choose a gauge such that the vector potential alone represents
the applied electromagnetic fields, and study the induced current. A given longitudinal
electric field produces the same current, whether it is by a vector potential or by a scalar
potential; thus the induced polarization current is gauge invariant. Explicit expressions
for transverse and longitudinal current density susceptibilities are obtained in (q,m)
space within the RPA.

In chapter 7, we develop a general result for another linear response tensor,
the polarizability density. The results are obtained for an isolated molecule in its
ground electronic state. We prove by direct perturbation theory that when a nucleus
in a molecule shifts infinitesimally, the resulting change in polarizability density is
determined by the same hyperpolarizability density that fixes the response to external
fields. This is a general quantum mechanical result. It is rather straightforward, but
previously unanticipated. It has applications in the analysis of interpretation of
intensities of vibrational Raman bands.

Chapter 8 contains a brief summary and conclusions. Mathematical identities

needed in the calculations are presented in two appendices.

2. THE GENERAL THEORY OF LINEAR RESPONSE

The formalism of linear response dreary is straightforward. We consider a
system that in the distant past (t—>-oo) was prepared in a State of equilibrium. Let us
suppose that in the distant past an external force was switched on adiabatically. As
a result, the system is no longer in stable equilibrium. On the other hand, internal
affairs in the system should not be much affected by a small force, and the lowest-
order response that moves the system to a new stable state compatible with the
imposed constraints can be connected to local equilibrium fluctuations. We must, first
of all, estimate when an external force can be considered small. In the most general
case, we can argue that the energy transmitted to a particle by the external force over
a characteristic distance in the system has to be small compared to its average
energy in local equilibrium [la].

This criterion is, in fact, very conservative in a situation where the external
force is switched on adiabatically. “Adiabatically” here means slow on the time scale
of the regression of local fluctuations. In this case the system only has to adjust at
any instant of time by an infinitesimal amount to the new external constraints.

If we follow the time evolution of the system via the statistical operator, this implies
that first-order, time-dependent perturbation theory should be adequate [1a]. Such
an approach leads to constitutive relations that are linear in the applied external
forces [1a]. It has therefore received the name “linear response theory”. The use

of first-order time dependent perturbation theory in the study of transport phenomena
in many-body systems was advocated in an early paper by Callen and Welton [ I] .

8

The following review is patterned on a paper of Kubo [2], where a rather general
framework was laid out for the calculation of the systems response to mechanical
forces within a Hamiltonian formalism [1a].

Linear response theory makes an assumption that the applied external force
must be such that it can be incorporated as part of the total Hamiltonian of the
system. This, of course, means that the results of linear response dreary are as
general as can be, and are, as with most general results in physics, not directly useful
for practical calculations [la]. However, linear response theory is eminently suited to
prove general features of nonequilibrium physics, such as positivity of transport
coefficients, validity of the Onsager reciprocity [3] relations, sum rules, and dispersion
relations. To do practical calculations based on linear response theory, one can either
employ equilibrium Greens function techniques, use kinetic equations, or resort to
semi-empirical models with experimental data as partial input. The latter approach
is often rewarding since linear response theory expresses transport coefficients
in terms of time-dependent correlation functions, which are quite often available fiom

experiments [1a]. However in the work presented in this thesis, the former

method is followed.

a. Quantum Mechanical Resmnse Theog

Let us consider an isolated system, the Hamiltonian of which is denoted by H.

The dynamical motion of the system determined by H is called the ‘ ‘natural motion”
of the system. We suppose that an external field F(t) is applied to the system, the

effect of which is represented by the perturbation term,

10

H’ (t) = -AF(t) (2.1)
The motion of the system is perturbed by the external force, but the
perturbation is small if the force is weak. The response is observed through the
change AB(t) of a physical quantity B. The problem is now to express AB(t) in terms
of the natural motion of the system.
The initial ensemble which represents statistically the initial state of the
system is Specified by the density matrix p satisfying [H, p] = O [2]. The motion of
the ensemble under the perturbation (2.1) is represented by p’(t), which obeys the
equation,
BOYD/at = 1/ il'f [H + H'O). MD] (2.2)
With the initial condition p’(-oo ) = p, we expand p’(t) as
p’(t) = p + A 90) (2.3)

In the linear approximation, the solution for Ap(t) is [2]
Ap(t) = -1/iIi IL“ exp (-i(t-t’)H/it) [A, p] exp(i(t—t')H/lt) F(t’)dt’. (2.4)
It is convenient to introduce the operator ax operating on another operator b by the
following definition

x
a b = [a, b] , (2.5)
for which one finds the rule [2]

x a ~a

ea b=e be . (2'6)
With this notation, equations (2.2) and (2.4) can be written as [2]

ante/aw lfifi (HX+H"‘(t» #0) (2.7)

11

Ap(t) = -1/m It,m exp (-i(t-t’)HX/ll) [A, p] F(t') dt’. (2.8)

The response AB(t) of the quantity B is statistically [2]

AB(t) = Tr Ap(t)B
= 4/15 Tr IL” [A, p] B(t—t’) F(t')dt’ (2.9)

where B(t) is the Heisenberg representation of B following the equation
dB(t)/dt = 1/il'r[B(t), H] . (2.10)
The response function is now [2]

vBAa) =4th Tr IA. 9er

= llih Tr p [A, B(t)]. (2.11)
The above Kubo formula [2] is valid at finite temperatures, because a trace over
the thermal distribution is taken. At zero temperature this is equivalent to taking the

expectation value in the ground state.

b. Double-Time Green Functions and Time Correl_ation Function_s

The Green functions in statistical mechanics are the appropriate generalization
of the concept of correlation functions. They are just as intimately connected with the
evaluation of observed quantifies, and they have well-known advantages when
equations are formulated and solved [4].

Following Zubarev [4] we define double-time retarded and advanced Green
functions Gr(t, t’) and Ga(t, t’) as follows:
Gr(t, o = -i 9(t-t’)<[A(t). B<t31> (2°12)

Ga(t, t') = +i 9(t'-t)<[A(t), B(t')]> (2.13)

12

where <.....> indicates that one should average over a thermal ensemble at finite
temperature, and at zero temperature it simply means averaging over the
Heisenberg ground state of the interacting system. A(t) and B(t) are the Heisenberg
representation of the operators A and B, expressed in terms of a product of particle
creation and annihilation operators (or a product of quantized field functions). The

l t > 0
theta function is defined by 9(1) = O K o

(For convenience, a system of units in which h=1 is used from here onward.)
Finally, [A, B] indicates the commutator or anti-commutator:

-l Fermion

[A,B]=AB-nBA,n= +1 Boson

The sign of t] is determined by the problem. Generally speaking, A and B are neither
Bose nor Fermi operators [4].

When the time arguments are the same, t = t’, the Green functions are not
defined, because of the discontinuous factor 9(t-t’). We note that in the case of
statistical equilibrium the Green functions Gr(t, t') and Ga(t, t’) depend on t and t’
only through (t-t’) [4].

For the time being we have introduced the Green functions purely formally, by
analogy with the quantum theory of fields. We shall satisfy ourselves now by concrete
examples that they are very conveniently applied in quantum statistics to problems
concerning a system of a large number of interacting particles. One can choose for
A and B operators of different kinds: for instance, Fermi or Bose operators and their

products, density operators or current operators. The choice of the operators A and B

13

is determined by the conditions of the problem [4].

The time correlation funtions used in statistical mechanics are averages over
the statistical ensemble of the product of operators in the Heisenberg representation
of the kind

(pBAa, t’) = <B(t’) A(t)> and (pAB(t, t’) = <A(t) B(t’)> (2.14)

In the case of statistical equilibrium the time correlation functions depend, as do the
Green functions, only on t - t’. But they are defined also when the times are the same,

t = t’. In quantum mechanics we need to take the symmetrized product of operators.

In subsequent work we restrict our attention to the zero temperature problem,
and we are interested in two types of external perturbations, namely those due to
scalar potentials and those due to vector potentials.

We can write down the perturbing Hamiltonians for electrons exposed to a
scalar potential perturbation or a vector potential perturbation respectively as
H’(t) = -e 1.13: p(r,t) q)(r,t) (2.15)

H’(t) = —l/c 1:13: J(r,t) - A(r,t). (2.16)

The induced particle density and paramagnetic term in the induced current density in

the linear approximation are

5< p(r.t)> = ie IL... dt’1d3r’ <[p(r.t). pen] > «m (2.17)

8< J(r.t)> = i/c IL... dt’ ld3r’ <[J(r.t). J(r’.t’)]> - A(I’.t’) (2-18)

Now in analogy with Eq. (1.11) we define retarded density and current

correlation functions as

l4

a(r t, r’ t') = -i 9(t-t') <[p(r,t), p(r’,t')] > (2.19)
m t. r’ t’) = -i 0(t-t’) <[J(r.t). J(r’.t’)]>. (2.20)

Then Eq. (2.17) and Eq. (2.18) may be written as

5< p(r,t)> = -e I+°°dr I d3r’ a(r t, r’ t') (p(r',t’) (2.21)
8<J(r,t)> = -l/c1+°°dt’1d3r’ x(r t, 1’ t') - A(r’,t'). (2.22)

where causal behavior is enforced by the retarded nature of or and x respectively.
Eq. (2.21) and Eq. (2.22) typify a general result that the linear response of an
operator to an external perturbation is expressible as the space-time integral of a
suitable retarded correlation function. [ Note: Eq. (2.18) does not give us the total
current, as there is an additional term proportional to -e2/mc A(r,t); see reference 11,
Eq. (52.12). ]

In subsequent work we wish to evaluate a and x for a specific model system,
namely the interacting electron gas with a uniform and exactly opposite charged back-

ground, so that the whole system is neutral.

3. COLLECTIVE OSCILLATIONS AND THE RANDOM PHASE
APPROXIMATION

In this section we give a short review of Pines and Bohm [5] theory of plasma
oscillations, and wish to develop a detailed physical picture of the behavior of the
electrons in a dense electron gas. We are concerned with the organization produced
by the Coulomb interactions. From the work of Pines and Bohm: ‘ ‘In a dense electron
gas, the particles interact strongly because of the long range of the Coulomb force; in
fact each particle interacts simultaneously with all the other particles. As a result the
equations of motion become extremely difficult to solve. The usual perturbation based
on the assumption of interaction between particles breaks down.... A collective
description provides a far better starting point for a solution than a description in
terms of individual particles [5]”.

‘ ‘Instead of following the motion of the individual particles, we describe the
gas in terms of the Fourier components of the electron density at each point in space.
These Fourier components are proportional to the density fluctuations in the electron
gas. We find that the density fluctuations can be split into two parts. One part
represents an organized oscillation with the characteristic plasma frequency, and is
clearly associated with the collective behavior of the system. The other part is
associated with the motion of the individual particles... For wavelengths greater than

a certain critical length 1D (the Debye length). the fluctuations are primarily
collective... For wavelengths smaller than AD, however, the fluctuations are primarily
associated with individual particle motion [5]”.

“As any electron moves through the assembly, the other electrons are pushed

15

16

away from it by the Coulomb repulsion. Each particle is thus surrounded by a cloud of

extent "Dv in which there is a deficiency of electrons, which is responsible for

screening the field of the particle in question [5] ”.

‘ ‘The splitup of the density fluctuation into collective and individual-particle
components may be viewed in the following way. The collective part includes the
effects of the long range of the Coulomb force which lead to the simultaneous
interactions of many particles. The individual-particle component represents the
density fluctuations arising from the randomly moving individual particles plus their,
comoving electron clouds, and thus includes the effects of the residual short-range
screened Coulomb force [5] ”.

‘ ‘Certain examples of collective behavior in an electron gas are well known
from the study of gaseous discharges. These are the organized oscillations of the
system as a whole, the plasma oscillations. The ions in a metal are also susceptible
to a collective description, and, in interaction with electrons, they give rise to sound
waves, whose properties canbe calculated with the collective method. In this way,
one can obtain an improved treatment of the so-called lattice-electron interaction,
which is important in the dreary of electrical conductivity [5]”.

“We begin by a study of the way in which the interactions in an assembly of
electrons bring about organized behavior and collective oscillations. We shall consider
an aggregate of approximately free electrons embedded in a medium of fixed positive
charges whose average density is equal to that of the electrons [5] ”. For the
purpose of these calculations this distribution of charge can be regarded as uniformly

smeared out, throughout the entire system. Hence, it merely serves to neutralize the

17

net electron charge.
Each electron in the assembly is acted on by the sum of the forces arising from

all of the other electrons plus that resulting from the smeared-out positive charge [5].
The potential energy of interaction between the ith and jth electrons, e2] 1 xi - xj l ,

may be expanded as a Fourier series in a box of unit volume with periodic boundary

conditions, which gives [5]
e2/lxi - le =4min1/1c2)e.xp[irr(xi . x9]. (3.1)
k

The equation of motion of the i‘h electron is given by [5]

dzxi ldt2=-(41te2i/m) Z’(k/k2)CXP[ik'(Xi-xj)l (3.2)
is

where the prime denotes a sum in which It = 0 is excluded. (The term with k = 0 takes
into account the uniform background of positive charge, and hence the overall charge
neutrality of the system.)

‘ ‘The range of the Coulomb potential is so great drat many-body collisions are
important... Under drese conditions, the electrons move together in organized fashion,
and one finds the well-known phenomenon of “plasma” oscillations of the system as
a whole [5].

The particle density in our box of unit volume is given by
p(x)=):.6<x-x.). (3.3)

The Fourier components pk of the density are given by

pk =1 dx p(x) exp(-ik-x) = 2i exp(-ik-xi) (3.4)

l8

and p(x) = 21k exp[ik-(x - xi)]. (3.5)
We note that p0 represents the mean electron density, n, and the pk widr

k at 0 describe fluctuations about drat mean density [5]. Furthermore, we note drat

the expectation value of pk widr k at 0 is equal to zero because of the translational

invariance of the system. “It is readily verified that the equation of motion (3.2) may

be re-expressed as [5]
dzxi/dt2= -(41ce21/m)2'k (It/k2) pk cxp(ik°xi). (3.6)
The pk drus determine the force acting on each particle.”
We now obtain the equations describing the time behavior of the pk. On
differentiating (3.4), we have [5]
dpk/dt = -i 2i (It'vi) exp(-ik-xi) (3.7)
dzpk/dtz = - 2. [(k'vi)2 + i(k°dvi/dt)] exp(-ik-xi). (3.8)
We obtain dvi/dt from the equations of motion (3.2), and dzpk/dt2 becomes [5]

d2pk/dt2 = _zi(k.vi)2 exp(-ik-xi) -zij [41te2/mk’ 2] k . kl [cxp[i(k’_ k)-xi]}CXp(-ik’.Xj)
k’ at 0

(3.9)
In the second term we split the sum over k’ into two parts. The first part, widr

k’ = k, is independent of the coordinate, xi, so that the sum overi yields 11, the total
number of particles [5]. The terms with k’ at k contain the phase factor exp[i(lt’- k).xi]

which depends on the position of the particles. These terms tend to average out to

zero, since there are a very large number of particles distributed very nearly in

19

random positions [5]. As a first approximation, we neglect such terms. This
procedure we call the random phase approximation [5]. Using the above

approximation we then obtain

dzpk/dt2 = - 2i (It-vi)2 exp(-ik-xi) -(41tne2/m) Xi exp(-ik-xi). (3.10)
The first term arises from the motion of the individual particles. For

sufficiendy small It it is clear that the first term can be neglected. Under these

conditions, Eq. (3.10) becomes [5]

dzpk/dtz + (41tne2/m)pk = 0. (3.11)
Thus, as a result of the Coulomb interaction, the electron density oscillates

widr the well-known ‘ ‘plasma’ ’ frequency, (up = (41tne2/m) 1/2. The excitation of a

particular pk corresponds to a wave-like density fluctuation, analogous to a sound

wave [5].

a. uivalence of th Self-Consistent Field A roach and the llective A roach
Rafi on the RPA

‘ ‘The electromagnetic properties of crystals have long been studied by
considering the time-dependent interaction of a single particle widr a self-consistent
electromagnetic field. This procedure seems plausible for studying the response of
electrons to any external perturbation, and Bardeen [6], Ferrel [7] and Linth [8]...
have used this ...approach ...in discussing such phenomena as the frequency and
wave-number dependence of the dielectric constant, plasma oscillations, and

characteristic energy losses in solids [10]’ ’. These and similar phenomena have also

20

been studied on the basis of a dielectric formulation of the many-body problem by

Noziéres and Pines [9]. The explicit relationship of the self-consistent field approach

(Lindhard) to the collective approach (N azieres and Pines) has been studied by

Ehrenreich and Cohen [10]. Here we give a short review closely following their work.
We consider the single-particle Liouville equation

i ante/at = 1H. 90)] (3. 12)

as describing the response of any particle of the system to the self-consistent

potential V(x,t), where p is the single-particle density matrix. The single particle

Hamiltonian in Eq. (3.12) is

H: H0 + V(x,t) , (3.13)
where H0 = p2/2m is the Hamiltonian of a free electron satisfying Schrtidinger’s
equation H01 p) = 8P1 p) , and l p) = exp ip-x , in a system of unit volume. We expand
the operator p in the form p = p(0) + p“). The unperturbed density matrix has the

property 9“» 1 p) = n(p)| p), where n(p) is the distribution function. We now Fourier
analyze V(x,t) in the form

V(x,t) = ZqV(q,t) exp -iq-x (3.14)

and linearize Eq. (3.12) by neglecting products of the type me. This approximation
is equivalent to first-order self-consistent perturbation theory [10]. Taking matrix

elements between states p and p+q, we thus obtain [10]
i3<p| 9"" MW = (pl [110. pm]|p+q) + (pl [V. 9(0)] | Mr)

= (e, - 2,...) <p|p<1>|p+q> + 1 n(p + q) - n(p) 1V(q.t). (3.15)

21

where (pl VI p+q) = V(q,t). The potential V consists of an external potential V0 plus

the screening potential Vs, which is related to the induced change in electron density
[10].

An = Tr {5(rrc - x)p(1)} = Zq exp -iq-x 2p, (p’l p(1)| p’+q) (3.16)
by Poisson’s equation:
V2 Vs = -41t An e2. (3-17)

Here 8(x’e - x) is the charge density operator, xe being the position operator and x
referring to a specific point in space [10]. We drus find [10]

mm = qup. <p’|p<‘)lp’+q>. (3.18)
where Vq = 41te2/q2. By substituting the above expression giving V8 for V in

Eq. (3.15), we obtain the Liouville-Poisson equation determining (pl pm I p-t-q) in

the absence of an external perturbation:

i 8(pl p“)|p+q)/8t=(ep-8M) (plpmlrmr)

+ v.1 :10) + q) - n(p) 1 2,. (p'l p“) I p’+q>. (3.19)
‘ ‘In solving problems by the SCF method, however, one can usually avoid the
explicit expression of V8 in terms of pm widrin the equation of motion by means of an
assumption about the time dependence of V(q,t). To illustrate this point, we
calculate the frequency and wave-number dependence of the longitudinal dielectric
constant §(q,to). We imagine drat the external potential V0(q,t) acts on the system

with time dependence exp (itat+nt), where t’)-90* corresponds to an adiabatic turning

on of the perturbation. This potential polarizes the system [10]”. From

22

electrodynamics we have the relation

P(q. t) = (U41!) [£01.00 - 1] E(q,t). (3.20)
The polarization P(q,t) is related to the induced change in electron density by

V-P = eAn or

-iqP(q.t) = eAn(q.t) (3.21)
and the electric field E(q,t) is given by

eE(q,t) = -qu(q,t). (3.22)
“Equation (3.15) is readily solved for (pl pa” p-t-q) by assuming that (p 1 pa” p+q)

and Vs(q,t) have the same time dependence as V0(q,t). The induced change in

electron density An(q,t) may then be calculated from Eq. (3.16) and §(q,o)) deduced

from the field equations (3.20, 21, 22). We find [10]”

§(q.m)=1-11m VQX n<P+<l)-n(p)
11-90" p

 

ep+q- ep- (1) + 111 (3.23)

“This result was first obtained by Lindhard [8], with the SCF medrod and later by
Nozieres and Pines [9] using a many-particle approach based on the random phase
approximation for a Fermi gas at zero temperature [10]”. The equivalence of the
SCF approach and the collective approach based on the RPA is clearly demonstrated
in Section 2 of Ehrenreich and Cohen’s paper. In the next chapter we follow a closely

related analysis to find the response of a many-electron system.

4. T'HELINEARRESPONSETOSCALARANDVECTOR POTENTIALS INTHE
RPA

There are many medrods of deriving the RPA equations. In this work the
equation of motion method is followed, because of its simplicity. The derivation by use
of equations of motion is also called the self-consistent field method [10].

The Hamiltonian of the system (HO) consists of the kinetic energy term and

the electron-electron interaction term. To this we add the external interaction term.

We can express H0 in terms of second-quantized particle creation c"' and annihilation

P
cpoperators[12]
= + + + 4.1
H0 Zepcpcp-t-l/ZZVQC p+qc k-qckcp ( )
P 11.113

where vq is the Fourier transform of the Coulomb potential. Spin indices are omitted

and a factor of two is introduced in the final expressions to account for the spin

summation.

For scalar potential and vector potential perturbations, we express the

external interactions as

HS ’0) = -e£q p(-q.t)q>(q.t) (4.2)
Hv ’(t) = —l/c):.‘l J(-q,t ) - A(q,t) (4.3)
where the particle density operator and the current density operator have the forms

p(q.t) = 2 p c“ M0) cp(t) (4.4)

J(q.t)=(-e/m)2,(p+ q/2)c+,.q(t)c,<t). (4.5)

23

24

It should be noted that the diamagnetic current, proportional to p(r,t) A(r,t) in 1' space, is
not included in Eq. (4.5).
In the homogeneous electron gas, the particle density operator (q at 0) has an
expectation value of zero unless there is a perturbation of the system. A perturbation
on the system of character ((1,0)) will polarize the electron system. The effect of such
a polarization of the system is to screen out the external potential at large distances.
In the linear screening model, the average of the particle density operator is
proportional to the potential causing the perturbation [12].
The equations of motion to be solved are

i dc+p + q cP/dt = [H, c+ ] (4-6)

p+q°p
(-ielm)d(p + q/2) 0+1) +q cht = (-e/m)(p + q/2) [I-I, c+p+qcp]. (4.7)

The four terms in H give four terms in the commutator with c+p+q CI). The

kinetic energy operator leads to the terms

[ 2 ch flick, c+p+q cp] = (8in - ep) c+p+qcp (4.8)
k
(-e/m)(p + q/2)[ 2 8" c+ kck , c+p+qcpl = (-e/m)(ep+q - ep)(p + q/Z) c+p+qcp (4.9)

It
for density and current respectively.

The external perturbations give the terms

'82 (P(q’ rt) [c+k_q' ckrc+p+q ’]

cp] = -e2 (P(q'.t) [0*
ql k ql

_ '9'
WHY cp c p+q°p+q

== -e¢(q.t)[c+pcp - C’p+qcp,q] (4.1011)

25

«:2 WI’ 0 (-e/m) (p + (1/2)[<:".‘_(I 0., c
q’ k

W‘lcll’

_ - _ I + _ +
- e ( em) (I) + (In) 211K“) [c Mq'cp c MCMJ

q
z -e (-e/m) (p + q/2) tp(q,t)[c+pcp - c" ] (4.10b)

MGM
and similarly,

-1/c (-e/m) 2 (k’ + (172) - A q. [63. cw”. , C+P+q cp]
kl q!

(-p+q /2’) A q:c+

—, _ _ ’ . +
— l/c ( elm) 2(p-1-q q [2) Aq. c PHICPHI'

q!

P‘Hl‘q CD

= -1/c (~e/m) (p+ q/Z) - Aq [cipcp - c+p+qcp+q]. (4.113)

-1/c(e/m)2(p + q/2)2 (k’ + q’l2) - A q, {a} cm, , c+M cp]
kl q!
= -1/c(e/m)2(p + q/2)); (p+q-q’/2) - A q, 6M .q, cp - (p+q'/‘2) - Aq. c+McP+q.
ql
== -l/c(e/m)2(p + q/2)(p+ q/Z) ' Aq [c"pcp - c+p+qcp+q]. (4.11b)

The summation over q’ is approximated by taking only the term q = q’ in
Eqs. (4.10) and (4.11). The terms with other values of q’ are neglected. It is
assumed that drey average to zero within the random phase approximation.

The commutator with the electron interaction term leads to two terms, each

with four operators [12]:

+
[l-Iec, c+p+q c=p] 2M], vq,[c p+q+qw+ k’ck’+qc p- c+p+qc k’ 1|,ck.cp.q.] (4.12)

"The result is approximated by pairing up the operators. We select values of

26

the summation variables which produce operators of the type we are seeking. These

are either combinations such as c+Pcp or c+p+qc p. There are drree combinations which

are interesting: q’ = -q; k’: p; k’= p+q [12]". These give the combinations, in the
case of the density [12]:
+ _ + _ + + _ + ,
[ch p+q CF] ' [c PCP c we 011“!“ “if" k’+q ck’ 2V‘l'c P+q+9'cl’+‘|}
I ql

c"INI cplivq .c Hp+q+q.cp+q+q -q2v ,c M CML (4.13)

q q

and in the case of the current:

_ + = _ + _ + + I
( e/m)(p + q/2)[H .0 M cpl ( e/mXp + q/2)[c pep c M CMJI vq El 0 1. “I ck
_ +
qu, c p+q+q’cp+q’)
ql
_ _ + +
(elm)(p+q/2)c p+q CPIqu,c p+q+qu+q+¢
q
- +
2 vq. c M'CMJ' (4.14)
q!

(We note that the commutator with the Coulomb interaction term remains the same in both

cases. When spin indices are included explicitly, the term v (1.2% c k’ ck, includes a sum

“I

over spin states, while the remaining terms have a single associated spin index identical to

4.
that for c P N cp.)

Now we take the ground state average of Eqs. (4.8, 10a, 11a, 13) and Eqs. (4.9, 10b,

- - + + '
11b, 14). In addrtron, the number operators c PCP and c “q cp+q are replaced by therr

averages n(p) and n(p + q) to obtain the result, in the case of the density:

27

+ .. _ + _ +
<[I-I, c p“. cp]> — (8pm ep)<c P+q cp> + [ S(p+ q) £(p)]<c >

M Cr)
+ [ n(p) - n(p + q) ltvq <p(q.t)>[1-G(q)} - e (P(q,t)
+ (e/mC) (p + q/Z) ° A(q.t)]. (4.15)

and in the case of the current:
<[I-I, (-e/m)(p + q/Z) C+P+q cp]> = (elmI - 8p) (~e/m)(p + q/2) <c+p+q cp>
+ [3(1) + q) - E(p)] (~e/m)(p + qfl) <61”+q C]?

+ [ n(p) - n(p + q) 11vq (-e/m)(p + q/2)<p(q.t) >1 10(q)}

-1/c (elm)2(p + mm: + an) . A(qn

+ (elm) e (P + q/Z) (P(q,t)l . (4.16)
where E(p) = {fivq n(p + q) is the exchange energy and the Hubbard approximation as
defined in Ref. (12) has been used to introduce the function G(q).

Since the density and current respectively satisfy the equations of motion:

i d< 0+9“! cp>ldt = co<c+p+q cp> (4.17)

id(-e/m)(p + q/2) (C+ C >/dt = oX-e/m)(p + q/2) <c+

Mr p we cp)’ (4'18)

where external potentials are assumed to be oscillating at a single frequency a), we finally

obtain the equations:

{to + 8,; ep+q+ 2(1)) - £(p+q)}<c" >

p+q°p
= I n(p) - n(p+q) ] [vq<p(q.t)>{1-G(q)} - 8 Wm) + (e/mc) (p + 11/2) - A(q.t) ]

(4. l9)

and

28

{to + 89- ‘m’ 8(1)) - £(p+q)l (-e/m)(p + (In) <c"M cp>

= [n(p) - n(p+q)] [vq(-e/m)(p + q/2)<p(q.t) >1 1-G(q)}

+e(e/m)<p + (W) wit) -1/c (e/m)2 (p + q/zxp + qn) - A111,»).
(4.20)

Eqs. (4.19) and (4.20) are consistent, since (4.20) can be obtained from (4.19) by
multiplication by (-e/m)(p + q/Z). Eq. (4.19) differs from the result in Chapter 5 of
Ref. 12 by the inclusion of the vector potential terms.

Now we can sum Eqs. (4.19) and (4.20) over p to obtain equations for the particle
density and the current density. Since the external potentials are assumed to be
oscillating with a single frequency component, as exp(-iort), we substitute for the
external potentials (p(q,to) exp(-itat) and A(q,o)) exp(-itot). Within linear response,

a homogeneous electron gas responds at the same wave vector and frequency as the

external perturbation, so we can replace <p(qt)> by p(q,(o) exp(-iort) and

<1 (q,t)> by J (q,to) exp(-iaJt) to obtain equations for the induced particle density
and current density respectively:
p(q.m) = E(q,to) vqi l-G(q)}p(q,co) - e E(qm) <P(q.co)

+ (elmC) 961.0» ° Mm» (421)

and
J ((1.0)) = (—e/m) thxn) vqt 1-G(q))p(q.co) + (elm) e Q(q.co) <p(q.w)

-1/c(e/m)2 5mm) - A(q,a)) (4.22)

where

29

2111.0» = Z [ “(9) ' “(P “1) J

p a) + 6,; 8M+ film-3(1) + q)

 

Q(q.co) = 2 [ [n(l’) - n(P + (1)] (p + (In) ]

p a) + 6,; 6M+ 23(1)) - 2(1) + q)

and

 

fi(q,co) = Z [ [n(P) - n(p + (1)] (p + q/2) (p + q/2) ]

(Ms-e +£(p)-£(p+q)

p 1) WI

(4.23a)

(4.23b)

(4.23c)

By neglecting the exchange energies and G(q). replacing the polarization term

E(q,tn) by the less accurate RPA result P1(q,a)), and similarly replacing Q(q,co) by Q1(q,a))

and 5mm) by n1(q,m), we obtain the induced density in the RPA:
P(q,co) = P‘(q.co) vq paw) - e P‘(q.m) (New)

+ (e/mc) Q‘(q.w) - A(q.m)

and the current density in the RPA:

J (em) = (-e/m) Ql(q.m) vq p(q.co) + (elm) e Ql(q.m) (Nam)

-1/c (elm)2 R‘(q.m) - A(q.m) .

Here Pl(q,or) is the RPA polarization diagram,
prmm = 2 [n(P) - 11(1) + (1)]

+ -
" m 8p 8W1

and similarly for Q1(q,co) and Rl(q,m).

Now we can solve Eq. (4.24) to obtain the particle density self-consistently.

(4.24)

(4.25)

(4.26)

30

- e Plum) (P(qm) + (e/mc) Q‘(q.o)) - A(q.co)

 

p(q.co) =
1-P1(q,(a)vq (4.27)

Now the RPA dielectric constant is defined as §RPA(q,a)) = l — P1(q,a)) vq.
This function was first evaluated by Lindhard [8], as a model for a dynamic dielectric
function. It is rather easy to derive and has a simple basis. It also predicts correctly a
number of properties of the electron gas such as the existence of plasmons [11, 16].
The real part of the dielectric function may be represented by a single formula (see, e. g.,
Ref. 12, Eq. 5.5.5). The imaginary part has a variety of functional forms for different

values of (q,(o) (see Ref. 12, Eq. 5.5.6). In certain limits, the imaginary part of

§RPA(q,to) is zero. In our subsequent work, its limiting behavior at zero frequency and in

the small momentum transfer limit will be used extensively.

Eq. (4.25) cannot be solved self-consistendy by itself, but it can be solved by
substituting the result for p(q,(o). Below, we analyze separately the response to a transverse
vector potential and to a longitudinal vector potential.

The transverse component of A(q,o)), denoted by At(q,a)), is unaffected by a change
of gauge. Also At(q,co) is independent of the longitudinal component Al(q,(o) and the
scalar potential (p(q,(a). We set Al(q,to) = q)(q,(a) = 0, and determine the response to a
nonvanishing At(q,tr)). For an electron gas that is homogeneous and isotropic prior to the
perturbation, Q(q,m) and Q1(q,co) are both vector functions with direction q. Then

J,(q,m) = -l/c (e/nn)2 nl(q,ce) - A,(q,m) , (4.28)

or equivalently,

31

J((q.co) = -l/c (elm)2 2 “(9) ' “(P + ‘1) (P + (l/ZXP + (1/2) ‘ At((l.(o)

 

P -
a) 4» SP ep H! (4.29)

This result is consistent with the observation that particle-hole excitations of transverse
character cannot decay via the longitudinal Coulomb interaction, as noted in Ref. 16; we

note that vq is not present in Eq. (4.29).

Changes of gauge interconvert the scalar potential and the longitudinal component

of the vector potential. The change from A, qr to A’, q)’ given by

A’ = A + VA (4.30a)
(p’ = (p - l/c alt/8t (4.30b)
in q, to space is

A’=A+iqA (4.31s)
(1” = (P + i We A (4.311.)

From Eqs. (4.21) and (4.27), we require

01mm) =Q((r.(o)- q/m (4.32)
in general for gauge-invariance of the field-induced change in the particle density, and
coP‘(q.w)=Q‘(q.co)-q/m (4.33)

for gauge invariance in the RPA.

If we consider the response to a perturbing scalar potential only, the longitudinal

component of the induced current density in the RPA is

J1((r.(o) = cz/m Q‘(q.(n) <p(q.co) / 11-P1(q.m)vq1 . (4.34)

The equation of continuity

32

(r - Jl(q.(o) = w p((tm) (4.35)
connects p(q,(1)) and Jl(q,a)). This equation is satisfied when the gauge-invariance
condition is met.

Alternately, we may consider the response to the longitudinal component of the
vector potential only. In this case, we obtain
mm) = -l/c (elm)2 (011(1chq Q‘((r.(n) - A.(q.co) tl-P‘tq.co)vq1-1

+ alarm) - Alum) (4.36)

We note drat the diamagnetic current is omitted in Eq. (4.36), as before. For gauge
invariance of J l(q,co) it is required that

(ammo) =B(q.(o) ° (r/m . (4.37)
and similarly for the corresponding functions in the RPA. From Eqs. (4.25), (4.33), and
(4.37) in the RPA, we obtain

Jl(q,to) = -l/c (e/m)2 81mm) - A,(q,co) / [l-P1(q,a))vq] . (4.38)
Comparison of Eqs. (4.27), (4.28), and (4.38) shows that dielectric screening afi'ects

Jl(q,a)) but not Jt(q,(n) in the RPA.

In the next chapter, we Show the form of the particle-density susceptibility, and then

in subsequent chapters we analyze the current-density susceptibility.

5. CHARGE DENSITY SUSCEPTIBILITY

The linear response of the interacting electron gas to an added impurity charge
was first evaluated by Langer and Vosko [13]. Screening of the scalar potential of an
impurity charge gives rise to Friedel [14, 15] oscillations at large distances. The Friedel
oscillations are real features of the charge distribution in metals containing impurities and
have been detected via broadening of nuclear magnetic resonance lines in dilute Cu alloys
by Rowland and Bloembergen [16a]. The experimental fact is drat the presence of one
foreign atom seems to eliminate about fifty Cu nuclei from the central resonance signal. If
straightforward Fermi-Thomas screening [11, 12] applied, one would have an electrical
disturbance limited to the near neighbor Cu nuclei alone.

Friedel oscillations are a consequence of the logarithmic singularity possessed
by the RPA dielectric function. The origin of this singularity is the sharpness of the Fermi

surface; one passes into a different physical regime when going fiom k < 2kF to k > 2kF

because one is passing into a region in which the momentum transfer can no longer
take an electron from one part of the Fermi surface to anodrer [11]. This disturbance is
therefore effective at rather large distances.

Since the polarization of the system satisfies the equation,
P(MD) = (1/41t)[8(k.(0) - l] ' E(k.(0). (5.1)
the charge density susceptibility is intimately related to the complex polarizability.
In the subsequent sections of this chapter we evaluate the charge density susceptibility
and the nonlocal polarizability density of the interacting electron gas as linear responses

to an arbitrary perturbing scalar potential.

33

34

a. Relation between the nonlocal mlarizabilig densig and the dielectric constant I l7|
The nonlocal polarizability density is a linear response tensor that gives the

polarization P(r, (1)) induced at a point r in a molecule by a perturbing electric field
E(r’, (1)) that acts at the point 1" [18-23]:

P(r, m) = I dr’ a(r, r’; to) - E(r’, m) . (5.2)
Thus a(r, r’; (1)) represents the distribution of polarizability widrin a molecule or an
extended quantum mechanical system. The purpose of this work is to gain information
about the nature and functional form of the static polarizability density, by determining this
property for a homogeneous electron gas at T = 0.

The induced polarization P(r, to) in Eq. (5.7) is related to the change in charge
density 8p(r, (a) due to E(r’, (a) by [22]
V - P(r, co) = - 6p(r, to) , (53)
so a(r, r’; or) provides a complete description of the response to perturbing fields that are
derivable from scalar potentials. The polarization can be interpreted as the density of the

induced dipole moment, but it also includes quadrupolar and higher-order response [23].

The nonlocal polarizability density was introduced by Maaskant and Oosterhoff in
a study of optical rotation in condensed media [18]. In their work, a(r, r’; to) is expressed
as a difference of two components that are individually divergent as a) —-) 0, and each
transition matrix element in the expression involves an operator specified only as an infinite
series [18]. For response to fields that are derivable from a scalar potential, a(r, r’; (a)
can be recast in a computationally tractable form, via a connection [22] between its

spatial Fourier transform and that of the charge-density susceptibility a(r, r’; co) [24-29].

35

The function a(r, r’; (1)) determines the change in charge density at point r in response to
a perturbing scalar potential acting at r’, within linear response. This connection gives the
longitudinal component of the polarizability density aL(r, r’; or) [22].

For small molecules, the spatial Fourier transforms of a(r, r"; or) (and drerefore
GLGt, k’;a))) have been computed via pseudo-state methods [28], a stationary principle
[25], and an Unsold-type approximation [27; see also Refs. 26 and 29]. The point—atom-
polarizability approximation (PAPA) [20, 30], truncated at the dipole terms, is equivalent
to approximating a(r, r’; (a) as a weighted sum of products of Dirac delta functions located
at the nuclei. Modifications of the PAPA model to allow for distributed point multipoles
[30, 31] can improve its approximation to the continuous polarizability distribution of the
actual function 60', r’; or) and drus extend the range of validity of the PAPA model.

In this section (cf. [17]), we analyze the static, longitudinal polarizability density
aL(r, r’; a) = 0), for a homogeneous electron gas at zero temperature. Our intent is to gain
a better understanding of o(r, r’; a) = 0), and to provide information for the development
of approximations to molecular polarizability densities.

In this section, we derive an equation for the longitudinal component of the nonlocal
polarizability density aLaB(r, r’; 0) via its connection to the charge-density susceptibility
01(r, r’; 0). In wave-vector space [22],
6L(k, k’; or) = k k’ n(k, 11'; to) / (k k’) , (5.4)
where k and k’ are unit vectors in the directions k and lt’, and
a(k, k’; to) = 1dr Idr' a(r, r’; to) e" i" - ' elk" " . (5.5)

In sum-over-states form, the charge-density susceptibility satisfies [24]

36

01(k.k’;m)= 2/112' (snow |p(-k) ln)(n | p(k’) IO)/(arn02-to2). (5.6)
n

In Eq. (5.6), the sum runs over the excited states 11, (Ono denotes the transition
frequency (E,1 — Eo)/h, p(k) is the Fourier transform of the electronic charge density

operator,

90‘) = - e 2 exp (i k - rj) , (5.7)
.1

and rj is the coordinate for the jth electron.

Because of the translational invariance of the homogeneous electron gas, a(r, r’; 0))
depends only on r - r’, and therefore a(k, k’; or) takes the form
n(k, k’; or) = (211)36 (k, to) so" - k) . (5.8)
The susceptibility a(k, (1)) gives the change in the k, a) component of the charge density
8(p(k, 01)) due to an external scalar potential 8¢°x(k, (a):
5(p(k. (0)) = a(k. (o) 5¢°"(k. 03) . (5.9)
Linear response theory relates the susceptibility a(k, or) to the retarded density-density

correlation function DR(k, to) [l 1]. In terms of the function TIR(k, to) defined by

11%, to) = 11-1 13%., (a):
6<p(k.m)> = (:2 nR(k.m)6¢“(k.m) . (5.10)

In diagrammatic dreary [l l], I'IR(k, (a) is the "retarded polarization part." From

Eqs.(5.4), and (5.8)-(5.10),

aL(k, k’; or) = (21:)3 e2 k k’ 11R(k, (n) 8(k’ - k) / (k k’) . (5.11)

37

Thus in coordinate space, the Static longitudinal polarizability density is

“Lab“, 1"; 0) = (210—3 e2 I dk ei ‘ ' ("'3 ka k5 nR(k, 0) 162 . (5.12)
We simplify Eq. (5.12) using [11]

nR(k. 0) c2 1-2 = (410-1 [E(k. or‘ —11 . (5.13)

and the Rayleigh expansion [32]

ei " ' 0"") =2? if (21+ 1) j,(1cx) P[(cos y) , (5.14)
[=0

where I r — r’ l= x, j [(kx) is the [th spherical Bessel function, and yis the angle between k

and (r - r’). In terms of the orientation angles 01., (pk for the vector k and the angles 0, (p

for (r — r’), the addition theorem for spherical harmonics gives [33]

I
P[(cosy) = 41: (21+ 1)‘12 Y,m(ek, (pk) Y,m*(e, (p) . (5.15)

m=-[
For an isotropic, homogeneous electron gas, after specializing the result from Eqs.

(5.12)-(15) to eLnu, r’; 0), we obtain

0Lu(r. r'; 0) = 1/3 (2n)'3lo dk 1.2 joacx) 180:. 0)"1 - 11

— 213 (210-310 dk k2 j2(kx) [e(k, 0)"l - 1] P2(cos 9) , (5. 16)

where j0(z) = 2’1 sin z and j2(z) = (3 z'3 - 2") sin 2 — 3 2‘2 cos 2. The analytical results

in Sec. g are based on Eq. (5.16) for aLzz(r, r"; 0). Other components of the polarizability

tensor may be computed analogously.

38

b. Asymptotic results for the ngnlgal mlarizabilig densig in the RPA | l7|

In the random phase approximation (RPA) [5, 10], the dielectric function is
[8,11,12,16]
E(k, 0) = 1 + 4 (:1 rs 1.1.2 (rtk2)‘1 g(k/kF) . (5.17)
In Eq. (5.17), at = (4/911:)1/3. The density of the electron gas is characterized by the
dimensionless ratio rs = r0 [80, where a0 is the Bohr radius and the volume per particle
V/N satisfies V/N = 411: I3 r03. The wavenumber kF of the highest occupied state in the
electron gas at T = 0 is given by It]; = l/(aro); kF is designated the Fermi wavenumber [l l].
The function g(z) is [8, 11, 12, 16]
g(z) = 1/2 - (22)'1 (1- 2%) In I (1 - 212) / (1 + 212) I . (5.18)
We find the form of aLzz(r, r’; 0) for large I r — r’ 1 following the medrod of Ref.

11. Since g(z) is an even function of z,

QLZZO', r’; 0) = (210-3 (6ix)‘1l_°°dkk eikx [e(k, 0)"l - 1]

- 2/3 (210'3 10L, dk k2 ((2i)'1 13 (too-3 — (loo-11 — an (new)
eikx [e(k, orl - 1] P2(cos e) , (5.19)
where ‘0 denotes the principal value of the integral (see the Appendix). After rewriting
g(k/kF) in the form [1 l]

g(k/kp) = 1/2 - (2 k n(p)-1 (k1.2 — 1.2/4)

xrimn_,0+1/21n{[(k-sz)2+n2]/[(k+2kF)2+n2]} ,(5.20)

39

we evaluate the integrals in Eq. (5.19) by complex contour integration around the contour

shown in Appendix A. In the upper half plane (in k), the integrands have simple poles near
ik-I-F where kn; is the Thomas-Fermi wavenumber given by kn: = («It's/101,2 k... In the
second integral in Eq. (5.24), the function 3 1:2 (too-3 eikx [E(k, 0)’l - 1] has an additional
pole at k = 0. Bodr integrands have branch points (in the function g) at r: 2kF :1: in, with

branch cuts originating at these points, as shown in Figure Al in Appendix A, which also
gives the details of the calculation. Results for integrals around the branch cuts C(1)

and C(2) in the upper half plane (of k) are:
1%) dkk“ eikx [e(k, 0)“1 — 1]
= 2M1 1: g kpn‘l x'2 exp (2ikFx) (4 + 1:)“2
x {-i+2[§(C-3/2+ln4kFx)-8+(2n+3)(1+§/4)]/[kFx(4+§)]+... }

(5.21)
and

Ice) dk k" eikx [e(k, or1 - 1]
= (—1)“+ll 2ml 1: g kph-1 x'2 exp (~21kFx) (4 + §)-2
x {—i-2[§(C-3/2+ln4kFx)-8+(2n+3)(1 +§/4)]/[kpx(4+§)]+ ...}

(5.22)
where n 2 —l, E, is defined by é a kHz/(2kg) E 2 (1 rs it”, and C is Euler’s constant

(C = 0.577216). The result obtained for shun, r’; 0) is

41-220; r’; 0) = (210-3 411/3 x-3 g (4 + §)-2 {cos(2kFx) + 2 (kpxrl (4 + §)-1

x sin(2kFx) [3 - g(c - 1/4) - a 1n(4kpx)] }

+ (2a)"3 811/3 x’3 t; (4 + 0'2 P2(cos e) {cos(2kFx) - 2 (kFx)'1 g

x (4 + §)'1 sin(2kFx) [c + 1/2 + ln(4kFx)] }

+ (210-3 n x'3 P2(cos e) , (5.23)
where x = Ir — r’ I . Fig. l [ 17] shows the asymptotic form of aLzz(r, r’; 0) from Eq. (5.23),
with rs = 4. The error due to terms omitted from Eq. (5.23) is of the order
(a x—5 + b x‘5 1n kFx).

In Eqs. (5.19) and (5.23), aLzz(r, r’; 0) has two components which transform

differently under rotations, one as Po and the other as P2. We have not found the integrals

needed for the P2 component in any earlier work. However, the integrals needed for the P0

component have previously been used to find the change in the charge density 5(p(x))
induced in an electron gas when a point charge Ze is inserted at an origin a distance x away

[11, 12, 13]:
6(p(x))= (210-3 Ze I d3k exp(ik-x) [e(k, 0)’l - 1]

= Ze/(4rr2 in]: k eib‘ [E(k, 0)‘l - 1] dk . (5.24)
Thus, comparison of asymptotic results for 5(p(x)) serves as a check on the P0 component
of Eq. (5.23). Specializing Eqs. (5.21) and (5.22) to n = 1, we obtain
8(p(x)> = 2e 1:“ 2t (4 + of x‘3 {cos(2kpx) - 2 (4 + o‘1 (14.x)-1

x sin(2kFx) [g (c - 1/4) - 3 + g In 4kFx] + } , (5.25)

41

in agreement with the result of Fetter and Walecka [l 1] for large x.
We note that Eqs. (5.21) and (5.22) can also be used to obtain the asymptotic form

of the charge-density susceptibility 01(r, r’; 0) in the RPA:

01(r, r’; a): (210-3 (zip-IL” k3 eikx [15(k,0)'l - 1] dk
= (210-3 161: 1.1.2 x'3 g (4 + §)-2 {cos(2kpx) — 2 (4 + §)-1 (kFx)"l

x sin(2kFx) [ g (c + 3/4) + 1 + g In 4kFx] + } , (5.26)
with x defined as above. The susceptibility a(r, r"; 0) gives the change in charge density
induced by a scalar potential of arbitrary spatial variation. This contrasts with the function
6(p(x)) previously computed in Refs. 11 and 13, since 8(p(x)) is the response specific to a
single added point-charge impurity.

The asymptotic form of aLu(r, r’; 0) contains oscillatory terms and a non-
oscillatory term (the last). Madrematically, the oscillatory, long-range terms in
aLu(r, r’; 0), 8(p(x)), and a(r, r’; 0) are produced by integrations along the branch cuts in
801, 0), widrin the RPA. The oscillations in oLn(r, r’; 0) are evident in Figure 1 [17].

The non-oscillatory, long-range (x'3) term in aLzz(r, r’; 0) has a different
mathematical and physical origin. It stems from the singularity of the integrand in the
P2-component at k = 0, and an identical term is present even at Thomas-Fermi level. In the
Thomas-Fermi approximation, the dielectric function is [12]
owns, 0) = 1 + RTFZAZ . (5.27)
When Eq. (5.27) is substituted into Eq. (5.16) for aLzz(r, r’; 0), the resulting integrands

have simple poles at k = 0 and k = a i kTF° Hence by complex contour integration, we

42

I
I. .
6

.‘
...

I

f O 0
’0...
of" (
t 5”]:

I
O

\\
‘ \~

 

Figure l. Asymptotic form of the nonlocal polarizability density in the
random phase approximation, from Eq. (23). The figure shows

“RP A a (211:)3 3/71: kF_3 103 oLu(r, r'; 0) as a function of

kF" a kF (r — r’)x and sz .=. 1‘1: (r - r')z, widr rs = 4. The range of
planed oRPA values is [-2.0, 2.0]. Friedel oscillations are evident
in the plot.

43

obtain

“L'I'FZZO'. 1"; 0) = (210-3 111/3 P2(cos 9) k’I‘F3

x { 3 (rum-3 — exp(-kTFx) [ 3 (k-I-Fx)'3 + 3 (RTFxr2 + awn-1]}
— (210-3 11/6 k-I-F3exp(-kTFx) (kTFxYl . (5.28)
The long-range (x'3), non-oscillatory component of the polarizability density in
Eqs. (5.23) and (5.28) reflects charge screening. For example, when a perturbing point

charge + Ze is added to the electron gas, the change in net charge inside a large sphere

centered on the perturber approaches — Ze, as the radius R1 of the sphere increases [11, 12].
Equation (5.2) relates the induced polarization P(r) to the induced change in charge
density; drerefore as Ri increases,

1 P(r) . dsi = —l 5p(r) dvi —> Ze , (5.29)
where the integrals run over the surface dSi and the volume dVi of the sphere of radius Ri°

Thus, for sufficiendy large Ri, the polarization on the surface of the sphere must vary as
Ri‘z. The non-oscillatory, x’3 terms in the Thomas-Fermi and RPA expressions for

aLzz(r, r’; 0) give rise to the required Ri'z component of the polarization.

We note that charge is conserved overall for the electron gas, so Eq. (5.11) holds. A
surface charge of Ze develops at the outer boundary of the electron-gas system, on a sphere

of radius RS. In the calculations, the limit Rs -) oo is taken before the limit R1 -) no.

For numerical results based on the RPA and a more accurate dielectric function

derived by Vashishta and Singwi [34, 35], see Ref. 17.

c. Fourier Transform of gmamic Charge Densig Susceptibilig
The dynamic charge-density susceptibility can be evaluated by extension of the

approach used in the previous sections. The susceptibility is determined by the following

integral over wave vector k and frequency a):

a(x, t) = (210-4 (21x)'l L“ dk L” d0.) k3 exp(ikx) exp(—ia)t) [e(k, to)'1 —1], (5.30)
where x = Ir - r’ I, t is the time elapsed between perturbation and response points, and
E(k, to) is the frequency-dependent dielectric function. Widrin the RPA, the dielectric
function is connected to the lowest-order polarization insertion 110(k, to) via

E(k, (0) = 1 - U00.) node, 0)), (5.31)
with U0(k) = 4ne2/k2. Below, we use scaled variables q and v, defined by q = k/kp and
v = mar/(likpz). Then for positive v,
Re 11%, v) = map/(41:2 112) {-1 + 1/(2q) [1 — (v/q - q/2)2]

x In I [l + (v/q - q/2)][1-(v/q - 2/q)] I

- mo 11 - (v/q + q/2121 1n 1 11 + (v/q + q/2)1/11 - (v/q + cm I).
(5.32)
The imaginary part takes on different functional forms, depending upon the value of q and
the relation between q and v:
i) Ifq>2, q2/2+q2v2q2/2-q, orifq<2andq+q2/22v2q-q2/z, then

1m 11%. v) = —mk../(4nqfiz)11-(v/q—q/2)21. (5.33)

45

ii) Ifq<2, andOSqu—q2/2, then

Im r1°(q, v) = — mka/(li2 21: q). (5.34)
ITO(q, v) is even in q, Re I'IO(q, v) is even in v, and Im [10(q, v) is add in v. The integral
over k can be found by complex contour integration in the q plane, with the contour closed
in the upper half-plane. Poles in the upper half-plane of q (with a finite displacement from
the real axis) do not contribute to the asymptotic value of the susceptibility density,
because the terms due to such poles decay exponentially with increasing x. There are

nonvanishing, asymptotic contributions from poles on the real axis, and from integrations

around the branch cuts of the log functions in Re I'IO(q,v). To locate the branch cuts, we
rewrite
In | [1 +(V/(1 - q/2)]/[l-(VI(1 - (1/2)]|

= tunfl _,01/21n 1 1(1+v/q-q/2)2+n21/1(1-v/q +q/2)2+n2]} (5.35)

and similarly for the second logarithmic term. There are four equations defining the origins

of the branch cuts:

a) l + v/q - q/2 = i it] (5.36)
b) 1 - (v/q — q/Z) = r: it] (5.37)
c) 1 + (v/q + q/2) = :1: it] (5.38)
d) 1 — (v/q + q/2) = a: in. (5.39)

When v < 1/2, the origins of the branch cuts lie just above and below the real axis, at points
with real components given by q(")i = l :t (l + 2v)m, qwi = —l i (l + 2v)m,
q(°)2t = —1 i (1 — 2v)1/2, q“):t = l i(1— 2v)1’2. Thus for small but non-zero v, there are

eight branch points in the upper half plane. The branch cuts are all taken parallel to the Im

q axis, widr those originating in the upper half plane running out to 1m q = + co; cuts
originating in the lower half plane run to Im q = -- on. As v approaches 1/‘2, two branch cuts
in the upper half plane coalesce, with real components of the origins at Re q = l; and a
second pair in the upper half-plane coalesces at Re q = —1. (Similarly, there are two pair-
wise coalescences in the lower half-plane). For v > 1/2, drere are four branch-cut origins
just above the real axis, four just below the real axis, and two (with vanishing separation as
r] —-> 0) in each quadrant of the complex q plane. We run branch cuts between the two
branch points in each quadrant that are well off the real axis, and choose the remaining

branch cuts as before. Then the integral to be evaluated takes the form

I: f(q) dq = — ct f(q) dq - 1C2 f(q) dq - 1C3 f(q) dq - Id f(q) dq + i n 21.. Kn

1 — Ics f(q) dq -- Ice f(q) dq - Icv f(q) dq -— Ice f(q) dq 1. (5.40)
asymptotically. Kn denotes the residue at the nth pole on the real axis, and we have neglect-
ed contributions from residues at poles with a finite displacement into the upper half-plane,
as explained above). The contours Cl-C8 run around the branch cuts (down on the right, up
on the left); integrals along the contours Cl-C4 always contribute in the expression above,

while integrals along CS-C8 contribute only when v < 13.
First, we focus on the integral along C1, widr origin q(a)+ = l + (1 + 2v)1/2:
— Cl f(q) dq = (27t)'4 (2 i x)"1 LL” do) I0 du (q(a)++ iu)3 exp[i (q(“)+ + iu) x]

x exp(—i(r)t) [eR(q(3)+ + iu, iu)-1 — eL(q(a)+ + in, or)"1], (5.41)

where t:R(q(“)+ + in, (0) denotes the value of the dielectric function to the right of the branch

cut, and eL(q(a)+ + in, (1)), the value to the left. These differ because of the difference in

47

the phase of the logarithm across the cut. The equation above simplifies furdrer to

- Cl f(q) dq = (210-4 (2 i x)_1 1L” d(1) lo du (q<3)++ iu)3 exp[i (11(3)+ + iu) x]
x exp(-i(1)t) U081“). + iu) [110R(q(a)+ + iu, v) - 110L(q(a>+ + iu, v)]
+ [[1 - 0081(1)+ + iu) 110L(q((1>+ + iu, v) ] [ 1 - U0(q(a)+ + in) 110R(q(a>+ + iu, v) 1].
(5.42)
Next, we observe that the difference I'IOR(q(“)+ + iu, v) — H0L(q(a)+ + iu, v) is linear in u,
to lowest order. We obtain the leading asymptotic term by retaining the u dependence here

and in exp[i (q(a)+ + iu) x], but in the remaining factors, we replace q(“)+ + in simply by the

value at the origin of the branch cut q(a)+. This gives a contribution from the Cl integral
equal to

- 1C1 f(q) dq = (211)-4 (2i x)-1 i (88).)3 exp[i q<a>.x 1 Uo(q<a>.)

L, (1(1) exp(-i(1)t) {[1 - U0(q(a)+) n0(q(a)+, v) ] l0 du 2143),, u) exp(-ux),

(5.43)

where I)(q(a)+, u) denotes the linear term in H°R(q(a)+ + iu, v) — H0L(q(a)+ + iu, v). The u
integral in this expression can be evaluated analytically. Then summing the contributions
from Cl-C8 (with n < 1/2) or Cl-C4 (with n > 1/2) gives the branch-cut contributions to
(r(x, (1)).

Contributions to (r(x, (1)) of a second type come from the poles of the integrand on

the real-q axis, where E(q, (1)) = 0. These poles are also the poles of I'l(q, (1)), so they

48

occur at the excitation energies of electron-gas states that are connected to the

ground state via the density operator [1 l]. The poles occur on the real q axis when

1=Uo(q)RcrI°(q.nq). (5-44)
and
Im 11%, ()q) = o . (5.45)

The dispersion relation (5.44) has the approximate solution (see Ref. 11, Sec. 15):

(2,, = 1- am [1 + 9/10 (q/q-I-p)2 + ] (5.46)
where up] is the plasma frequency:
up, = (4 it ne2/m)1/2 (5-47)

and q": is the Thomas-Fermi wavenumber,

(ITF = (6 1|: n62/8F0)1/2 . (5.48)
For a specified real frequency (1), we substitute 0.) for Qq in Eq. (5.46), then solve for q. We

designate the positive solution q*. If (IQ and q* satisfy the condition lflq l > hq*kp/m +
hq*2/2m, then 1m 110(q“, Qq) vanishes, and there are poles on the real q axis at :hq*, for drat

value of (1). The residues at these poles contribute to the asymptotic value of the integral
in Eq. (5.40). The poles correspond to undamped plasma oscillations, which occur
widrin the RPA. (We note drat for numerical purposes, it may also be necessary to in-
clude contributions fiom poles that are slightly displaced fiom the real axis, aldrough the
formal asymptotic contribution of these poles vanishes.)

In this section, we have shown how to reduce the evaluation of a(x, t), asymptotic
in x, to a single remaining quadrature over (1). Numerical (or analytical) investigation of

the ar-integral is reserved to future work.

6. CURRENT DENSITY SUSCEPTIBILITY

There has been considerable'interest in quantum derivations of the conductivity
using the density matrix formalism. Kubo [2] has given a formal solution and Kohn
and Luttinger [36] have shown how the Boltzmann equation appears in a certain
approximation. Among the earlier work, we mention papers by Dresselhaus and
Mattis [37] and Mattis and Bardeen [38], related to an anomalous skin effect in
normal metals. For a fiee electron gas in the absence of a magnetic field, expressions
for conductivities are given in Mattis and Dresselhaus’s paper mentioned above. In
the Coulomb gauge an expression for the current density is obtained by taking into
account particle-like excitations, as in Mattis and Bardeen’s paper.

The response of an insulator to a weak external electromagnetic field of long
wave-length was studied by Ambegaokar and Kohn [39] from a many-particle point
of view. They treated the Coulomb interaction between electrons to all orders of
perturbation theory and analyzed the structure of the corresponding Feynman graphs.
In their analysis they defined proper and improper polarization graphs. The proper
polarization graphs occur for both transverse and longitudinal fields. Improper
polarization graphs occur only in the response to a longitudinal field. We remark drat
there has to be a close connection of their graphical analysis to our equation of motion.

Here we study the paramagnetic current density susceptibility in a gauge
wherein the scalar potential (1) is zero, and the vector potential alone represents
the applied electromagnetic field. Then the longitudinal electric field can be
expressed in terms of the longitudinal vector potential. The gauge invariance condition

gurrentees that the a longitudinal electric field gives rise to the same current, whether it is

49

50

described by a vector potential or by a scalar potential.

a. Tmsvme Current Densig Susceptibilig

Eq. (4.26) serves as the starting point of our evaluation of the transverse
current density susceptibility. The momentum transfer q is fixed along the z direction,
and by considering the x component of the induced current density and replacing the
summation by integration over p, we obtain the transverse susceptibility density
xxx(q/z\,(1)) as

)c,.,.((1’z\.m)=(e/m)2 d3p n(p) _ n(p) (1) Sin 9 Cos (102

(amp-(51ml (1)+ep .q- ep (6.1)

 

The transverse susceptibility simplifies to

 

 

pF 1‘
-(e/m)2 0! p4 tip 0 I Sin3 8 d0 1 1
(21:)2 ar+eq-(pq/m)Cos 0 +in ar—eq-(pq/m)Cos 9 HT]

(6.2)

where eq = q2/2m, 'n is a small positive quantity, and an extra factor of 2 is introduced

to account for the spin summation. By evaluating the angular integral followed by the
momentum integral (Appendix B), we obtain the following results for the transverse

susceptibility.

51

xxx(q.(1)) = -(e/m)2/(21c)2 2mp3p/3

— (In/(l)3 (CD-+8.I + in)2{1/2[p2p -(m/q)2(eq+a) + i102] 1n eq+§+0+iTl
eq-§+(o+in

+ (eq+co + in)(p,.m/q) }

+ ((0-8,, + in)2{ 1/21p2p -(m/q)2(eq-w - i102] 1n “hi-“mi
E(r-fi-(D-ifl

+ (sq—co — inprm/q) }

 

+(m/q) 1/41p4p -(m/q)4(eq+co + i104] 1n “WW"
aq-§+w+in

+ 1/4(m/q)4(eq+m + in)‘{ 2&(eq+co + in) + ZINE/(8.1m + in)]3 }

+1/4lp4p —(m/q)"(eq-(o - in)"] In w‘m'i“
sq-é-(n-in

+1/4 (uvq)4(eq—w - in)4{2§/(eq-(° - in) + ”315"qu ‘ “m3 }

(6.3)

where F; = qu lm. By taking into account the identity (B.3) the above equation can be

further simplified in the limit 11—90".

52

b. n ' ' nt ni usc tibili

Eq. (4.30) serves as the starting point of our evaluation of the longitudinal
current density susceptibility. By considering the 2 component of the induced current
density and replacing the summation by integration over p we obtain the longitudinal
current susceptibility density xzz(qQ,(1)) as

xu((1/z\.(o)= (elm)2 d3p n(p) n(p) [p2 Cos 29 + q2/4]

(map-2pm w ' Err-(1'81)

 

 

§RpA(qr(0)

+ (elm)2 L131) n(p) n(p) P(lCOS 9

-— +
(D-I-Ep-E'Hq WEN-8p

 

 

The numerator of the longitudinal susceptibility simplifies to

 

 

 

 

P1: +1
NLS(q£‘,cu)=_-2_ OJ p4 dp _1_1x2dx 1 _ 1
(211)2 meq-(pqx/m)+ in m—eq-(pqx/m) + it]
. PF +1
+2_q OI p3dp _1_lxdx 1 + 1
(21:)2 (meg-(memoir (D-Eq-(qu/m)+iTl
+ (12 P‘(qz.co) I4 (6.5)

where P1(q,(o) is the RPA polarization diagram and x = Cos 9. By evaluating the angular
integral followed by the momentum integral (Appendix B), we obtain the following result

for the numerator of the longitudinal susceptibility denoted by NLS(qz,(1)):

53
NLS(qz.m) = —2 K2102 2mp3n/3
+(m/q)3(co+eq + in)2{ 1/2[P21= —(m/q)2<eq+co + i102] 1n [Mmm
sq-§+<o+in
+ (eq+w + in)(ppm/q)}

+(m/CI)3(£0-8q + in)2{1/2tpzp —<m/q)2(eq—w — in)2] 1n [ swam]
eq-é—(o—in

+ (sq-w - in)0>Fm/q)}

-(mZ/quq + in){ 1 ”(1)21: "(m/Q)2(€q+0) + i102] 1n [Wmifl]
eq-émin

+ (eqw + in)(ppm/q)}

+(m2/Q)(0)—€q +in){1/2[p2p —(m/q)2(eq-co - i102] 1n [em-é—m—in]
eq—g—co—i'n
+ (sq—o) - in )(me/q) }

+ q2 P1(q,to) /4
(6.6)

Here again we can make use of the identity (B.3) and take the limit 11-)0” for

further simplification.

]

7. CHANGES IN ELECTRONIC POLARIZABILITY DENSITIES DUE TO

SHIFTS IN NUCLEAR POSITION

The nonlocal polarizability density 00‘; r’, (1)) gives the co-frequency
component of the polarization induced at point r in a molecule by an external electric
field F(r’, to) acting the point r’, within linear response [18-23]. This property reflects
the distribution of polarizable matter within the molecule; it represents the full
response to external fields derived from scalar potentials of arbitrary spatial
variation. Thus'0(r; r’, 0)) is a fundamental molecular property. It has applications
in theories of local fields and light scattering in condensed media [20], and in
approximations for dispersion energies [22], and collision-induced polarizabilities
[23, 41] of molecules interacting at intermediate range.

The hyperpolarizability density B(r; r’, (0’, r”, 0)") gives the polarization
induced at r by the lowest-order nonlinear response to a field of frequency of acting

at r’ and a field of frequency on” acting at r”.
The electronic polarization Pim(r, 0)) induced in a molecule by an external field
F(r, (0) depends on the polarizability density a(r; r’, (u), the hyperpolarizability
density B(r; 1", (0’, r”, to”) and higher-order nonlinear response tensors:
Pind(r-, 0)) = I dr’ am I“, or) - F(r’, to)
+ 1/2 1603' I dr’ dr” B(r; r’, (u - to', r”, (0’) : F(r’, to - on F(r”, m’) +. .

(7.1)
This work focuses on the changes in the frequency-dependent molecular

polarizability density when a nucleus shifts infinitesirnally. In Ref. 42 it is shown

54

55

that the derivative of the static polarizability 867043(0)/3RIY is related to the

nonlinear response tensor B(r; r’, O, r”, 0). This accounts for the connection between
the polarizability derivative and the quadratic electric field shielding tensor at nucleus

I, noted by Buckingham and Fowler [43].

The purpose of this work is to prove that the relation between
3%th r’, 0)/<3RI7 and the nonlinear response tensors generalizes to the frequency-
dependent case.
From Eq. (7.1), if a molecule is placed in a static external field Fs(r), its
reaction to an additional external field P(r, 0)) [44] can be characterized by the
effective polarizability density f(r; r’, to; F3), given by
o°(r; 1‘30); F) = a(r; r’, to; F8 = 0) + I dr” B(r; r’, to, r”, 0) - Fs(r”) +. . . (7.2)

The permutation symmetry of the B hyperpolarizability density has been employed to

obtain this result.

A shift 5R1 in the position of nucleus I in a molecule changes the nuclear
Coulomb field acting on the electrons. In this section, we prove directly that the
resulting change in polarizability density is determined by the same hyperpolariza-
bility density B051“; r', (0’, r”, to”) that fixes the response to external fields.
Specifically, we show
agents r’, 0))[6R1a = I dr” [3515“; r’, to, r”, 0) 2I Two", R1), (7.3)

where ZI is the charge on nucleus I and T8a(r"' R1) is the dipole propagator,

1043011, 1') = Vm v,3 ( I R1 - r H). (7.4)

56

The proof is via time dependent perturbation theory. We first calculate the
nonlinear polarization of an isolated atom in an electromagnetic field. The atom may
be characterized by a complete set of unperturbed eigenstates In) satisfying a

Schrbdinger wave equation with energy eigenvalues En. In the presence of the
perturbing Hamiltonian H’, the new ground state is denoted by IW ). In!) is

expressed as an expansion in In) with coefficients an(s), where 3 refers to the order of

of perturbation:

llll>=2,,,s an“) In). (7.5)
The 311(8) are determined by the Schrodinger equation to satisfy

8311(s+1)(t)/at = (in)1 2", (n lH’I n’) an,(s) + (an-1 (n IHOI n) an<s+1> (7.6)
subject to the initial condition

anw) (t=0) = ans , an“) (t=0) = o for s at o (7.7)

where the subscript g refers to the unperturbed ground state. The polarization P is

the expectation value of the polarization operator in the perturbed ground state.

P=<v|P|v)/<\v|v) (7.8)
The perturbing Hamiltonian has the form,
H’= — I d3r- E‘”(r) - P(r) Cos tot. (7.9)

To include all terms proportional to (E“’)2, the perturbation expansion must be

pursued to second order, so that the first three terms of Eq. (7.5) need to be
considered:

57

Iv) = 2,, (13,8 + an“) + an”) In) (7.10)
The coefficients are found from Eqs. (7.6), (7.7) and (7.9) and are

2.5%) = (-1/II) m4)“ [ exp Km“! + co)t + exp Krona - (1))t ]

 

 

(tong + co) (a)n8 — to) (7.11)
an(2)(t) = (1/li2) 2m (H’m)nm(H"°2)mg exp mung + (91 + 012)t
(mmg + (1)1)(Cong + «)1 + 032)
+ exp mung - (01 + (1)2)t + exp mung + (01 -— (1)2)t
(tomg — (01)((0ng - 001 + (02) (a)mg + (1)1)(Cong + ct)l — 012)

+ exp i((l)nE — m1 — (1)2)t ]
(a)mg - (01)((Dng — (01 - (02) (7.12)

Evaluation of the expectation value in Eq. (7.8) gives the polarizability

G(I'; r’, to) and hyperpolarizability B(r; r’, (.01, r”, (02) respectively as [45]

am r'. o) =(-1i)-1 2', [ (we <11“),g + <H“’>E mug]

(cang - 0)) (cang + to) (7.13)

 

B“: I", (01, r”, (02) = K(-m0'; (01’ (02)('fi)-2 112 2",2’ m [ Emmaz>m m’ml)n§

 

+ (11%,? at“)... (on, + (may, (F)... <H*°‘>,,]
(0mg + m2)(mng + (no) (a)mg + m2 )(cong — (1)1 ) (7.14)

where m6 = (01 + (02.
The symbol I denotes a permutation operator: for example, 112 denotes the

average of all terms generated by simultaneously permuting the frequencies (01 and

 

58

(1)2, and the corresponding operators. The notation 2’ denotes a summation from
which the ground state is excluded. The barred notation (Hm)mu is shorthand

for the expression aim)“m - (H7588. The numerical coefficients K depend on the

nonlinear optical process of interest. Their evaluation has been discussed in

Ref. 48.

A shift 5RI in the position of nucleus I in a molecule changes the nuclear
Coulomb field acting on the electrons. This shift can be considered as a static

perturbation, and any perturbed state can be written as

lat) = In) + 2,, Im)(m|H’|n)
——(En rm) . (7.15)

Now we expand all states and energy denominators in the polarizability
expression, Eq. (7 . 13), to first order in the nuclear shift perturbation, and neglect the

higher order terms. In addition we replace the difference between the first order cor-
rection to the energy (AF.n - ABS) by ( (H’(r”))nn - (H’(r”))gg). The resulting
effective polarizability density can be written as,

gem r’,co)= 0(r; r’, 0)) +oP(r-; r’, to) + . .. (7.16)

0%; r’. c0)= (-B)'2 I dr”E(r”) '2n’ (P(r))gn (H"“’(I"))..g (H’(r”)),,n

 

(cons - (0)2

-<P(r))g,. (H ""(r’)),.g <H’(r’)>gg

 

(00,.g - c0)2

59

+ 2', 2' m <1>(r»,,,.<H"“(r3).,,,,<H'(r')>.,,.g + m8<r7>ng<H’<r’3>E<P(r»mn
cumg (tong - or) (0mg (tong — to) ’

 

+ 2’, 2mg ,, (H"’(I°)>..E(P(r))gl,,<H’(r”)>mn + (H“"(r’)),..,,,<P(r)>g,.(H’(r”)>um

(0m (cong — 0)) comm)n8 - to)
1

 

 

+ [mung - (0) replaced by (con8 + (.0) ][complex conjugate of the rest ] . (7.17)

 

The last two terms in aP(r; r’, co) can be rewritten by splitting the-sum into

two parts, the first part with m at g and the second part with m = g.

Fifth = 21.2 'nta n<H “(r ’)>ng(P(r))m<H’(r")>rnn ‘2 'n(H 4”“ 7>ng<P("»gg<H 1'75!-

term comn (tong - (0) runs (cong - to)

 

(7.18)

Sixth = X’nX’mRH“(r’))mg(P(r))gn(H’(r”))nm «Z’,,<H“"(r7),;g<P(r))g..<H’(r’3),.g
term wmn (mug — 0)) mng (mug - (1))

 

(7.19)

We can rewrite Eq. (7.18) by permuting m and n as,

Fifth = 21.2 ’mai’”<r)>m§<P<r))E<H ’(r’wnm - 23.01“(r7>ns<1’('»ss<“’("7>sn
term (um (a)m8 - 0)) tong ((l)“g - 0))

 

(7.20)

By adding the first two terms in Eq. (7.19) and Eq. (7.20) we obtain the single term
Z’nZ’mfl, (H"’(r’)),,.§(P(r))gn<H’(I"3)mm

(a)mg — (.0)(0)ng - to)
By adding the above expression and the first term of aP(r; 1", 0)), we can rewrite

aP(r; r", (u) as

60

am; r'. a» = (40-21 dr" E (r') - 23.23,, (P(r))E<H’(r"))nm(H “(l-um;
(comg - 0mm“8 - m)

 

+ (P(r)),mm ”(runmarc’wm + <H‘°(r 9).. <H’(r’3>gm<P(r)>mn

ms (was ' E) mms(‘°ns ' “’5

(D

 

+ “(film — to) replaced by ((1),.g + to) ][complex conjugate of the rest]

(7.21)
The expression in brackets is identically equal to B(r; r’, to, r”, O) (with all
permutations counted). This proves directly that the resulting change in polarizability

is determined by the same hyperpolarizability density B(r; r’, (1)1, r”, (02) that fixes

the response to external fields. Thus we prove the validity of Eq. (7 .2) and Eq. (7.3).

8. SUMMARY AND CONCLUSIONS

We have unified the asymptotic expansions of linear response functions of an
interacting Fermi gas in its ground state with a uniform positive charge background,
so that the whole system is neutral. The results obtained are valid within the random
phase approximation.

In chapter 4 we demonstrated the possibility of studying the induced current
density by use of the equation-of-motion method, as a linear response to a vector
potential. We separated the transverse and longitudinal current-density susceptibilities,
and in accord with physical expectations, there is no screening in the transverse
current density susceptibility.

In chapter 5 we have given a generalization of the Langer and Vosko result
by evaluating the charge density susceptibility and the nonlocal polarizability density for a
scalar potential of arbitrary spatial variation. The static longitudinal component of the

nonlocal polarizability density is related to the dielectric function of the electron gas. The
asymptotic analysis shows that 0Lu(r, r’; 0) has two long-range components within the
random phase approximation. One is associated with Friedel oscillations in the charge
density, and it reflects the impossibility of constructing a smooth function with the
restricted set of wavevectors for the states in the electron gas that are unoccupied at T = 0
(k > kF) [11, 12, 16]. The second long-range component is an I r — r’ I’3 non-oscillatory
term due to charge screening. Because of its physical origin, the presence of this term has

significant implications for calculations of aL(r, r’; 0) for individual molecules: it raises

the possibility of long-range, intramolecular polarization effects. As an extension of

61

62

the above work, we have reduced the calculation of the asymptotic, dynamic charge
density susceptibility (within the RPA) to a single quadrature over frequency.

In chapter 6 we have obtained explicit expressions for current density
susceptibilities within the RPA in a gauge where in the scalar potential is zero.

In chapter 7 we obtained a connection between linear and nonlinear response
tensors, via a direct perturbation theory of general validity. We proved directly that the
electronic charge distribution responds to the change in Coulomb field due to a shift in the
position of the nucleus via the same hyperpolarizability density that describes its

response to external fields.

APPENDIX A
This appendix provides mathematical details on the derivation of the asymptotic
form of the nonlocal polarizability density aLzz(r, r"; 0) in the RPA, as given in Eq. (5.23).

It also provides auxiliary results for the derivation of Eqs. (5.25), (5.26), and (5.28), and
brief comment on the numerical evaluation of the RPA and VS polarizability densities. In
deriving Eq. (5.23), we start with Eq. (5.16), which we first express in terms of integrals
over the full real axis from -oo to co, and then evaluate by complex contour integration. To
convert Eq. (5.16) to Eq. (5.19), we have used the fact that g(z) is an even function of its

argument. In Eq. (5.16), the integrand containing j2(kx) is well-behaved as k —) 0, since
limk _) 0 (kx)"l sin kx = 1. In contrast (kx)’l exp(ikx) diverges as k —-) 0. The convergent
integral in Eq. (5.16) can be expressed in terms of the Cauchy principal value of the integral

containing (kx)-l exp(ikx), however, as in Eq. (5.19). The Cauchy principal value of the

integral is designated by p; it is defined as a symmetric limit:

to L, dk k2 jzasx) «cikx mo

—8 N

lime _, 0L” dk k2 j2(kx) oi"x F(k) +1 8 dk k2 j2(kx) eikx F(k) . (A. 1)
The function g defined in Eq. (5.20) has branch points at :1: 2kF :t in. By choosing

the branch cuts shown in Figure 1, we ensure that g is real on the real axis, as required
[11]. Also with this choice, g is real on the imaginary axis, in the limitn —+ 0.
The integrals in Eq. (5.19) are evaluated by complex contour integration, with the

contour shown in Frg. A. 1. The only singularity enclosed by the contour is a simple pole,

63

lmk
1

C(2) C(1)

III/IIIIIIIIIIIIIIIIIIII

---- ------ -_-_-_-_---
---- - --- --- -
rain—7
4

 

 

 

 

 

-2kF + in 2kF + in

2.6% A
x . Rek

6*

\l

I‘
‘)fivvvuvwvzznt

 

 

 

 

"iZI‘f: " i1]

to
r
'n
I
3

4uvuuuvuvuUvvwwuanmnnnnannnnnnannnnnb(

 

 

Figure A.l. Contour in the complex It plane, used for evaluation of the nonlocal
polarizability density in the RPA. Branch cuts are shown as striped
lines; the symbol x marks the branch cut origins at the points
iZkFtin, apoleatk=0, andapole neari k1?-

65

located near ik'rF (see below). The integral over the large semicircular portion of the
contour vanishes in the limit as R -> no, where R is the radius of the semicircle. The integral

of ku eikx [E(k, 0)"1 - 1] along the smaller semicircle around k = 0 (with radius 1:, K -—> 0)
has the value in: when n = -1; for larger n, the integral along the smaller semicircle vanishes.

' Hence, by the residue theorem,

to L” dk 1tn eih [e(k, 0)-1— 1] + i rt is",l + 10(1) tiltlrn eikx [6(k, 0)-1 - 1]

+ C(2) dk k“ eikx [e(k, or1 - 1] = 21: i Kl , (A.2)
where K1 is the residue at the enclosed pole. The contours C(1) and C(2) run around the
branch cuts, in the sense shown in Fig. 1. Since
[8(k.0)’1-ll = -k1p2ga</kp)/lk2+k1ng(k/kp)l . (A3)
the integral around C(l) is

Ice) dk k“ oi"x (eat, 0)"1 - 1]

= 1 R1122 In dv [ (21s,; + iv)“+2 exp [ i (ZkF + iv) x ] (gg - 81)

x [ (2kF + iv)2 + it“:2 gL 1'1 [ alt}: + iv)2 + kn} gR ]‘1 } (A.4)

and similarly for C(2)

Ice) dk 1t" en“ [E(k, orl — 1]

= i W In du ( (-2kF + in)n+2 exp [ i (-2kF + in) x ] (gg - gL)

x [ {-ZkF + in)2 + 1t“.2 gL ]‘1 [ (-2kF + in)2 + lo",2 gR ]‘1 } . (A.5)

In Eqs. (AA) and (A5), gR and BL denote the values of g at points immediately to the right

or left of the branch cuts. For the integral along C(l),

gR — gL = - (4ka + iv2) 1t] [8 (zltF + iv) k1,] (A.6)
and for the integral along C(2)
gR - gL = (—4kpu + iu2) rt/ [8 (-2kF + iu) kF] . (A.7)

Because of the factors of exp(-vx) and exp(-ux) appearing in Eqs. (AA) and (A5)
respectively,the dominant contribution to the integrals comes from small v or u, when x is
large (i.e., from points near the branch origins). Laplace’s method is not directly applicable
to the integrals, though, due to the logarithmic singularities remaining in gR and gL [46].
Therefore, we expand the integrands for small v or u, retain the terms of the two leading
orders, and then use the analytical theory given by Olver to obtain asymptotic expansions

(in x) for the integrals with logarithmic singularities [46]. We also use [47, a]

lo vllnve‘Vde = 2x-3(3/2-c-lnx), (A.8)

where C is Euler’s constant. Algebraic simplification yields Eqs. (5.21) and (5.22). Errors
in these equations due to the first omitted term in the asymptotic expansion are of the

order
(a x‘4 + b x'4 ln kpx).
To complete the calculation, we must determine the residue K1 of the integrand at

the pole in the upper half plane. This pole occurs at kp, the root of

k, =ik1~sgm<kpikp) . (A9)

67

and the residue at the pole is

Kl = -1/2 kpn'l kn} g(kp/kF)exp(i1tpx) . (MD)

We recast Eq. (B9) in terms of the new, real variable cp defined by kp = i cp k’I‘F- We
also define C =(4ars/1t)m. Then cp satisfies

cp2 = g(icpC) . (A. 1 1)
In the limit 11 —> 0, evaluation of g at points on the imaginary axis yields

g(icpt) = 1/2 + (ope-1[1+(cp02/4larctanlcpt/21 . (A12)
Hence the location of the pole depends on rs. When rs = 2.0, cp = 1.056; when rs = 4.0,

cp = 1.114; and when rs = 6.0, cp = 1.172. The residue K1 is

K1 = —1/2 (rely-1 kn,“+1 g(iepc) exp(-ckaFx) . (A.13)
Therefore, the contribution to the integral hour the pole in the upper half-plane decays
exponentially with x, and for large x it is negligible relative to the error in the asymptotic
expansions of the integrals along C(1) and C(2).

Use of Eqs. (5.21), (5.22), and (A2) yields Eqs. (5.23), (5.25), and (5.26) in the

main text. We obtain Eq. (5.28) with the use of auxiliary integrals

L, k (k2+ltTF2)‘1sinltx dk = m exp (-kTFx) (A.14)
L, (k2 +k1-F2)-1 cos lot dk = warn.) exp (—k-I-Fx) (A15)
and

Jo [(1 (k2 + kWh-1 sin lot dk = w(2k1,,2)[1- exp (—kTFx)] . (A.16)

68

APPENDIX B
The integral needed to evaluate the transverse susceptibility is [47, b]
1|:

J Sin3 0 d9_ = (2m?) + (1mm - (a/b)21 1n[ Mb]

. a+b Cos 9 a-b

(3.1)

The equality needed to evaluate all resulting momentum integrals is [47, c]

Idx xan In (ax+b) = [l/(m-1-1)][x""'l - (-h/a)m+1] ln (ax+b)

m+1
-[1/(m+l)](-b/a)“1+1 2 (1/r)(-ax/b)r
r=l
(B.2)
We have used the following identity in order to simplify the current susceptibility

densities:

ln(a+i11)=ln I a | +i tan1(n/a) (B.3)

1.

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