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The electron liquid at any degeneracy

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The electron liquid at any degeneracy
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Wilson, Brian Gregory, 1956-
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v, 235 leaves : ill. ; 28 cm.

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Fermions ( jstor )
Flux density ( jstor )
Kinetic energy ( jstor )
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Dissertations, Academic -- Physics -- UF
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Thesis:
Thesis (Ph. D.)--University of Florida, 1987.
Bibliography:
Includes bibliographical references (leaves 224-234).
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Also available online.
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Typescript.
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Vita.
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by Brian Gregory Wilson.

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Full Text









THE ELECTRON LIQUID AT ANY DEGENERACY By



BRIAN GREGORY WILSON





















A DISSERATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY


UNIVERSITY OF FLORIDA 1987














TABLE OF CONTENTS

PAGE

ABSTRACT................................................. ....... iv

CHAPTER

I INTRODUCTION.. ............................. ................. 1

II BACKGROUND THEORY................................... 7

The System......................................... 8
Basics of Linear Response........................... 9
The Dynamic Density Response ....................... 11
Relation to Density Correlation Functions............ 16
Further Relations .................................. 17
Distribution Functions.............................. 19

III QUANTAL FUNCTIONAL EXPANSIONS ........................... 27

The QOZ Relation................................... 27
Classical Generating Functionals.................... 32
Quantal Generating Functionals...................... 38
Density Functional Approach......................... 45

IV THE BASIC QHNC EQUATION.................................. 51

The QHNC Effective Potential ....................... 54
Numerical Calculations.. ........................... 57

V LOCAL FIELD CORRECTIONS.... ............................. 69

Mean Field Theories ................................ 71
The STLS Method.................................... 73
The Relaxation Function............................. 78
Scalar Products of Operators ....................... 81
Microscopic Theory ................................. 86

VI NUMERICALLY EVALUATION S(Q). ............................ 93

The Lindhard Function............................... 97
Convergence Acceleration............................ 103









VII THERMODYNAMIC PROPERTIES................................. 111
Thermodynamics with Slater Type
Effective Potentials ............................. 116
Interaction Energy Results.......................... 118
Kinetic Energy Results.............................. 120

VIII RESULTS OF THE EXTENDED QHNC EQUATIONS .................. 131

Method of Solution................................. 133
Quantal Hartree Results............................. 141
The Zwanzig Equation ............................... 144
Blending........................................... 147

IX DYNAMICS OF THE LFCF.................................... 189

Calculating the Memory Function..................... 197
Results............................................ 200

X CONCLUSIONS............................................. 219

REFERENCES...................................................... 224

BIOGRAPHICAL SKETCH............................................ 235































-iii-













Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy




THE ELECTRON LIQUID AT ANY DEGENERACY


Brian Gregory Wilson


May 1987


Chairman: Charles F. Hooper, Jr. Major Department: Physics




The dielectric formulation of the many-body problem is applied to the study of the static correlations in electron liquids at non-zero temperatures. The intermediate coupling effects arising from the exchange and Coulomb correlations are taken into account through a local-field correction factor obtained from quantal extensions of classical fluid integral equations. This results in a unified theory which can describe the thermodynamic and dielectric properties of the










-iv-









electron liquid covering the limits of the zero temperature jellium to the classical one-component plasma.

Detailed numerical calculations are provided for the

self-consistent set of equations describing the static structure factor. The variation of interaction energy per particle versus temperature at constant density is shown to exhibit a peaked structure near absolute zero. This feature is not observed using effective potential approaches in classical fluid equations. A comparison of zero temperature results with those obtained from the Greens function Monte-Carlo method is good; however, agreement worsens when dynamical effects of the local-field correction factor are included using the Mori continued fraction method.



























-v-














CHAPTER I

INTRODUCTION


The one component plasma (OCP) is a system consisting of a single species of charged point particles together with a uniform oppositely charged rigid background to ensure charge neutrality. It was first introduced by Sommerfeld as an approximation to metals when band structure effects are negligible. By coupling the standard adiabatic (Born-Oppenheimer) approximation with an assumption of a weak electronion pseudopotential interaction, it can be shownl-3 that the thermodynamics of a simple metal is the sum of that due to a one component electron plasma and that due to a system of classical ions interacting via a state-dependent pair potential. Viewed as a model fluid, the OCP exhibits the characteristics which distinguish real coulomb systems, such as plasmas and electrolyte solutions, from neutral fluids. These include the phenomena of screening and plasma oscillations, which arise from the long-range nature of the coulomb interaction. Thus the OCP is a standard approximation for both astrophysical and laboratory fusion plasmas,4 where the electrons are treated as a polarizable background for the gas of ions. Further applications lie within the local density approximation of the density functional theory of electronic structure, where the excess chemical potential of a one-component electron plasma plays the role of an




-1-





-2



exchange-correlation potential in the one-electron Schroedinger equation.

It is thus not surprising that there has been a large amount of work expended in obtaining accurate theoretical descriptions of both the zero temperature fermion OCP (degenerate electron liquid) and the high temperature classical OCP (electron gas).5-9 Active research is ongoing in three main areas. First, in the course of study of linear dispersion of plasmons in the simple organic polymers polyacetylene and polydiacetylene, it has proved important to contrast the properties of the exchange hole, and its screening, in quasi-one-dimensional solids with those in isotropic three-dimensional jellium (classical OCP).10 Secondly, one has been able to determine experimentally, using inelastic X-ray scattering, the dynamical structure factor S(q, w) for the electrons in a few metals.11-13 It is of particular interest that the experimental S(q, w) show a tendency of two-peak structure for q within the particle-hole continuum, a feature believed to be a property of the uniform electron liquid not presently accounted for accurately.14-19 Lastly renewed attempts are being made to quantify the intermediate degeneracy regime of the fermion plasma.20-22

There are, in fact, many physical systems whose electrons exist in partially degenerate states and are inadequately described by either zero temperature or classical formalisms. For example, the finite temperature thermodynamic and dielectric properties of the electron liquid are needed for a proper treatment of the equation of state of high temperature liquid metals forund in shock wave experiments23 and experiments on liquid metals expanded to near their critical





-3


points,24-25 of high density inertially confined fusion plasmas, and of the finite temperature exchange-correlation potentials used in density functional calculations of atomic properties at high pressure and temperature.26 Semidegenerate electron liquids are also found in semiconductors where the low densities involved lead to correspondingly low Fermi temperatures. The finite temperature equation of state of the electron liquid is also needed for a quantitive explanation of the miscibility gap in solutions of alkali metals in their alkali-halide melts.27

The problem of the OCP at intermediate degeneracy has only
recently received attention because of complications not present in the extremes of a classical or a ground state quantum description. For example at zero temperature diagrammatic,28 self-consistent distribution function methods29-31 and pseudo integral equation methods,32-37 together with recent Quantum Monte Carlo38-40 calculations, have combined to present an accurate description of the paramagnetic electron liquid.41-42 But at finite temperature we no longer deal with a unique ground state (invalidating the Monte Carlo approach) nor may we utilize the ground state energy variational principle (invalidating Fermi Hypernetted Chain approaches).* Diagrammatic approaches may be generalized to finite temperatures43



Fermi Hypernetted Chain calculations of quantum fluids9 are
actually variational calculations of the ground state using an adjustable Slater-Jastrow type wavefunction and should not be
confused with the classical integral equation method per se.Rather
the name stems from a diagramatic expansion of the ground state
distribution function which is regrouped into a structure of the
same form as the classical HNC equation.





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but are perturbation expansions reliable only in the regime of weak coupling.44
On the other hand the classical limit may be handled by a variety of techniques, including the Monte Carlo method,45-50 the molecular dynamics method,51 and various approximate methods involving integral equations, most notable amongst these the Hypernetted-Chain equation52-58 and modifications thereof.59-62 Here quantal effects introduce two complications. First a quantal calculation of the pair distribution function is hampered fundamentally by the fact that the canonical partition function can no longer be factored into a solely configuration-integral part and a part involving solely momentum coordinates. Secondly the thermodynamic description will also depend on the momentum distribution (or off diagonal single body density matrix)63-64 and this information is not trivially contained in the pair distribution function.

Various attempts have been made to overcome these complications within the framework of the successful methods available for classical fluids. Wigner-Kirkwood expansions65-66 may be used for those cases where quantum corrections to the classical partition function are perturbative, but their convergence as a series in temperature or Plancks constant is slow (if existent) and is restricted to short ranged potentials;67 it is, in fact, an ill defined expansion for the OCP.68 Published work often centers around diffraction effects and assumes Boltzmann statistics.

Previous calculations of the OCP at intermediate degeneracy and coupling have been of several types. The most obvious course has been





-5



to approximate the true quantum mechanical Slater sum which appears in the canonical partition function with the corresponding classical expression involving a temperature and density dependent effective potential. Pokrant69 and others70-71 have developed a finite temperature variational principle to approximate the Slater sum as a product of an ideal fermion Slater sum and that of pairwise additive potentials. The energy is calculated by approximating three-body distribution functions in terms of pair distribution functions, which are obtained from the HNC equation using the effective potentials. Zero temperature correlation energies from this method are, however, in substantial disagreement with the parameterized fittings of the quantum Monte-Carlo results.41-42

Several other groups have studied the thermodynamic21-22'72-75 and dielectric76,77 properties of the OCP by using finite temperature perturbation theory within the random phase approximation. In particular Perot and Dharma-Wardana21 and Kanhere et al.22 have presented closed form parameterizations of the exchange-correlation energy and chemical potential within the random phase approximation (RPA) for the paramagnetic and spin polarized cases. However the RPA is a weak coupling theory that results in considerable error at metallic densities.

Dandrea et al.20 attempt to overcome the limitations of the RPA by including a static local field correction factor (LFCF) within the framework of the dielectric formulation. They use an approximate form for the LFCF which has two adjustable parameters. The first is fixed by assuming a value for the pair distribution function at the radial





-6



origin. The second is obtained from an appropriate interpolation between known Monte-Carlo results at zero temperature41-42 and in the classical regime47-50 for the bulk modulus. Tanaka et al.78 approximate the LFCF at all temperatures by the static zero temperature LFCF of Singwi-Tosi-Land-Sjolander (STLS) self-consistency scheme.29-31

This thesis concerns itself with a unified theory capable of

describing the dynamic density response function of the OCP throughout a broad range of densities and temperatures: from a fully degenerate quantum plasma at absolute zero temperature to a nondegenerate classical gas. It differs from the approach of Dandrea and Ashcroft or Tanaka et al. in two important respects. First, that the theory be particularly applicable to electron liquids at metallic densities, which are not weakly coupled, perturbative local field corrections to the random phase approximation are replaced altogether by a selfconsistent field determined by a quantal extension of classical integral equations. Formally we can define a LFCF that, unlike Dandrea et al., requires no known results, and unlike Tanaka et al., is temperature dependent; indeed it may be viewed as a generalization of the STLS LFCF of zero temperature and reduces to the proper classical result. Secondly, dynamical dependencies of the LFCF can be rigorously included. This has been shown to be a mechanism for modeling the double peaked shape of the dynamic structure factor observed in metals.11-19 It is also an essential theoretical featurestatic LFCFs cannot simultaneously satisfy the compressibility sum rule and the third frequency moment sum rule for the density response.79














CHAPTER II

BACKGROUND THEORY


This thesis will concern itself solely with the OCP as a Fermi liquid. The most basic characteristic of a liquid is that it possesses short range order as opposed to the long range periodicity of a crystalline solid. Since the structure of a crystal is determined experimentally by observing the Bragg reflection of X-rays, it is natural to seek a quantitative description of the liquid structure via the intensity of X-ray, thermal neutron, or light scattering.3,80-81 Of particular interest then are the fluctuations of space and time dependent densities which describe, at long wavelength, the cooperative motion of many particle systems. For this purpose define the particle density correlation function as

S (r ', t) : <6n(, t) Sn(6, o)>
nn eq

where 6n(r,t) is the excess particle density operator


Sn(r, t) = n(r, t) <6n(r, t)>eq

and the equilibrium average is indicated by the brackets <...> eq Specifically, even though eq = 0, there will be spontaneous, usually small, fluctuations on a local scale. The density correlation function is a measure of these fluctuations.




-7-





-8



In this chapter we will begin our investigation into the

structure of quantum fluids by reviewing the general properties of the time correlation functions related to the particle density correlation function. These include 1) various symmetries (for example time reversal), 2) positivity properties which are related to the dynamical stability of the system, 3) fluctuation dissipation theorems which connect spontaneous fluctuations and energy dissipation in a thermally equilibriated system, 4) Kramers-Kronig relations which express causality, like the well known one between the index of refraction and the absorption coefficient, and 5) sum rules which provide restrictions that any approximate microscopic theory must fulfill.


The System


Because we will be dealing with a many particle system from a quantum statistical point of view, we will employ the second quantization representation of quantum mechanics even though we will mostly be working in a canonical ensemble where the number of particles is fixed.

For our equilibrium "system" we shall define in the Schroedinger (time dependent state) representation the unperturbed Hamiltonian


H = T + V


in terms of field creation and annihilation operators.43 The kinetic energy operator is





-9


h2 82
T = J dI ~I+(I) {- 2nn -2 (I) 8r1

while the interaction energy operator is given by

2
S f dl dlI k'(I) (I) { e '(I) t(I) Irl r21

Roman numerals were used as shorthand notation for labels; that is "I" stands for the position coordinate as well as the discreet spin label of particle 1.

If there is an external (possibly time dependent) field present, there is in addition a perturbation to the hamiltonian of the form

SH(t) = f drn(r) 6V(r, t)

where we introduced the electron density operator

n(X) = i '+(I) T(I)
spins

Basics of Linear Response82-85


In the external field the hamiltonian is explicitly time dependent and given by

H(t) = H + 6H(t)

in the Schroedinger representation where the operator n(r) is time independent. The time dependance is carried by the density matrix, or ensemble operator p(t) which describes the state of the system such that the average of n(r) is

= tr p(t) n(r) tr p(t) = 1





-10



What follows is entirely parallel to traditional derivations of time dependent perturbation theory in elementary quantum mechanics texts. We have to solve the time evolution equation for the density matrix

i at p(t) = [H(t), p(t)] = [H, p(t)] + [6H(t), p(t)]

(note the commutator is reversed from the Heisenberg equation of motion for operators) subject to the initial condition

p(t = m) = Peq [H, p eq] = 0

The initial condition expresses the fact that the system is stationary before the external field is turned on--we also require that the external field decay sufficiently rapidly as time reaches infinity.

For manipulations it does not matter what p eq is but since the system starts from thermal equilibrium we take


Peq = P-BH/tr p-BH

If we assume a time dependance to the density matrix of the form

p(t) = Peq + 6p(t)

then the solution to the time evolution equation to first order in 6p and linear in the external field is

Sp(t) = dr e-iH(t r)/f [6H(), eq e+iH(t )/


(This is easily verified by differentiation remembering that time appears both in the limits and in the integrand.) From this equation the induced change in the average density





-11



6 tr p(t) n(r) tr Peq n(r) is given by

t
6 = f dt' f dr' < [n(? t), n(F' t')]>eq 6V(6' t')

t
zi f dt' J d'' x"(rt; r' t') 6V(6' t') (2.A)
-w

where [A, B] = AB BA is the commutator bracket. In the above equation are the Heisenberg operator for the unperturbed system

n(r' t) = et/ n(r) e-iHt/

and we have introduced the dynamic density response function (please note the double prime superscript) defined by


x"(rt; 't') = < En(t), n('t')]>eq

This equation is the fundamental result of linear response theory. It shows that the density response to a small external potential is the averaged commutator rather than the correlation function S(r,t) as one might expect. However we shall see that the two are intimately related.


The Dynamic Density Response Function


In equilibrium the system is space translationally invariant. We adopt the convention that forward spatial Fourier transforms have a minus one signature and write





-12



x"(E) = d(r -r) e x"(r r')

= < 1 [n (t), nj(t')]> k k

Here we have used the definition of the spatial transform of the density operator

-ik-r
n (t) = S dr e n(rt)

which can be expressed in terms of plane wave creation and annihilation operators as

n (o) = f dP a+ 'a+
K a (21) P+K,a from which it follows that

n n -K K

Time translation invariance of the dynamic density response

function is easily established from the definition of a Heisenberg operator and the cyclic nature of the trace. We adopt the notation that forward temporal Fourier transforms have a plus one signature:


x"(Kw) = f d(t t') ei(t t') x"(K, t t')


It is easy to show that this space and time Fourier transform is real (since the function is a commutator of hermitian operators), an odd function of frequency (since the equilibrium state is invariant under time reversal and parity), and depends only on the magnitude of k (spatially isotropic). It can also be shown that

ex"(Kw) > 0





-13



in a stable system.85 This is just a statement that a dissipative many body system takes more energy out of the external field than it gives back.

In addition to the density-response function it is convenient to introduce certain interrelated functions. The complex response function (please note the tilda) is defined in transform space as

x(Kz) = f S x"(kw)
n w-Z

This is an analytic function of the complex variable z as long as imaginary z does not equal zero (on the real axis it has a branch cut). For z above the real axis it reduces to the temporal Laplace transform of the density response:


x(Kz) = zi S dt eizt x"(Kt)
0

while below the real axis

o izt
= (- zi) f dt e x"(Kt) This follows from the identity

iwt izt
co L e e t>o nn
f 21i w z 0 t < o


o t>o
t >o Inn z < o
-izt
e t
which can be proven using the Cauchy integral formula in the complex omega plane. (For t>O we can close the contour on top-exp(iwt) being bounded-and the contour is in the positive sense. For t




-14



negative sense. We then just sum the residues, the pole being included depending on the sign of imaginary z.)

The physical response may be defined in transform space as the limit of the complex response as we approach the real frequency axis from above


x(Kw) = lim x(K, z = w + ic) = lim f do x"(Kw')
C0 CO _. ir w' -(w + ic) Using the identity (P denoting Cauchys principal value)

= P + ir 6(x)
X+ic X


the physical response can be expressed in terms of its real and imaginary parts as

x(kw) = x'(Kw) + ix"(Kw)

where the imaginary part is just the previously defined density response and the real part (note the single prime) is


x' (Ko)) = P f d x"(Kw)
-wnw'an even function of frequency, is a manifestation of the KramersKronig relations.86

The utility of the physical response function is that it links

the induced change in the time-space Fourier component of the density with the time-space Fourier component of an arbitrary external potential

6 = x(Kw) 6(Kw)





-15



One may also note that this same relation may be easily obtained via eq. 2.A using the definition of the physical response in real (untransformed) space, namely

x(rt) = zi e(t) x"(r,t)

The response to a static external potential may be derived from linear response theory by assuming that the potential is adiabatically switched on starting from the remote past

-E(t)
SV(Ft) = V() e(t) t < o o t > o At t = 0 the external potential is at full strength and we find


6 = zi J d fJ dI' x"(r r', -) e-EZ 6n(')
0

or

6 = x(K) 6V(K)

where the static response is given by zero frequency limit of the physical response

x(K) E x(K, = o) = lim x(K, z = ic) C-o
A note on this last equality. Formally we have lim x(K, z = ic) = lim (Pf x"(Kw) + ix"(K, w= o) s0o E-o -C W -but because x" is a real odd function of frequency the imaginary term vanishes and we can drop the "principal part" condition.





-16



Relation to Density Correlation Functions


At this point we should explicitly confirm the statement that the density response is intimately related to the density correlation function, the latter being the experimentally pertinent quantity. Now because the Fourier transforms of correlation spectra are related by the equation83


S eq e -it dt ePO fc e-it dt it follows directly that

< e-i"t 1 e-Of -wt
1 < [A, B(t)]> e dt (1 -e ) eq e dt and so in particular (using the fact that x"(Kr) is odd in frequency) x"(Kw) ~ (1 e-f ) S(Kw)

where S(k,w) is the time-space Fourier transform of the time dependent density correlation function. We note that S(r,t) is Fourier transformable because when r and/or t are very large, then n(r,t) and n(O,O) are statistically independent


eq eq eq


and so S(r,t) as defined as a correlation of excess density vanishes.

The above relation between the density response and density correlation functions is an explicit example of what is called the fluctuation dissipation theorem84 which relates two physically distinct quantities of fundamental experimental significance: 1)





-17



Spontaneous fluctuations, which arise even in the absence of external forces from the internal motion of the constituent particles. Described by S(k,w) these fluctuations give rise to scattering of neutrons or light. 2) Dissipative behavior--all or part of the work done by external stirring forces--is irreversibly disseminated into the infinitely many degrees of freedom of thermal systems.

The fluctuation-dissipation theorem shows that S(k,w) is not

quite symmetric in w, being a little stronger at positive frequencies than at negative. Indeed, since wX"(Kw) is always even in w, it is generally true that

S(K, -w) = e-fK S(K, w)

This result makes sense from a neutron scattering point of view. Positive frequency means the neutron has lost energy to the system (by creating an excitation of energy hw) while negative frequency describes a process in which the neutron has picked up energy from the system (by destroying an excitation). Of course, to destroy an excitation you must first have one, and their relative abundance is given by exp[-bhw]. This dissymmetry of the scattering intensity, proportional to S(k,w) is only pronounced at low temperatures, kt
-- it is absent classically.


Further Relations79


Besides its relation to the time dependent density correlation function the density response satisfies certain sum rules, usually presented in the form of moments of the density response:





-18



<<)>> E f J { x"(Kc)} dw


The j = 0 or zeroth moment follows from analyticity of x"(k,w).

D x"(Kw) = x(K, o) = x(K)


All odd order moments vanish because x" is odd in frequency. The remaining higher order moments can be derived from the expression

ia8 1
(t) x"(rt; 't') = < ) n(, t) n('t')]> taken at equal times t = t'; this means that

I dw J x"(Kw) I d( e') ei(r r')



<(i)J+1 [[...En(rt), H] ..., HI, n(r't)]> The right hand side contains a sequence of equal time commutators which can in principle, and sometimes in fact, be exactly calculated. For example the second order moment yields the Thomas-Reiche-Kuhn or F-sum rule equivalent

CK2
f d wX"(Kw) = eK
Ira m

For the electron gas the fourth moment has also been calculated.87-89

The sum rules are often used to provide coefficients for an expansion of x(k,z) for large z from the definition


x(Kz) = J d xK( <> <>
(Kz)f Z z 2 4+





-19



From its derivation, which expands (1 w/z)-1 = 1 + (w/lz) + (w/z)2 + ... it is clear this expansion can only be asymptotic, i.e. valid only when Izl is large compared to all frequencies in the system, which means all frequencies for which x"(k,w) is not effectively zero.

An interesting feature of the sum rules is their existence. There is no reason why the thermodynamic average of the multiple commutator should exist to all orders, and indeed although the sixth moment has been calculated in the classical fluid limit90 it may diverge for the degenerate electron gas in certain regions.91 As the sixth moment determines the w-6 term in an inverse frequency expansion, its divergence would herald the presence of a term of the form k4w-11/2 in S(k,w) at long wavelength for the high frequency behavior of the spectrum. Great caution is thus indicated in the use of moment sum rules beyond the first three mentioned.


Distribution Functions


It is natural to describe the structure and thermodynamic

properties of liquids in equilibrium by employing static distribution functions. Most treatises on fluids introduce such quantities first and later generalize to include time dependance as a prelude to exploring non-equilibrium statistical mechanics. Here we have taken a reverse path; we have already related time dependent density correlation and density response functions and it is our intent now to relate these with the static distribution functions common to liquid theory.





-20



The most basic static distribution function is the single

particle distribution function. If we define the number density operator in N particle configuration space as

n
n(r) = X 6C- ii)
i=l

(the ri are quantum operators) then the single particle distribution is given by

p(r) = N f
p() = = N J W(r 1, r2 ". rN) dr2 .. dN

where the configurational part of the partition function is

Q =f W(r .. rN) drl .. drN

and the configuration probability W(rl, ... r n) is given by the diagonal slater sum


W(rl, ... rnlrl, ... rn)


where, in general, the off diagonal Slater sum is defined as


3N -Hop W(r rN I ...rN) = N 3N rr" N) e -So ;i r N )


In the above equation the index "i" denotes any complete set of properly symmetrized N-particle basis states and we have introduced the DeBroglie thermal wavelength

2 1/2
T m





-21



In the classical limit the probability function "W" is just the Boltzmann factor of the potential energy of the configuration; at zero temperature it consists solely of the ground state wavefunction.

The information contained in the single particle density distribution is minimal, for in the case of a translationally invariant Hamiltonian it is a constant equal to the average number density of the system. However we introduce it here because relevant thermodynamic properties we will consider require the information of the off-diagonal single particle distribution function63-64

p( I ') = N (r r2... N .. ) d ... d Q 1 2"' Ni r2 rNdN The physical significance of the off-diagonal distribution function is that its spactial Fourier transform provides us with the momentum distribution of the interacting system.

The pair distribution function is defined by
2 I
p g(? ) = N(N 1) f H d-r3 .. rN = < 6(r r) 6(r' r>
p g(r Q 3 N i
lsi

and for a translationally invariant system is of the form g(r-r'). The pair distribution function in turn is related to the static structure factor through the relation


S(K) = I + p f ei1' (g(r) 1) dr

Within a factor of density the static structure factor is simply the time independent density correlation function.


S( <(n() ) (n(') )>
S(F ') =P





-22



Note that if we had defined the static structure factor in terms of the density operator instead of the excess density we would replace g(r) 1 by g(r) in the formula relating the two. This amounts to including the experimentally unmeasurable "forward scattering."

The definition of the static structure factor is immediately generalized to form the dynamic structure factor, essentially the previously introduced time dependent density correlation function (again within a constant factor of density) by replacing the quantum operators ri with their time dependent equivalents in the Heisenberg representation. Bearing in mind that ri(t) and rj(t) are operators which in general do not commute (except at t = 0), we easily obtain several important properties of the dynamic structure factor for translationally invariant systems:


1) S(-r, t) = S*(r,t) where S* is the complex conjugate of S.
2) In the classical limit the imaginary part of S(r, t) vanishes.
3) S(Kw) iwtik
3 2n dt e f d e-ikr S(rt)
is real in both the classical and quantal cases.
4) The elastic sum rule


S(K) = J S(Kw) dw


is an immediate consequence of the definitions.


All the relationships amongst the various functions discussed are best conveyed diagrammatically (See Fig. 2.1). Note that in the upper





-23


left corner we have introduced a new function C(k,w) called the Kubo or Relaxation function, about which we will have much to say later. For now it simply expresses the difference between the (dynamic) physical response and the static response function.

The point of our review of the response, correlation, and
distribution functions, so concisely summarized in Fig. 2.1, is that if we wish to generalize the classical many-body methods of calculating g(r) into the quantal regime, there are two possible relations open to us. First there is Scl(k) as the classical limit of Sqm(k) (obviously) or, as


F 6n(r I U) 1
6 u(') x(K)

where the 10 denotes evaluation at constant density/zero external potential, and the F denotes the Fourier transform with respect to the spactial variable r-r'.

Let us contrast the two alternatives. The quantum mechanical static structure factor Sqm(k) is essentially statistical in nature; it is strongly dependent on the dynamics of the particles, depending on S(k,w) Imag x(Kw) over the full range of frequency. On the other hand x(K) depends on the single zero frequency limit of x(Kw), and is essentially mechanical, arising from the Schroedinger equation by our linear response derivation, which makes but implicit reference to statistical questions. This is clearly indicated by the F-sum rule-which holds in any stationary





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x(kw)






x(kw) = X(K) + iwM3C(k) y(Kw) Im x(Kw) lim W o S(kw) =2 (1 + cot h ) y(kw)






x(K) = J y(Kw) dw






x(K) S(K) = J S(Kw) dw Figure 2.1


Relations between the linear density-density response function and the structure factor.





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ensemble--expressing nothing but the conservation of particles and the elementary commutator of position and momentum.

This dissertation will concern itself with the integral equation methods of solving the many body problem. This route seems particularly promising for treating quantal fluids in that for the classical limit there exists a wide variety of integral equations for g(r) which have been investigated and give results in good agreement with experimental results.93'3 The purpose here is to extend this procedure into the quantal regime. A fundamental ansatz of this treatise is that one would expect the more accurate integral equation approximation to correspond with the less sensitive function. The extension of classical integral equations into the quantal regime via the density response function is presented in Chapter II.

With this fundamental ansatz, however, there is a fundamental drawback: From x(K) one obtains an integral equation not for gqm(r) (which is used in the computation of thermodynamic quantities) but rather n(r/v), that is the distribution function in the presence of an external potential which has the form of the interparticle interaction. While it is true that in the classical limit n(r/v) = nog(r), this is not true in the quantal case, since we are neglecting exchange and recoil effects. The external potential can be thought of as arising from a distinguishable particle of infinite mass and therefore fixed in position. An extreme example occurs for the case of very weak interactions between fermions. In such a case n(r/v)/n0 is approximately unity everywhere, whereas g(r) has approximately the ideal fermion value of 1/2 at the origin.





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Thus our task is twofold: first to generate an accurate quantal integral equation for n(r/v)/no and second to generate an approximate g(r) from knowledge of n(r/v)/n In essence our second task is to go from the bottom left of Fig. 2.1 to the bottom right along the paths drawn in this figure. We cannot go along the diagonal, as this path is one directional (we lose information when we integrate y(k,w) over w in order to obtain x(k)). We must first traverse over the upper left path of Fig. 2.1; we must obtain an accurate approximation to the Relaxation function. This is taken up in subsequent chapters.














CHAPTER III

QUANTAL FUNCTIONAL EXPANSIONS


In this chapter we will establish integral equations93 for the quantity n(rlv)/n (the density distribution in the presence of an external potential of the form of the interparticle potential) through the functional expansion method.94-95 Because in the classical limit these equations reduce to the familiar Hypernetted-Chain (HNC) or Percus-Yevick (PY) equations for the pair distribution function, we will review the classical derivation with the aim of establishing corresponding quantal generalizations.


The OOZ Relation


In classical fluids the Ornstein-Zernicke (OZ) relation plays an important role in treating integral equations for g(r). Thus in this section we derive a quantal extension of the OZ relation (QOZ).

We start from the known functional dependance of the classical canonical partition function on pairwise additive potentials. This allows us to write the functional derivative of the density distribution as96


n( n 6(r r') + n2 {g(Ir r') 1} = F {n S(K)}
6 Ur') o




-27-





-28



The I denotes evaluation when the arbitrary static external potential U(r) is turned off, that is at n(r) = no (n ; the average density). The (negative power) script F denotes the (inverse) Fourier transform on the spatial variable r r', and S(k) is the previously defined static structure factor


S(K) = 1 + no f dr e- ik {g(r) }

Now consider the functional

6 BU(r')
6 N(r")

This is a vacuous quantity unless such a functional of density can actually be constructed.97 If such a creature does not exist, we define it through the chain rule of functional calculus:


f dr" 6 BU(F) 6n(F") =
6 n(") 0 6 U(r')I

It follows that the inverse functional derivative is

S- BU(r') F-1 1 ( )
Sn(r") o o

where we have split up the expression for the derivative into two pieces (this should be considered an arbitrary process) with the first piece chosen to be the non-interacting (g(r) = 1) result and the second piece (c(r)), called the direct correlation function, representing the deficit from from the non-interacting result. The functional chain rule as expressed in terms of c(r) takes the form





-29



h(r) = c(r) + no f c(Ir r'l) h(r') dr'


This is the classical Ornstein-Zernicke (OZ) relation, where h(r) = g(r) 1 is called the pair correlation function.

Furthermore, in the classical limit where there are no exchange or recoil effects (the momentum integrations factor from the configurational piece of the partition function) one can rigorously derive the bootstrap relation96


n(f I U = v) = no g(F)

where the external potential U(r) has the form of the interparticle interaction V(r). This bootstrap relation allows us to reformulate the OZ relation in terms of the inhomogeneous density distribution as follows:


f d?' 6 BU(f) n(F' I U = v) no = c(F) (3.1)
6 n( ')

A quantal extension of the OZ relation is straightforward;98 we will follow many of the above steps.

First, from linear response theory, we saw that in the quantal case

6n(r I U) = F1 {n x(K)}
6 BU(r) 0 K

where x(k) differs by constant factors from the static response function previously defined in Chapter II. Defined here it is conveniently dimensionless; note that only the product Bx(k) remains finite as temperature goes to zero.





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Again the chain rule of functional differentiation requires that


6 KU(r') F {1 L 1 { 1 "
6n(" I U) = K n x(K) n- ) (3.2)


but here we have decomposed the inverse derivative into two new (and as yet unspecified) functions. These are determined as follows.

As we wish c(r) to reduce to the classical direct correlation function in the proper limit, we need only require J(k) = 1 in the classical limit. We furthermore make the ansatz that


f d' 6 U(F) n(1' I U = v) no = c(r) (3.3)
6n(r') o


also holds for the quantal case.

We have seen that this equation is strictly true in the classical case; by combining the two above equations we constrain the function c(r), and so simultaneously J(k), by the relation


FK{n(? I U = v) n } = 1 (3.4)


In hindsight we now see the motivation of requiring eq. 3.3 be fulfilled: as J(k) tends to unity in the classical limit we have merely imposed the condition that we recover the bootstrap relation in the classical limit.

The above equation also determines J(k) (and so c(r)) in the general quantum limit, for if the system described by n(r/v) and x(k) were that of non-interacting fermions, then in the limit that U(r) vanishes we see that 3(k) must equal xo(k). This is an appealing result, for it amounts to defining the quantal direct





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correlation function by separating out the non-interacting result, just as we did in the classical case. Henceforth we will implicitly use this result, which will be referred to as the quantal bootstrap relation (QBS).

As a consequence of this now specified decomposition of the

inverse functional derivative eq. 3.2, the functional derivative chain rule expressed in terms of the quantal direct correlation function becomes

( 1) = M{c(F)} + n f d?'(M{c(I ?'I)}) (n((V) 1)
o o where M{...} is an operator defined by

FK{Mf(F)} = xo(R) FK{f(F)}

This is the quantal generalization of the OZ relation (denoted QOZ for short). Note that this equation can be cast into the classical form by defining analogue functions:

gan(F) n( Iv) han) an


can(?) E M{c()}

The classical equivalents of these functions all have "physical" interpretations that are only loosely identified with the analogue. (For example gan should not be confused with the true or quantum mechanical pair distribution function.) To distinguish classical analogues we shall henceforth employ the superscript "an."





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Classical Generating Functionals


The heart of the functional expansion method of deriving integral equations lies in the ability to construct a functional W[n] of the density distribution n(rlu) in a nonuniform system which is reasonably insensitive to the function n(rlu).99 (For other points of view, see Bakshi100 and Choquard.101) If such is the case we can approximate the functional W[n] by making a functional "Taylor series" expansion, for example about the average density no, and truncate after the first (linear) term

W[n] = W[n ] + df'( 6w (n(F' I U) no) Sn(&')

The OZ relation then forms a closed set of two equations in two unknown functions. As an illustration we will consider a set of functionals and the integrals equations they generate in classical statistical mechanics.

First consider the functional


n(F I U)
W[n] = ln
no eu()

The denominator forms what is called the reference distribution. In order to make the functional variationally insensitive, it is constructed to approximate the numerator as close as possible. Here we have taken the classical non-interacting Boltzmann factor. The first functional derivative is easily found to be





-33



6w 6(?-?') 6 U(?)
6n(?') n(r I U) 6 n(?')

and from the definition of the direct correlation function:

6 BU(r) 6r- r') c
6n(F') no we find

6w i c( ')
6n(F')

The functional expansion of W[n] as the potential U(r) is changed from the intitial value 0 to its final value

In n(r I V) = J d?'(n(' I v) n ) 6(- I )
n e- )V(?) 6n(')

along with the bootstrap relation n(r/v)/no = g(r) and the OZ relation then combines to give the HNC equation

g(F) = e- RV(F) ey(7)

where y(r) = h(r) c(r) is called the nodal function.

The PY equation


g(f) = e- RV(F) [1 + y(F)]

follows from the same steps by starting with the functional

W[n] = n( I U)
no e- BU(r)

Because the HNC functional is the logarithm of the PY functional, it is the more insensitive functional of the two from variations. One might be tempted to conclude that it would therefore produce the





-34



superior approximate integral equation. However the validity of the HNC equation (or any other derived by a functional expansion method) depends on whether the cumulative effects of the higher order derivatives in the functional expansion are negligible. In actuality their effect is to push the form of the equation towards that of the PY (note how the form of the PY equation is the small y(r) expansion of the exponential y(r) term of the HNC). This is evidenced by the fact that for purely repulsive potentials the PY and HNC equations of state bracket the "exact" Monte-Carlo or molecular dynamics computer simulation results.102 What we do know from diagrammatic analysis52-58 is that the exact closure equation to the OZ relation for g(r) is of the form


g(?) = e- RV(r) + y(F) + B(F)

where the unknown "bridge function" B(r) is the sum of bridge diagrams of two point functions. The evaluation of the bridge diagrams in terms of higher functional derivatives103'61 or other attempts to account for bridge effects will be discussed in Chapter V. Here our intent is to generate zeroth order integral equations from variationally insensitive functionals, thereby minimizing the effects of the higher order terms that we neglect in the Taylor functional expansion.

It is well known that the PY equation has no solution for a one component plasma system.104 Numerically this arises from the fact that in the PY equation for g(r) the long range tail of the potential is not compensated by that of the nodal function y(r), as it is in the HNC equation.105 This is symptomatic of a physically meaningful





-35



inadequacy, namely; in real systems the particles do not feel the "bare" long ranged potential but a collectively screened or short ranged potential.

An improvement on the PY generating functional would be to

replace the bare potential appearing in the Boltzmann factor of the reference distribution by an appropriately screened one. For example


n(r I U)
W[n] =
RUH(r)
n e

where the Hartree potential is defined as

BUH(F) = BU() + no f dr' (1U(? ?') (g(&') 1)


As the Boltzmann factor now more closely describes the physical distribution n(r/v) this functional is variationally less sensitive than the usual PY


6w ( ') n(r I U) 6 UH(F)
6n(r') BUH(r) BUH(r) Sn(r')
no e no e


Using the bootstrap relation in the definition of the Hartree potential we see that

6- RUH() 6 BU(F)U(r
= + J d5 (S )
6n('') Sn(i') 6n(#')

(n(S) no) J dS BU(P 5) 6(?' s) which when evaluated in the limit v = O(n(rlv) = n ) and using the definition of the direct correlation function (eq. 3.4) yields





-36



6 PUH(r) s(F r')
S (r = c( ') BU(r r')
6n(r') no
0

and so

w = c(r ') + BU(r r')
6n(F')

The linear expansion of the functional W[n] thus generates the equation

n(P I U)
n(? I U) = 1 + f d?' (n(F' I U) n ) (c(F F') + BU(F F'))
BUH(r)
n e


which can be re-expressed with the help of the bootstrap equation and the OZ relation as

IVH(r)
g(r) = e {1 + IBVH(F) + y(F) V(F)}

Note that in this equation the long range part of the potential is cancelled by the long range part of the nodal function, and all other functions appearing in this equation are screened. This equation, which we shall term the PYH equation (for Percus-YevickHartree) is equally applicable to long and short range potentials, unlike the PY equation.

If we try to improve the HNC equation by introducing the screened Hartree potential into its generating functional


W[n] = ln n(T I U)
BUH(F)
no e





-37



we find that its first variation is the same as in the above PYH derivation and (following many of the same steps) the functional expansion leads to the equation


BVH () y(r) + RVH(r) BV() e- RV(F) ey()
g(?) =e e = e

namely the original HNC equation. In other words the HNC equation has the striking property of being invariant to (Hartree) screening of the reference distribution in its generating functional.

From a related point of view it has also been shown106 that the HNC equation is the limit in a series of integral equations that can be systematically generated starting from the PY equation, among others.

Finally we should note that the HNC equation can be derived from an even more interesting point of view. We ignore higher order functional derivatives and assert that the best functionals for our closure purpose are insensitive to variations, and this property shows up in the magnitude of its derivative. This can be minimized by using an even more realistic screened potential for the reference distribution. In fact it is easy to show that if we modify the Hartree potential by replacing the convoluted bare potential by the direct correlation function

Uc() 7 BU(r) f dr' c(P ?') (g(F') 1)

then the derivative of both the HNC and PY like functionals

S= ln n(r -rU) n(r I U)

n e- BU Tr) n e- U (T)
no e n





-38


actually vanishes. They in turn both yield the same equation


g(f) = e (F)

which one will now notice, via the OZ relation, is simply the HNC equation.


Quantal Generating Functionals


Corresponding quantal equations can be obtained from the above classical generating functionals if we replace the reference distributions--the Boltzmann factor n exp{-bU} describing classical non-interacting particles in the presence of an external potential U(r)--by its quantal extension n*(rlu)--the density distribution of non-interacting fermions in an external potential. (Henceforth non-interacting fermion system quantities will be distinguished by a star.) That is we replace the reference distribution by a single particle Hamiltonian approximation. As in the classical presentation of the above section the functionals are then linearly expanded and evaluated in the limit where the external potential is set equal to the interparticle interaction potential.

We have previously derived a quantal extension of the pair correlation function

FQ{no c(F)} = 1

from the ansatz that the exact classical relation

f6 BU() ) (n(F' IU = V) n ) di' = c(F)
6n(r')





-39



remains valid in the quantal case, alternatively expressed by the ansatz that the classical bootstrap relation is extended as


FQ{n(F(U = n) n} = x( 1

By making use of these relations and defining the analogue potential

SUan(r) n*(P I U)
e = a


a closed set of equations is formed with the QOZ relation. Expressed in terms of previously introduced analogue functions, these integral equations bear a close correspondence with their classical counterparts, and indeed reduce to those counterparts in the appropriate limit.

As a first example we consider the functional

n(r I U)
W[n] = ln n*(r I U)

Its first variation is

Sw n*(r I U) 6(? F') n(r I U) Sn*(r I U)
n(r I U) {n*(r I U) [n*(r I U)]2 6n(r')
Sn(r') In*(r U)] This can be simplified in the following manner:

F6n*(r I U) (F{Sn*(r I U))(F{ 6 BU(r")) (n x*(Q))( Sa n(F') 6 RU(F") 6n(F') n x(Q

= 1 n Fa{c an

where it should be remembered that the analogue pair correlation function differs from the quantally extended "physical" pair correlation through the operator






-40



M{f(F)} = F~Q{x*(Q) F{f(i)}}

From the above considerations we see that

6n*(r I U) nan
6n(F') 0

and so

6Sw an
= c (r-r')
Sn(?')

By Taylor expanding the functional W[n] about U(r) O to v(r), we find

n n*(r I V) = f d?' (n(r I v) n ) ( 6 Sn(r')

or

gan e- Van () ean()

This equation we will call the Quantal Hypernetted Chain or QHNC equation.

Following similar steps with the functional

n(r I U)
W = n*(r I U)

we can arrive at


gan (F) = e- an) + yan()}

or what shall be referred to as the QPY equation.

It is of no trivial importance that the QOZ, QPY and QHNC

equations are of the exact same form as their classical counterparts (this in addition to the property of reducing to the classical equation in the limit). This is because phenomenological ways of incorporating the effects of truncating classical functional





-41



expansions at the linear term may still be of use. [In the classical HNC case such a truncation is equivalent to neglecting bridge diagrams in the Mayer expansion of non-ideal gases.] This will be considered further in Chapter V.

It should also be stressed again that these integral equations, even if they could be made exact, are for gan(r) = n(r/v)/n0 and not the pair distribution function. This is evidenced by the fact that if we turn off interparticle interactions, Van(r) vanishes and the solutions of either the QHNC or the QPY equation yield gan (r) =

1. This is correct for the density distribution but the fermion pair distribution has an exchange hold. This difference is taken up in the next chapter.

The exact correspondence between quantal analogue and classical integral equations is lost when one considers screening effects on long range potentials. Rigorously incorporating screening effects is complicated in the quantal case by symmetry effects. It can be shownl07 that the best single particle Hamiltonian (best in the sense that the statistical average over the square of the difference of the exact and single particle Hamiltonian is minimized) is provided by a Hartree-Fock potential, not the Hartree. On the other hand we will see in the next section that as far as density distributions are concerned there exists an exact effective potential single particle Hamiltonian. In the quantal integral equation approach we will consider here screening of the reference potential will ignore exchange effects; we will see in the next section that they are actually incorporated in the closure of the linear functional expansion, namely the QOZ relation.





-42



When we screen the reference distribution of the QPY generating functional


W = n( U)
n*(r I UH)

with the Hartree potential

OUH(?) = BU(?) + J d?'(MU(F' F) (n(?' I U) no) its linear expansion requires the variation

S r 6n*(r I UH)
Sn(n') 1 n0 no 6n(r' I U)
o o

This can be evaluated by invoking the chain rule over the Hartree potential, which in Fourier space becomes

Sn*(? I UH) Sn*(F I UH) 6 BUH(F")
FQ{- F } Fa
6n(r' I U) 6 RUH(") a 6n(F' I U)


6 OUH(F")
nx*(Q) F{ } 6n(F' I U)

Now the variation of the Hartree potential yields

6BUH() I 6 U)
6n(F') 6n(F') or

6 3BUH() 1
Fat Sn(F') n x(Q) Fa{BU(? F')} from which a few simple manipulations yield





-43


6 w can(F-r') + F-1 x*(Q) F{BU(F-F')} = can(r-r') + M{BU(G-')} 6n(r') FQ

Plugging into the functional expansion of W[n] we obtain

n(F I V) = 1 + J d?'(can(r F') (n(F' I v) n0)
n*(r I VH)

+ f dr'(M{U(r r')}) (n(r' I v) no) In terms of analogue functions and employing the QOZ relation this becomes

V(an
gan = e [1 + an () + F Q 1 {x*(Q) F{n( I v) no} F{BV()}}

SVan
= e H + yan(T) + M{f dT'(RV(T T')) (n(F' I V) no)}]

BRan
= e [1 + yan(F) + M{BVH(F) BV(F)}]

This equation will be referred to as the QPYH equation.

Unlike the classical case, if we screen the QHNC generating functional, we do not recover the QHNC equation. A now familiar procedure yields instead the result


g (r) y(?) + M{IVH(?) BV(?)}
gan() = e e

to be known as the QHNCH equation. We see that for both the QPYH and QHNCH equations long range tails cancel but the potential enters in a nontrivial manner.

Lastly, in the classical case we saw how the HNC equation could also be derived by constructing a screened potential such that the first variation of the generating functional vanished. The same





-44


procedure can be applied to the quantal case. Define an effective potential

BUc(F) = BU(F) f d?'(c(F F')) (n(F' I U) no)

(where c(r) is the quantal pair correlation function and not the analogue quantity); then both the QHNC- and QPY-like generating functionals


W = In n(r I U) W n(T I U)
n*(F I Uc) n*( I Uc)

have vanishing first variations and both yield the same equation:


g- Van(r)
an(F) = e c

This equation is distinct from the QHNC equation but like the QHNC equation reduces to the classical HNC equation in the limit. For this reason it shall be termed the QHNC2 equation.

Interestingly enough the QHNC2 equation can be obtained from an entirely different approach. Instead of assuming the quantal bootstrap relation and utilizing Percus' method we can arrive at the QHNC2 result directly from Kohn-Sham-Mermin density functional theory.107-110 From there we can recover the quantal bootstrap relation (our starting ansatz) and so justify our variant equations by making what we will see as a reasonable assumption concerning the effective potentials of many body systems.





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The Density Functional Approach


Following the zero temperature approach of Hohenberg and
Kohnl08 and Kohn and Shaml09 for the ground state properties of an electron gas in an external potential U(r), Mermin showed that110 i) the Helmholtz free energy A of an inhomogeneous electron gas in thermal equilibrium is a unique functional A[n] of the one electron density n(r), and ii) the grand potential

0[n] = A[n] p J n(r) d

is minimum for the equilibrium density n(r). We employ the grand canonical rather than a canonical ensemble because macroscopic quantities derived from a non-interacting Fermi system are more easily calculated and the fundamental variable, namely the density distribution, is a physical observable that should be the same from either ensemble. The free energy is customarily decomposed as A[n] = I J v( F') n(F) n(F') d? d?' + J U(F) n(r) dF + A*[n] + Axc[n]

to separate out respectively 1) solely classical electron-electron interaction contributions, denoted by v(r), 2) the external potential contribution, 3) the contribution A.En] that a non-interacting Fermi system with density n(r) would still exhibit, and 4) all other effects are grouped to define the unkown exchange and correlation functional Axc[n]. The minimum property of the grand potential leads to the functional equation72-73

6A,[n]
--- + Ueff() = (3.5)
6n(r)





-46


where u is the chemical potential and Ueff(r) is defined by

6A En]
Ueff(F) = U(F) + f d? V(F F') n(F) + XC 6n(?)

Equation 3.5 is equivalent to the equation for non-interacting electrons in this effective potential; that is we have n(rIU) = n*(rlIUeff). The non-interacting fermion distribution n*(rlU) is given by the following system:


(- 2 A + Ueff) i = ci = M = 1 units

n*(F I Ueff) = f(Ci) I ti(r) 12


where the sum over the index "i" refers to all bound and continuum eigenstates and f(e) is the Fermi-statistics occupation number

f() = {l + exp[B(ci p)]}-1

dependent on the temperature and the chemical potential. The latter is determined as usual from the total number of electrons by spatially integrating over n*(r).

For the jellium model of the electron gas the external potential must not only include the source of the external potential Uo but the effect of the neutralizing background as well

U(F) = U o() I dF' no V(F F') so that the effective potential is

6A [n]
Ueff(F) = Uo(F) + f d?' V(r F') (n( I Uo) n + xc 6n(r)





-47



At large radii, the displaced electron charge vanishes, and any long range tail of the source potential is cancelled on the assumption of perfect screening, so that the assymptotic value of the effective potential is SAxc
Ueff(r n sn


Applying eq. 5 at large radius gives

6A, 6A
S6n0 =- 6n or, equivalently

no = / 3/2 1/


where the standard definition of the Fermi-Dirac functions has been used:


I () i y* dy
S o 1 + ey-n

It is convenient to shift the zero point of our energy scale to define a new effective potential which goes to zero at infinity


Ueff(F) = U o() + f d?' V(i F) (n(? I Uo) n )



A+ ( xc 6Axc n no


Up to this point our equations are exact; the exchange

correlation part of the free energy functional is, however, quite unknown. But if we make a functional expansion to first order





-48


6Axc SAxcI
Sn n S ) + f K(S ?') (n(?'l U ) n ) dr'
n n

K(F T') ( xc

Sn(r) Sn(F')


we obtain an approximate integral equation in terms of the unknown kernal K(Ir-r'I):

n(? I V ) no = n*( I Ueff) n

Ueff = Uo + r dT'{V(F r') + K(r r')} {n(r' I Uo) n} Since the unknown kernal is independent of the external potential U(r), we can determine this function by considering the case of a very weak external potential where the linear response formula can be employed on both sides of the first of the above equations with impunity. The left hand side yields

6n(Q) = FQ{n(r I Uo) n} n0 x(Q) U (0) while the right hand side gives

FQ{n*(r I Ueff) no} noBx*(Q) Ueff(Q)


= n Bx*(Q) {U (0) + [V(Q) + K(Q)] Sn(Q)} Using the last equation we can show that the Fourier transform of the unknown function K(r) is

K(Q) = (- -( -1-_) V(Q)
no0 x(Q) x*(Q)





-49



Due to this assignment the effective potential reduces in form to the previously defined screened potential


U (F) = Uo () f d?' ( c(? -?') (n(r I Uo) nO) and we obtain the relation

n(r I U ) = n*(r I Uc)

Note that the source external potential is completely arbitrary, and so this result is a generalization of the QHNC2 equation, which follows immediately when the external potential is taken to be the interparticle interaction. From this generalized result we can work backwards and obtain the quantal bootstrap relation, and so justify the other (QPY, QHNC, etc) equations we have obtained.

This is done by now making the assumption that the average one body potential felt by a constituent electron due to a fixed test charge Se, namely


Ve-t =Vtest d'( ')) (n(' I Vtest) n) where

V e*6e
test r

is equal to the average one body potential felt by a free test charge 6e, arising from a fixed electron in an electron gas, namely

Vt-e = (6e) W(i)

V2 W(F) = 4n{ eS6() + e[n(? I U = e2/r n ]}





-50



In the limit of small Se we can again invoke linear response with impunity and solve for


FQ{n(? I U = V) n } = -1 x*(Q)

thus recovering the quantal bootstrap relation.

We see that our density functional approach, aside from

justifying the quantal bootstrap ansatz of the Percus method, gives physical insight to its meaning. Even though it is not an exact approach, it clearly points out its limitation, as we approximated the variation of the exchange-correlation free energy functional by a first order functional Taylor expansion. This type of truncation is a property shared by the Percus method and ultimately may be responsible for the self-consistency of the two approaches.














CHAPTER IV

THE BASIC QHNC EQUATION


An alternative ansatz for the extension of classical integral equations into the quantal regime lies in the construction of effective pair potentials. These incorporate the bulk of quantal effects; for example in multicomponent plasmas quantal effects prevent the collapse of particles in an attractive coulomb field. More generally the effective potential yields finite values for the pair distribution factor g(r) in the limit of zero interparticle separation

r.

Such classical pseudopotentials have been constructed by

requesting an approximate account of the N-body Slater sum11-112 or, in an approach which can in principle be implemented exactly but is more limited in its physical range of applicability, by reproducing the two-body Slater sum113-118

BVeff(r) Hel
e (r) = (42 /m)3/2 (4.1)

Hrel f 2 + V(r)
2 m

It can be shown that this Binary Slater Sum approximation correctly represents the derivative of the classical pseudopotential for the N-body problem in the limit r tends to zero.119




-51-





-52



As shown in the previous chapter, the QHNC equation for n(rlv) is also equivalent to the classical HNC equation with an effective potential defined by

BVeff(r) n*(r I V)
e
no

where n*(rlv) is the density distribution of non-interacting fermions in the presence of an external potential of the form of the interparticle interaction--the bare coulomb potential. Normally this entails the solution of the Kohn-Sham-Mermin equations.108-110 That is calculating n*(rlv) is the solution of the following system:120-121


(- 2V + V) i = ci i


n*(r) = f(ci) i 1'i (4.2)


where f(ei) is the Fermi-statistics occupation number

f(ci) = {1 + exp[B(ci p)]}depending on the temperature and the chemical potential fixed by the total number of particles. The index "i" runs over all continuum and bound (if existent) states.

The treatment of the delocalized states proposed by Friedel for impurities at zero temperaturel2 can be applied to the nonzero temperature case too. The eigenfunctions F KLM of eq. 4.2 for energy


sK = 2m





-53


are

TKlm(r) = AKim *Kl(r) Ylm(r)

Kl(r) is normalized by requiring that it have the asymptotic form

K1(r) sin[Kr e i + n(K)]

n1(K) being the phase shifts in the potential V(r) and 1n(K = = 0. Overall normalization is accomplished by requiring that the wavefunctions vanish at a spherical wall of very large radius R. We assume that R is so large that most of the integrand is approximated by its asymptotic form:


1=1 AK1 2 j dr sin2[Kr e i + 1]
0

with the boundary conditions

KR + 1l(K) = e = m m = 0, 1, 2, 3, ....

incrementing m by unity for a given L yields the density of states (including spin multiplicity) as

2R 1 a
91(K) = (1 +R aK nl(K))

while the normalization factor for very large R is given by

AKI = v2/R (1 -0( ))

(These same results can be derived for a long range coulombic potential.123 cf Eq. 36.23) A straightforward derivation gives for the density


n*(r) n = Xf(cb) b b ~ f f(c) dK I (2e + 1) [ 2K1- K 2 (Kr)]
b Ir 0 1





-54



where subtracting off the uniform density result accelerates convergence of the integral and summation. We note that in general a potential may support a number of bound states and we have included them also.


The OHNC Effective Potential


We thus see that the QHNC pseudopotential is essentially a

one-particle Schrodinger equation calculation. This is a feature in common with the Binary Slater Sum. It is important to contrast the two however. The Binary Slater sum (BSS) is used to calculate the pair distribution function g(r) while the QHNC equation yields n*(rlv); this difference is important in consideration of quantum effects. It should be noted that in the BSS approach the Schrodinger equation is solved with the use of the reduced mass. The matrix element in eq. 4.1 can be evaluated by inserting a physical basis set of properly symmetrized or antisymmetrized wavefunctions, depending on the statistics of the two particles. Parallel spin versus antiparallel spin contributions can be calculated separately. In the BSS approach the effective potentials are density independent. In the QHNC approach the Kohn-Sham system of equations can also be written in the form of a matrix element

n*(r I V) =

where the operator "f" is defined as

f = (n + exp B[H ull-1





-55



H is the coulomb hamiltonian operator, mu the chemical potential and we have included a variable "eta" to be set equal to unity to describe Fermi statistics. In the QHNC approach the external potential has no spin attribute, and the matrix element is evaluated using a complete basis set with no regard to statistics; the statistics are built into the functional form of the matrix element. The QHNC effective potentials are implicitly dependent on the density through the chemical potential.

All these subtle differences are manifestations of the fact that in the QHNC approach g(r) must be obtained from n*(rlv) by a further calculation. One obvious advantage into this splitting of tasks is that the BSS pseudopotential breaks down at zero temperature, whereas the QHNC remains valid.

A surprising feature of the QHNC effective potential is that it is analytically calculable. This allows one to bypass the computationally time consuming Kohn-Sham-Mermin system of equations. The starting point for the solution is the fact that the effective potential is related to the I 2 limit of the off diagonal density matrix for fermions in an external coulomb potential:107


P(r I 1 = < l I f I 2

Spherical symmetry reduces the functional dependance of the off diagonal density matrix does from the full six variables of r1 and 2 to only the magnitudes r1, r2 and the angle between rl and 2. This is easily seen if we expand the matrix element using a complete set of spherical coulomb wavefunctions:





-56


SKI(r) 2K2
mKlm = Y (r) =2m

2 2
d S + [K2 2m (1 + 1) S = o
dr 6 r

Using the addition theorem of spherical harmonics then yields (2 1 + 1) PI(rl I r2)
P(11 I 4n r1 r2 P1(cos()

where the 1-wave radial density matrix components are defined by


l(rl I r2) = (spin dK) 22 SK1(rl) SK1(r2) e 2m +


Furthermore, because the potential is coulombic, there is an additional symmetry: the matrix element "f" also commutes with the Runge-Lenz vector.124 This reduces the functional dependance of the off diagonal density matrix to the two variables "x" and "y":

x = (r1 + r2)/2 y = (r2 + r2 2rl r2 cose) /2/2

The solution then follows the steps of Hostler125-126 and Storer127 except that the Bloch equation satisfied by the off-diagonal canonical density matrix is replaced by a Fermi statistics generalization

[(H p) (1 +n a + a] P(r1 I r2) = o

(Here H acts only on the r1 coordinates and we set eta equal to unity at the end of the calculation.) Because of the aforementioned symmetries, the solution to this equation reduces to a functional form identical with the s-wave radial density matrix component (however in the variables x and y):





-57



P(rI 1 r2) 1 ay Po(x + y I x -y) The diagonal limit rl = r2 = r is given by


n*(r) = p( ) 8 ay2 Po(x + ) x y) )


The second derivatives of the radial coulomb wavefunctions can be eliminated in favor of the radial wavefunctions themselves by employing the radial coulomb differential equation. He thus obtain

n*(r) = sUn f dK (K, r)
212 eB I2K2 r
22m ] + 1

2 2m
J(K, r) { dp SKo(r) (2 K2) I SKo(r)2 2rfir



Numerical Calculations


The effective potential, obtained from the ideal fermion density distribution in a coulomb potential, will be calculated from the analytic formula derived above. For comparison with the classical HNC equation, we will scale lengths in units of the interparticle separation x r/a where


no = (ra3 )Temperature/density points will be expressed by the classical coupling parameter


r =
a





-58



along with a quanticity temperature (two divided by the temperature in rydbergs)


B* E me4/fi2

In the classical limit all functions are solely dependent on the parameter r6; we expect any dependance on the quanticity temperature to indicate the pervasiveness of quantum effects.

The calculations were performed corresponding to the temperature density points of tabulated results provided by Pokrant69 for later comparison. These consisted of values at constant density, as measured by the dimensionless ground state coupling parameter

rs = ala


where ao is the bohr radius, as well as a dimensionless degeneracy temperature

t = Kt/EF

where Ef is the ground state Fermi energy. The results are presented in the following graphs.

In Fig. 4.1 we present plots of the ideal fermion density in the presence of an external coulomb potential. For comparison these are plotted together with two approximations. The first is merely the classical limit

n*(r I V)
no

The second is the Thomas-Fermi approximation.107 It is obtained by expanding the diagonal density matrix element "f" using a complete set





-59



of plane waves. If we then neglect the non-commutativity of the potential and kinetic energy operators in the exponential (the neglected terms can be systematically calculated by Trotter formulas128 and shown to vanish for large r) we then obtain

n*(r I V) F 1/2 (3p r/x)
n F1/2(3P)

where the Fermi-Dirac integral of order p is defined by the integral


Fp r r(P + 1) f dy -x o 1 + e-x

(In this formula the symbol r denotes the gamma function.) It is seen that the exact result quickly merges with the Thomas-Fermi approximation.

By using the relation

dF (x)
dx Fp-1(X)

we find that asymptotically the Thomas-Fermi approximation has the form F 1 (Op)
n*(r I V) r 2
n x F (1P)


This is indeed the correct asymptote of the analytic solution as can be verified by looking at the appropriate expansion of the spherical coulomb wavefunctions.129 This tells us that the effective potential in the QHNC approximation has a long range l/r tail--but with a strength modified from the classical theory. The QHNC tail contains the additional factor





-60


F 1 (p)/F P)
2 2

not found in the Binary Slater Sum approach. In Fig. 4.2 we present plots for the QHNC effective potential. For comparison the corresponding effective potentials of the Binary Slater Sum approximation are presented in Fig. 4.3.

The approach to solving the HNC system of equations with long tail potentials is well known--we follow the convention of Ng105 and renormalize the constituent functions by subtracting out an analytically transformable long range function of the form

z erFc(ar)
r

We employed the Ng method of solution (essentially a picard iteration with an accelerated convergence procedure) although it should be mentioned that two new algorithms have recently been published130-131 that employ a hybrid Picard/Newton-Raphson iteration procedure which promises faster convergence.

The results of the integral equation solver, that is the analogue pair distribution function n(r/v)/no, are presented in Figs. 4.4-4.8. This distribution is of theoretical interest in its own right, for example in the study of impurities in metals or positron annihilation rates.132 However we will limit our concern to the approximate calculation of the pair distribution function. A method of calculating g(r) based in part on the information contained in n(rlv)/no, and a comparison of results with the Binary Slater Sum approximation and that of Pokrant will be taken up in the next two chapters.






-61





















0.

storer



.... boltzman .1 -. rs- 2.





'0. tau 2g.8 OD .6-,. J"
4.)- tau- i 1.68


tau- 6.5e

tau- 6. t

-1.0 i
0. .5 1.0 1.5 2.0 2.5 3.0 3.5 x-r/a












Figure 4.1 Plots of n*(rlv)/no-1 versus x = r/a for a density of rs = 2 at temperatures of kt/Ef of .1, .5, 1.0, 2.0, 20.0. Storer denotes the analytic solution, following Russian literature the Thomas-Fermi approximation has been denoted Hartree, and Boltzmann is the classical approximation.






-62





















1.0

storer

.8 har tree



S r-. 2.88 tau-20.88

4 tau- 2.88
x
tau- 1.88

2 -- tau- 8.58 tau- 8. 18

0 .I I I
0. .5 1.0 1.5 2.0 2.5 3.0 3.5

x-r/a





Figure 4.2

Plots of the QHNC effective potential versus x = r/a for a density of rs 2 at temperatures of kt/Ef of .1, .5, 1., 2., 20. We have plotted the potential in the form


xr Veff(x)

A purely classical coulombic interaction would correspond to a constant values of unity. Storer represents the analytic solution; Hartree is the Russian literature denotation of the Thomas-Fermi Approximation, which is truncated near the origin to prevent divergences.






-63


















effective potential

1.2


slater sum




1.00
tau-5.00




tau-2.00 tau- 1.00


tau-0. 50 .2
tau-*. 10


0. .5 1.0 1.5 2.0 2.5 3.0 x-r/a







Figure 4.3 Plots of the Binary Slater Sum effective potential versus x r/a for a density of rs = 2 at temperatures of kt/Ef of .1, .5, 1.,
2., 3., 5. We have plotted the potential in the form

x V ff(K)

A purely classical coulombic interaction would correspond to a constant values of unity. Note that as we approach classical temperatures the curve reaches this constant value quickly, and in fact first overshoots this value.







-64



















static structure factor




hnc

-. free chihera
teu-.18
-. 4 rs-2. e









-.8




-1.0I
0. 2. L. 6.

k-q*a










Figure 4.4

Plots of density times the Fourier transform of h(r) = g(r) -1 versus k = q*a (q the reciprocal space to r) for a density of rs = 2 at temperatures of kt/Ef of .1. Free denotes the ideal fermion pair distribution function, HNC the results of the QHNC calculation where g(r) represents the analogue pair distribution n(r/v)/no. Chihara denotes the results of a yet to be discussed construction of g(r) from the analogue g(r).







-65




















static structure factor




hnc

free
-.2

Chhare
tau-a.5 S
4 rs-2. 8
-,




-.6




-.8




-1.0
0. 2. q. 6.

k-q-a










Figure 4.5

Plots of density times the Fourier transform of h(r) g(r) -1 versus k = q*a (q the reciprocal space to r) for a density of rs = 2 at temperatures of kt/Ef of 0.5. Free denotes the ideal fermion pair distribution function, HNC the results of the QHNC calculation where g(r) represents the analogue pair distribution n(r/v)/no. Chihara denotes the results of a yet to be discussed construction of g(r) from the analogue g(r).







-66




















static structure factor

0.

hnc


-.2 ---- f ree

ch her a
tau l.89
S4, rs 2. 88




-.6




-.8



-1.0 -0
0. 2. 4. 6.












Figure 4.6

Plots of density times the Fourier transform of h(r) = g(r) -1 versus k = q*a (q the reciprocal space to r) for a density of rs = 2 at temperatures of kt/Ef of 1.0. Free denotes the ideal fermion pair distribution function, HNC the results of the QHNC calculation where g(r) represents the analogue pair distribution n(rlv)/no. Chihara denotes the results of a yet to be discussed construction of g(r) from the analogue g(r).







-67


















static structure factor






-.2- tr ee cahira
tau-2.88
.9 rs-2. e


-.

-.6




-.8




-1.0
0. 2. L. 6.

k-q*a










Figure 4.7

Plots of density times the Fourier transform of h(r) = g(r) -1 versus k = q*a (q the reciprocal space to r) for a density of rs = 2 at temperatures of kt/Ef of 2.0. Free denotes the ideal fermion pair distribution function, HNC the results of the QHNC calculation where g(r) represents the analogue pair distribution n(r/v)/no. Chihara denotes the results of a yet to be discussed construction of g(r) from the analogue g(r).






-68



















static structure factor


---------------------hnc

-.2- rree Chi hara
tu-8. 88
rs-2.00


0

-.6




-.8




-1.0 i I
0. 2. 4. 6.

k-q-a










Figure 4.8

Plots of density times the Fourier transform of h(r) = g(r) -1 versus k = q*a (q the reciprocal space to r) for a density of rs = 2 at temperatures of kt/Ef of 8.0. Free denotes the ideal fermion pair distribution function, HNC the results of the QHNC calculation where g(r) represents the analogue pair distribution n(r/v)/no. Chihara denotes the results of a yet to be discussed construction of g(r) from the analogue g(r).














CHAPTER V

LOCAL FIELD CORRECTIONS




We have seen in the previous chapter how classical integral equations generalized into the quantal region can yield accurate results for the quantity n(rlv), namely the density distribution of interacting particles in the presence of an external potential of the form of the interparticle interaction. This in effect is equivalent to knowledge of the static response function x(k). If from this information the dynamic response x(k,w) could be extracted, then we could easily calculate the static structure factor or pair distribution function via the fluctuation dissipation theorem


S(K) = f S(Kw) dw = f Im Bx(Kw) dw
e 1

These operations would constitute the quantal generalization of the classical bootstrap assignment g(r) = n(rilv)/nO.

To demonstrate that such a program is plausible, we note that the analytical properties of the dynamic susceptibility make it possible to write an exact "dispersion" relation for x-1(k,z) of the form133


x (KZ) = x- (K) + 2 [Z2 + iZK2 G(KZ)] pK






-69-





-70



This particular relation, which is one of many possible representations, introduces an unknown function e(k,z) which is analytic in the upper half plane. The physically meaningful dynamic response is given in the limit z = w + ie, and in this limit the real and imaginary parts of

O(K, Z = w + ic) = O'(Kw) + iQ"(Kw)

are, like the dynamic susceptibility itself, related by the Kramers-Kronig relations. The function e'(k,w) is known to be a positive, even function of frequency, as follows from the known parity of x'(k,w) and x"(k,w) and the fact that S(k,w)> 0. The utility of the function e(k,z) is that it interpolates smoothly between the known results for x'(k,z) in the limits of large and small z, and being less sensitive than x(k,z) itself, is more amenable to approximation.

By subtracting out a corresponding expression for a noninteracting Fermi system we can express the dynamic susceptibility as xo(KZ)
x(KZ) = (5.1)
1 Veff (KZ) x (KZ)


Here we have grouped all of the actual physics into a dynamic effective potential Veff(k,z), actually a functional of the static response x(k) and the unknown function D(k,z). We will show through the use of the Mori projection operator technique134 that under reasonable assumptions the dynamic effective potential can be approximated by the static (z = 0) limit of the above equation

1 1
Veff(KZ) (x(K) xK)





-71



It should be noted that although x (k,z) in eq. 5.1 is

usually taken to be the free response function, this choice is not a unique one; xo(k,z) plays a role of a reference system in terms of which the expression for x(k,z) is constructed.135-136 A possible improvement in the procedure may be achieved by using an adjusted choice of x (k,z) that includes self-energy effects, for example.14-19'137 This method of improvement will not be taken up in this thesis in lieu of other avenues of extension. We will retain the free response as our reference system; this will allow us comparison with certain mean field theories.


Mean Field Theories


Mean Field Theories bypass the dependence of veff(k,z) on x(k) and directly try to relate Veff to the bare interparticle potential by solving approximately the many body problem. (Something we have already done via our integral equations.) The efective potential is usually taken to be "static," that is independent of z, and is commonly expressed in the form

Veff(K) V (K) [1 G(K)]


where V (k) is the Fourier transform of the bare interparticle potential and G(k) is called the local field correction factor. A review of such local field function theories can be found in Kugler.138 The many body part of the problem is generally solved by the Vlasov kinetic equation139 or, what amounts to the same thing, appropriately decoupling the equations of motion for the density





-72



fluctuations.140 Whatever route, at some point collisions between particles are neglected and the particles are assumed to move in a mean or average field based on the collective motions of all the other particles. The physical significance of the correction factor is thus to incorporate some of the effects of short range correlations between individual particles not taken care of by the average field.

Mean field theories for the most part have been limited to the ground state quantal electron gas. These have disagreed on the form of the local field correction factor on three main points.

First some workers141-145 have been led to the conclusion that G(k) is a universal function of k/kf (kf the Fermi wave vector), while others146-149'29-31 contend that G(k) is density dependent.

Second, there is wide diversity of opinion regarding the value of G(k) in the limit of large wavevector k.150-151 At this limit, a constant 1/3, irrespective of density, has been given by Geldart and Taylor,142 Rajagopall144 and others,143,145-147 while that of Togio and Woodruffl48 and Kugler38 use 2/3. On the other hand Singwi et al.29,30 derive this value to be l-g(O) in terms of the pair distribution function g(r) at the origin, and that of Vashista and Singwi31 is


1 g(o) ap(a )

(where a is an adjustable parameter determined self-consistently) in contrast with the result of Niklasson149 who gives 2/3(1 g(O)) to this limit.

Thirdly, there are several authors138,146-147,142 who have shown the local field factor G(k) has a maximum around k = 2kf.





-73



Kugler138 among them further indicates the slope of G(k) has a logarithmic singularity at k = 2kf in addition to a maximum. On the contrary, G(k) of Singwi and collaborators29-31 has neither maximum nor singularity in its slope around k = 2kf. It should be noted that the behavior of G(k) in the large wave vector region is quite extremely sensitive to small differences in x(k) so that it offers a stringent test for the distinction of various theories.

In the case of the degenerate electron gas, there is neither

computer simulation nor experiment to be used as a criterion to check these diverse conclusions concerning G(k). However, at present the prescription of Singwi et al.29-30 is regarded as fairly successful in treating a degenerate electron gas and is relatively easy to program numerically. It encompasses in the classical limit, as do the integral equation methods of Chapter II, the Hyper-Netted Chain equation, which is known to give the best fit to the pair distribution function of computer simulations. For this reason the derivation of the Local field correction factor in the STLS theory will be presented here as a basis for later comparisons.


The STLS Method


Following Singwi8 we will derive the density response function by following the time evolution of the off diagonal single particle density function


p (x I x" t) -a < ( t) t(',t)>





-74


If we employ a Heisenberg equation of motion approach with a Hamiltonian containing an external potential Vext(x,t), we obtain


{i mA A + VH(it) VH(i't)} pa( I '; t)


S d [V(R ") V(' ")] <,F; (xt) n(R"t) T (R' t)> where the Hartree potential

VH(it) = Vext(it) + f di' V ( i") p(i", t)

depends on the interparticle interaction potential Vo and the average single particle density distribution

p(xt) = E < I +(xt) T (Xt)>


We were able to regroup the terms of the equation of motion into its present form by defining the cumulant bracket

=
Sa c


a a a a Our equation of motion is equivalent to an infinite set of equations of motion for observable physical quantities with classical equivalents. This follows from the fact that the Wigner distribution function96,152

f (Rt) = J d? e- ipr/i <+ ( 1 t) (R + t> pa at) (R+ r, t)> 2





-75



has properties analogous to the classical phase space distribution f(r,p;t). In particular it can be used to construct physical observables such as the single particle density distribution


p(Rt) = I f (Rt)
pa

the particle current density

(Rt) = m- P f (Rt)
po
po
and the kinetic stress tensor
'(Rt)=m- P P f (Rt)
pa
pa

By expanding the equation of motion for the off diagonal density distribution about its diagonal, that is in powers of r = x' x, the coefficients of the first few powers of r yield the beginning of a hierarchy of one particle equations. First we have the usual continuity equation


at p(Rt) + *= (Rt) = o

second the equation of motion for the current density

m t J(Rt) = ~.k(t) p(Rt) VH(t) I d IV(R R)


The last term in the above equation describes through the

correlation function c all the complicated effects of the Paull and Coulomb hole surrounding each electron (in the presence of the external field). In lieu of an exact evaluation of this term or main aim is to extract a local field correction. Noting that


c = n(Rt) n(it) {g(R, i; t) 1}





-76



where g(r,x;t) is the non-equilibrium pair correlation function, the simplest approximation is to replace this function by its equilibrium value g(r x), thus accounting for the local depletion of charge density but neglecting the dynamics of the hole.

This approximation replaces the Hartree potential by a local effective potential given by


Veff(O) = VH(Kw) G(K) p(Ew) with

G() = (23 q[S -) 11 (5.2)
(2r)3 q

Relating ii (k,w) to thas esectave potential An the same manner as a free particle calculation, namely as ( P f (p K/2) fo(p + K/2) ir (Kw) = 2 V (Kw)
OC pa p*R/m + i e
then the continuity and current density equations together yield

x(Kw) = x (Kw)/l 4re2 (1 -G(K)) x (Kw)


where xo(k,w) is the Lindhart polarizability153 of an ideal Fermi gas

f (p K/2) f (p + K/2)
xo(K) = X o
pa W p R/m + ic

Although G(k) contains the unknown structure factor S(k) of the electron fluid, this can be determined by requiring consistency with the structure factor obtained through the use of the fluctuation dissipation theorem. Of use numerically is the little known direct relation between G(q) and the pair distribution function g(r)154





-77



G(Q)= 1 Q f dr g(r) jl(Qr)
o

and by using orthogonality of the spherical bessel functions

2r2
g(r) = 1 f o dQ Q G(Q) jl(Qr)
o

These results are extremely useful because they enable one to generate a G(q) directly from the pair correlation function and to test G(q) by constructing the corresponding g(r).

In the STLS scheme both exchange and correlation corrections to the Hartree field are automatically taken into account through its self-consistency. Moreover it is straightforward to recover the Hubbard141 result

1 K2
G(K) 2 2
K2 + K2

by substituting for the structure factor in eq. 5.2 its Hartree-Fock value. The Random Phase Approximation, which is the dynamic extension of the Debye-Huckel theory, can be viewed in this scheme as the approximation that g(r,x;t) = 1, and leads to the result G(k) = 0.

At zero temperature a further constraint can be used to check the validity of any approximate microscopic theory. The compressibility "sum rule" states that the long wavelength limit of the static response should agree with the result obtained for the compressibility from differentiation of the ground state energy. (This and other ground state properties of the quantal electron gas are covered in Ichimaru.5 Both the Hubbard and STLS equations violate this





-78


constraint. This has led to modifications of the Hubbard correction
factorl155-157

1 K2
2
G(K) =
K + aKF

and of the STLS scheme by Vashista and Singwi31

G(K) = (1 + a ) {- 1f [S(K Q; p) 11]} (5.2)


where the additional free parameter "a" is to be varied such that compressibility sum rule is nearly satisfied. However it has been shown79 that if the correction factor G(k,w) is static, it is impossible to satisfy the compressibility sum rule and the third frequency sum rule simultaneously (the f sum rule being denoted as the first moment).


The Relaxation Function


As an alternative approach to calculating the correction factor G(k,w), which unlike the methods covered so far easily generalizes to finite temperatures, we shall consider approximations to the Kubo function provided by the Mori continued fraction method.134

In Chapter I the Kubo function was identified from the difference of the dynamic and static density response functions:

x(KZ) = x(K) + iRZ C(KZ)

In this section we shall demonstrate that this follows from the definition of the Kubo function as the equilibrium averaged density auto-correlation function





-79


Co XH -XH
C,(r r', t t') j dX




where <.... > stands for the thermal expectation average.

It is easily shown that the Kubo function vanishes at large

times; spatial Fourier transformability is insured by the subtracted term . The intimate relationship of the Kubo function with the density correlation function can by exposed by writing the Kubo function in the form


C(r F', t t') = where we defined the Kubo transform of an operator as

1H
J fJ dX e J eH


By taking matrix elements diagonal in the Hamiltonian of the Kubo transformed operator, it is easily seen that the Kubo function reduces in the high temperature limit to the density correlation function. (The temperature = 0 limit however is ill-defined). Other notations abound in the literature; in lieu of the Kubo transform we shall employ the semicolon bracketl4

1 XH -XH
dX Tr {peq e Ae B}
0eq

Using time translation invariance and the cyclic property of the trace one finds that the Kubo function has the property

ia C(r r', t) = x"(r r', t)
atrD





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The time Fourier transform of this equation

c(r r', w) = 2 (r xr- r'

yields as a consequence that the static response is related to the time = 0 limit of the Kubo function


x(K) = f X W =f 3C() = C(t = o)

As a further consequence we see that, unlike the intimately related density correlation function, the frequency moments of the Kubo function are those of the response function x" without encumbering factors of coth(bhw/2).

Taking the Laplace transform of the time derivative equation yields the relation referred to in Chapter I

x(KZ) = x(K) + iZ C(KZ) (5.3) alternatively this can be viewed as simply the integration by parts of the Laplace transform of the Kubo function (note the tilda)


C(rZ) = f dteizt C(rt) In Z > o
0

= J do C(W') 1 dc x"(rw) In Z 0
21i m' Z =3 0 i m(w Z)

The last equality follows from the integral representations of the complex and static response in terms of the response function.

Based on analytic properties the Laplace transform of the Kubo function can always be represented as85


C(Z) = i n-I
z + i D(z) (5.4)
C(Z = z + 1 D(z)- X (5.4)





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with D(z) an analytic function for Imaginary (z) not zero. It is the function D(z) that we will investigate further and which we shall see can be expressed in the form of a continued fraction that may be truncated at some appropriate point of approximation.


Scalar Products of Operators


We start by considering in detail the time evolution of physical observables. In quantum mechanics operators satisfy the equation of motion


at A(t) [A(t), H] = iLA(t)

which defines a linear Liouville "super-operator," that is an operator on operators. If we write this out in the form of matrix elements, taken for a complete set $i of quantum states, the first half of the above equation reads

a 1
at AMN(t) = 1 (Am(t) Hvn HA (t)) (summation on repeated indices implied) or

AMN(t) = i { HM Hvn mp} A M(t)


If one now treats the pairs mn or uv as one index each, then the Auv(t) can be stacked to form a column vector, while the Liouville super-operator within the braces of the above equation can be put into the form of a matrix. The point to all of this is that the usual conception of quantum mechanics as a Hilbert space of say N dimensions





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can also be easily envisaged as a super-Hilbert space of N2 dimensions, where the dynamics occur through a linear matrix operator.

In this super space we can define a bracket such that


(FIG)1 f X =
0

Although often omitted the factor one over beta is retained in this definition in order that the bracket have the same dimensions as the operator elements, and that the bracket be well defined in the beta equal infinity. The dagger representing Hermitian conjugate within the thermal expectation brackets allows us to write the spatial Fourier transform of the density-density Kubo function as


C(Kt) = 9 (nK(t) I nK(o))

(noting that is zero for nonzero wavevector.) In addition this scalar product satisfies the defining properties of a Hilbert space:


(FIG)* = (GIF)


(FIF) > 0


(aF + bGIH) = a(FIH) + b(GIH) a,b complex constants

i.e. linear in bra (antilinear in ket)


Furthermore it has the remarkable property





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(FIG) = (G+IF+)


This formalism affords us the flexibility of manipulations borrowed from familiar quantum mechanics. For example the auto-correlation function Caa(t) can be written as


Cn(t) = = =

=

The Hermiticity of the Liouvillian follows from the time translational invariance of the brackets. (Henceforth we will drop the argument zero, understanding A = A(t = 0).)

Our aim is to find an exact description of the evolution of the auto-correlation function Caa(t), in particular for the case where A represents the density operator. In achieving this a geometric interpretation of the problem proves to be very helpful. In the above equation we can interpret the operator exp/-itL/ as rotating the vector IA>, and except for a normalization constant, the correlation function C(t) is the component of this rotated vector parallel to the original direction. It is therefore suggestive to introduce a projection operator


P = IA> -l

into the formal expression for the Laplace transform of the correlation function





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C(Z)

Z LQ Z LQLP Z L

Here we have used a formal operator identity to reexpress the inverse of z LQ. This allows us to manipulate the Laplace transform as follows. The first term of the above equation yields

= = C(t = o)


as can be easily demonstrated by expanding in powers of l/z and using the fact that QIA> = 0. As to the second term we insert the definition of the projection operator

i 1 Al IA > = IA > C-l(t = o) (Z) PZLIA>I A>


C(Z) i C (t o)
Z
C'(t = o)

Noting that 1C(t = 0) = x we finally arrive at the form of eq.

5.4; additionally we now have explicit knowledge of the unknown function D(z). This function is reminiscent of the "self energy" in quantum mechanics.158 It can be simplified further because

=


= i
= i





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Since the second term vanishes as z tends to infinity, the first term, which we have separated off, is the self-energy in the infinite frequency limit. In this sense it bears a correspondence to the Hartree-Fock part of the self energy in Green's function theory.159 The second term is referred to in non-equilibrium statistical mechanics as the memory function (it is nonzero for small z/large time).

The fundamental result of this section is that this memory function is itself of the structure of a time auto-correlation function, namely of A. There are two differences however: first, only the part QIA> which is orthogonal to IA> enters in this correlation, and second, the dynamical operator QLQ is not the full Liouville operator but has projected from it the part which determines the intrinsic fluctuations of the variable A. In other words, if the dynamics of the property A is of interest and we call the many other degrees of freedom the bath, then the part QIA> determines the coupling of A to the bath, and QLQ describes the internal dynamics of the bath which feeds back to impose its behavior on A itself.

As the memory function is a form of auto-correlation function itself, we can follow the same procedure as above by projecting out the time evolution of QIA> perpendicular to its original direction. This procedure can be repeated ad infinitum to obtain a continued fraction representation of the memory function.

We should also note that an extension of the method is to

generalize it to the multivariable case, where the dynamic property of interest is not a single fluctuating property of the system but a set of independent observables A, A2, ... An. When calculating the





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density-density response it is customary to include the longitudinal component of the current density as well as the number density. Such an extension will be considered later.


Microscopic Theory


We will now use the methods of the above section to

microscopically derive the dynamic response. Let us turn our attention therefore to two functions, the off diagonal dynamic response with momentum variables p and p':

o
xpp, (dZ) f dt e- < 1[n t nIQ> and the relaxation function


App,(QZ) E f dt e-Zt

where the off diagonal single particle density operator is

n pQ() a+ a
PP-QZ ap+2

(For simplicity of notation we will omit explicit consideration of spin.) Of course the actual dynamic response we are interested in is given by

x(QZ) = ( xpp,(QZ)
pp

and the off diagonal relaxation and response functions are related through a simple integration by parts, in the same manner as their diagonal counterparts (see eq. 5.3)





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A ,(QZ) = I {Xp,(Q) Xpp (QZ)} (5.5)

According to the continued fraction approach presented in the

preceding section this relaxation function may be expressed in the form

A pp,(QZ) = X (P) P") (P" I x(Q) I P') (5.6)
p" Z is + T (Z)
where we have defined operators with matrix elements given by

(p I x(Q) I p') E = Xpp,(Q) (5.7) (p iW I p') E (p" x (Q) I p') (5.8)
p

(p I (Z) p') I

(pl -l(Q) I p') p" Z QLQ

We see that the off diagonal momentum variables serve the utilitarian purpose of matrix indices. As we saw in the previous section the operator (z) represents the back effects of the bath on the dynamics of n ; in what amounts to neglecting collisions between particles, we will assume that (z) = 0 (our first approximation). This allows us to use the identity


Z i Z- i

and so combine eqs. 5.5-5.7 to yield

xpp,(QZ) = n (p 1 p") (p" I i x I p') (5.9)
p z- i p
Let us consider first the operator x(q) or equivalently the function x p,(q). This function contains more information than that contained in the static density response function x(q) supplied





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for example, from the integral equations of Chapter II. This is easily seen by the relation

S2 2 pp,(Q) = i = x(Q) (5.10)
N p p,

That is the off diagonal information is integrated out. Thus to obtain an appropriate form for the off diagonal response xp ,(q) we must rely on its more fundamental definition in terms of a temperature Green's functions37 whose Dyson's equationl0 is approximated by assuming the vertex function "r" is dependent on the wavevector q only. As we will show this is correct in the classical limit and is approximately true in general for fermions.161 This (second) approximation reduces x p,(q) to the form

Xpp (Q) dp (Q) 6pp, d (Q) Tr(Q) dp,(Q) (5.11) where

GRet (c) Ref c
1 __P+Q_ pdp(Q) = f de am p+
p e Be o 1 o 1
ap+Q + p+Q (c) -pQc -Q (c) and

ret 1
G et(c)
p Cp + W p()


is the retarded single body Greens function, co the kinetic
p
energy of a free particle and p (c) the self-energy.
The function d (q) is easily evaluated for two cases. If the self-energy Zp(c) is independent of c we have


d (Q) f(Ep + 0/2) f(Ep 0/2) p + Q/2 Q/2





-89


where

E = Co + p p p p
and

f(c) = 1/[I + eB(c p)3

is the Fermi distribution (note that a grand canonical ensemble is necessary for this portion of the evaluation only). In the other case, namely the classical limit, d (q) reduces to the normalized Maxwell-Boltzmann momentum distribution function #mb( )


lim dp(Q) = MB(P)
-0o

and so eq. 5.11 agrees in the classical limit with N iQri N iQr lim N = < e 6(p pi) I e 6(p' pj)>C5.12) E o i=l j=1

= OMB() 6(p p') + 4MB(P) {S(Q) 1} (MB(P')

where <...>c denotes the classical average over the canonical ensemble. This verifies the classical limit of the vertex function as being dependent on the wavevector q only and identifies it as S(q)-l.

In general the vertex function can be identified by requiring agreement with eq. 5.10. We obtain


x, (Q) dp(Q) Spp' + dp(Q) } {} d(Q) (5.13) where we have introduced a reference static density response by defining

xo(Q) X dp(Q)





-90


In the classical limit xo(q) = 1 and x(q) = S(q) so we recover the classical result eq. 5.12. In the quantal limit when we neglect self-energy effects, xo(q) is simply the static limit of the Lindhart or RPA polarizability.8
Next let us consider the matrix elements of the operator W .
First we note that the above results for the operator x(q) allow us to write

S "I
SI (Q) I dp(Q) xo(Q) x(Q) N also we note that we can exactly evaluate


= a {Pp-Q/2 Pp+Q/2 pp, E bp(Q) p (5.14) in terms of


p = =- f dc imag Get (c)
p P = eBe 1

namely the momentum distribution of particles in the interacting system. We have introduced the function b (q) for notational convenience.

Inserting these results into eq. 5.8 yields the result

(p I is I p') = i 0p 6pp, n CQm(Q) bp(Q) (5.15) Here we have defined an effective frequency "omega"
i0pq Q bp(Q)/d (Q)

which is motivated by the fact that in the limit of zero self-energy, where

pp = f(p)





-91


we may write

1 0 0 i
pQ = 4 {Cp+Q/2 p-Q/2} = Q/n
We have also introduced a quantal direct correlation function

noCQ (Q) = x(Q (5.16)


in agreement with the results of Chapter II. We note again that in the appropriate limit this function reduces to the classical direct correlation function. It now follows from eq. 5.15 that


(p I {z i}- I p') = (z+ip S p z+p pp,

b(Q) 1 nCm(Q) 1
N z+iLQ 1 + n QM(Q) Xo(Qz) z + ip) (5.17) where the reference dynamic response defined as

X Qb (Q)
x (QZ) N p z + iQQ
p pQ

is consistent with the reference static response defined earlier. In general xo(q) is a complicated functional of the self energy. But if we make the assumption (third approximation) that


pp = f(c )

then this reference response reduces to the ideal fermion or Lindhart dynamic response function. We will see later on that although the pair distribution function changes readily from its ideal fermion form as interparticle interactions are "turned on" the momentum distribution is in fact an extremely insensitive function, and so this assumption is in fact adequate.





-92



By using the results eqs. 5.17, 5.8, 5.14 and summing equation over the variables p,p' a few easy manipulations yield the result Xo(Qc)
x(Qu) = (5.18)
1 noCom(Q) Xo(Q) (5.18)


We note that the corresponding classical result, where the effective potential has been replaced by the direct correlation function, has been independently derived elsewhere162'139 via a variational criterion to find approximate eigenfunctions and eigenvalues of the classical Liouville operator.

The above equation is the main result of this chapter. Through it and the fluctuation dissipation theorem we are able to obtain the pair distribution function from the corresponding analogue function provided by the integral equations of Chapter III, and in a manner where all approximations are systematically accounted for.














CHAPTER VI

NUMERICALLY EVALUATING S(q)




This section details the procedures for calculating the pair distribution function g(r) utilizing the results for the "analogue" pair distribution function n(rlv)/no output from the quantal integral equations discussed in Chapter II.

We employ the fluctuation dissipation theorem

d(fo) 2
S(Q) = Jfd 2d 2 e Im Bx(Q, Z = + ic) (6.1)
-_ 2n 1 -e

together with the approximation derived in the last chapter

x(Qz)
x(Qz) = (6.2)
1 n C Q(Q) Xo(Qz)

where the quantum mechanical direct correlation function can be obtained from the analogue pair correlation function directly in spatial Fourier transform space as

CQm(Q) = Can(Q)/xo(Q)

It must be remembered that it is the analogue pair correlation function that one obtains from the classical form of the Ornstein-Zernicke relation in the QHNC system of equations. Note also that we have employed the notation of Chapter III: the density response function


-93-





-94


is defined with an additional factor of -1/beta*density not found in Chapter II or commonplace in the literature.79 As defined here the ideal density response is dimensionless


X(Qz) =g dk f(K + 0/2) (K 0/2)
0 p (2) 3 r3fz [Bc(K + Q/2) c(K Q/2)] (6.3) g denotes the spin multiplicity, rho the density, f the Fermi momentum distribution and E(q) the kinetic energy of wavevector q. For arbitrary complex frequencies "z" the ideal response has both a real and imaginary part. Using the identity

lem 1 P ir6(x)
c-o x + ic

we see that along the real frequency axis there is a discontinuity; approaching the real frequency axis from above (below), we have

xo(Q, z o+ ic) R(Q) = xix ()

Under the approximation we have derived for our local field correction factor (eq. 5.16, 5.18), we see that the interacting response shares this property as well by virtue of the fact that Cqm is purely real. This allows a simpler numerical evaluation of the fluctuation dissipation theorem as follows:153 The integral over the entire real frequency axis involving the imaginary piece of the complex response can be rewritten as a complex variable contour integration


S(Q) = d(z) 2 Bi(Qz)
S2 1 e-iz 2i





-95



where the contour cl is given by Fig. 6.1.


Imz




Rez =



Figure 6.1

Integration contour in complex plane.



By closing the loops in the upper and lower half of the complex plane with an infinite radius arc (over which the integrand contributes nothing) we can then proceed to deform the contour to that of Fig. 6.2.


Imz




Rez = w



Figure 6.2

Integration contour in complex plane. This loop encloses the poles at


z = m = O, + + 2, ....
Rfi




Full Text

PAGE 1

THE ELECTRON LIQUID AT ANY DEGENERACY By BRIAN GREGORY NILSON A DISSERATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1987

PAGE 2

TABLE OF CONTENTS PAGE ABSTRACT iv CHAPTER I INTRODUCTION 1 II BACKGROUND THEORY /. 7 The System 8 Basics of Linear Response 9 The Dynamic Density Response 11 Relation to Density Correlation Functions 16 Further Relations 17 Distribution Functions [ 19 III OUANTAL FUNCTIONAL EXPANSIONS 27 The QOZ Relation 27 Classical Generating Functionals 32 Ouantal Generating Functionals 33 Density Functional Approach 45 IV THE BASIC QHNC EQUATION 51 The QHNC Effective Potential 54 Numerical Calculations 57 V LOCAL FIELD CORRECTIONS 69 Mean Field Theories 71 The STLS Method '.'.'.'.'.'.'.'.['.'.'.'.]'. 73 The Relaxation Function 78 Scalar Products of Operators 81 Microscopic Theory 86 VI NUMERICALLY EVALUATION S(Q) 93 The Lindhard Function 97 Convergence Acceleration 103

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VII THERMODYNAMIC PROPERTIES Ill Thermodynamics with Slater Type Effective Potentials 116 Interaction Energy Results 118 Kinetic Energy Results 120 VIII RESULTS OF THE EXTENDED QHNC EQUATIONS 131 Method of Solution 133 Quantal Hartree Results 141 The Zwanzig Equation 144 Blending 147 IX DYNAMICS OF THE LFCF 189 Calculating the Memory Function 197 Results 200 X CONCLUSIONS 219 REFERENCES 224 BIOGRAPHICAL SKETCH 235

PAGE 4

Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy THE ELECTRON LIQUID AT ANY DEGENERACY Brian Gregory Wilson May 1987 Chairman: Charles F. Hooper, Jr. Major Department: Physics The dielectric formulation of the many-body problem is applied to the study of the static correlations in electron liquids at non-zero temperatures. The intermediate coupling effects arising from the exchange and Coulomb correlations are taken into account through a local-field correction factor obtained from quantal extensions of classical fluid integral equations. This results in a unified theory which can describe the thermodynamic and dielectric properties of the -i V-

PAGE 5

electron liquid covering the limits of the zero temperature jellium to the classical one-component plasma. Detailed numerical calculations are provided for the self-consistent set of equations describing the static structure factor. The variation of interaction energy per particle versus temperature at constant density is shown to exhibit a peaked structure near absolute zero. This feature is not observed using effective potential approaches in classical fluid equations. A comparison of zero temperature results with those obtained from the Greens function Monte-Carlo method is good; however, agreement worsens when dynamical effects of the local-field correction factor are included using the Mori continued fraction method. -V-

PAGE 6

CHAPTER I INTRODUCTION The one component plasma (OCP) is a system consisting of a single species of charged point particles together with a uniform oppositely charged rigid background to ensure charge neutrality. It was first introduced by Sommerfeld as an approximation to metals when band structure effects are negligible. By coupling the standard adiabatic (Born-Oppenheimer) approximation with an assumption of a weak electronion pseudopotential interaction, it can be shown^"^ that the thermodynamics of a simple metal is the sum of that due to a one component electron plasma and that due to a system of classical ions interacting via a state-dependent pair potential. Viewed as a model fluid, the OCP exhibits the characteristics which distinguish real coulomb systems, such as plasmas and electrolyte solutions, from neutral fluids. These include the phenomena of screening and plasma oscillations, which arise from the long-range nature of the coulomb interaction. Thus the OCP is a standard approximation for both astrophysical and laboratory fusion plasmas,^ where the electrons are treated as a polarizable background for the gas of ions. Further applications lie within the local density approximation of the density functional theory of electronic structure, where the excess chemical potential of a one-component electron plasma plays the role of an

PAGE 7

-2exchange-correlation potential in the one-electron Schroedinger equation. It is thus not surprising that there has been a large amount of work expended in obtaining accurate theoretical descriptions of both the zero temperature fermion OCP (degenerate electron liquid) and the high temperature classical OCP (electron gas).^"^ Active research is ongoing in three main areas. First, in the course of study of linear dispersion of plasmons in the simple organic polymers polyacetyl ene and polydiacetylene, it has proved important to contrast the properties of the exchange hole, and its screening, in quasi-one-dimensional solids with those in isotropic three-dimensional jellium (classical OCP).^'^ Secondly, one has been able to determine experimentally, using inelastic X-ray scattering, the dynamical structure factor S(q, w) for the electrons in a few metals. ^^"^^ It is of particular interest that the experimental S(q, w) show a tendency of two-peak structure for q within the particle-hole continuum, a feature believed to be a property of the uniform electron liquid not presently accounted for accurately. ^^"^^ Lastly renewed attempts are being made to quantify the intermediate degeneracy regime of the fermion plasma. There are, in fact, many physical systems whose electrons exist in partially degenerate states and are inadequately described by either zero temperature or classical formalisms. For example, the finite temperature thermodynamic and dielectric properties of the electron liquid are needed for a proper treatment of the equation of state of high temperature liquid metals forund in shock wave experiments^^ and experiments on liquid metals expanded to near their critical

PAGE 8

-324-25 points, of high density inertially confined fusion plasmas, and of the finite temperature exchange-correlation potentials used in density functional calculations of atomic properties at high pressure and temperature. Semidegenerate electron liquids are also found in semiconductors where the low densities involved lead to correspondingly low Fermi temperatures. The finite temperature equation of state of the electron liquid is also needed for a quantitive explanation of the miscibility gap in solutions of alkali metals in their alkal i-hal ide melts. The problem of the OCP at intermediate degeneracy has only recently received attention because of complications not present in the extremes of a classical or a ground state quantum description. For example at zero temperature diagrammatic, self-consistent distribution function methods^^"^^ and pseudo integral equation methods, ^^"^'^ together with recent Quantum Monte Carlo^^"^ calculations, have combined to present an accurate description of the paramagnetic electron 1 iquid.'*^"'*^ But at finite temperature we no longer deal with a unique ground state (invalidating the Monte Carlo approach) nor may we utilize the ground state energy variational principle (invalidating Fermi Hypernetted Chain approaches).* Diagrammatic approaches may be generalized to finite temperatures^^ Fermi Hypernetted Chain calculations of quantum fluids^ are actually variational calculations of the ground state using an adjustable Slater-Jastrow type wavefunction and should not be confused with the classical integral equation method per se. Rather the name stems from a diagramatic expansion of the ground state distribution function which is regrouped into a structure of the same form as the classical HNC equation.

PAGE 9

-4but are perturbation expansions reliable only in the regime of weak 44 coupling. On the other hand the classical limit may be handled by a variety of techniques, including the Monte Carlo method/^"^ the molecular dynamics method, and various approximate methods involving integral equations, most notable amongst these the Hypernetted-Chain equation^^"^^ and modifications thereof .^^"^^ Here quantal effects introduce two complications. First a quantal calculation of the pair distribution function is hampered fundamentally by the fact that the canonical partition function can no longer be factored into a solely configuration-integral part and a part involving solely momentum coordinates. Secondly the thermodynamic description will also depend on the momentum distribution (or off diagonal single body density matrix)^^"^^ and this information is not trivially contained in the pair distribution function. Various attempts have been made to overcome these complications within the framework of the successful methods available for classical fluids. Wigner-Kirkwood expansions^^"^^ may be used for those cases where quantum corrections to the classical partition function are perturbative, but their convergence as a series in temperature or Plancks constant is slow (if existent) and is restricted to short ranged potentials;^^ it is, in fact, an ill defined expansion for 68 the OCP. Published work often centers around diffraction effects and assumes Boltzmann statistics. Previous calculations of the OCP at intermediate degeneracy and coupling have been of several types. The most obvious course has been

PAGE 10

-5to approximate the true quantum mechanical Slater sum which appears in the canonical partition function with the corresponding classical expression involving a temperature and density dependent effective potential. Pokrant^^ and others^^"^^ have developed a finite temperature variational principle to approximate the Slater sum as a product of an ideal fermion Slater sum and that of pairwise additive potentials. The energy is calculated by approximating three-body distribution functions in terms of pair distribution functions, which are obtained from the HNC equation using the effective potentials. Zero temperature correlation energies from this method are, however, in substantial disagreement with the parameterized fittings of the quantum Monte-Carlo results. '^^"^^ Several other groups have studied the thermodynamic^^"^^*^^"^^ and dielectric^^'''^ properties of the OCP by using finite temperature perturbation theory within the random phase approximation. In particular Perot and Dharma-Wardana^^ and Kanhere 22 et al. have presented closed form parameterizations of the exchange-correlation energy and chemical potential within the random phase approximation (RPA) for the paramagnetic and spin polarized cases. However the RPA is a weak coupling theory that results in considerable error at metallic densities. 20 Dandrea et al attempt to overcome the limitations of the RPA by including a static local field correction factor (LFCF) within the framework of the dielectric formulation. They use an approximate form for the LFCF which has two adjustable parameters. The first is fixed by assuming a value for the pair distribution function at the radial

PAGE 11

-6origin. The second is obtained from an appropriate interpolation between known Monte-Carlo results at zero temperature^^ ""^^ and in the classical regime^'^"^^ for the bulk modulus. Tanaka et al.'^^ approximate the LFCF at all temperatures by the static zero temperature LFCF of Singwi-Tosi-Land-Sjolander (STLS) self-consistency scheme. ^^"^^ This thesis concerns itself with a unified theory capable of describing the dynamic density response function of the OCP throughout a broad range of densities and temperatures: from a fully degenerate quantum plasma at absolute zero temperature to a nondegenerate classical gas. It differs from the approach of Dandrea and Ashcroft or Tanaka et al in two important respects. First, that the theory be particularly applicable to electron liquids at metallic densities, which are not weakly coupled, perturbative local field corrections to the random phase approximation are replaced altogether by a selfconsistent field determined by a quantal extension of classical integral equations. Formally we can define a LFCF that, unlike Dandrea et al requires no known results, and unlike Tanaka et al., is temperature dependent; indeed it may be viewed as a generalization of the STLS LFCF of zero temperature and reduces to the proper classical result. Secondly, dynamical dependencies of the LFCF can be rigorously included. This has been shown to be a mechanism for modeling the double peaked shape of the dynamic structure factor observed in metals. ^^"^^ It is also an essential theoretical featurestatic LFCFs cannot simultaneously satisfy the compressibility sum rule and the third frequency moment sum rule for the density response.

PAGE 12

CHAPTER II BACKGROUND THEORY This thesis will concern itself solely with the OCP as a Fermi liquid. The most basic characteristic of a liquid is that it possesses short range order as opposed to the long range periodicity of a crystalline solid. Since the structure of a crystal is determined experimentally by observing the Bragg reflection of X-rays, it is natural to seek a quantitative description of the liquid structure via the intensity of X-ray, thermal neutron, or light 3 80—81 scattering. Of particular interest then are the fluctuations of space and time dependent densities which describe, at long wavelength, the cooperative motion of many particle systems. For this purpose define the particle density correlation function as S^pCr ?'. t) = <6n(r, t) 5n(r. o)>^^ where 6n(r,t) is the excess particle density operator 6n(r, t) = n(r, t) <6n(r, t)> eq and the equilibrium average is indicated by the brackets <. > eqSpecifically, even though gq = 0, there will be spontaneous, usually small, fluctuations on a local scale. The density correlation function is a measure of these fluctuations. -7-

PAGE 13

-8In this chapter we will begin our investigation into the structure of quantum fluids by reviewing the general properties of the time correlation functions related to the particle density correlation function. These include 1) various symmetries (for example time reversal), 2) positivity properties which are related to the dynamical stability of the system, 3) fluctuation dissipation theorems which connect spontaneous fluctuations and energy dissipation in a thermally equilibriated system, 4) Kramers-Kronig relations which express causality, like the well known one between the index of refraction and the absorption coefficient, and 5) sum rules which provide restrictions that any approximate microscopic theory must fulfill. The System Because we will be dealing with a many particle system from a quantum statistical point of view, we will employ the second quantization representation of quantum mechanics even though we will mostly be working in a canonical ensemble where the number of particles is fixed. For our equilibrium "system" we shall define in the Schroedinger (time dependent state) representation the unperturbed Hamiltonian H = T + V in terms of field creation and annihilation operators. The kinetic energy operator is

PAGE 14

-9T = J dl ^""(I) {} ^(I) while the interaction energy operator is given by V = ^ J ; dl dll f^il) = tr p(t) n(f) tr p(t) = 1 spins Basics of Linear Response 82-85

PAGE 15

-10What follows is entirely parallel to traditional derivations of time dependent perturbation theory in elementary quantum mechanics texts. We have to solve the time evolution equation for the density matrix ifi ^ p(t) = [H(t), p(t)] = [H, p(t)] + [6H(t), p(t)] (note the commutator is reversed from the Heisenberg equation of motion for operators) subject to the initial condition P(t = — ) = [H. p^q] = 0 The initial condition expresses the fact that the system is stationary before the external field is turned on— we also require that the external field decay sufficiently rapidly as time reaches infinity. For manipulations it does not matter what p is but since eq the system starts from thermal equilibrium we take Peq = P ^^'^ P If we assume a time dependance to the density matrix of the form p(t) = pgq + 6p(t) then the solution to the time evolution equation to first order in 6p and linear in the external field is 5p(t) = ^ / dx e-^"^t -^^/^ [6H(x). p ] e-^^'H^t _oo eq (This is easily verified by differentiation remembering that time appears both in the limits and in the integrand.) From this equation the induced change in the average density

PAGE 16

-116 = tr p(t) n(r) tr p n(r) is given by 5 = ; df J dr' <1 [n(r t), n(r' t')]> 6V(r' f ) _00 Q t = zi ; dt' ; df' x"(ft; f' t') 6V(r' t') (2. A) where [A, B] = AB BA is the commutator bracket. In the above equation are the Heisenberg operator for the unperturbed system n(?' t) = e^"*/^ nCr) e"^"*/^ and we have introduced the dynamic densitv resp onse function (please note the double prime superscript) defined by x"(ft; r'V) [n(ft), n(f't')]>gq This equation is the fundamental result of linear response theory. It shows that the density response to a small external potential is the averaged commutator rather than the correlation function S(r.t) as one might expect. However we shall see that the two are intimately related. The Dvnamic Densitv Response Fiinrtinn In equilibrium the system is space translational ly invariant. We adopt the convention that forward spatial Fourier transforms have a minus one signature and write

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-12x"(k) = J d(? -?) e"^"^*^"^ "'''^ x"(r V < k f^^t)' n!(t')]> k k Here we have used the definition of the spatial transform of the density operator n^(t) = ; dr e"^"^*^ n(?t) k which can be expressed in terms of plane wave creation and annihilation operators as njo) = i; aj ^a^ ^ from which it follows that n ^ = n^ -K K Time translation invariance of the dynamic density response function is easily established from the definition of a Heisenberg operator and the cyclic nature of the trace. We adopt the notation that forward temporal Fourier transforms have a plus one signature: 00 x"(Kto) = ; d(t t') e^"^* ^'^ x"(K. t t') 00 It is easy to show that this space and time Fourier transform is real (since the function is a commutator of hermitian operators), an odd function of frequency (since the equilibrium state is invariant under time reversal and parity), and depends only on the magnitude of k (spatially isotropic). It can also be shown that x"(Kco) > 0

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-13QC in a stable system. This is just a statement that a dissipative many body system takes more energy out of the external field than it gives back. In addition to the density-response function it is convenient to introduce certain interrelated functions. The complex response function (please note the tilda) is defined in transform space as ^(Kz) = f ^ ^^^^ ^ ir CO z -.00 This is an analytic function of the complex variable z as long as imaginary z does not equal zero (on the real axis it has a branch cut). For z above the real axis it reduces to the temporal Laplace transform of the density response: 00 x(Kz) = zi ; dt e^^* x"(Kt) 0 while below the real axis 0 = (zi) J dt e^^^ x"(Kt) This follows from the identity M i"t dco e Ziri co-z~ 0 t 0 J 7 = 4. ^ Inn z > 0 —00 0 t > 0 T ,'-,4. Inn z < 0 e"^^^ t < 0 which can be proven using the Cauchy integral formula in the complex omega plane. (For t>0 we can close the contour on top-exp(iwt) being bounded-and the contour is in the positive sense. For t<0 we must close the contour on the bottom, the contour now being in the

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-14negative sense. We then just sum the residues, the pole being included depending on the sign of imaginary z.) The physical response may be defined in transform space as the limit of the complex response as we approach the real frequency axis from above 00 x(Kco) = lim x(K. 2 = CO + ie) = limf — /"i'^"'^ e-o e-o -co ^ + Using the identity (P denoting Cauchys principal value) = P 1 iir 5(x) the physical response can be expressed in terms of its real and imaginary parts as x(ku) = x'(Kco) + ix"(Ku) where the imaginary part is just the previously defined density response and the real part (note the single prime) is x'(K(o) = PJ dcoLx;^ _^ ir u 0) an even function of frequency, is a manifestation of the KramersKronig relations. The utility of the physical response function is that it links the induced change in the time-space Fourier component of the density with the time-space Fourier component of an arbitrary external potential 5 = x(Ka)) 6(Kco) #

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-15One may also note that this same relation may be easily obtained via eq. 2. A using the definition of the physical response in real (untransformed) space, namely x(rt) = zi e(t) x"(r,t) The response to a static external potential may be derived from linear response theory by assuming that the potential is adiabatical ly switched on starting from the remote past 6V(rt) = ^<'^> I I 0 t > 0 At t = 0 the external potential is at full strength and we find 00 6 = zi J dx ; dr' x"(r r', x) e"" 6n(r') 0 or 5 = x(K) 6V(K) where the static response is given by zero frequency limit of the physical response X(K) = x(K, CO = 0) = lim x(K, z = ie) e-O A note on this last equality. Formally we have lim x(K. z = ie) = lim f ^ = (P f ^co iHlMi^ ^ ^ ^ ^ e^O e"*o -00 but because x" is a real odd function of frequency the imaginary term vanishes and we can drop the "principal part" condition.

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-16Relation to Density Correlation Functions At this point we should explicitly confirm the statement that the density response is intimately related to the density correlation function, the latter being the experimentally pertinent quantity. Now because the Fourier transforms of correlation spectra are related by 83 the equation r _ e"^"* dt = e"^^ f e"^"* dt -00 _oo it follows directly that J <|:^ [A. B(t)]> e-^"* dt = j:^ (1 e"!^^) J e"^"^ dt and so in particular (using the fact that x"(K(o) is odd in frequency) x"(Kco) = ^ (1 e""^^) SCKco) where S(k,w) is the time-space Fourier transform of the time dependent density correlation function. We note that S(r,t) is Fourier transformable because when r and/or t are very large, then n(r,t) and n(0,0) are statistically independent „ = „ *-M "q eq and so S(r,t) as defined as a correlation of excess density vanishes. The above relation between the density response and density correlation functions is an explicit example of what is called the fluctuation dissipation theorem^^ which relates two physically distinct quantities of fundamental experimental significance: 1)

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-17Spontaneous fluctuations, which arise even in the absence of external forces from the internal motion of the constituent particles. Described by S(k,w) these fluctuations give rise to scattering of neutrons or light. 2) Dissipative behavior— all or part of the work done by external stirring forces—is irreversibly disseminated into the infinitely many degrees of freedom of thermal systems. The fluctuation-dissipation theorem shows that S(k,w) is not quite symmetric in w, being a little stronger at positive frequencies than at negative. Indeed, since tox"(Ku) is always even in w, it i s general ly true that S(K, -(o) = e"''^'^ S(K, co) This result makes sense from a neutron scattering point of view. Positive frequency means the neutron has lost energy to the system (by creating an excitation of energy hw) while negative frequency describes a process in which the neutron has picked up energy from the system (by destroying an excitation). Of course, to destroy an excitation you must first have one, and their relative abundance is given by exp[-bhw]. This dissymmetry of the scattering intensity, proportional to S(k,w) is only pronounced at low temperatures, kt
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-1800 (o^ E ; X"(K{o)} do) _00 The j = 0 or zeroth moment follows from analyticity of x"(k,w). f ^ = X(K. 0) = X(K) _00 All odd order moments vanish because x" is odd in frequency. The remaining higher order moments can be derived from the expression (^)'' x"(rt; ?'t') = <^ [(i 1^)"^ n(r. t) n(r't')]> taken at equal times t = t'; this means that r x"(Kco) = J d(? f') e^^*^'' '''^ _00 <(j;)^''^ [[...[n(rt), H] .... H] n(r't)]> The right hand side contains a sequence of equal time commutators which can in principle, and sometimes in fact, be exactly calculated. For example the second order moment yields the Thomas-Rei che-Kuhn or F-sum rule equivalent 00 2 r do) „ii/L^ \ e K J ^ wx (Kco) = — — -CD ^ "> For the electron gas the fourth moment has also been calculated. ^^"^^ The sum rules are often used to provide coefficients for an expansion of x(k,z) for large z from the definition vri^y'* r" x"(K(o) (/ -•'^Trco-z -^ 2 4~ + — ^ _oo 2 z

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-19From its derivation, which expands (1 w/z)~ = 1 + (w/z) + (w/z) + ... it is clear this expansion can only be asymptotic, i.e. valid only when Izl is large compared to all frequencies in the system, which means all frequencies for which x"(k.,w) is not effectively zero. An interesting feature of the sum rules is their existence. There is no reason why the thermodynamic average of the multiple commutator should exist to all orders, and indeed although the sixth 90 moment has been calculated in the classical fluid limit it may 91 diverge for the degenerate electron gas in certain regions. As the sixth moment determines the w~^ term in an inverse frequency expansion, its divergence would herald the presence of a term of the 4 -11/2 form k w in S(k,w) at long wavelength for the high frequency behavior of the spectrum. Great caution is thus indicated in the use of moment sum rules beyond the first three mentioned. Distribution Functions It is natural to describe the structure and thermodynamic properties of liquids in equilibrium by employing static distribution functions. Most treatises on fluids introduce such quantities first and later generalize to include time dependance as a prelude to exploring non-equilibrium statistical mechanics. Here we have taken a reverse path; we have already related time dependent density correlation and density response functions and it is our intent now to relate these with the static distribution functions common to liquid theory.

PAGE 25

-20The most basic static distribution function is the single particle distribution function. If we define the number density operator in N particle configuration space as n n(r) = I 6(f r.) i = l ^ (the r. are quantum operators) then the single particle distribution is given by p(r) = = N ^ J" W(rp .. r^^) dr2 .. df^^ where the configurational part of the partition function is 0 = J W(r^ rj^) df ^ drj^ and the configuration probability WCr^ ... r^^) is given by the diagonal slater sum W(r^, ... r^Ir^, ... r^^) where, in general, the off diagonal Slater sum is defined as N(r^ r^ I r| ...r') = N X^'^ Z ?t(r^ .. f^) e" P
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-21In the classical limit the probability function "W" is just the Boltzmann factor of the potential energy of the configuration; at zero temperature it consists solely of the ground state wavefunction. The information contained in the single particle density distribution is minimal, for in the case of a translational ly invariant Hamiltonian it is a constant equal to the average number density of the system. However we introduce it here because relevant thermodynamic properties we will consider require the information of the off-diagonal single particle distribution function^"^"^^ p(r I r') = N 1 J W(r^ r^... r,^ I r j r^ r,^) df^ ... dr^ The physical significance of the off-diagonal distribution function is that its spactial Fourier transform provides us with the momentum distribution of the interacting system. The pair distribution function is defined by g(r ?') = N(N 1) 1 J N d?. .. r. = and for a translational ly invariant system is of the form g(r-r'). The pair distribution function in turn is related to the static structure factor through the relation S(K) = 1 + p J e^^*'' (g(r) 1) dr Within a factor of density the static structure factor is simply the time independent density correlation function. S(r -?')=! <(n(r)

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-22Note that if we had defined the static structure factor in terms of the density operator instead of the excess density we would replace g(r) 1 by g(r) in the formula relating the two. This amounts to including the experimentally unmeasurable "forward scattering." The definition of the static structure factor is immediately generalized to form the dynamic structure factor, essentially the previously introduced time dependent density correlation function (again within a constant factor of density) by replacing the quantum operators r. with their time dependent equivalents in the Heisenberg representation. Bearing in mind that r.(t) and rj(t) are operators which in general do not commute (except at t = 0), we easily obtain several important properties of the dynamic structure factor for translational ly invariant systems: 1) S(-r, t) = S*(r,t) where S* is the complex conjugate of S. 2) In the classical limit the imaginary part of S(r, t) vanishes. 3) S(K(o) E^f dt e-^"^ ; dr e"^''*^ S(rt) _00 is real in both the classical and quantal cases. 4) The elastic sum rule 00 S(K) = J S(Ka)) dto —CO is an immediate consequence of the definitions. All the relationships amongst the various functions discussed are best conveyed diagrammatical ly (See Fig. 2.1). Note that in the upper

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-23left corner we have introduced a new function C(k,w) called the Kubo or Relaxation function, about which we will have much to say later. For now it simply expresses the difference between the (dynamic) physical response and the static response function. The point of our review of the response, correlation, and distribution functions, so concisely summarized in Fig. 2.1, is that if we wish to generalize the classical many-body methods of calculating g(r) into the quantal regime, there are two possible relations open to us. First there is s''^(k) as the classical limit of $^""(1^) (obviously) or. as P 6n(r I U) 6 pU(f') = ^ X(K) where the Iq denotes evaluation at constant density/zero external potential, and the F denotes the Fourier transform with respect to the spactial variable r-r' Let us contrast the two alternatives. The quantum mechanical static structure factor S^'"(k) is essentially statistical in nature; it is strongly dependent on the dynamics of the particles, depending on S(k,w) ~ Imag x(Kw) over the full range of frequency. On the other hand x(K) depends on the single zero frequency limit of x(Ku), and is essentially mechanical, arising from the Schroedinger equation by our linear response derivation, which makes but implicit reference to statistical questions. This is clearly indicated by the F-sum rule-which holds in any stationary

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-24xCkco) X(kco) = x(K) + itoPCCku) yiKta) E Im x(Kco) ir (0 lim (0 0 SCkco) = ^ (1 + cot h ^) Y(kco) x(K) = ; y(Kw) d(o X(K) S(K) = J S(Kto) dco Figure 2.1 Relations between the linear density-density response function and the structure factor.

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-25ensemble— expressing nothing but the conservation of particles and the elementary commutator of position and momentum. This dissertation will concern itself with the integral equation methods of solving the many body problem. This route seems particularly promising for treating quantal fluids in that for the classical limit there exists a wide variety of integral equations for g(r) which have been investigated and give results in good agreement With experimental results. The purpose here is to extend this procedure into the quantal regime. A fundamental ansatz of this treatise is that one would expect the more accurate integral equation approximation to correspond with the less sensitive function. The extension of classical integral equations into the quantal regime via the density response function is presented in Chapter II. With this fundamental ansatz, however, there is a fundamental drawback: From x(K) one obtains an integral equation not for g^"^(r) (which is used in the computation of thermodynamic quantities) but rather n(r/v), that is the distribution function in the presence of an external potential which has the form of the interparticle interaction. While it is true that in the classical limit n(r/v) = n^qir) this is not true in the quantal case, since we are neglecting exchange and recoil effects. The external potential can be thought of as arising from a distinguishable particle of infinite mass and therefore fixed in position. An extreme example occurs for the case of very weak interactions between fermions. In such a case n(r/v)/nQ is approximately unity everywhere, whereas g(r) has approximately the ideal fermion value of 1/2 at the origin.

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-26Thus our task is twofold: first to generate an accurate quantal integral equation for n(r/v)/nQ and second to generate an approximate g(r) from knowledge of n(r/v)/n^. In essence our second task is to go from the bottom left of Fig. 2.1 to the bottom right along the paths drawn in this figure. We cannot go along the diagonal, as this path is one directional (we lose information when we integrate y(k,vi) over w in order to obtain x(k)). We must first traverse over the upper left path of Fig. 2.1; we must obtain an accurate approximation to the Relaxation function. This is taken up in subsequent chapters.

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CHAPTER III QUANTAL FUNCTIONAL EXPANSIONS In this chapter we will establish integral equations for the quantity n(r\y)/n^ (the density distribution in the presence of an external potential of the form of the interparticle potential) through the functional expansion method. ^'^"^^ Because in the classical limit these equations reduce to the familiar Hypernetted-Chai n (HNC) or Percus-Yevick (PY) equations for the pair distribution function, we will review the classical derivation with the aim of establishing corresponding quantal generalizations. The OOZ Relation In classical fluids the Ornstein-Zernicke (OZ) relation plays an important role in treating integral equations for g(r). Thus in this section we derive a quantal extension of the OZ relation (QOZ). We start from the known functional dependance of the classical canonical partition function on pairwise additive potentials. This allows us to write the functional derivative of the density distribution as^^ = n^ 6(? ?) + n^^ ^g(,^ r'l) 1} = F"^ {n^ S(K)} 0 6n(r I U) 6 3U(r') -27-

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-28The denotes evaluation when the arbitrary static external potential U(r) is turned off, that is at n(r) = n (n ; the 0 0 average density). The (negative power) script F denotes the (inverse) Fourier transform on the spatial variable r r', and S(k) is the previously defined static structure factor S(K) = 1 + n^ ; dr e"^'^*'^ {g(r) 1} Now consider the functional S BU(r') 6 N(f") This is a vacuous quantity unless such a functional of density can actually be constructed.^^ If such a creature does not exist, we define it through the chain rule of functional calculus: 6 n(r") Sn(r") 6 3U(r') = 6(r r') It follows that the inverse functional derivative is 6 BU(r') 6 n(r") 1 K ^n S(kT> = r Cdr' ?"l) 0 0 where we have split up the expression for the derivative into two pieces (this should be considered an arbitrary process) with the first piece chosen to be the non-interacting (g(r) = 1) result and the second piece (c(r)), called the direct correlation function, representing the deficit from from the non-interacting result. The functional chain rule as expressed in terms of c(r) takes the form

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-29h(r) = c(r) + J c(lr r' I) h(r') dr' This is the classical Ornstein-Zernicke (OZ) relation, where h(r) = g(r) 1 is called the pair correlation function. Furthermore, in the classical limit where there are no exchange or recoil effects (the momentum integrations factor from the configurational piece of the partition function) one can rigorously derive the bootstrap relation^^ n(r I U = v) = n^ g(r) where the external potential U(r) has the form of the interparticle interaction V(r). This bootstrap relation allows us to reformulate the OZ relation in terms of the inhomogeneous density distribution as fol lows: 6 n(r') n(r' I U = v) n^ = c(r) (3.1) 0 A quantal extension of the OZ relation is straightforward;^^ we will follow many of the above steps. First, from linear response theory, we saw that in the quantal case Sn(r I U) 6 3U(r) ^ = I^K^ {% X(K)} where x(k) differs by constant factors from the static response function previously defined in Chapter II. Defined here it is conveniently dimensionl ess ; note that only the product (5x(k) remains finite as temperature goes to zero.

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-30Again the chain rule of functional differentiation requires that 6 BU(r') 6n(r" I U) = ^K^ ^=k^K ^J(K)> ^"'> (3.2) but here we have decomposed the inverse derivative into two new (and as yet unspecified) functions. These are determined as follows. As we wish c(r) to reduce to the classical direct correlation function in the proper limit, we need only require J(k) = 1 in the classical limit. We furthermore make the ansatz that J ^ 6n(r') n(r' I U = v) n^ = c(r) (3.3) also holds for the quantal case. We have seen that this equation is strictly true in the classical case; by combining the two above equations we constrain the function c(r), and so simultaneously J(k), by the relation F,{n(r I U V) n^) = xili ^^ ^^ In hindsight we now see the motivation of requiring eq. 3.3 be fulfilled: as J(k) tends to unity in the classical limit we have merely imposed the condition that we recover the bootstrap relation in the classical limit. The above equation also determines J(k) (and so c(r)) in the general quantum limit, for if the system described by n(r/v) and x(k) were that of non-interacting fermions, then in the limit that U(r) vanishes we see that J(k) must equal This is an appealing result, for it amounts to defining the quantal direct

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-31correlation function by separating out the non-interacting result, just as we did in the classical case. Henceforth we will implicitly use this result, which will be referred to as the quantal bootstrap relation (QBS). As a consequence of this now specified decomposition of the inverse functional derivative eq. 3.2, the functional derivative chain rule expressed in terms of the quantal direct correlation function becomes ^ 1) = M{c(?)} + n ; dr'(M{c(lr r'l)}) (^i^^^ i) 0 where M{...} is an operator defined by F|^{Mf(r)} = Xq(K) F^{f(?)} This is the quantal generalization of the OZ relation (denoted QOZ for short). Note that this equation can be cast into the classical form by defining analogue functions: gan(-) = nirj_vi ^an^-^ ^ ^an^-^ ^ 0 c^"(r) E M{c(r)} The classical equivalents of these functions all have "physical" interpretations that are only loosely identified with the analogue. (For example g^" should not be confused with the true or quantum mechanical pair distribution function.) To distinguish classical analogues we shall henceforth employ the superscript "an."

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-32Classical Generating Functionals The heart of the functional expansion method of deriving integral equations lies in the ability to construct a functional W[n] of the density distribution n(rlu) in a nonuniform system which is reasonably insensitive to the function n(r|u).^^ (For other points of view, see Bakshi^ and ChoquardJ^^) If such is the case we can approximate the functional W[n] by making a functional "Taylor series" expansion, for example about the average density n^, and truncate after the first (linear) term W[n] = WCn^] + ; dr'( 6w 6n(r') ) (n(r' I U) n^) The OZ relation then forms a closed set of two equations in two unknown functions. As an illustration we will consider a set of functionals and the integrals equations they generate in classical statistical mechanics. First consider the functional W[n] = In U> e-PU(r) The denominator forms what is called the reference distribution. In order to make the functional variational ly insensitive, it is constructed to approximate the numerator as close as possible. Here we have taken the classical non-interacting Boltzmann factor. The first functional derivative is easily found to be

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-336w ^ 6(r r') 6 SU(r) 6n(r') n(r I U) 6 n(r') and from the definition of the direct correlation function: 6 m?) 6n(r') 6(r r') = ^ c(r r' ) 0 we find Sw 6n(r') = c(r r' ) 0 The functional expansion of W[n] as the potential U(r) is changed from the intitial value 0 to its final value In "^^ = ; d?'(n(?' I V) nj (6w ) along with the bootstrap relation n(r/v)/nQ = g(r) and the OZ relation then combines to give the HNC equation g(r) = ee^^"^^ where yd) = h(r) c(r) is called the nodal function. The PY equation g(?) = ePV^^^ [1 Y(-r)] follows from the same steps by starting with the functional n^ eBecause the HNC functional is the logarithm of the PY functional, it is the more insensitive functional of the two from variations. One might be tempted to conclude that it would therefore produce the

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-34superior approximate integral equation. However the validity of the HNC equation (or any other derived by a functional expansion method) depends on whether the cumulative effects of the higher order derivatives in the functional expansion are negligible. In actuality their effect is to push the form of the equation towards that of the PY (note how the form of the PY equation is the small y(r) expansion of the exponential y(r) term of the HNC). This is evidenced by the fact that for purely repulsive potentials the PY and HNC equations of state bracket the "exact" Monte-Carlo or molecular dynamics computer 1 02 simulation results. What we do know from diagrammatic 52—58 analysis is that the exact closure equation to the OZ relation for g(r) is of the form g(r) = e" '^^^''^ Y^r) + B(r) where the unknown "bridge function" B(r) is the sum of bridge diagrams of two point functions. The evaluation of the bridge diagrams in terms of higher functional deri vati ves^^*^^ or other attempts to account for bridge effects will be discussed in Chapter V. Here our intent is to generate zeroth order integral equations from variationally insensitive functional, thereby minimizing the effects of the higher order terms that we neglect in the Taylor functional expansion. It is well known that the PY equation has no solution for a one component plasma system. Numerically this arises from the fact that in the PY equation for g(r) the long range tail of the potential is not compensated by that of the nodal function yd), as it is in the HNC equation. This is symptomatic of a physically meaningful

PAGE 40

-35inadequacy, namely; in real systems the particles do not feel the "bare" long ranged potential but a collectively screened or short ranged potential An improvement on the PY generating functional would be to replace the bare potential appearing in the Boltzmann factor of the reference distribution by an appropriately screened one. For example W[n] = "^'^ l5U„(r) "o ^ where the Hartree potential is defined as 3U^(r) = 3U(r) + n^ / dr' (3U(r ?') (g(r') 1) As the Boltzmann factor now more closely describes the physical distribution n(r/v) this functional is variational ly less sensitive than the usual PY _6w_ ^ Sir ?) n(r I U) ^ ^^h^"^^ 6n(?') 3U.(r) 3Um(?) 6n(r') Using the bootstrap relation in the definition of the Hartree potential we see that LlJMIl ^ s m?) ^ ^ ^5 ( S m? Sn(r') 6n(r') 6n(r') (n(S) n^) ; dS fJU(? 5) 6(?' s) which when evaluated in the limit v = 0(n(r|v) = n^) and using the definition of the direct correlation function (eq. 3.4) yields

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-366 3U^(r) = ^^'^n "^'^ c(r r') 3U(r ?') 0 6n(r') and so = c(r ?') + 3U(r ?') 6n(r') Q The linear expansion of the functional W[n] thus generates the equation "^^ = 1 + ; d?' (n(?' I U) n^) (c(? ?') + mr ?')) which can be re-expressed with the help of the bootstrap equation and the OZ relation as BV.(?) g(r) = e {1 + pV^C?) + y(?) pvCr)} Note that in this equation the long range part of the potential is cancelled by the long range part of the nodal function, and all other functions appearing in this equation are screened. This equation, which we shall term the PYH equation (for Percus-Yevi ckHartree) is equally applicable to long and short range potentials, unlike the PY equation. If we try to improve the HNC equation by introducing the screened Hartree potential into its generating functional W[n] = In U) 3U^(r) "o

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-37we find that its first variation is the same as in the above PYH derivation and (following many of the same steps) the functional expansion leads to the equation namely the original HNC equation. In other words the HNC equation has the striking property of being invariant to (Hartree) screening of the reference distribution in its generating functional. From a related point of view it has also been shown^^ that the HNC equation is the limit in a series of integral equations that can be systematically generated starting from the PY equation, among others. Finally we should note that the HNC equation can be derived from an even more interesting point of view. We ignore higher order functional derivatives and assert that the best functional for our closure purpose are insensitive to variations, and this property shows up in the magnitude of its derivative. This can be minimized by using an even more realistic screened potential for the reference distribution. In fact it is easy to show that if we modify the Hartree potential by replacing the convoluted bare potential by the direct correlation function W^ir) = (3U(r) / dr' c(r ?') (g(?') i) then the derivative of both the HNC and PY like functional H = In -^U) ,, n(? I IJ) g(r) = e BV^(r) Y(r) + 3V^(r) 3V(r) n 3Ujr) n 0 e 3U (?)

PAGE 43

-38actually vanishes. They in turn both yield the same equation 3V ^r) g(r) = e ^ which one will now notice, via the OZ relation, is simply the HNC equation. Quantal Generating Functionals Corresponding quantal equations can be obtained from the above classical generating functionals if we replace the reference distributions— the Boltzmann factor n^expf-bU} describing classical non-interacting particles in the presence of an external potential U(r)— by its quantal extension n*(rlu)— the density distribution of non-interacting fermions in an external potential. (Henceforth non-interacting fermion system quantities will be distinguished by a star.) That is we replace the reference distribution by a single particle Hamiltonian approximation. As in the classical presentation of the above section the functionals are then linearly expanded and evaluated in the limit where the external potential is set equal to the interparticle interaction potential. We have previously derived a quantal extension of the pair correlation function V"o ^("^^> -rk)-^ from the ansatz that the exact classical relation J ( 6 SU(r) 6n(r' ) ) (n(r' I U = V) n^) dr' = c(r)

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-39remains valid in the quantal case, alternatively expressed by the ansatz that the classical bootstrap relation is extended as By making use of these relations and defining the analogue potential g3U^"(?) ^ n*(? I U) "o a closed set of equations is formed with the QOZ relation. Expressed in terms of previously introduced analogue functions, these integral equations bear a close correspondence with their classical counterparts, and indeed reduce to those counterparts in the appropriate limit. As a first example we consider the functional W[n] = In "^"^ I n*(r I U) Its first variation is _-5w_ n*(r I U) r 6(? r') n(r I U) Sn*(r I U) n(r I U) ^n*(r I U) ^^.^^ ^^^2 6n(r") > This can be simplified in the following manner: 6n(r') 5 3U(?") ^ 6n(r') ^'''n^ xCQ) ) = 1 FJc^"} where it should be remembered that the analogue pair correlation function differs from the quantal ly extended "physical" pair correlation through the operator

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-40-1 M{f(r)} = F-'{x*(0) F{f(r)}} From the above considerations we see that 6n*(r I U) 6n(r') = 6(r r') n^c^^Cr r') and so 6w 5n(r') = c'"(? r') By Taylor expanding the functional W[n] about U(r) = 0 to v(r), we find In "(r I V) p n*(r I V) = J ^"^^ I "o^ ^ 6w 6n(r') or This equation we will call the Quantal Hypernetted Chain or QHNC equation. Following similar steps with the functional u n(r I U) ^ n*(r I U) we can arrive at or what shall be referred to as the QPY equation. It is of no trivial importance that the QOZ, QPY and QHNC equations are of the exact same form as their classical counterparts (this in addition to the property of reducing to the classical equation in the limit). This is because phenomenologi cal ways of incorporating the effects of truncating classical functional

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-41expansions at the linear term may still be of use. [In the classical HNC case such a truncation is equivalent to neglecting bridge diagrams in the Mayer expansion of non-ideal gases.] This will be considered further in Chapter V. It should also be stressed again that these integral equations, even if they could be made exact, are for g^"(r) = n(r/v)/nQ and not the pair distribution function. This is evidenced by the fact that if we turn off interparticle interactions, V^"(r) vanishes and the solutions of either the QHNC or the QPY equation yield g^"(r) = 1. This is correct for the density distribution but the fermion pair distribution has an exchange hold. This difference is taken up in the next chapter. The exact correspondence between quantal analogue and classical integral equations is lost when one considers screening effects on long range potentials. Rigorously incorporating screening effects is complicated in the quantal case by symmetry effects. It can be shown^^'^ that the best single particle Hamiltonian (best in the sense that the statistical average over the square of the difference of the exact and single particle Hamiltonian is minimized) is provided by a Hartree-Fock potential, not the Hartree. On the other hand we will see in the next section that as far as density distributions are concerned there exists an exact effective potential single particle Hamiltonian. In the quantal integral equation approach we will consider here screening of the reference potential will ignore exchange effects; we will see in the next section that they are actually incorporated in the closure of the linear functional expansion, namely the QOZ relation.

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-42When we screen the reference distribution of the QPY generating functional ^ ^ n(? I U) n*(r I U^) with the Hartree potential 3U^(r) = (5U(r) + J dr'(pU(r' r) (n(r' I U) n^) its linear expansion requires the variation 6w 6n(r') 6(? ?) 1 V "o "o 6n(r' I U) This can be evaluated by invoking the chain rule over the Hartree potential, which in Fourier space becomes 6n*(r I U^) 6n(r' I U) 6n*(r I U^) 6 pU^(r") S 3U^(r") 6n(r' I U) = "oX*(0) Fq{ 6 3U^(r") 6n(r' I U) Now the variation of the Hartree potential yields 6 3U^(r) 6n(r') 6 BU(r) 6n(r') |iU(r r') or 6 3U^(r) Sn(r') } = n^x(Q) 'a K{mr ?')} from which a few simple manipulations yield

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-436w = c^"(r-r') + Fq^ x*(0) F{(iU(r-r')} = c^"(r-r') + M{|3U(r-r')} 0 ^ 5n(r') Plugging into the functional expansion of W[n] we obtain "^"^ = 1 + ; d?'(c^"(? r') (n(?' I v) n ) n*(? I V^) 0 + ; dr'(M{flU(r ?')}) (n(r' I v) n^) In terms of analogue functions and employing the QOZ relation this becomes g (r) = e [1 + Y (r) + Fq {x*(0) F{n(? I v) nj F{(JV(?)}} = e [1 + Y (r) + M{; dr'(pV(r ?')) (n(r' I V) n^)}] = e [1 + Y (r) + M{3V^(r) 3V(?)}] This equation will be referred to as the QPYH equation. Unlike the classical case, if we screen the QHNC generating functional, we do not recover the QHNC equation. A now familiar procedure yields instead the result g^"(?) e' '^""^''^ ^^""^ ^ M{3V^(r) pvc?)} to be known as the QHNCH equation. We see that for both the QPYH and QHNCH equations long range tails cancel but the potential enters in a nontrivial manner. Lastly, in the classical case we saw how the HNC equation could also be derived by constructing a screened potential such that the first variation of the generating functional vanished. The same

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-44procedure can be applied to the quantal case. Define an effective potential (5U.(r) = (5U(r) J dr'(c(r r')) (n(r' I U) n^) I. 0 (where c(r) is the quantal pair correlation function and not the analogue quantity); then both the QHNCand QPY-like generating functional s W = In U> H^_ni^_LUi_ n*(r I U^) n*(r I U^) have vanishing first variations and both yield the same equation: an This equation is distinct from the QHNC equation but like the QHNC equation reduces to the classical HNC equation in the limit. For this reason it shall be termed the 0HNC2 equation. Interestingly enough the 0HNC2 equation can be obtained from an entirely different approach. Instead of assuming the quantal bootstrap relation and utilizing Percus' method we can arrive at the QHNC2 result directly from Kohn-Sham-Mermi n density functional theory. From there we can recover the quantal bootstrap relation (our starting ansatz) and so justify our variant equations by making what we will see as a reasonable assumption concerning the effective potentials of many body systems.

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•1 -45The Density Functional Approach Following the zero temperature approach of Hohenberg and Kohn^*^^ and Kohn and Sham^^^ for the ground state properties of an electron gas in an external potential U(r), Mermin showed that^^ i) the Helmholtz free energy A of an i nhomogeneous electron gas in thermal equilibrium is a unique functional A[n] of the one electron density n(r), and ii) the grand potential Q[n] = A[n] vi J" n(r) dr is minimum for the equilibrium density n(r). We employ the grand canonical rather than a canonical ensemble because macroscopic quantities derived from a non-interacting Fermi system are more easily calculated and the fundamental variable, namely the density distribution, is a physical observable that should be the same from either ensemble. The free energy is customarily decomposed as A[n] = 1 J v(r ?') n(r) n(r') dr dr' + J U(r) n(r) dr + A*[n] + A [n] X c to separate out respectively 1) solely classical electron-electron interaction contributions, denoted by v(r), 2) the external potential contribution, 3) the contribution A*[n] that a non-interacting Fermi system with density n(r) would still exhibit, and 4) all other effects are grouped to define the unkown exchange and correlation functional A^^[n]. The minimum property of the grand potential leads to the functional equation 6A*[n] — + U f^(r) }i = 0 (3.5) 6n(r)

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-46where u is the chemical potential and U^^^d) is defined by SA rn] Ug^^(r) = U(r) + J dr V(r ?') n(r) + 6n(r) Equation 3.5 is equivalent to the equation for non-interacting electrons in this effective potential; that is we have n(rlU) = n*(rlUg^^). The non-interacting fermion distribution n*(rlU) is given by the following system: (^ A + Ug^^) = e. fi = M = 1 units n*(r I Ug^^) = I f(e.) I ^.(r) 1^ where the sum over the index "i" refers to all bound and continuum eigenstates and f(e) is the Fermi-statistics occupation number 1-1 f( e) = {1 + exp[p(e. li)]}' dependent on the temperature and the chemical potential. The latter is determined as usual from the total number of electrons by spatially integrating over n*(r). For the jellium model of the electron gas the external potential must not only include the source of the external potential but the effect of the neutralizing background as well U(r) = U^Cr) J dr' n^ V(r ?') so that the effective potential is Ueff(r) = U^Cr) + ; dr' V(r r') (n(? I U^) n^ +

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-47At large radii, the displaced electron charge vanishes, and any long range tail of the source potential is cancelled on the assumption of perfect screening, so that the assymptotic value of the effective potential is 6A W^^">= 6n xc Applying eq. 5 at large radius gives 6A* 6A. 6n. = ^ xc 6n or, equivalently v2 „3/2 n„ = — 3 ^ I^/^Pii) 0 2 ir where the standard definition of the Fermi-Dirac functions has been used: I (n) E J y* 0 1. e^-^ It is convenient to shift the zero point of our energy scale to define a new effective potential which goes to zero at infinity Ugff(r) = Uq(?) + J d?' V(? r) (n(r I U ) n ) 6A + (xc 6n 6A. ) (xc 6n ) Up to this point our equations are exact; the exchange correlation part of the free energy functional is, however, quite unknown. But if we make a functional expansion to first order

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-486A. xc 6n 6A ) = (xc 5n ) + ; K(r r') (n(r' I U^) n^) dr' 5 A K(r r') E (xc 6n(r) 5n(r') we obtain an approximate integral equation in terms of the unknown kernal K(lr-r'l): n(? I V^) n^ = n*(-r I U^^^) n^ ^eff = ^0 -f" dr'{V(r r') + K(r ?)} {n(?' I U^) n}^ Since the unknown kernal is independent of the external potential U(r), we can determine this function by considering the case of a very weak external potential where the linear response formula can be employed on both sides of the first of the above equations with impunity. The left hand side yields 6n(0) E FQ{n(? I U^) n^} ^ n^p x(Q) U^CQ) while the right hand side gives FQ{n*(r I Ug^^) n^} : n^3x*(Q) U^^^CQ) = n^^x*(Q) {Uq(0) + [V(0) + K(0)] 8n(Q)} Using the last equation we can show that the Fourier transform of the unknown function K(r) is K(0) = (V x(0) x*(0) ) V(Q)

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-49Due to this assignment the effective potential reduces in form to the previously defined screened potential Uj.(?) = Uq(?) ; d?' (1 c(? -?') (n(r I U^) n^) and we obtain the relation n(r I U^) = n*(r I U^) Note that the source external potential is completely arbitrary, and so this result is a generalization of the QHNC2 equation, which follows immediately when the external potential is taken to be the interparticle interaction. From this generalized result we can work backwards and obtain the quantal bootstrap relation, and so justify the other (QPY, QHNC, etc) equations we have obtained. This is done by now making the assumption that the average one body potential felt by a constituent electron due to a fixed test charge 6e, namely Vt =Vtest ^ d-r'(^ c(-r r')) (nCr' I V^^^^) n^) where is equal to the average one body potential felt by a free test charge Se, arising from a fixed electron in an electron gas, namely V^-^g = (5e) l^(r) W(r) = 4Tr{ e8(r) + e[n(r I U = e^/r n ]}

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-50In the limit of small Se we can again invoke linear response with impunity and solve for F^{n(r I U = V) n^} = 1 thus recovering the quanta! bootstrap relation. He see that our density functional approach, aside from justifying the quanta! bootstrap ansatz of the Percus method, gives physical insight to its meaning. Even though it is not an exact approach, it clearly points out its limitation, as we approximated the variation of the exchange-correlation free energy functional by a first order functional Taylor expansion. This type of truncation is a property shared by the Percus method and ultimately may be responsible for the self-consistency of the two approaches.

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CHAPTER IV THE BASIC QHNC EQUATION An alternative ansatz for the extension of classical integral equations into the quantal regime lies in the construction of effective pair potentials. These incorporate the bulk of quantal effects; for example in mul ti component plasmas quantal effects prevent the collapse of particles in an attractive coulomb field. More generally the effective potential yields finite values for the pair distribution factor g(r) in the limit of zero interparticle separation r. Such classical pseudopotential s have been constructed by requesting an approximate account of the N-body Slater sum^^^~^^^ or, in an approach which can in principle be implemented exactly but is more limited in its physical range of applicability, by reproducing the two-body Slater sum^^^"^^^ (r) (JH^^^ e = (A-^'^^/my'^ (4.1) "2'^ = ^ v2 + V(r) It can be shown that this Binary Slater Sum approximation correctly represents the derivative of the classical pseudopotential for the N-body problem in the limit r tends to zero.^^^ -51-

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-52As shown in the previous chapter, the QHNC equation for n(rlv) is also equivalent to the classical HNC equation with an effective potential defined by P^eff^^^ n*(r I V) e = — — — "o where n*(r|v) is the density distribution of non-interacting fermions in the presence of an external potential of the form of the interparticle interaction—the bare coulomb potential. Normally this entails the solution of the Kohn-Sham-Mermin equations That is calculating n*(r|v) is the solution of the following system:120-121 (^ v2 V) ^. = c. f. n*(r) = I f(e.) (4.2) i 111 where f(e.) is the Fermi-statistics occupation number f(e.) = {1 + exp[3(c. \i)]y^ depending on the temperature and the chemical potential fixed by the total number of particles. The index "i" runs over all continuum and bound (if existent) states. The treatment of the delocalized states proposed by Friedel for impurities at zero temperature can be applied to the nonzero temperature case too. The eigenfunctions i^^^^ of eq. 4.2 for energy

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-53are (|>j^^(r) is normalized by requiring that it have the asymptotic form ^^^^(.r) ^ sin[Kr e J + n^CK)] n-jCK) being the phase shifts in the potential V(r) and n^(K = >) = 0. Overall normalization is accomplished by requiring that the wavefunctions vanish at a spherical wall of very large radius R. We assume that R is so large that most of the integrand is approximated by its asymptotic form: 1 = I A^^ 1^ ; dr sin^CKr e J + n^] with the boundary conditions KR + T]^iK) = e J = mir m = 0, 1, 2, 3. .... incrementing m by unity for a given L yields the density of states (including spin multiplicity) as 9i(K) =f ^1 ^ rIkHiCK)) while the normalization factor for very large R is given by A^^ = v27r (1 -0(^)) (These same results can be derived for a long range coulombic potential .^^^ 36.23^ ^ straightforward derivation gives for the density n*(r) = I f
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-54where subtracting off the uniform density result accelerates convergence of the integral and summation. We note that in general a potential may support a number of bound states and we have included them al so. The OHNC Effective Potential We thus see that the QHNC pseudopotential is essentially a one-particle Schrodinger equation calculation. This is a feature in common with the Binary Slater Sum. It is important to contrast the two however. The Binary Slater sum (BSS) is used to calculate the pair distribution function g(r) while the QHNC equation yields n*(r|v); this difference is important in consideration of quantum effects. It should be noted that in the BSS approach the Schrodinger equation is solved with the use of the reduced mass. The matrix element in eq. 4.1 can be evaluated by inserting a phvsi cal basis set of properly symmetrized or anti symmetrized wavefunctions depending on i the statistics of the two particles. Parallel spin versus antiparallel spin contributions can be calculated separately. In the BSS approach the effective potentials are density independent. In the QHNC approach the Kohn-Sham system of equations can also be written in the form of a matrix element n*(r I V) = where the operator "f" is defined as f = {n + exp m p]}"^

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-55H is the coulomb hamiltonian operator, mu the chemical potential and we have included a variable "eta" to be set equal to unity to describe Fermi statistics. In the QHNC approach the external potential has no spin attribute, and the matrix element is evaluated using a complete basis set with no regard to statistics; the statistics are built into the functional form of the matrix element. The QHNC effective potentials are implicitly dependent on the density through the chemical potential. All these subtle differences are manifestations of the fact that in the QHNC approach g(r) must be obtained from n*(r|v) by a further calculation. One obvious advantage into this splitting of tasks is that the BSS pseudopotential breaks down at zero temperature, whereas the QHNC remains valid. A surprising feature of the QHNC effective potential is that it is analytically calculable. This allows one to bypass the computationally time consuming Kohn-Sham-Mermin system of equations. The starting point for the solution is the fact that the effective potential is related to the r^ = limit of the off diagonal density matrix for fermions in an external coulomb potential : p(f^ I r^) = Spherical symmetry reduces the functional dependance of the off diagonal density matrix does from the full six variables of r^ and 12 to only the magnitudes r^ and the angle between r^ and T^. This is easily seen if we expand the matrix element using a complete set of spherical coulomb wavefunctions :

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-56dL c ri^2 2m (1 + 1) ^ X S + [K ~^ ~ I 0^ S = 0 Using the addition theorem of spherical harmonics then yields where the 1-wave radial density matrix components are defined by 00 Pl(r^ I r2) = ; (spin J dK) — ^-^ S*^(r^) S^^^r^) ^ ^ 2m ^ ^ Furthermore, because the potential is coulombic, there is an additional symmetry: the matrix element "f" also commutes with the 1 24 Runge-Lenz vector. This reduces the functional dependance of the off diagonal density matrix to the two variables "x" and "y": 7 7 ^ 17 X = (r^ + r2)/2 y = (r^ + r2 2r^ cose)"^/2 The solution then follows the steps of Hostler^^^"^^^ and Storer^^^ except that the Bloch equation satisfied by the off-diagonal canonical density matrix is replaced by a Fermi statistics generalization [(H 1^) n + n + fj^] p(r^ I r2) = 0 (Here H acts only on the r^ coordinates and we set eta equal to unity at the end of the calculation.) Because of the aforementioned symmetries, the solution to this equation reduces to a functional form identical with the s-wave radial density matrix component (however in the variables x and y):

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-57The diagonal limit r] = r2 = r is given by n*(r) p(? 1 7) = ^ p (X + y) x ^ ay y) y=o The second derivatives of the radial coulomb wavefunctions can be eliminated in favor of the radial wavefunctions themselves by employing the radial coulomb differential equation. We thus obtain n*(r) = ^ J dK ( 1 2ir' J(K, r) = { 3 rfi] + 1 ) J(K. r) d? ^Ko^^^ 2m 2 K^) I S^^(r)|2} Numerical Calculations The effective potential, obtained from the ideal fermion density distribution in a coulomb potential, will be calculated from the analytic formula derived above. For comparison with the classical HNC equation, we will scale lengths in units of the interparticle separation x = r/a where n^ = u a3)-l Temperature/density points will be expressed by the classical coupling parameter

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-58along with a quanticity temperature (two divided by the temperature in rydbergs) 13* E fime^/fi^ In the classical limit all functions are solely dependent on the parameter r^; we expect any dependance on the quanticity temperature to indicate the pervasiveness of quantum effects. The calculations were performed corresponding to the temperature density points of tabulated results provided by Pokrant for later comparison. These consisted of values at constant density, as measured by the dimensionl ess ground state coupling parameter r^ = a/a^ s 0 where a^ is the bohr radius, as well as a dimensionl ess degeneracy temperature t = Kt/Ep where is the ground state Fermi energy. The results are presented in the following graphs. In Fig. 4.1 we present plots of the ideal fermion density in the presence of an external coulomb potential. For comparison these are plotted together with two approximations. The first is merely the classical limit n*(r I V) The second is the Thomas-Fermi approximation. It is obtained by expanding the diagonal density matrix element "f" using a complete set

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-59of plane waves. If we then neglect the non-commutativity of the potential and kinetic energy operators in the exponential (the neglected terms can be systematically calculated by Trotter 1 23 formulas and shown to vanish for large r) we then obtain 'j n*(r I V) ^1/2 (Pv^r/x) I "o ^1/2^^^^ where the Fermi-Dirac integral of order p is defined by the integral 0 1 + e-' (In this formula the symbol r denotes the gamma function.) It is seen that the exact result quickly merges with the Thomas-Fermi approximation. By using the relation dF (X) we find that asymptotically the Thomas-Fermi approximation has the form F ^ (3n) n*(r I V) ~ 1 I 2 ^ 1 ^^V"^ ^ 2 This is indeed the correct asymptote of the analytic solution as can be verified by looking at the appropriate expansion of the spherical 1 29 coulomb wavefunctions. This tells us that the effective potential in the QHNC approximation has a long range 1/r tail—but with a strength modified from the classical theory. The QHNC tail contains the additional factor

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-60F ^ ((5p)/F ^ "2 "^2 not found in the Binary Slater Sum approach. In Fig. 4.2 we present plots for the QHNC effective potential. For comparison the corresponding effective potentials of the Binary Slater Sum approximation are presented in Fig. 4.3. The approach to solving the HNC system of equations with long tail potentials is well known — we follow the convention of Ng^^^ and renormalize the constituent functions by subtracting out an analytically transformable long range function of the form ^ erFc(ar) We employed the Ng method of solution (essentially a picard iteration with an accelerated convergence procedure) although it should be mentioned that two new algorithms have recently been published^^^"^^^ that employ a hybrid Picard/Newton-Raphson iteration procedure which promises faster convergence. The results of the integral equation solver, that is the analogue pair distribution function n(.r/yj)/r\^, are presented in Figs. 4.4-4.8. This distribution is of theoretical interest in its own right, for example in the study of impurities in metals or positron annihilation 1 32 rates. However we will limit our concern to the approximate calculation of the pair distribution function. A method of calculating g(r) based in part on the information contained in n(r|v)/n and a 0 comparison of results with the Binary Slater Sum approximation and that of Pokrant will be taken up in the next two chapters.

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-610. XT/a Figure 4.1 Plots of n*(r|v)/no-l versus x = r/a for a density of rg = 2 at temperatures of kt/Ef of .1, .5, 1.0, 2.0, 20.0. Storer denotes the analytic solution, following Russian literature the Thomas-Fermi approximation has been denoted Hartree, and Boltzmann is the classical approximation.

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tau8.10 0. -fT 1.0 1 — ,5 2.0 x-r/a 2.5 3.1 3.5 Figure 4.2 Plots of the QHNC effective potential versus x = r/a for a density of rg 2 at temperatures of kt/Ef of .1, .5, 1., 2., 20. We have plotted the potential in the form I Veff('^> A purely classical coulombic interaction would correspond to a constant values of unity. Storer represents the analytic solution; Hartree is the Russian literature denotation of the Thomas-Fermi Approximation, which is truncated near the origin to prevent divergences.

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-63Figure 4,3 Plots of the Binary Slater Sum effective potential versus x = for a density of rj = 2 at temperatures of kt/Ef of .1, .5, 1., 2., 3., 5. We have plotted the potential in the form I Veff
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-64static structure factor m o Figure 4.4 Plots of density times the Fourier transform of h(r) = g(r) -1 versus k = q*a (q the reciprocal space to r) for a density of rc 2 at temperatures of kt/Ef of .1. Free denotes the ideal fermion pair distribution function, HNC the results of the QHNC calculation where g(r) represents the analogue pair distribution n(r/v)/no. Chihara denotes the results of a yet to be discussed construction of g(r) from the analogue g(r).

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-65static structure factor hnc free ctitnara tau-a.se r3-2.ee Figure 4.5 Plots of density times the Fourier transform of h(r) = g(r) -1 versus k = q*a (q the reciprocal space to r) for a density of rj = 2 at temperatures of kt/Ef of 0.5. Free denotes the ideal fermion pair distribution function, HNC the results of the QHNC calculation where g(r) represents the analogue pair distribution n(r/v)/no. Chihara denotes the results of a yet to be discussed construction of g(r) from the analogue g(r).

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-66static structure factor k-q-a Figure 4.6 Plots of density times the Fourier transform of h(r) = g(r) -1 versus k = q*a (q the reciprocal space to r) for a density of rg = 2 at temperatures of kt/Ef of 1.0. Free denotes the ideal fermion pair distribution function, HNC the results of the QHNC calculation where g(r) represents the analogue pair distribution n(r/v)/no. Chihara denotes the results of a yet to be discussed construction of g(r) from the analogue g(r).

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-67static structure factor k-q-a Figure 4.7 Plots of density times the Fourier transform of h(r) = g(r) -1 versus k = q*a (q the reciprocal space to r) for a density of rg 2 at temperatures of kt/Ef of 2.0. Free denotes the ideal fermion pair distribution function, HNC the results of the QHNC calculation where g(r) represents the analogue pair distribution n(r/v)/no. Chihara denotes the results of a yet to be discussed construction of g(r) from the analogue g(r).

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-68static structure factor k-q-a Figure 4.8 Plots of density times the Fourier transform of h(r) = g(r) -1 versus k = q*a (q the reciprocal space to r) for a density of rj = 2 at temperatures of kt/Ef of 8.0. Free denotes the ideal fermion pair distribution function, HNC the results of the QHNC calculation where g(r) represents the analogue pair distribution n(r/v)/no. Chihara denotes the results of a yet to be discussed construction of g(r) from the analogue g(r).

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CHAPTER V LOCAL FIELD CORRECTIONS We have seen in the previous chapter how classical integral equations generalized into the quantal region can yield accurate results for the quantity n(rlv), namely the density distribution of interacting particles in the presence of an external potential of the form of the interparticle interaction. This in effect is equivalent to knowledge of the static response function x(k). If from this information the dynamic response x(k.,w) could be extracted, then we could easily calculate the static structure factor or pair distribution function via the fluctuation dissipation theorem S(K) = ; S(Kco) d
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-70This particular relation, which is one of many possible representations, introduces an unknown function e(k,z) which is analytic in the upper half plane. The physically meaningful dynamic response is given in the limit z = w + ie, and in this limit the real and imaginary parts of G(K, Z = to + ie) = 0'(Kco) + ie"(K(o) are, like the dynamic susceptibility itself, related by the Kramers-Kronig relations. The function e'(k,w) is known to be a positive, even function of frequency, as follows from the known parity of x'(k,w) and x"(k,w) and the fact that S(k,w)> 0. The utility of the function e(k,z) is that it interpolates smoothly between the known results for x'(k,z) in the limits of large and small z, and being less sensitive than x(k,z) itself, is more amenable to approximation. By subtracting out a corresponding expression for a noninteracting Fermi system we can express the dynamic susceptibility as Here we have grouped all of the actual physics into a dynamic effective potential Vg^^(k,z), actually a functional of the static response x(k) and the unknown function D(k,z). We will show through the use of the Mori projection operator technique^^^ that under reasonable assumptions the dynamic effective potential can be approximated by the static (z = 0) limit of the above equation x(KZ) = (5.1) 1 V eff (KZ) x (KZ) V eff (KZ) = ( X(K) Xq(K) )

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-71It should be noted that although x^{k,z) in eq. 5.1 is usually taken to be the free response function, this choice is not a unique one; x^Ck.z) plays a role of a reference system in terms of which the expression for x(k,z) is constructed. A possible improvement in the procedure may be achieved by using an adjusted choice of y.^(k,z) that includes self-energy effects, for 14-19 137 example. This method of improvement will not be taken up in this thesis in lieu of other avenues of extension. We will retain the free response as our reference system; this will allow us comparison with certain mean field theories. Mean Field Theories Mean Field Theories bypass the dependence of v^^^Ck.z) on x(k) and directly try to relate V^^^ to the bare interparticle potential by solving approximately the many body problem. (Something we have already done via our integral equations.) The efective potential is usually taken to be "static," that is independent of z, and is commonly expressed in the form Vg^^(K) = Vq(K) [1 G(K)] where V^(k) is the Fourier transform of the bare interparticle potential and G(k) is called the local field correction factor. A review of such local field function theories can be found in 1 og Kugler. The many body part of the problem is generally solved by the Vlasov kinetic equation or, what amounts to the same thing, appropriately decoupling the equations of motion for the density

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-72fluctuations J'*^ Whatever route, at some point collisions between particles are neglected and the particles are assumed to move in a mean or average field based on the collective motions of all the other particles. The physical significance of the correction factor is thus to incorporate some of the effects of short range correlations between individual particles not taken care of by the average field. Mean field theories for the most part have been limited to the ground state quanta! electron gas. These have disagreed on the form of the local field correction factor on three main points. 141_145 First some workers have been led to the conclusion that G(k) is a universal function of k/k^ (k^ the Fermi wave vector), 146_i4g 29-31 while others contend that G(k) is density dependent. Second, there is wide diversity of opinion regarding the value of 1 50-1 51 G(k) in the limit of large wavevector k. At this limit, a constant 1/3, irrespective of density, has been given by Geldart and Taylor. Rajagopal^^^ and others, ^^^'^^^"^^^ while that of Togio and Woodruff^^^ and Kugler^^^ use 2/3. On the other hand 29 30 Singwi et al derive this value to be l-g(O) in terms of the pair distribution function g(r) at the origin, and that of Vashista 31 and Singwi is 1 g(o) ap(^) (where a is an adjustable parameter determined self-consistently) in contrast with the result of Niklasson^'*^ who gives 2/3(1 g(0)) to this limit. Thirdly, there are several authors^^^'^^^"^^^'^^^ who have shown the local field factor G(k) has a maximum around k = 2k^.

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-73Kugler among them further indicates the slope of G(k) has a logarithmic singularity at k = 2k^ in addition to a maximum. On the 29-31 contrary, G(k) of Singwi and collaborators has neither maximum nor singularity in its slope around k = 2k^. It should be noted that the behavior of G(k) in the large wave vector region is quite extremely sensitive to small differences in x(k) so that it offers a stringent test for the distinction of various theories. In the case of the degenerate electron gas, there is neither computer simulation nor experiment to be used as a criterion to check these diverse conclusions concerning G(k). However, at present the 29-30 prescription of Singwi et al is regarded as fairly successful in treating a degenerate electron gas and is relatively easy to program numerically. It encompasses in the classical limit, as do the integral equation methods of Chapter II, the Hyper-Netted Chain equation, which is known to give the best fit to the pair distribution function of computer simulations. For this reason the derivation of the Local field correction factor in the STLS theory will be presented here as a basis for later comparisons. The STLS Method o Following Singwi we will derive the density response function by following the time evolution of the off diagonal single particle density function P (X I X' ; t) E < ^^(xt) t (x't)> a a CT i 1

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-74If we employ a Heisenberg equation of motion approach with a HamiUonian containing an external potential Vg^^Cx.t), we obtain 3t 2m ^" ^H^^^^ V^Cx't)} p^(x I x'; t) = r d [V(x X") V(x' X")] <^''^(xt) n(x"t) t (x' t)> where the Hartree potential V^(xt) E Vg^^(xt) + ; dx' V^(x x") p(x". t) depends on the i nterparti cl e interaction potential and the average single particle density distribution p(xt) = E < T 'f'^Cxt) t (xt)> S We were able to regroup the terms of the equation of motion into its present form by defining the cumulant bracket <'f'^(xt) n(x"t) ^ (x't)> E a a C <^'"^(xt) (x't)> a a a a Our equation of motion is equivalent to an infinite set of equations of motion for observable physical quantities with classical equivalents. This follows from the fact that the Wigner distribution function^^'^^^ fp^(Rt) = ; d? e" '^^'^'^ <

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-75has properties analogous to the classical phase space distribution f(r,£;t). In particular it can be used to construct physical observables such as the single particle density distribution p(Rt) = I f (Rt) po the particle current density J(Rt) = m"^ ^ P f (Rt) ^ pa pa and the kinetic stress tensor r(Rt) = m"M P P f (Rt) pa pa ^ By expanding the equation of motion for the off diagonal density distribution about its diagonal, that is in powers of r = x' x, the coefficients of the first few powers of r yield the beginning of a hierarchy of one particle equations. First we have the usual continuity equation 1^ p(Rt) + VJ (Rt) = 0 second the equation of motion for the current density m 1^ J(Rt) = V^(Rt) p(Rt) VV^(Rt) J dx VV(R x) ^ all the complicated effects of the Pauli and Coulomb hole surrounding each electron (in the presence of the external field). In lieu of an exact evaluation of this term or main aim is to extract a local field correction. Noting that ^ = n(Rt) n(xt) {g(R, x; t) 1}

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-76where g(r,x;t) is the non-equilibrium pair correlation function, the simplest approximation is to replace this function by its equilibrium value g(r x), thus accounting for the local depletion of charge density but neglecting the dynamics of the hole. This approximation replaces the Hartree potential by a local effective potential given by 2 Vg^^(R) = V^(Ko)) ^ G(R) p(Ku) wi th G(R) = i ; [S(K q) 1] (5.2) Relating Tr^p(k,w) to thas eaaectave potentaal an the same manner as a free particle calculation, namely as P P. f (p K/2) f (p + K/2) pa to p.^/m + ie then the continuity and current density equations together yield xCKco) = X-(Kco)/l ^ (1 -G(K)) x^CKco) 0 where Xo(k,w) is the Lindhart polarizabi 1 i tyl53 of an ideal Fermi gas f^(p K/2) f (p + K/2) XqCKco) = I : 5 pa CO p^/m + ie Although G(k) contains the unknown structure factor S(k) of the electron fluid, this can be determined by requiring consistency with the structure factor obtained through the use of the fluctuation dissipation theorem. Of use numerically is the little known direct relation between G(q) and the pair distribution function g(r)^^'*

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-7700 G(0)= 1 0 J dr g(r) j^CQD 0 and by using orthogonality of the spherical bessel functions 2r 2 0 g(r) = 1 ; dO 0 G(0) j^(Qr) 0 These results are extremely useful because they enable one to generate a G(q) directly from the pair correlation function and to test G(q) by constructing the corresponding g(r). In the STLS scheme both exchange and correlation corrections to the Hartree field are automatically taken into account through its self-consistency. Moreover it is straightforward to recover the Hubbard^^^ result by substituting for the structure factor in eq. 5.2 its Hartree-Fock value. The Random Phase Approximation, which is the dynamic extension of the Debye-Huckel theory, can be viewed in this scheme as the approximation that g(r,x;t) = 1 and leads to the result G(k) = 0. At zero temperature a further constraint can be used to check the validity of any approximate microscopic theory. The compressibility "sum rule" states that the long wavelength limit of the static response should agree with the result obtained for the compressibility from differentiation of the ground state energy. (This and other ground state properties of the quantal electron gas are covered in Ichimaru.^ Both the Hubbard and STLS equations violate this G(K) =

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-78constraint. This has led to modifications of the Hubbard correction factor'"-'" G(K) = -ir^ X r + aKp and of the STLS scheme by Vashista and Singwi^l G(K) = (1 + a |-) {^ ; ^ ^ [S(K Q; p) 1]} (5.2) where the additional free parameter "a" is to be varied such that compressibility sum rule is nearly satisfied. However it has been 79 shown that if the correction factor G(k,w) is static, it is impossible to satisfy the compressibility sum rule and the third frequency sum rule simultaneously (the f sum rule being denoted as the first moment) The Relaxation Function As an alternative approach to calculating the correction factor G(k,w), which unlike the methods covered so far easily generalizes to finite temperatures, we shall consider approximations to the Kubo function provided by the Mori continued fraction method. In Chapter I the Kubo function was identified from the difference of the dynamic and static density response functions: X(KZ) = x(K) + ipz C(KZ) In this section we shall demonstrate that this follows from the definition of the Kubo function as the equilibrium averaged density auto-correlation function

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-79, > >^H -XH C^j^Cr r' t t') E ^ ; dX 0 where < > stands for the thermal expectation average. It is easily shown that the Kubo function vanishes at large times; spatial Fourier transformability is insured by the subtracted term . The intimate relationship of the Kubo function with the density correlation function can by exposed by writing the Kubo function in the form C(r ?'. t t') = where we defined the Kubo transform of an operator as J E 1 ; dX e^" J e"^" 0 By taking matrix elements diagonal in the Hamiltonian of the Kubo transformed operator, it is easily seen that the Kubo function reduces in the high temperature limit to the density correlation function. (The temperature = 0 limit however is ill-defined). Other notations abound in the literature; in lieu of the Kubo transform we shall employ the semicolon bracket^"^^ P XH -XH E ^ ;^ dX Tr {p^^ e A e B} Using time translation invariance and the cyclic property of the trace one finds that the Kubo function has the property ^ ft "^^^ = p ^"^"^

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-80The time Fourier transform of this equation c(r r'. CO) = I X" ^'^ ~ yields as a consequence that the static response is related to the time = 0 limit of the Kubo function ^(K) ^ J d ^„ moi ^ J. 3C(a,) = pC(t = 0) As a further consequence we see that, unlike the intimately related density correlation function, the frequency moments of the Kubo function are those of the response function x" without encumbering factors of coth(bhw/2). Taking the Laplace transform of the time derivative equation yields the relation referred to in Chapter I X(KZ) = x(K) + ipz C(KZ) (5.3) alternatively this can be viewed as simply the integration by parts of the Laplace transform of the Kubo function (note the tilda) 00 C(rZ) = ; dte^^^ C(rt) In Z > o 0 P d(o_ C((o') 1 p dco x"(r(o) Tn 7 n J 2Tr. CO' Z 3 J iri to(co Z) ^ Z 0 The last equality follows from the integral representations of the complex and static response in terms of the response function. Based on analytic properties the Laplace transform of the Kubo function can always be represented as^'^

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-81with D(z) an analytic function for Imaginary (z) not zero. It is the function D(z) that we will investigate further and which we shall see can be expressed in the form of a continued fraction that may be truncated at some appropriate point of approximation. Scalar Products of Operators We start by considering in detail the time evolution of physical observables. In quantum mechanics operators satisfy the equation of motion ^ A(t) = j:^ [A(t). H] E iLA(t) which defines a linear Liouville "super-operator," that is an operator on operators. If we write this out in the form of matrix elements, taken for a complete set $. of quantum states, the first half of the above equation reads f? AMM(t) = |t (A„ (t) H H„ a (t)) ot MN Tn mv vn mv vn (summation on repeated indices implied) or If one now treats the pairs mn or uv as one index each, then the A|jy(t) can be stacked to form a column vector, while the Liouville super-operator within the braces of the above equation can be put into the form of a matrix. The point to all of this is that the usual conception of quantum mechanics as a Hilbert space of say N dimensions

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-82can also be easily envisaged as a super-Hi Ibert space of N dimensions, where the dynamics occur through a linear matrix operator. In this super space we can define a bracket such that (FIG) E ^ ; X = is zero for nonzero wavevector.) In addition this scalar product satisfies the defining properties of a Hi Ibert space: (FIG)* = (GIF) (FIF) > 0 (aF + bGIH) = a(FlH) + b(GlH) a.b complex constants i.e. linear in bra (antilinear in ket) Furthermore it has the remarkable property

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'1 -83(FIG) = (G^'IF"') This formalism affords us the flexibility of manipulations borrowed from familiar quantum mechanics. For example the auto-correlation function C^^(t) can be written as aa Cj^(t) = = = = The Hermiticity of the Liouvillian follows from the time translational invariance of the brackets. (Henceforth we will drop the argument zero, understanding A = A(t = 0).) Our aim is to find an exact description of the evolution of the auto-correlation function C,^(t), in particular for the case where A aa represents the density operator. In achieving this a geometric interpretation of the problem proves to be very helpful. In the above equation we can interpret the operator exp/-itL/ as rotating the vector |A>, and except for a normalization constant, the correlation function C(t) is the component of this rotated vector parallel to the original direction. It is therefore suggestive to introduce a projection operator P = |A> "^
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-84C(Z) =
Z LQ Z LQ "-^ Z L L LP ^ A> Here we have used a formal operator identity to reexpress the inverse of z LQ. This allows us to manipulate the Laplace transform as follows. The first term of the above equation yields = 1 = 1 C(t = 0) as can be easily demonstrated by expanding in powers of 1/z and using the fact that Q|A> = 0. As to the second term we insert the definition of the projection operator — I A > = I A > T^r-hrr < A Z L ~ I A > = I A > C~^t = 0) (Z) Z L multiply our whole equation for the Laplace transform C(z) by z and rearrange: C(Z) = 3. -1 Z C'(t = 0) PC (t = 0) Noting that pC(t = 0) = x we finally arrive at the form of eq. 5.4; additionally we now have explicit knowledge of the unknown function D(z). This function is reminiscent of the "self energy" in quantum mechanics It can be simplified further because 1 2 LQ I = = i i

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-85Since the second term vanishes as z tends to infinity, the first term, which we have separated off, is the self-energy in the infinite frequency limit. In this sense it bears a correspondence to the 1 RQ Hartree-Fock part of the self energy in Green's function theory. The second term is referred to in non-equilibrium statistical mechanics as the memory function (it is nonzero for small z/large time) The fundamental result of this section is that this memory function is itself of the structure of a time auto-correlation function, namely of A. There are two differences however: first, only the part QIA> which is orthogonal to IA> enters in this correlation, and second, the dynamical operator QLQ is not the full Liouville operator but has projected from it the part which determines the intrinsic fluctuations of the variable A. In other words, if the dynamics of the property A is of interest and we call the many other degrees of freedom the bath, then the part OIA> determines the coupling of A to the bath, and QLQ describes the internal dynamics of the bath which feeds back to impose its behavior on A itself. As the memory function is a form of auto-correlation function itself, we can follow the same procedure as above by projecting out the time evolution of QIA> perpendicular to its original direction. This procedure can be repeated ad infinitum to obtain a continued fraction representation of the memory function. He should also note that an extension of the method is to generalize it to the multivariable case, where the dynamic property of interest is not a single fluctuating property of the system but a set of independent observables A^, A2, ... A^. When calculating the

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-86density-density response it is customary to include the longitudinal component of the current density as well as the number density. Such an extension will be considered later. Microscopic Theory We will now use the methods of the above section to microscopically derive the dynamic response. Let us turn our attention therefore to two functions, the off diagonal dynamic response with momentum variables £ and £' : 00 ^P' (dZ)=-l;^dt e<|l^ CnpQ(t). np.Q]> and the relaxation function 00 App,(QZ) E ;^ dt e"^* where the off diagonal single particle density operator is (For simplicity of notation we will omit explicit consideration of spin.) Of course the actual dynamic response we are interested in is given by and the off diagonal relaxation and response functions are related through a simple integration by parts, in the same manner as their diagonal counterparts (see eq. 5.3)

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-87PP" Xpp.(OZ)} (5.5) According to the continued fraction approach presented in the preceding section this relaxation function may be expressed in the form App,(OZ) (P) 1 P") (P" I X(0) I P') (5.6) Z io) + ? (Z) where we have defined operators with matrix elements given by (p I i(Q) I P') =EXpp.(Q) (p I ito I p') E (p" I x"^Q) I p') (5.7) (5.8) (p I ni) p') = 1 (P" I x-''(0) I P') p" Z QLQ P ^ We see that the off diagonal momentum variables serve the utilitarian purpose of matrix indices. As we saw in the previous section the operator ^{z) represents the back effects of the bath on the dynamics of n ; in what amounts to neglecting collisions between particles, we will assume that 'i'(2) = 0 (our first approximation). This allows us to use the identity _J 1 r = 2 I ^ Iw + 1/ Z iu Z iw and so combine eqs. 5.5-5.7 to yield 1 p") (p" I io) X I p') (5.9) Z lu Let us consider first the operator x(q) or equivalently the function Xpp,(q). This function contains more information than that contained in the static density response function x(q) supplied

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-88for example, from the integral equations of Chapter II. This is easily seen by the relation N I I, Xpp.(O) = I = x(0) (5.10) P P That is the off diagonal information is integrated out. Thus to obtain an appropriate form for the off diagonal response x (q) we must rely on its more fundamental definition in terms of a temperature Green's function^^^ whose Dyson's equation^^^ is approximated by assuming the vertex function "F" is dependent on the wavevector q only. As we will show this is correct in the classical limit and is approximately true in general for fermions.^^^ This (second) approximation reduces x (q) to the form ^pp'^^^ = ^p^^^ ^pp' I^^^Q^ %'^Q^ ^5.11) where .Ret ^ ^Ref 00 do(0) = WZ de -TT am — and pret. V ] p ~ 0 e Ep + |i Ep(e) is the retarded single body Greens function, e the kinetic P energy of a free particle and Z (b) the self-energy. The function d^(q) is easily evaluated for two cases. If the self-energy z (e) is independent of e we have d (Q) = 1 P + 0/2 P Q/2 P P + Q/2 ""p 0/2

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-89where -p -p • -p c 0 v and f(e) = 1/[1 + el^^" is the Fermi distribution (note that a grand canonical ensemble is necessary for this portion of the evaluation only). In the other case, namely the classical limit, d (q) reduces to the normalized Maxwel 1-Boltzmann momentum distribution function (j)|^j^(p): lim 1 dp(Q) = ^g(p) and so eq. 5.11 agrees in the classical limit with 1 1 N iOr. N iOr = •t'MB^P) (P P') + MB^P^ ^S^Q) 1> *MB^P'^ where <->^ denotes the classical average over the canonical ensemble. This verifies the classical limit of the vertex function as being dependent on the wavevector q only and identifies it as S(q)-1. In general the vertex function can be identified by requiring agreement with eq. 5.10. We obtain where we have introduced a reference static density response by defining XqCQ) = I I d (Q) P

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-90In the classical limit x^Cq) = 1 and x(q) = S(q) so we recover the classical result eq. 5.12. In the quantal limit when we neglect self-energy effects, x^Cq) is simply the static limit of the Lindhart o or RPA polarizabi lity. Next let us consider the matrix elements of the operator W First we note that the above results for the operator x(q) allow us to write also we note that we can exactly evaluate <^o' "p'o" = U pp.o/2> ^pp' = V^^ W ^'-^^^ in terms of namely the momentum distribution of particles in the interacting system. We have introduced the function b (q) for notational conveni ence. Inserting these results into eq. 5.8 yields the result (P I ico I p') = i flpQ 5pp. n^ CqJQ) Ibp(O) (5.15) Here we have defined an effective frequency "omega" %Q bp(Q)/dp(0) which is motivated by the fact that in the limit of zero self-energy, where Pp = f(^p)

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-91we may write % = i ^^p+Q/2 ^?-Q/2> = ?, P • We have also introduced a quantal direct correlation function in agreement with the results of Chapter II. We note again that in the appropriate limit this function reduces to the classical direct correlation function. It now follows from eq. 5.15 that (P I {zi^}-^ I p') = ) pq bp(Q) _J npCQ^(Q) ^ ^ N > ^z.iLpQ^ ^1 n^QQ,(Q) x,(Oz)> (5.17) where the reference dynamic response defined as 1 is consistent with the reference static response defined earlier. In general x^Cq) is a complicated functional of the self energy. But if we make the assumption (third approximation) that p =
= f(e) ^p p' p ^ p^ then this reference response reduces to the ideal fermion or Lindhart dynamic response function. We will see later on that although the pair distribution function changes readily from its ideal fermion form as interparticle interactions are "turned on" the momentum distribution is in fact an extremely insensitive function, and so this assumption is in fact adequate.

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-92By using the results eqs. 5.17, 5.8, 5.14 and summing equation over the variables a few easy manipulations yield the result X.(Qco) We note that the corresponding classical result, where the effective potential has been replaced by the direct correlation function, has 1 CO 1 "30 been independently derived elsewhere via a variational criterion to find approximate eigenfunctions and eigenvalues of the classical Liouville operator. The above equation is the main result of this chapter. Through it and the fluctuation dissipation theorem we are able to obtain the pair distribution function from the corresponding analogue function provided by the integral equations of Chapter III, and in a manner where all approximations are systematically accounted for.

PAGE 98

CHAPTER VI NUMERICALLY EVALUATING S(q) This section details the procedures for calculating the pair distribution function g(r) utilizing the results for the "analogue" pair distribution function n(r|v)/n^ output from the quantal integral equations discussed in Chapter II. We employ the fluctuation dissipation theorem 00 S(Q) ^ J. ^ diMi 2 = CO + ie) (6.1) together with the approximation derived in the last chapter x.(Qz) x(Qz) = ^ (6.2) where the quantum mechanical direct correlation function can be obtained from the analogue pair correlation function directly in spatial Fourier transform space as Cq^(O) = C^"(0)/Xq(0) It must be remembered that it is the analogue pair correlation function that one obtains from the classical form of the Ornstein-Zernicke relation in the QHNC system of equations. Note also that we have employed the notation of Chapter III: the density response function -93-

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-94is defined with an additional factor of -1 /beta*densi ty not found in 7Q Chapter II or commonplace in the literature. As defined here the ideal density response is dimensionless (Oz) ^ r H0/2) ~ 0/2) ^0^"^^^ p J (2Tr)2 1^^^ ^^^^^ + Q/2) 3e(K Q/2)] ^^'"^^ g denotes the spin multiplicity, rho the density, f the Fermi momentum distribution and E(q) the kinetic energy of wavevector q. For arbitrary complex frequencies "z" the ideal response has both a real and imaginary part. Using the identity lem J = P V i^S(x) e-'O X + ie we see that along the real frequency axis there is a discontinuity; approaching the real frequency axis from above (below), we have Xq(Q, z = 0) ie) = Xq(Q(o) IXqCQo) Under the approximation we have derived for our local field correction factor (eq. 5.16, 5.18), we see that the interacting response shares this property as well by virtue of the fact that C qm is purely real. This allows a simpler numerical evaluation of the fluctuation dissipation theorem as follows:^^^ The integral over the entire real frequency axis involving the imaginary piece of the complex response can be rewritten as a complex variable contour integration c(0) r d(f>z) 2 Bx(Oz) 2. ^ ^-3.z 2i

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-95where the contour c, is given by Fig. 6.1. Imz Rez = CO Figure 6.1 Integration contour in complex plane. By closing the loops in the upper and lower half of the complex plane with an infinite radius arc (over which the integrand contributes nothing) we can then proceed to deform the contour to that of Fig. 6.2, Imz Rez (0 Figure 6.2 Integration contour in complex plane. This loop encloses the poles at 2Trmi 3f. z = =^ m = 0, + 1 + 2,

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-96arising from the exponential in the denominator, from which we evaluate the contour integral by using the Cauchy Residue theorem. We obtain 00 S(Q) = I x(Q. fxo = ^) This formulation is numerically advantageous for two reasons. Firstly the relation^^ V(Q) tanh(^) S(Qto) = V(Q) Im x(Q,z = + ie) = ^"^^9 e(Q.Z Jco + ie) illustrates the phenomena of plasmon resonance; that is, for real frequencies and small wavevectors q the dynamic density correlation function S(q,w) behaves like a delta function in frequency due to the vanishing of the imaginary part of the dielectric function epsilon. The summation formula avoids those complications arising from numerical quadratures over delta peaked integrands. Secondly along the imaginary frequency axis the ideal response is purely real and an even function of (purely imaginary) frequency. For finite temperatures this allows us to write 00 S(Q) = t{Px(Qo)} + 2t I (5x(Q, 2TrmTi) m=l where tau = 1/beta. In the zero temperature (tau = 0) limit the product "3x" remains finite, and the sum is a function sampled at infinitesimally closely spaced points. Using Gregory's formula^^^ this summation can be written in terms of a definite integral plus correction terms

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-97(2x) I 3x(Q.2TrTi + iJ(2TrT)) = 00 = z J 3x(Q,iv) dv + ^ px(Q.2irxi) + 0(t^) Zk-z ^ There are examples^^'^ where the tau = 0 limit (integrating through a continuous sum) of this asymptotic expansion does not equal the result 1 i J &x(Q, iv) dv 0 due to cancellations involving the lower limit of the quadrature and the first correction term. However later numerical results will validate the assumption that (under the local field approximation we derived) the correct zero temperature form is 1 S(0) = I 3x(0iv) dv The Lindhard Function Numerical evaluation of the expressions that we have just derived for S(q) depend critically on the evaluation of the free response along the imaginary frequency axis for arbitrary temperatures and densities. At this point let us investigate further some of the mathematical properties of this function. For convenience we introduce a frequency dependent parameter nu by the expression .2 where

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-98, .2 1/2 ^ = ^mkt ^ is the thermal wavelength and w" is the magnitude of the purely imaginary frequency. Note that nu has dimensions of a wavevector squared. Upon carrying out the angular integrations in eq. 6.3, we obtain 3X (Ov) = l(^) ^ f ^ f(K) ^ in {Vi^^ioi-^^} The zero temperature limit of the right hand side is an analytic function obtained by integrating by parts 1 /2rn. 3 4ir Q .^^ .. dfv ^ 'KP (2ir)^ ^0 The remaining quadrature over the Lindhard function u, 2K + 4K^0^ 0^ rVL^IoLLEKQl!, 0 4 "to o o/ ^ 40^ + [Q^ 2K0]'^ ^ {tan-l tan-' 2^-^} (6.5, is trivial at zero temperature where the derivative of the Fermi distribution is simply a delta function in wavevector k at the Fermi surface. To evaluate the static (nu = 0) free response the integration by parts keeps the integrand finite; note however there is a logarithmic singularity in its slope at q = 2k. This makes the numerical evaluation at finite temperatures non-trivial, especially for low temperatures where the temperature dependance is rather weak and comes from a narrow region around the Fermi surface. We further note that

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-99regarding summation over frequency values (which occurs in our approximation for the local field correction factor) the convergence is very slow at low temperatures and large wavevectors. For this reason several attempts to analytically evaluate the finite temperature free response have been made. Khanna and Glyde^^^ and Gouedard and Deutsch^^ have presented an exact evaluation along the real frequency axis by summing the residues of the poles of the Fermi distribution. Their analysis may be extended to arbitrary complex frequencies; however the series is slow to converge and the zero temperature limit requires the summation over a continuum of infinitesimal ly spaced poles. It is interesting to note how the logarithmic singularity in the slope of the Lindhard (zero frequency free response) function^^^ 2Kp 4Kp Q + 2Kp ^ "Q~ ^" ^qT-2K^^ dissolves when finite temperature effects are included: 2Kpa 4K^Y 0^ + (0 + 2Kpa)^ ^ ~n~ + y In -^-^ ^ + 0(t ) AQ"" AK^b^ + (0 2Kpa)'^ where T = KT/cp Y = p/sp a = ^^Y + 2^Y^ + ir^T^ h 1 1,2 22 Wasserman et al.^^'' presented an evaluation (again along the real frequency axis) based on a combination of the Sommerfeld

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-100expansion and a Laplace transform representation of the Fermi distribution 168 These results can be generalized to arbitrary complex frequencies by extending the simpler approach of Kanhere et 22 al We write 3x(0V) = l(^) ? 1 1 LGi}^ + 0) G(7r 0)] 2\2' p 2/ 20 where G(0) = G(Q) = J dK Kf(K) In 0 0 + 2K 0 2K Following Kanhere et al we obtain G(y) = (z^ y^/4) In V + 2z y 2z 24.2 0^ 6z' 2z + y' 2 2 t F (^^ V ) in terms of the dimensionl ess variables y = q/k^, t = l/beta*e^ and 2 z = beta*u/e|r. Here we have introduced F^(x) = ; dO In 0 0 + X 0 X eQ. 1 a real function of a complex variable (note the absolute value braces.) By expressing x in terms of its real and imaginary pieces, integrating by parts and expanding the resulting logarithm we find that this integral has an exact representation in terms of exponential integral functions of complex arguments 00 N+1 F (x) = Real { I ^ e"^^ E,(NZ) e"^^* F,(NZ*)} N=l ,169 The approach of Dharma-Wardana is to note that the Fermi function can be written as an integral over a dummy chemical potential variable

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-101f(e, 11. T) = ; 0(v e) A(ni, v, T) dv 0 where theta is the step function and A(vi, V. T) = {2T[1 + cosh(>i v)/T]}"^ is the temperature weight function. Inserting this representation of the Fermi function into the integral definition of the free response and exchanging the order of integration results in a quadrature over the dummy chemical potential of the temperature weight function and the analytic zero temperature free response evaluated at the dummy chemical potential. Although a systematic comparison of the various methods was not undertaken, with the machine word size and speed of the Cray machine available it was found that a numerical evaluation of the integration by parts formula (eqs. 6.4 6.5) was adequate when supplemented by certain asymptotic expansions. For the nonzero frequency case several asymptotic expansions can be constructed from the Lindhard kernal (eq. 6.5) depending on the relative magnitudes of the variables k, q, nu. The expansion we found most accurate was o^,/^w^ l/'2mv /I 1 K^ K^ ,^ Px(Ov) ~ {^ „ + ^ — r— + } (6.6) ^ K" ofK 3a-^ K for large Here the double brackets k" denote the spherical averaged moments of the Fermi distribution function. For the zero frequency case we found that for large wavevectors

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-1022 (5X(Q, 0) ~ + — 7 — + ... } whi le for smal 1 q values Px(Q, 0) = 2^^^ { — — + — — + } ^ fi K ''^ K We note that the finite temperature moments k" can be given either by a Sommerfeld expansion^ ''^ or by exact analytic expressions in terms of hypergeometric functions. Graphs of the free response function for differing wavevector, frequency and temperature at a typical metallic density characterized by r^ = 2 are presented in the following pages. The first figure set presents a plot versus wavevector q, where the solid line is for the zero frequency response and the dashed curves are for increasing frequency. We note that zero frequency is a special case with differing limiting form near the wavevector origin. Next we present the free response as a function of frequency for a few selected wavevectors. The dashed line represents the leading term in the asymptotic expansion of eq. 6.6. We note that for larger wavevectors the free response dies away in frequency very slowly and the asymptotic expansion is increasingly more representative. Finally to show the effects of finite temperatures we plot the zero frequency response versus wavevector at several temperatures keeping density constant. The feature to keep in mind is the disappearance of the logarithmic singularity in the slope. These plots are followed by graphs of the free response as a function of frequency for a typical wavevector at various temperatures. We note that the asymptotic expansion is most representative at the lower temperatures.

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-103Convergence Acceleration Because the free response decays slowly with frequency at the larger wavevectors, two procedures were found necessary to produce accurate results in computing S(q). The first procedure is to subtract off the corresponding formulas for the noninteracting distribution S^^(q). This results in S(0) S^"(0) = ; — — dv at zero temperature while at finite temperatures (KT) pC^"(Q) 3x^(Q0) fixAQO) ^^o^^^m^ S(Q) S^"(0) = ^ + (2 KT) I -^-r^^ 1 PC^"(Q) n,= l pC^"^Q> PXq(QO) '^^o^^^m^ where v^^ represents the temperature spaced frequency points along the imaginary axis. The noninteracting distribution can be accurately computed separately and the difference now decays in frequency as the free response squared We next note that for large q and or frequencies the denominators in these expressions quickly reach a value of unity. For this reason we can accelerate convergence by Y~ -p subtracting a reference S (q) defined at zero temperature as cref,^. pC^"(0) 1 r.^REF.^ .,2 ^ = 3Xq(Q0) ^^'^^^ while at finite temperature

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-104The reference response function in these formulas is any expression that closely approximates the actual free response and allows an exact evaluation of the summation/integration in frequency appearing in the above formulas. At zero temperature the reference response can be taken as the exact closed form expression for the free response itself. The integral over frequency of its square is a universal function of q/k.^ (k^ the Fermi wavevector). However it is not known in a closed form expression. This universal function was numerically evaluated and fitted piecewise by polynomial approximations. The large q tail was found to be accurately represented by the first two terms in an asymptotic expansion of the form a/q"2+b/q"4+. At finite temperatures we do not have a closed form expression to work with so we used the first term in the asymptotic expansion of eq. 6: A Q + ("2) ^l^^^ni z + m A where This allows the summation over the reference response function squared to be written at any temperature as I [px'^^^O n )]2 = ^ ^(Z) m=l f,^

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-105where (x) = I 2^2 2 m=l (x'^ + m^r is a universal function. This function has an exact summation. It can be obtained from the partial fraction expansions of cothz and cothz (use the fact that cschz = icsc(iz)). We have (cschz) = "2 + 2 { — 2 2 — 2 2~2 "*' Z (Z+ir) (Z+4ir) 1 1 72 2 72 y, 2 2 = 3 + 2{-2 + ._2 2,2 ^ ••'> Z (Z+ir) (Z+4ir) 00 and (cschz)^ + ^ cothz = 21^ I — = — ^ 5-r ^ m=-oo [Z^ + (mrr An analytic expansion about the origin can be written in terms of the Riemann zeta function of integer argument zeta^ (x) = + (5 1) 2x^(5. 1) + 3x^(^„ 1) -.... Its asymptotic form can be obtained by using Gregory's formula^ to express the sum as an integral plus correction terms. By itself it leads to a bad representation, but by expanding all the correction terms in powers of x and regrouping one obtains a very accurate asymptotic expansion Ax^ 2x^ x^ We note in passing that using the approximate form of eq. 7 for the reference response at zero temperature results in

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-10600 which clearly indicates the utility of subtracting off the reference S(q) breaks down for small wavevectors.

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-107teinperature I mdhard function zero freg hu0. IBryB hu0.25rya hw B.Saryd q !l/bohrl Figure 6.3 One over temperature times the dimensionless free response function [1/ryd] is plotted as a function of wavevector q [1/bohr] in the limit of zero temperature and density of = 2. The solid line represents zero frequency; the dashed lines represent (imaginary) frequencies of 1.0, 0.5 and 0.25, 0.10, in rydbergs.

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-1 OStemperature lindhard function q-1.00 q-1.50 '0. 1. 2. 3. 4. 'a. 1. 2. 3. 4. hw Irydl Figure 6.4 One over temperature times the dimensionless free response function [1/ryd] is plotted as a function of (imaginary) frequency [ryd] in the limit of zero temperature and density of = 2. The solid line represents values for wavevectors of 1.0, 1.5, 2.0, and 2.5 [1/bohr] while the dashed line represents an approximation based on an asymptotic expansion.

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-109teiYiperature Mndhard function zero freq rs2.00 Figure 6.5 One over temperature times the dimension! ess free response function [1/ryd] is plotted as a function of wavevector [1/bohr] at zero frequency and density rc = 2. The plots show the variation with temperatures of 0.0, 0.25, 0.50 and 1.0 in units of Fermi energy.

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Figure 6.6 One over temperature times the dimensionl ess free response function [1/ryd] is plotted as a function of (imaginary) frequency [ryd] at a wavevector q = 1.0 [1/bohr] and density = 2. The plots show the variation with temperatures of 0.0, 0.5, 1.00 and 2.00 in units of Fermi energy. The dashed line represents an asymptotic approximation which is independent of temperature.

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CHAPTER VII THERMODYNAMIC PROPERTIES In this chapter we will present results for g(r) based on the QHNC QOZ equations and the approximation for the local field correction factor derived in the last chapter. These will be compared with the results obtained under the binary Slater Sum approximation and those of Pokrant. In classical statistical mechanics knowledge of the pair distribution function g(r) alone is sufficient for the calculation of most desired thermodynamic quantities, ^^'^^^ for example the internal energy per particle e = I kT + I ; dr g(r) V(r) the pressure ^ = 1 ; df g(r) V(r) or the compressibility = 1 + p ; dr [g(r) 1] = 1 p J dr C(r) 3p For the one component plasma g(r) must be replaced by g(r) 1 in all formulas whenever g(r) multiplies the interaction potential v(r). This is because the OCP is actually a two component system in which -111-

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-112the second component is taken to be completely structureless. It has a state-dependent potential energy uCr^ =1 1 N p J dr !^ The actual potential v(r) entering the thermodynamic formulas is that of a Coulomb potential minus its zero wavevector spatial Fourier component due to the neutralizing background, which is equivalent under spatial integrals to h(r) = g(r) 1 times a pure coulombic form. Furthermore the compressibility must be calculated in terms of the pair correlation function in which c(r) must be replaced by c(r) + bv(r) to insure finiteness. This can be derived from a microscopic field fluctuation calculation.^^ The normal derivation of the compressibility in terms of the fluctuation in number density of a grand canonical ensemble is invalidated by virtue of the fact that the potential energy in the grand partition function is state-dependent. ^^^"^^^ The quantal extension of the mechanical equations can be obtained by scaling methods similar to those used to derive the classical pressure and internal energy equations^^"^^ but that now require the off-diagonal density matrix. For example PVol = ^ lim_^ ; dr p^ p(f | 7. ) 1 p2 j ^-g^^^ ^(r) where p(hat) is the quantum momentum operator. On the other hand the compressibility equation has a purely statistical derivation that should remain equally valid in quantum mechanics. As the pair correlation function asymptotically reduces to the effective

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-113175 potential the criterion of finiteness illustrates that the analogue pair correlation of the QHNC equation should not be used to calculate the compressibility. Rather the proper quantal extension of the pair correlation function would be C(r) = F-^{1/Xq(0) F {C^"(r)}} ~ ^ in agreement with the development presented in Chapter II. The deviation from the classical formulas becomes apparent if we write the mechanical state equations in the compact form E = Ejp,,L '^^'^ ^ and from the quantum mechanical virial theorem64 |(PVOL PVol^^^^"-) = AK X < H >^ Here the brackets subscripted by lambda denote the thermal average over a hamiltonian at intermediate coupling H = T + XV; H is the virial operator (equal to minus one half V for coulomb interactions) and AK = . X 0 is the excess kinetic energy. The excess kinetic energy is a measure of the deviation of the momentum distribution of particles from the pure Fermi-Dirac distribution. In principle it can be obtained from the pair distribution function computed at different coupling strengths as f = f J%X (3 1^ ^) J d? g(f 3. X) vCr) this expression follows from combining the Maxwell relation

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-114with the quantum statistical definition of the free energy g-|5F(X) ^ ^ g-3(T + \V) and is equivalent to X F(X) = F(X = 0) + ; dX ^ 0 ^ In classical statistical mechanics the functional dependence of the pair distribution function upon the temperature and coupling strength enters through the single (OCP) parameter r = pe^/a and so the excess kinetic energy vanishes identically. In this way the excess kinetic energy also serves as a measure of the quantum effects on the pair distribution function. It is important to note that in the QHNC theory the momentum distribution is not directly obtainable— in principle we must integrate over the coupling constant in order to obtain thermodynamic properties. It is fortuitous that in the classical HNC-OZ system of equations the integration over the coupling ratio can be done analytically and the result is expressed solely in terms of fully-coupled distribution functions. This cannot be carried over into the quantal case because the effective potential depends on temperature and coupling in a nontrivial manner and the QHNC-QOZ equations are for the analogue pair distribution; the coupling constant integration must be performed over the physical pair distribution.

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-115At zero temperature the algorithm for the excess kinetic energy is considerably simpler. It is equivalent to an integration over different density values at full coupling: £ ^ 2^ ^ 1 J. s ^ V(rJ + .916} dr^ [Ryd] r s r 0 ^ This formula is easily enough derived from the relations T g?E(r^) y-'rJr rlE (r^) s s s which can be obtained by combining the definition E = T + V and the virial theorem 2 T + V = r ^ s dr, s (the right hand side is just the virial 3*pressure*volume) while noting dvSE ^> In addition at zero temperature we have two more constraints available to us. Since the minimum value of T is the noninteracting Fermi limit we have _3_ rx 2.21 ^ ^ 9 (r^E) > 0 Ferrel^^^ has a more restrictive condition ^ r V < 0 ar^ s In principle the procedure is to compute accurately values of the interaction energy per particle

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-116Pot E ^ p J dr ^ [g(r) 1] at varying densities and temperatures, following either the zero or finite temperature algorithms for calculating the excess kinetic energy as a means of obtaining thermodynamic quantities. However, there is an often overlooked drawback. The coupling constant algorithm for computing the energy of an interacting system can be 1 78 shown to be rather unreliable when it is used with approximate wavefunctions or pair correlation functions. Thus, among other practical considerations such as machine time, an alternative theory for approximately calculating the interacting momentum distribution might be preferable. In the next section we will compare results for the interaction energy per particle obtained from Pokrant,^^ the binary Slater Sum, and QHNC methods. Values for the excess kinetic energy obtained under the BSS or Pokrant methods will be compared to establish the magnitude of their contribution to thermodynamic quantities and their amenability to approximation. In either the BSS approximation of the the theory of Pokrant the thermodynamics are derived from the classical formulas involving a state-dependent configuration potential energy Thermodynamics with Slater Type Effective Potentials e |3U^(r'\ (i, p)

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-117This is just the diagonal Slater sum computed from a set of properly symmetrized plane wave basis functions (labeled by the 3N dimensional wavevectors k") where the "A" is an operator anti symmetrizing the position coordinates of the spatial function to its right. (We employed the commutivity of the anti symmetrizer with the hamiltonian and its idempotency property.) The expression for the free energy follows as .Tre-". (x3Nk!)-' ;d-re"^"T while the Maxwell relation for the internal energy yields, under the assumption that the effective potential is pairwise additive, is ^ = + f ; dr g(r) ^ [3U^(r. (J. p)] To properly compute this integral we should now split the pairwise additive interaction potential terms as The first piece, or direct part, arises from the non-commuti vi ty of the kinetic and potential energy operators in the exponentiated Hamiltonian sandwiched between simple product-ordered plane wave states (arising from the unity operator in a permutation expansion of the anti symmetrizer operator). Because the potential energy operator for the OCP is just a coulombic minus its zero wavevector component of the spatial Fourier transform, the direct part of the effective potential is minus this component also. (This property is invariant under the multiple commutations with the momentum operator.) As such the direct part has a 1/r tail (the contribution of the classical

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-118limit) and the integral over the product of g(r) and the direct potential (minus a Fourier component) should be replaced by g(r) 1 times the real space form of the direct potential (i.e. including all Fourier components). The second piece, or exchange part, arises from the exponentiated Hamiltonian sandwiched between simple product plane wave states ordered in index k" but with the spatial coordinated r" permuted out of order. This results in a function of the r" that must have a zero wavevector spatial Fourier component, regardless of the form of the potential, as is obviously the case when the potential is 1 79 absent. As such g(r) should not be replaced by g(r) 1. The BSS results were computed from the formula M-fjd^ {[g(r)-l]|pU^.g(r)|3U^} where the direct effective potential came from a calculation of two antiparallel spin electrons while the exchange effective potential arose from parallel spin electrons. A similar correction should be made to the results of Pokrant; however, the resolution of his effective potential into direct and exchange portions is tacitly more complicated. We note in passing that the same correction appears in the pressure formula ^ = 1 t ; df {[g(r) n r 1^ 3Up g(r) pu^} Interaction Energy Results Table 7.1 lists the interaction energy per particle obtained by Pokrant, the BSS effective potential using the above refinement, and

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-119from the QHNC local field correction factor. For comparison the same quantity is also calculated using the analogue pair distribution function obtained directly from the QHNC equations; that is, upon neglecting recoil and exchange effects. That the effective potential of Pokrant reduces to the BSS effective potential is reflected in the high temperature behaviors. At zero temperature the results of Pokrant are in fair agreement with those of Ceperley and Alder, as AO IS easily verified using the parametrization. r ZJA ^916 .2846 ^-i E = p ^r: [ryd] (for r > 1) r^^ 's 1+1 .0529 vr^ + .3334 r^ ^ A short comparison of quantum Monte Carlo results with those of Pokrant is presented in Table 7.2. For several r values the s agreement for the interaction energy per particle is within one half percent. Thus the deviation of the BSS data from that of Pokrant illustrates the breakdown of the binary approximation at low temperatures. The data calculated from the analogue pair distribution function lie below the results of Pokrant at high temperatures but at low temperatures approach those of the BSS approximation. This clearly illustrates the error in approximating g(r) by g^"(r) as is pervasively done in plasma theory. However the physical pair distribution function obtained from the local field correction factor calculation does not yield an improvement; the results are uniformly higher than Pokrant and there is an unusual dip around the temperature origin. These results, although disappointing, should hardly come as a surprise because, up to this point, we have concerned ourselves solely

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-120with the most rudimentary of the quantal extensions of the classical fluid equations. The QHNC equation does not incorporate the most important physical effect of many particle systems interacting with long range forces, to wit the screening of the effective potential. The results of more refined quantal integral equations will be presented in the next two chapters in lieu of laboring the results of the QHNC equation here. Instead we will consider more closely the thermal properties of the excess kinetic energy. Kinetic Energy Results A comparison of BSS and Pokrant values for the excess kinetic energy is presented in Table 7.3. Although the kinetic energy provides a more stringent test than the correlation energy (by virtue of the variational principle for the ground state energy), it is nevertheless surprising that the zero temperature results of Pokrant differ from those of the quantum Monte Carlo by fifteen per cent or more. (See Table 7.2.) The total energies indeed are in fair agreement (one or two percent) but this is because the excess k.e. is a fraction of the magnitude of the interaction energies and because the errors in the two quantities are self-correcting. This shows also that using the virial theorem to produce pressure values yields inaccurate results. (One must use a method which takes advantage of the relative accuracy of the total energy— i.e. integrating energy over inverse temperature to obtain the Helmholtz free energy and then differentiating with respect to volume.) For much the same reasons the internal energies of the BSS approximation reproduce the results

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-121of Pokrant down to surprisingly low temperatures; however, individually the interaction and excess kinetic energies differ considerably. In spite of the vagaries, features of the temperature-dependence of Pokrants excess kinetic energy (which are the only finite temperature results available) should be accurate. The feature that the excess kinetic energy is positive at zero temperature is elementarily explained because 2 AK J dp p^[f(p) f (p)] 0 and fgCp) is a step function. Statistics constrains that any change in the distribution must be accomplished by an increased occupation at a higher momentum value at the expense of a decreased population at a lower value. A typical plot of the interacting momentum distribution is presented in Fig. 7.1. The existence of a discontinuity at the Fermi surface (Migdal's theorem^^^"^^^) is modeled by the small r^ RPA theory of Daniel and Vosko,^^^ extended by Gel dart et al.,^^^ and at large r^ by Belyakov.^^ We note that this discontinuity disappears only when the momentum distribution is constructed out of orthogonal ized Wigner orbitals at the high r 191-192 ^ Mmi t. At finite temperatures the momentum distribution can in principle be calculated from the one-particle Green's function^^^ given the self-energy (related to the two-particle Green's function). At zero temperature the self-energy is closely approximated by the Hartree-Fock self energy plus a self-energy due to single particle

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-122coupling to density fluctuations, the main contribution of which arises from coupling to the plasmon field J ^'^"^^^ To model temperature effects we start with the finite temperature evaluation of the Hartree-Fock self-energy following the steps of 1 59 Kadanoff and Baym. We include spin degrees of freedom in the equation of motion for the single-particle Green's function 2 2 6(1 T) i I ; drg V^^(r^ g}^\i 2 1' 2+) o Here the second term on the right hand side is equivalent to the self-energy as defined in terms of the two-particle Green's function, the arrows denote spin labels (a parallel equation exists for the spin-down Green's function) and the shorthand argument "1" stands for the space-time argument of particle one, etc. The Hartree-Fock approximation consists of decoupling the equations of motion by assuming g{^^ = G^d 1') G^(2 2'') as particles with opposite spin are distinguishable and g{^^ = G^d 1') G^(2 2"') G^d 2"") G^(2 1') with a similar equation for the down-spins. Because the form of the potential is independent of spins we write

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-123^2 ^2 6(1-T) t2 = fi = 6(r^ J V(r^ r3)[-i G^(3 3) iG^(3 3)] V(r^ r^) [iG^Cl 2)] ^2 = ^1 Now the density distribution is given by the diagonal limit of the retarded Green's function n^(r t) = i G^(rt, rt) and is a constant for uniform systems. As a result the spatial Fourier transform of the Green's function satisfies the ideal Green's function differential equation [ifi It ^T^P^^ ^T^P' ^'^ = where the energy eigenvalues _2 ^T^P^ = L ^ ^T^P^ (7.1) are given in terms of the HF self-energy -.1 E^(p) = (n^ + n^) J dr3 ^ir^) ; V(p p') n^(p) (7.2) V(p) is the momentum (spatial Fourier) transform of the interaction potential. It is the only surviving term due to the presence of a neutralizing charred background for the OCP. The differential equation for the ideal Green's function implies that the momentum distribution is of the form

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-1243(E.(p) p) n^(p) = 1/{1 + e ^ } (7.3) where the chemical potential is fixed by the total particle number densi ty. The above eqs. 7.1-7.3 are to be solved sel f-consi stently for the finite temperature momentum distribution. However, near zero temperature the convergence can be quite slow. This is evidenced by the fact that at zero temperature f(p) has a step discontinuity which is hard to achieve through a Picard iteration, while the eigenvalues reduce to the Hartree-Fock ground state energy which diverge at small 28 r^. A near zero temperature approximation can be derived analytically^^^ but it is interesting to note that the same finite temperature system of equations can be obtained from a free energy variational principle: the value F = E TS is a minimum at constant temperature under the constraint that the total number of particles is fixed. Aside from the kinetic energy contribution to the internal energy, the entropy can also be expressed in terms of the quantum mechanical one-body Wigner distribution''^^ as — TS = kT J {f In f + (1 f) In (1 f)} (2Tr)'' (which for spatially uniform systems is the momentum distribution). The potential energy depends on the two-body Wigner distribution functions, which however can be split into anti symmetrized products of one-body Wigner functions, incorporating exchange effects, and irreducible two-body operators. For the OCP the exchange piece can be calculated in terms of the momentum distribution in the closed form

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-125(2Tr) dK 4Tre 2 K + 0 K Q 2 ) f(k) f(0) In ( Retaining only this contribution and varying the free energy with respect to the momentum distribution yields the system of equations derived by the Green's function method. However in this approach we may take an approximate form for the momentum distribution, dependent on a number of adjustable parameters which we vary to minimize 1 96-1 97 F. This allows for a fast accurate solution near absolute zero for the excess kinetic energy. The result of the temperature-dependent HF approximation is presented in Table 7.4. The data differ markedly from that obtained by the method of Pokrant. The excess kinetic energy properly tends to vanish in the classical limit under both approximations, but the magnitude and location of the extreme values differ. Under the HF approximation the excess kinetic energy vanishes identically at absolute zero as we are left with a step function momentum distribution. A positive excess kinetic energy as exhibited by Pokrant would require the self-energy beyond the HF theory (as in Fig. 7.1), That in turn would involve a coupling constant integration of density fluctuations. This underscores again the need for accurate values of the pair distribution function, and so we will next concentrate on improving the quantal integral equation results by incorporating screening effects.

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-126Table 7.1 A comparision of interaction energies per particle in rydbergs for a density of rg = 2 for varying temperatures tau = kt/Fermi energy. -Potential /N Tau Pokrant BSS QHNC g(r) QHNC gan^r) .1 .600 .633 .609 .632 .2 .597 .631 .614 .629 .5 .580 .607 .616 .602 .8 .557 .576 .600 .570 1 .0 .541 .556 .584 .550 1 .33 .515 .527 .557 .520 1 .60 .496 .506 .536 .499 471 H / 1 t / O oU/ .4/ 1 .455 .461 .488 .454 3.20 .412 .417 .440 .410 5.33 .346 .349 .364 .342 8.00 .296 .297 .308 .291 10.0 .270 .271 .279 .265 12.5 .245 .246 .253 .241 16.67 .216 .216 .221 .211 20.0 .198 .199 .203 .195 25.0 .179 .179 .182 .175

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-127Table 7.2 A comparison of zero temperature quantum Monte Carlo results to those of Pokrant for the interaction energy per particle and excess kinetic energy for density values computed by Pokrant. Ts Quantum Monte Carlo Pokrant -Pot dKE -Pot dKE 1.0 1.112 .07629 1.117 .089 2.0 .5982 .04985 .602 .059 3.39 .3739 .03379 .376 .041

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-128Table 7.3 A comparison of excess kinetic energies in rydbergs from Pokrant and the BSS methods for a density of r^ = 2 for varying temperatures tau = kt/Fermi energy. The ideal fermion kinetic energy per Nkt minus the Maxwell Boltzman value of 3/2 is also given. Tau Pokrant BSS Ideal Fermion dKE dKE KE/Nkt -3/2 .1 .060 .223 4.742831e 0 .2 .058 .212 1.957581e 0 .5 .046 .198 5.432847e-l .8 .031 .103 2.734860e-l 1.0 .023 .0811 1 .967396e-l 1.33 .013 .0570 1 .2887646-1 1 .60 .007 .0428 9.788451 e-2 2.0 .002 .0286 7.017671e-2 2.29 -.001 .0220 5.732930e-2 3.20 -.005 .00967 3.476111e-2 5.33 -.008 -.00100 1.619181e-2 8.0 -.008 -.00509 8.810031e-3 10.0 -.007 -.00473 6.305060e-3 12.5 -.007 -.00523 4.512104e-3 16.67 -.006 -.00454 2.930145e-3 20.0 -.005 -.00432 2.229816e-3 25.0 -.004 -.00407 1 .595610e-3

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-129Table 7.4 A comparison of excess kinetic. energies (dKE = KE KE^deal) and chemical potentials (du = u u^^eal) in rydbergs from a Hartree-Fock decoupling of the one particle Green's Function and that of Pokrant for a density of = 0.5 at varying temperature, tau = kt/Fermi energy. Also included are values of the kinetic energy and chemical potential of the ideal fermion gas in rydbergs. Tau Pokrant dKE HF dKE HF du Ideal KE Ideal u .025 -.010 -2.442 8.862 14.725 .05 .126 -.034 -2.439 8.930 14.702 .1 .124 -.112 -2.425 9.197 14.610 .2 .103 -.312 -2.362 10.188 14.211 .3 -.476 -2.246 11.598 13.474 .4 -.575 -2.098 13.252 12.381 .5 -.020 -.620 -1 .941 15.052 10.948 .6 -.632 -1.789 16.945 9.206 .7 -.624 -1.650 18.901 7.184 .8 -.095 -.605 -1 .524 20.903 4.908 1.0 -.119 -.556 -1.313 24.998 -0.316 1 .33 -.136 -.464 -1 .058 31 .917 -10.585 1 .60 -.139 -.418 -0.909 37.665 -20.247 2.00 -.137 -.353 -0.750 46.266 -36.264 2.29 -.132 -.317 -0.664 52.541 -48.953 2.67 -.125 -.278 -0.577 60.797 -66.753 3.20 -.116 -.237 -0.489 72.356 -93.489 4.00 -.103 -.194 -0.395 89.863 -137.363

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-130Figure 7.1 Average momentum distribution in the electron gas as calculated from the random phase approximation, for = 2, 3, 4, and 5 at zero temperature.

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CHAPTER VIII RESULTS OF THE EXTENDED QHNC EQUATIONS In this chapter we will consider results from two quantal extensions of classical integral equations. These were derived in Chapter III and both incorporate the effects of screening the longranged coulomb interactions. Like the QHNC method, in the appropriate limit, they also reduce to the classical HNC equations. One system, which we shall call the Quantal Hartree system, is based on an effective potential of the form ^^eff^^^ e = n*(r I VM)/n^ H 0 n*(r|v^) is the density distribution of ideal fermions in the presence of an external Hartree screened potential V^(r) = V(r) + p J d?^ V( I ? r' I ) {g^"(?') 1} v(r) is the bare interparticle potential in the interacting system (coulombic). The analogue pair distribution function n(r|v)/n 0 which appears in the convolution of the Hartree potential is to be solved self-consistently with the equation -131-

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-132together with the classical Ornstein-Zernike equation in terms of analogue distribution functions. Notice that, unlike the classical OCP potential or the QHNC effective potential, that which appears here is short-ranged. Instead the long-ranged nature of the coulombic many-body problem appears in the analogue bridge function B*"(r) = F"^{PXq(0) F{V^(r) V(r)}} and implicitly in the OZ relation. A misnomer, the analogue bridge function does not correspond to a quanta! generalization of classical bridge diagrams. However it appears in various integral equations in the same manner as the classical bridge function would, hence its name. For example, there is the Hartree screened Quantal Percus-Yevic equation g^"(r) = e~^^'^^^'^ {1 Y^"(r) + B^"(r)} We note in passing that both the effective potential and analogue bridge function remain finite at zero temperature. The other system of equations under consideration we shall call the quantal Zwanzig system. It is based on an effective potential -3V^..(r) e = n*(r I V,)/n^ z 0 wherein the externally imposed Zwanzig potential the bare interaction under the convolution is replaced with the analogue pair correlation functionl39.162 V^(r) = V(r) + p ; dr' C^"( I r ?') h^"(r')

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-133The analogue pair distribution function is solved self-consistently with the OZ relation and the closure an Xff^^^ g^"(r) = e This system of equations has been studied extensively at zero 1 98 temperature by Chihara. Because the effective potential is short-ranged the coulombic influence on the solutions enters only implicitly through the OZ equation. The long-range nature of the analogue pair correlation and nodal functions must be inferred from the fact that the Zwanzig closure can be written in the same form as the Hartree closure, except that the Zwanzig screened potential appears in the definition of the analogue bridge function, while ignoring the fact that the nodal and bridge functions cancel identically. Method of Solution Both the Hartree and Zwanzig equations are self-consistent field calculations for the distribution of interacting electrons in the presence of an elementary test charge; both were solved using a Picard iteration by mixing input and output to accelerate convergence. However, there are subtle differences which preclude concurrently reviewing their methods of solution. We will concentrate on the Hartree solution first. The Hartree system converged conditionally on the quality of the initial guess for the analogue pair distribution. Occasionally the solution at one temperature/density was an unacceptable seed for another. To construct an initial guess ab initio we imposed the

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-134constraint that the electrons redistribute themselves so as to completely shield the test charge. It is found that many-body effects screen the bare coulomb potential into an exponential damped form^^ e? r/d r ^ where the generalized screening length (8.1) \ = 4Tr e^ ^ n(vi) d2 dp reduces to the Debye screening length in the classical limit ^ = (4Tr p e^/kT)^^^ and the Thomas-Fermi screening length 1 •J 1/6 2 _L ,3^x 4 me in the completely degenerate limit. The ansatz was made that h*"(r) could be modeled by the distribution of non-interacting fermions in the Thomas-Fermi approximation of a modified Yukawa potential: a(r) is a trial function of the form l-exp(-const*r) This form results in divergences at small r of the potential which mimic the multiple commutators of the coulombic 1/r pole with the momentum operator found in Wigner-Kirkwood^^ corrections to the T-F approximation. The constant was chosen to ensure total screening, that is ph^"(0 = 0) = J dr ph^"(r) = 1

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-135A plot of an initial guess for h^"(r) constructed in this manner (for r^ = 2 and kt = 1 Fermi energy) is presented as an illustration in Fig. 8.1 It is surprising that this ansatz is satisfactory, because asymptotically it is of the form (dashed curve of Fig. 8.1) h^\r) ~ -4 e4irpd'^ (even at zero temperature) whereas physically h^"(r) exhibits Friedel oscillations^"^ at zero temperature of the form Ze 2g cos (2KpX) ^ ~ ir 2 ^ ^ ? 9 (8.2) (4 + V x-^ 2Kj-^ d^P^ These arise from the sharp Fermi surface; it its not possible to construct a smooth function out of a restricted set of wavevectors q < k^. This explanation does not depend on the strength of the external perturbation, so the results should be implied also by calculations based on the dielectric function method in linear response. There it is often claimed that the oscillations are a consequence of the singularity of the dielectric function at q = 2k^, but (except in the limit of very large radius) this is not strictly true because the slightest blurring of the Fermi surface due to thermal excitation is enough to destroy the singularity but does not remove the 73 200 oscillations. (At large radius R damping of the oscillations requires a blurring over a range of k of order pi/2R-see eq. 8.2.) In fact thermal effects result in an exponential damping of the oscillations.

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-136That the starting ansatz remains good down to zero temperature reflects the unimportance of the asymptotic form in r-space relative to a general agreement in momentum space with the actual solution. The iterative cycle to solve for h*"(r) is described as follows. Given an initial guess for h^"(r) the Hartree potential is computed in Fourier space as using the OZ relation an ^COUL^Q^ 1 p C (Q) the coulomb potential is then expressed as the sum of longand shortranged pieces (V^ = + ^^^^ possess analytic transforms p2 2 Vg(r) = erfc (ar) Vg(r) = ^ erf (ar) In the decomposition alpha is an arbitrary parameter to be conveniently chosen in a manner described later. This allows the Hartree potential to be similarly decomposed into a sum of the shortranged and "extra" potentials (V^ = where V^(Q) = [pC*"(0) V^(0) + Vg(0)] / [1 pC^"(Q)] This extra potential is transformed numerically into real space for use in solving the Schroedinger equation in the Kohn-Sham system of equations. Fiqures 8.2 and 8.3 are plots of "extra" potentials at a temperature of kt = one Fermi energy at densities r^ = 1.0 and 2.00. These were computed using the solution values of h^"(r). It can generally be said that higher r^ values decay more slowly; at

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-137the same temperature all have comparatively the same maximum value which decreases with higher temperature. The Hartree potential at constant density for varying temperatures is plotted in Fig. 8.4; the only major difference being that higher r^ values decay more slowly. Plots of the total Hartree potential at varying densities for the same temperature are presented in Fig. 8.5. The interesting feature is that the plots are all of the exact same shape—however the x-axis was scaled by the temperaturedependent generalized screening length. Again it must be emphasized that this is a feature of the solution h^"(r). The effective potential entering into the system of integral equations is related to the logarithm of the density distribution of non-interacting fermions in the presence of the Hartree potential. Unlike the QHNC effective potential this has no analytic solution, and we must resort to a numerical solution of the Kohn-Sham system of equations. Computationally this is a very expensive process, for although n*(r|v^) soon reaches its TF asymptotic limit, the continuum wavefunctions upon which it is built must be calculated to much larger radial values in order to ensure proper normalization. An alternative approach based on the work of March et JO, 203 ^5 receiving renewed interest, deserves comment. The expession n*(r|v) can be exactly given as the energy integral over the Fermi distribution of a function that satisfies a third order differential equation in r. Current applications satisfy the boundary conditions in terms of free field solutions via a perturbative solution in orders of the potential. The first order solution is the (zero temperature) Mott equation^^^

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-138n ^ (2Kp I r s I) I r s |2 where -2 T^(x) = X ^[SINx xCOSx] This can be shown to give negative values near the radial origin. For such reasons the (albeit time-consuming) Kohn-Sham equations were preferred for their overall robustness. (For example the algorithms for solving the Schroedinger equation are well known.) Figures 8.6-8.10 present solutions of the KS equations at zero temperature for varying densities. The results merge with the asymptotic TF approximation at smaller radii with decreasing r^ values, or, from Figs. 8.11-8.13, with increasing temperature (as expected). The corresponding effective potentials to Figs. 8.6-8.13 are presented in Figs. 8.14 and 8.15. As only the ratio of the effective potential over kt remains finite at zero temperature, an analogue to a classical potential was defined by multiplying the ratio by 1 rydberg. We see that with decreasing r^ the effective potential weakens and becomes more short-ranged. The effective potential is only weakly dependent on temperature. The analogue bridge function, which arises as a consequence of screening, is also obtained from the initial h^"(r) via the Hartree screened potential. Its calculation is performed by writing B"(0) = 3x^(00) [V^(Q) V(,Q^L.^O)] = |3Xq(0. 0) [V^(0) Vg(0)] = {3Xq(Q, 0) Vj^(Q) (3Xq(Q. 0) 3x^(00)) Vg(Q)} Px(OO) Vg(Q)

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-139This splits the bridge function into shortand long-ranged pieces. The short-ranged bridge function is numerically transformed into two parts; the first part involving the static Lindhard function with the "extra" potential, the second part the difference of the static Lindhard function from its zero wavevector value times the long-ranged potential. In the course of this decomposition each individual part is rearranged into a machine finite form. A typical output for the calculation of the components of the short-ranged bridge function is illustrated by Fig. 8.16 for r^ = 2 at zero temperature. The corresponding short-ranged bridge function is shown in Fig. 8.17. The long-range piece of the bridge function is identical to the long-range tail of the direct correlation function. This is easily demonstrated as follows. The Fourier transform of the starting ansatz for h*"(r) yields for small wavevectors ph (Q) = r 1 + (Qd)*^ which implies through the Ornstein-Zernike relation pC^"(Q) = — (Qd)^ and so at large radii C^"(r) ~ e^ Px(OO) The arbitrary constant appearing in our long-range reference function is chosen such that a short-ranged direct correlation function pC^"(Q) = p C^"(0) + p3Xq(00) Vg(0) (8.3)

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-140is, besides being finite and short-ranged, relatively small in magnitude in order to reduce numerical errors that occur in numerical Fourier transformations. We found that taking the constant equal to the screening length (see eq. 8.1) divided by the square root of two adequately met this purpose. As an illustration the (solution) shortranged pair correlation function for r^ = 2 at zero temperature is plotted in Fig. 8.18. The value at the origin corresponds to approximately one tenth the value of the long-range contribution. With the effective potential and analogue bridge functions specified, the OZ and closure equations are then solved iteratively for a new short-ranged direct correlation function in the manner 105 described by Ng. Then the solution for the new (analogue) shortranged direct correlation function serves as a new input for calculating a new Hartree screened potential. The cycle is repeated until the rms value of the difference of input and output short-range c^"(r) { J dr I c^UT.c^n ,2jl/2 is smaller than the rms magnitude of c^"(r) by a factor of one part in ten to the fifth power. The iteration solely in terms of short-ranged functions avoids the divergent results that plague the self-consistent solution of the KS equation, due to the long-range nature of the coulomb 206—207 121 interactions. To date several other procedures have been proposed to avoid numerical instabilities; for example Popovic et al.^^ and Zaremba et al.^^^ have used a parameterized electrostatic potential to fulfill the Friedel sum rule, while

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-141Manninen et al have made use of the Poisson equation substructed contribution of the Thomas-Fermi potential. The procedure followed 1 00 here is based on the work of Chihara. Ouantal Hartree Results The solution analogue pair correlation function obtained from the quantal system of equations was used to construct the local field correction factor in the manner developed in Chapter V. This was used to calculate the physical pair distribution function as described in Chapter VI. The results represent a significant improvement over the QHNC data reported in Chapter VII. Fig. 8.19 presents the zero temperature interaction energy per electron for six densities between r =1.0 s and 3.39. Also plotted are the interaction energies obtained from the high-density expansion of the correlation energy. ^^^"^^ ^ ^CORR '^^^^ ^" ^s -^^^ ^ -^^^ ''s ^" ^s and the low density expansiontl 9] .876 2.65 2.94 s s The range of densities investigated was selected to bridge the intermediate density gap between these known results. The quantum Monte Carlo results do not differ visually on the graph and a comparison of Quantal Hartree with MonteCarlo values, along with those provided by Pokrant, is presented in Table 8.1. The quantal Hartree values have a maximal deviation of 1.6 percent at the

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-142highand low-density ends, being above and below the Monte Carlo results, respectively. The three points supplied by Pokrant are uniformly one half percent above the Monte Carlo data. Figures 8.20-8.25 present the analogue and pair distribution functions corresponding to the data of Table 1. Referring to Fig. 26, we see that the pair distribution function remains positive at the radial origin— unlike the RPA or Hubbard theory. A comparison with the STLS results can be made through their g(r) values at the origin: STLS Hartree 1.0 .24 .23 2.0 .04 .18 3.0 .04 .15 The Hartree results decay smoothly with r^, and remain much larger in magnitude. Results from the quantum Monte Carlo approach or that of Pokrant are not available for comparison. The quality of the analogue/pair distribution functions can be expressed quantitatively by the degree to which each fulfills the requirement of perfect screening. For the density distribution in the presence of an external test charge ; n(r I v) n^ dr = p ; dr h*"(r) = ^^"^ ^ — V~ 1} = 1 O-o 1 pC^"(0)

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-143due to the OZ relation because the analytic long-range piece of c^"(q) diverges at the wavevector origin. Therefore any deviation from this value is a measure of the numerical accuracy of results. The same cannot be said of the physical pair distribution. In addition deviations can be attributed to the range of validity of the local field correction factor assumed. Moments for both h(r) and h^"(r) were computed in real space and are presented in Table 8.2. Except for the two largest r^ values the analogue moment was fulfilled to within hundredths of a percent, the physical to under one percent. This verifies both the accuracy of the numerical procedure and the validity of the LFCF. The degradation of the two higher r^ values can be attributed to coarser grid sizes precipitated by slowlydecaying effective potentials. This phenomenon was also encountered 1 QR and explained in terms of phase shift calculations by Chihara. The variation of the interaction energy per electron with temperature is presented in Figs. 8.27-8.29 where comparisons are made with data at the densities supplied by Pokrant. The most striking feature is the hump that occurs near the temperature origin, which becomes relatively more pronounced at the higher r^ values. Because this feature 1) does not appear if one were to compute the interaction energy from h^"(r) (see Fig. 8.30), and 2) is also exhibited in the QHNC equations, its existence may lie in the approximation for the LFCF assumed. This will be examined in the next chapter. The variation of the pair distribution function with temperature is illustrated by Figs. 8.31-8.34 for a density of r^ = 2.0. At the low temperature side of the hump, the physical pair distribution

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-144function lies above the analogue distribution (see also Fig. 8.22 which is the zero temperature plot), at the peak of the hump they are nearly coincident, while above the hump the physical distribution lies below the analogue. Table 8.3 presents a comparison of the analogue/physical screening moments with temperature. It is seen that the analogue function fulfills the perfect screening criterion to within thousanths of a percent all the way down to zero temperature. The physical distribution retains this accuracy at the high temperature end, but suffers a comparatively large degradation at the low temperature end below the hump. These two features reinforce the idea that the hump is an unrealistic feature inherent in our present approximation of the LFCF. It is also important to note that at r = 0 the physical pair distribution function goes negative at intermediate temperatures before resuming positive values with the advent of a classical description. The onset of such a degradation occurs at smaller temperatures for higher values of r^. Thus, one of the criteria used to judge the success of zero temperature theories, namely the positivity of the pair distribution function, may be an illusury feature with regard to a more complete temperature dependent theory. The Zwanziq Equation In the Zwanzig system of equations we subtract out a long-range reference function that (unlike the quantal Hartree system) does not possess an analytic transform (compare with eq. 8.3).

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-145pC^"(0) = pC^"(0) + p(JXq(0. ) Vg(Q) Furthermore, instead of iterating in terms of c^"(r), we define new functions (of dimension rydbergs) that correspond to the analogue pair and nodal functions '^^^ (5Xq(Q, 0) ^^^^ PXq(Q, 0) along with corresponding definitions involving the short-ranged analogue decompositions. The iterative solution starts with an initial guess for the short-ranged c(bar) in wavevector space, from which we obtain the short-ranged quasi nodal function by the OZ relation C(0) Px^(0. 0) C (Q) p V (Q) t CO) = ^ s^^^ 1 3Xq(Q, 0) C(0) When the screened Zwanzig potential is resolved into short-ranged and excess parts, the excess potential is given explicitly in terms of the short-ranged quasi nodal function V^CQ) = V^(Q) r^(Q)/p This is numerically transformed to real space to solve the Kohn-Sham equations. This results directly in a new h^"(r), without having to involve a classical fluid integral equation routine. The iterative loop is closed by taking the Fourier transform of the new h*"(r) and the old short-ranged quasi nodal function to obtain a new guess for the short-ranged c(bar) from the definition

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-146The iteration continues until the rms difference of the input and output short-ranged c(bar) in real space is within an acceptable tol erance. Like the quantal Hartree system the rate of convergence depends on the quality of the first guess. Unlike the Hartree system it was found that in lieu of solutions from other temperature and density points the best starting input was a null function. One subtle difference in the Hartree and Zwanzig systems arises from the fact that distribution functions are of a necessity transformed from real to wavevector space and back again in the iterative process. This occurs less frequently in the Zwanzig system—no classical hnc set of equations must be solved in the overall iterative loop— and so one possible source of numerical error is eliminated. Fourier transforms were accomplished using a Fast Fourier Transform algorithm. Due to discretization and wrap-around effects the asymptotic form of an FFT transform differs from an analytic transform, and taking inverse FFT transforms of functions dependent on analytic transforms may yield slight errors. Such an operation occurs only once in each system (involving a distribution function and the analytic Lindhard function). However calculating the pair distribution function through the LFCF requires c(bar) in wavevector space as input; this is supplied by the Zwanzig system directly, the Hartree system requires another mixed transform.

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-147It is amazing that in spite of the overt difference of the two systems of equations or the subtle differences mentioned above, the results obtained by independent calculation of the Zwanzig system agreed with those obtained by the Hartree system within three significant figures for both the analogue and physical pair distribution functions at all temperatures and densities presented. This can only be ascribed to the fact that both systems of equations were derived from the Percus generating functional method on the basis of a common quantal bootstrap ansatz— that is, the same nonlocal density approximation to the exchange correlational functional in density functional theory (refer to Chapter III.) That is in the regions of temperature and density considered, the main effects were pointedly quantal and arose in the same physical yet mathematically inequivalent ways. Blending In classical statistical mechanics the analysis of the density expansion of g(r) has led to two "exact" equations which, however, involve three unknowns: h(r) = c(r) + p J dr' h( I r r' I C(r') g(r) = exp{pV(r) + h(r) C(r) + B(r)} The first of these is the Ornstein-Zernike relation defining the direct correlation function c(r) in terms of h(r) = g(r) 1. In the second, a closure equation, V(r) is the pair potential and B(r) is the sum of "bridge" or "elementary" graphs in the diagrammatic analysis of

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-148two-point functions. Though the same analysis leads to a formal relationship between B(r) and g(r), it involves an infinite sum of highly-connected diagrams and so cannot be utilized in practice. This apparent absence of a simple functional connecting of B(r) to g(r) prevents this scheme from being fully closed. The various integral equations derived in Chapter III were an attempt to derive the majority of the effects of the bridge function by starting from physically sensible generating functionals, the remainder of the effects being lost in the truncation of the generating functional expansion. In the practice of classical statistical mechanics all that one can do to obtain a more realistic g(r) is to take two (approximate) integral equations that bracket known (computer simulation) results, and infer from their form what the true bridge function must look like. The only guidance is provided by the fact that if we had the true g(r) then the same equation of state should result from both the virial (pressure) and compressibility equations. (Based on scaling arguments one can show that the virial and internal energy equations are equivalent for simple power law potentials.) A number of authors ^' '^'^'-^ has proposed imposing this requirement directly to the generation of the integral equation. Stell^^^ has shown that the results of Rowlinson^^^ and Lado^^^ can be obtained by a functional expansion method based on a linear combination of the PY and HNC approximations. Hutchinson and 102 Conkie studied a functional of the form 5 W[n] = ^ {(n(? I U) e"'^^^''^) n^^}

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-149where the parameter S is varied until consistency is found. Note that S = 1 yields the PY functional, and by taking the limit as S tends to zero, the functional for the HNC approximation is recovered. A 215 related approach is to identify thermodynamical ly consistent truncations of distribution function hierarchies. Thermodynamic consistency is used in the Reference-HNC pic approach which approximates the B(r) with the bridge function of a short-range (reference) potential. Following Rosenfeld and 59 60 Ashcroft, who proposed and extensively documented the view that B(r) should be essentially the same function for all potentials, the reference potential is viewed as an adjustable function. In practice it is chosen to be a hard sphere system with variable radius, which is 172 217 optimized by requiring that it minimize the free energy, a condition that greatly increases the internal consistency. Note that the true bridge function is a short-ranged function even for systems 103 218 interacting via the long-ranged coulomb potential. ^'^'-'" In the quantal regime the approach we investigated relied on two independently generated integral equations which are assumed to bracket the true solution and which can be spanned by the inclusion of some adjustable function. For classical systems with repulsive potentials the PY and HNC equations have this desired property and the spanning has been 2 1 9 accomplished with a mixing of the two equations. The requirement of thermodynamic self-consistency has provided accurate reproduction 220 of Monte Carlo results. Recent work has considered a blending of the PY and HNC equations in the form

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-150g(r) = e {1 + } where f(r) varies from zero at the origin (where the above equation reduces to that of the PY) to a value of unity at radial infinity (where the above equation becomes the HNC approximation). The virtue of this algorithm is that a simple form for the blending function, such as f(r) = 1 e~ ^ (8.4) suffices and the adjustable parameter used to achieve thermodynamic consistency was found to be approximately independent of the coupling parameter. Even more remarkable was that, for simple inverse power potentials, a simple inverse relationship was found between the adjustable parameter and the power of the potential. Quantal extensions of the classical HNC and PY equations have a qualitatively similar structure. The classical equations can be smoothly obtained from their quantal extensions by letting 1) x*(q) —a wide function of wavevector q~be broadened to a constant value of unity, and 2) letting hbar tend to zero in the fluctuation dissipation theorem. Because of this similarity blending was attempted with the hybridization of the Hartree screened quantal HNC and PY equations: -|5V" .(r) JY^"(r) + B^"(r)] f(r) g(r) = e { W ^ ^} It was found that varying the mixing parameter, alpha, appearing in eq. 8.4 from a value of 1000. (Essentially pure quantal Hartree) down to 1.0 resulted in changes in the interaction energy of less than

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-151one part in one thousand for the density and temperature regime presented. This result is not unexpected in light of the Zwanzig system of equation results.

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-152Figure 8.1 A typical initial guess that was generated ab initio for rc 1 .0 and tau = 1 .0

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Figure 8.2 Plots of the Hartree screened potential minus an analytical short-ranged reference potential. The result is short-ranged and is presented at tau = 1.0 for densities of rg = 1.0.

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-154vh(r) minus vs(r) [ryd] Figure 8.3 Plots of the Hartree screened potential minus an analytical short-ranged reference potential. The result is short-ranged and is presented at tau = 1.0 for densities of rg = 3.39.

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-155Figure 8.4 Plots of radius times the screened Hartree potential at constant density rg = 1.0 and temperature tau = 0, 0.4, 0.8.

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-156Figure 8.5 Plots of radius times the screened Hartree potential at constant temperature tau = 1.0 and rg = 3.39, 2.0 and 1.0.

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Figure 8.6 Plots of the density distribution of noninteracting fermions in the presence of an external Hartree potential of zero temperature for density = 3.39.

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-158Figure 8.7 Plots of the density distribution of noninteracting fermions in the presence of an external Hartree potential of zero temperature for density = 2.5.

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-159density: n(r|v)/nO I ' I ' I ' J is in b.hr -rs=2.00 -tiu=0.00 _|l>9h=ta9jBptatic ->rifin= 7.2SS653e-02 Figure 8.8 Plots of the density distribution of noninteracting fermions in the presence of an external Hartree potential of zero temperature for density rs = 2.0.

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-160density: n(r|v)/nO I ' I ' I ' J i. in bohr -r=1.50 -lu=0.00 CM m 3^ Figure 8.9 Plots of the density distribution of noninteracting fermions in the presence of an external Hartree potential of zero temperature for density = 1 .5.

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-161Figure 8.10 Plots of the density distribution of noninteracting fermions in the presence of an external Hartree potential of zero temperature for density = 1 .0.

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-162dens i ty : n( r | v )/nO 1-02 • • CM CO rr LO Figure 8.11 Plots of the density distribution of non-interacting fermions in the presence of an external Hartree potential at constant density rc = 2.0 and temperature tau = 0.2.

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-163density: n(r|v)/nO ~^ \ ' \ ' \ i i" bohr -rs=2.00 -tta=0.40 jriiin= 7.923085e-02 • • (SI cn zr Figure 8.12 Plots of the density distribution of non-interacting fermions in the presence of an external Hartree potential at constant density re = 2.0 and temperature tau = 0.4.

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-164Figure 8.13 Plots of the density distribution of non-interacting fermions in the presence of an external Hartree potential at constant density = 2.0 and temperature tau = 0.8.

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-165Figure 8.14 Plots of the analogue potential employed in the fluid integral equations at zero temperature for densities rc = 3.39, 2.5, 2.0, 1.5, 1.0, respectively. Only the potential divided by kt remains finite at zero temperature: For comparison with classical potentials we multiplied by 1 ryd.

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-166analogue potential(r) [ryd] 4.0 3.5 3.0 n 1 i \ T 1 1 r lefined for b(ref) 'HTtt]= l.OOOOOOe+OO r is in bohr -rs=2.00 .ttu=0.20 ttu=0.40 ttu=O.SO tiu=0.80 t>u=l.(IO (Nl Figure 8.15 Plots of the analogue potential employed in the fluid integral equations at constant density = 2.0 for temperatures tau = 0.2, 0.4, 0.8, respectively. Only the potential divided by kt remains finite at zero temperature. For comparison this potential is that quantity multiplied by 1 ryd.

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-167Figure 8.16 The construction of the short-ranged analogue bridge Function from the inverse Fourier transform of (top) the static Lindhard function times the Fourier transform of the excess potential and (bottom) the inverse Fourier transform of the difference of the static Lindhard function minus its wavevector origin times the reference analytic long-range potential. Illustrated for = 2.0 and zero temperature.

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Figure 8.17 The total short-ranged analogue bridge function constructed from the parts of Fig. 8.16. at = 2.0 and zero temperature.

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Figure 8.18 Illustration of a typical short-ranged analogue direct correlation function for =2.0 and zero temperature.

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-170interaction energies per particle qhnc ~> ^ \ — n 1 1 012. 3. 4. 5. rs-a/ao Figure 8.19 Plot of the zero temperature results for the interaction energy per particle obtained from the quantal Hartree equation versus density. Analytic highand low-density asymptotic forms are also included.

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-171Figure 8.20 Plot of the physical and analogue pair distribution function at zero temperature for densities rj = 1.0. The physical pdf was computed under a static LFCF.

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-172physical/analogue h(r) Figure 8.21 Plot of the physical and analogue pair distribution function at zero temperature for densities rg = 1.5. The physical pdf was computed under a static LFCF.

PAGE 178

-173Figure 8.22 Plot of the physical and analogue pair distribution function at zero temperature for densities = 2.0. The physical pdf was computed under a static LFCF,

PAGE 179

-174Figure 8.23 Plot of the physical and analogue pair distribution function at zero temperature for densities = 2.5. The physical pdf was computed under a static LFCF.

PAGE 180

-175phys i cal/anal ogue h(r) Figure 8.24 Plot of the physical and analogue pair distribution function at zero temperature for densities = 3.0. The physical pdf was computed under a static LFCF.

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-176Flgure 8.25 Plot of the physical and analogue pair distribution function at zero temperature for densities rg = 3.39. The physical pdf was computed under a static LFCF.

PAGE 182

-177Figure 8.26 Pair correlation function in the degenerate electron plasma two values of rg in the range of metallic densities according to various theories of correlations.

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-178Figure 8.27 Plots comparing the interaction energies versus temperature obtained from the Quanta! Hartree equations along with those of Pokrant, at densities rg = 1.0.

PAGE 184

-179interaction energies per particle Figure 8.28 Plots comparing the interaction energies versus temperature obtained from the Quanta! Hartree equations along with those of Pokrant, at densities = 2.0.

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-180Figure 8.29 Plots comparing the interaction energies versus temperature obtained from the Quantal Hartree equations along with those of Pokrant, at densities = 3.39.

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-181Table 8.1 A comparison of interaction energies per particle at zero temperature for varying densities. Monte-Carlo -Potenti al Hartree [ryd] %diff Pokrant %dif1 1 .0 1 .112 1 .1293 1.5 1.117 .56 1.5 .7731 .78274 1 .2 2.0 .5982 .60021 .34 .602 .64 2.5 .4905 .49028 .04 3.0 .4171 .41228 1.16 3.39 .3739 .36783 1 .63 .376 .56

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-182Figure 8.30 Plot comparing the interaction energy versus temperature as obtained from the pdf along with that which would be obtained from analogue pdf at density = 2.0. The results of Pokrant are also supplied.

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-183phys i cal/anal ogue h(r) -01 J — — I — I — 1 — I — I — I — I — I I I I I I I I I I I I I I ^ CSJ CO 3^ LO Figure 8.31 Plots of the physical and analogue pair distribution functions at constant density = 2.0 for temperature tau = 0.2. The physical pdf was computed under a static LFCF.

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-184Figure 8.32 Plots of the physical and analogue pair distribution functions at constant density = 2.0 for temperature tau 0.4. The physical pdf was computed under a static LFCF.

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-185Figure 8.33 Plots of the physical and analogue pair distribution functions at constant density rg = 2.0 for temperature tau = 0.6. The physical pdf was computed under a static LFCF.

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-186phys ical/analogue h(r) • • (XI tn =r LD Figure 8.34 Plots of the physical and analogue pair distribution functions at constant density = 2.0 for temperature tau = 0.8. The physical pdf was computed under a static LFCF.

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-187Table 8.2 Total screening moments of the analogue and physical pair distribution functions obtained from the Quantal Hartree equations at zero temperature and varying densities. r^ Analogue Physical 1.0 -1.000207 -0.994870 1.5 -1.000313 -1.009535 2.0 -1.000701 -0.985722 2.5 -0.999976 -0.992360 3.0 -0.994812 -0.966804 3.39 -0.992707 -0.964524

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-188Table 8.3 Total screening moments of the analogue and physical pair distribution functions obtained from the quantal Hartree equations at a density of rs = 2.0 at varying temperatures tau = kt/Fermi energy. Tail 1 au Ana 1 ogue Physi cal 1.0 -0.9999538 -0.9999548 0.8 -1 .000045 -1.000018 0.5 -1 .000118 -1.000069 0.4 -1 .0000151 -1 .000097 0.2 -0.9999595 -0.9982337 0.0 -1 .000699 -0.9857216

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CHAPTER IX DYNAMICS OF THE LOCAL FIELD CORRECTION FACTOR We saw in the previous chapter that the zero temperature results obtained from the Quanta! Hartree system of equations were in fair agreement with the Quantum Monte Carlo results, but at finite temperatures an anomalous hump in the interaction energies per particle was observed near the temperature origin. This feature was found to be exhibited by other systems of integral equations. It was not present if we calculated the interaction energies with the analogue pair distribution, and was accompanied by a degradation in the value of the perfect screening moment. These features indicate a breakdown of the validity of the static local field correction approximation made in Chapter V. In the present chapter we will investigate a refinement of the LFCF which incorporates dynamic effects. As our starting point we will again consider the auto correlation of density fluctuations C(Q, t) = ^ (n^Ct) I nq(o)) That is the "Kubo function" in the scalar product notation presented in Chapter V. Recalling the definition of the Liouvillian time evolution operator and the identity -189-

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-190(LFIG) = I <^ [F, G*]> (9.1) we find that the Kubo function is related in real time to the density response function as ^ C(Q, t) = zi x"(0. t) while the dynamic and static physical responses are respectively given by x(Q. t) = e(t) ^ C(Q. t) (9.2) x(0) = x(0. z = (0 + ie) = C(0, t = 0) We employ the convention that tilde denote temporal Laplace transforms. The response is defined as in Chapter I to be consistent with the literature and not in the dimensionless form of Chapter III. The dynamics of the Kubo function will be explored through the continuity equation d^ 2 ^ C(Qt) = Q'^ J(0. t) dt"^ using the memory function formalism^^^ to describe the autocorrelation of the longitudinal current J^Qt) = ^ (Lnq(t) I Lnq(o)) As an autocorrelation function J(0,t) satisfies the Mori equation ^ J(Qt) + J df M(0. t f) J(0. f) = 0 0 where the memory function M(0,t) remains unspecified and t>0. However its zero time behavior can be obtained by differentiating the Mori equation once

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-191h2 t ^ J(Ot) + M(0, t = 0) J(Q. t) + J dt' J(Ot') Jt M(Q, t t') = 0 dr 0 ^ from which E M(Q. t = 0) = ^ J(Ot) df^ / J(Q. t = 0) t=o The large time behavior of the memory function is ascertained by splitting it up into its asymptotic limit plus a piece that vanishes at large times M(t) = + M^(t) We then take the Laplace transform of the Mori equation J(t = 0) iZ J(Z) 4^+ J(Z) M^(Z) = 0 (for imaginary z>0) and consider expanding this equation in powers of z; for example letting 3(1) = J(Z = 0) + Z J^^\z) + Z^ J^^^Z) + ... The first two terms can be specified by considering the solution to the continuity equation expressed in the form 2 C(Qt) = x(Q) ; dt' J(Qt') {t t'} (9.3) 0 A sufficient condition that the Kubo function vanish at infinite times is provided by the assumption 00 00 J dt J(Qt) =0 J dt t J(Qt) = x(Q)/Q^ 0 0 Therefore, because

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-192J(Z) M^(Z) must be of least order z (only if M^(q,z) were a constant would it have a 1/z behavior near the origin), we see that the zeroth order term of the Laplace transformed Mori equation yields = i J(t = 0) / J^^^Z = 0) Furthermore the autocorrelation of the longitudinal current can be evaluated exactly at t = 0 through eq. 9.1. It is essentially a manifestation of the f-sum rule: 0(Q. t = 0) = S m We are now in a position to discuss approximations for the complex response function; this function inserted into the fluctuation-dissipation theorem results in the dynamic structure factor. Now it is clear from eqs. 9.2 and 9.3 that the complex response is directly related to the Laplace transform of the longitudinal current autocorrelation as which by virture of the Laplace transform of the Mori equation becomes x(Qz) = —o 2 :: (9.4) "o 2 + "o^ ^" M2(0.z)) Here we have introduced a unit normalized memory function M2(q,t) by the definition M(Qt) = M„ + (Mq M^) M2(0t)

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-193where the scale frequencies are given by "o^ = = to^/x(0) "oo^ E = (L^ I nq)/(L I L n^) = to^/£o^ In the above equations we have employed the double brackets to denote frequency moments of the density response. We recall that by definition M2(q, t = 0) = 1 and this implies that M2(Q. z <) = i so that eq. 9.4 gives the proper value of the fourth frequency moment of the density response. The condition M2(q, t = infinity) = 0 implies that iz M2(Qz) vanishes as z tends to zero, yielding the correct static compressibility. For intermediate times we have no real knowledge of M2(q,t) and it is here that one has to make a reasonable ansatz. We make the approximation that is is given by M2^(q,t), the noninteracting fermion memory function. A dynamic local field correction factor can be obtained by inverting eq. 9.4 1 1 "co 1 X(Qz) ^^^^ a3S> ^^^^ 2F and then subtracting the corresponding equation for a system of ideal fermions

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-194x(Oz) x^(Qz) ^(Q) ^0^^^ co2 x(Q) x,(0) (-iz M2p(0z)) In the above derivation we have used the fact that M^ = = pQ^/m The remaining scale frequencies are given explicitly by^^ where is the ideal energy-wavevector relation, Wp is the plasma 87 221 frequency and • 2 (0^(1 P(Q)) =^S dr g(r) [1 cos Qz] ^ V(r) for neutral fluids with general non-transformable potentials, while = co^ + ^ ; dr [g(r) 1] {1 cos Qz} ^ V(?) ^ 8z for the OCP or long ranged potentials. Several comments can now be made about the dynamic local field correction factor V(k,z)^^^'^^^ appearing in the following equation x(Kz) = x.(KZ) 1 V(K2) Xq(Kz) First, V(k,z) can be written in the form V(Kz) = V^^CK) + [Vp^(K) V3^(K)] [iz M2p(Kz)]

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-195where V^^(k) is the static local field correction factor we have previously employed, equivalent to the quantum mechanical direct correlation function. Secondly, because V^^Ck) and V^y(k) are purely real, and from the known analytic properties of the dynamic ideal fermion response (temperature Lindhard function), we see that X(K, Z = CO ie) = x'(Kto) ix"(Ka)) and so the analysis carried out for the static LFCF in Chapter VI remains in force, and we can calculate the fluctuation dissipation theorem from values along the purely imaginary frequency axis. Thirdly, the dynamic LFCF merely changes the static form of V^^Ck) over to that of V^y^t^) at larger (imaginary) frequencies. The general dynamic dependence can be straightforwardly derived by making the gaussian approximation for the memory function 1 a^ t^ M2(Kt) = e ^ where in general "a" is a function of wavevector k. (Note that this satisfies the necessary temporal boundary conditions.) We find that for purely imaginary frequencies z = i nu (nu real) U iz M^iKz) = ^ ^ I e ^ erfc(— ^) V 2a ~ (1 -"2) as V < V = a V 2 as V = 0 Finally we note that the dynamically mixed LFCF

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-1964 e^i ( r) 5 depends on the excess kinetic energy and the pair distribution function itself (contained in P(q)). As this latter quantity is what we hope to obtain from the calculation, we see that this algorithm must in principle be calculated self-consistently. In practice P(q) depends only weakly on the pair distribution function itself and so we may reasonably approximate it by h^"(r). This is easily demonstrated as follows. For the OCR the volume integral for P(q) can be reduced to the one-dimensional radial integral P(0) = 2 ; dr[g(r) 1] J ^ (Qr)^ (Qr)-^ We note that the terms in the braces are related to the Bessel function of order 5/2. This is an oscillatory function which is essentially damped out by its first node, which occurs at qr = 5.763. For q much smaller than the scale of h(r) we can series expand the braces and obtain where Pot is the interaction energy per particle 1 J Pot = ^ p J dr [g(r) 1] ^ while for large q the only contribution to the integral occurs from around the radial origin, and we may pull the slow dependence of h(r) out from under the integral by series expanding it about its value at the origin. To lowest order we obtain

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-197P(0) = f If = 0) 2Mn r=o This asymptotic value is quickly reached and P(q) does not depend critically on the shape of h(r) but rather on the single numbers h(r = 0) and Pot, which are fairly well approximated by h*"(r) values. As for the dependance on the excess kinetic energy, it can be shown^^ that the successful (zero temperature) STLS theory can be derived in part by assuming that its effect is negligible. This approximation has also been made by other authors. ^^^"^^^ However, the fact that it would be the dominant contribution to V^^Cq) at large wavevector makes this approximation suspect, and we shall discuss the effect of its inclusion later. Calculating the Memory Function Defining the Lindhard function as in Chapter III, that is Px^COco) = ? ; ^ ^> Im CO > 0 we saw from the previous section that for values of w" along the imaginary frequency axis {w" M(Oco")} In units of a frequency parameter nu defined as

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-198the dimensionless function R(q,co") = a)"M(q,to") can be calculated as r 1 /fl^v (^_J^\\ ff^Li\ __]_\ ^3x^(0v) ^2m^ ,^2 ^4ni^ 2 3x(0)^ R(Qv) = 2 2~2 {2 p} ^ ^} The braces group the inverse of the static and dynamic Lindhard function with the inverse of their respective leading asymptotic terms. The function PXq 2m 2 for a density of r^ = 2.0 and at zero temperature is presented in 224 Fig, 9.1. It is non-analytic about the Fermi wavevector, at the wavevector origin it assumes the value f,^ K 2'" K-2 while at large wavevectors it approaches a constant value of 2 K^ 3 K (Here double brackets denote the spherical wavevector average over the Fermi distribution). These limiting forms are equally applicable at finite temperatures where Fig. 9.2 (r^ = 2 and tau = kt/Fermi energy = 0.2) illustrates that a slight increase in temperature modifies the cusp. The braces involving the dynamic Lindhard function exhibit similar temperature-sensitive features which make calculating R(q,w) non-trivial. Specifically, the small

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-199V = 40^ Q + V asymptotic expansion of the dynamic Lindhard function quickly diverges at finite temperatures, furthermore, after cancelling the reciprocal of the leading term what is left in the numerator of R(q,w) closely approximates the denominator. The physically meaningful quantity R(q,w) 1 has thereby suffered a quick loss of significant figures. (It must be remembered that the dynamic Lindhard function is calculated by a numerical quadrature and starts out with only a finite number of significant figures.) The quality of the numerical results for the memory function becomes increasingly questionable with temperature. At zero temperature the memory function is analytically calculable, and the salient features are illustrated in Figs. 9.2-9.3. The effects of small but finite temperatures (kt/Fermi energy = 0.2) corresponding to Figs. 9.2-9.3 are presented in Figs. 9.4-9.5. Figure 9.2 plots the dimensionl ess memory function (at = 2 and zero temperature) versus wavevector in bohr for increasing imaginary frequencies of 1 2 and 3 ryd. Higher frequencies exhibit a higher shoulder at the Fermi wavevector. Finite temperatures eliminate the shoulder (Fig. 9.4), and higher frequencies take on a gaussian apperance. The zero temperature frequency dependence is illustrated by Fig. 9.3 for increasing values of wavevector. Finite temperature effects (Fig. 9.5) slow the convergence to the asymptotic value of unity but otherwise alter no general features.

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-200Results The quantal Hartree system of equations was re-run at zero temperature for varying densities employing the dynamic local field correction factor while neglecting the excess kinetic energy contribution. A comparison of the obtained interaction energy per particle is presented with the Quantum Monte Carlo results in Table 9.2. It is seen that the results from the dynamic LFCF are uniformly one to two percent greater than the Monte Carlo data. This represents a degradation in the quality of results vis a vis that obtained from the static local field correction factor. On the other hand, the quality of the pair distribution function as measured by the criterion of perfect screening is seen in Table 9.2 to be uniformly improved. The pair distribution functions themselves are presented in Figs. 9.6 and 9.10. Overall the values of the pair distribution function are smaller at the origin than they were under the static LFCF. These values are compared along with the analogue values at the origin in Table 9.3. Curiously enough the static LFCF values are uniformly larger than the analogue values. This was the case for temperatures on the low side of the temperature "hump" discussed in Chapter VIII. Under the dynamic LFCF no such general statement can be made. Finite temperature effects on the interaction energies per particle were next investigated for those low temperatures where the memory function could still be considered accurate. It was found that the values generally followed the hump obtained from the static LFCF but with values shifted by an amount equal to that at zero temperature.

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-201The inclusion of the excess kinetic energy contribution to the dynamic LFCF produced pair distribution functions that had undesirable features near the radial origin. This is illustrated in Figs. 9.11-9.12 at zero temperature for two densities supplied with an artificially small value of 0.01 ryd for the excess kinetic energy. More realistic values based on quantum Monte Carlo data resulted in a greater divergence at the origin. From the data in Table 9.4 it is seen that, in addition to negative values of the PDF at the origin, the interaction energies per particle suffered in comparison with the Monte Carlo data with no improvement of the total screening moment. These unsatisfactory corrections comprise the main result of this chapter: namely, that a higher order approximation in the Mori memory function formalism, in such a way that the first and third frequency moment sum rule is simultaneously satisfied, is in fact not a fruitful course to follow. Rather, diagrammatic analysis at zero 1 9 temperature has shown the importance of modeling the finite lifetime of the excited electron and hole states. These lifetime effects are not taken into account when one chooses a real local field correction factor as was done here. On the other hand past attempts at incorporating such lifetime effects within the framework of the 14-15 Mon theory have led to violations of the continuity equation. One should also note that the Mori memory function formalism, which is essentially a truncation of a continued fraction expansion, can be viewed as having a basis in the belief that the high frequency expansion of the complex response might be possible up to infinite order. The following fact, however, is worthy of particular

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-202attention. An asymptotic form of Imag xCq.z) for a degenerate electron liquid is given by Click and Long^^^ as p 6 2 Imag x(Ql) = vCQ) ff^ ^ q > „ E (9.5) ir a^ (mco) It arises from two-pair excitations of second-order perturbation. When substituted into the fifth moment integral the result is quite divergent r* 5-11/2 J dtd U CO 00 (l) c (w^ is the minimum value of w for which eq. 9.5 holds with sufficient accuracy.) This divergence would appear when truncating (in a Mori continued fraction sense) at the next level beyond that employed in this chapter to derive the dynamic LFCF. A priori this is not sufficient to discount the accuracy of the dynamic LFCF derived in this chapter. With hindsight one may infer that the divergence heralds the degradation of the dynamic LFCF results over those obtained using a static LFCF obtained from the Mori method.

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-203Table 9.1 Interaction energies per electron [ryd] obtained from a pair distribution function calculated under the dynamic Local Field Correction Factor approximation from the Quantal Hartree equations. Results are for zero temperature with neglect of excess kinetic energy contributions to the LFCF. Comparison is made with Quantum Monte Carlo results. -Pot Monte Carlo Quanta! Hartree %diff 1.0 1.112 1.1358 2.1 1 .5 .7731 .79013 2.2 2.0 .5982 .60870 1 .7 2.5 .4905 .49904 1 .7 3.0 .4171 .42154 1.0

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I -204Table 9.2 Total screening moments for the pair distribution function calculated under the dynamic local field correction factor approximation from the Quantal Hartree equations. Results are for zero temperature with neglect of the excess kinetic energy contribution to the LFCF. Comparisons is made to the total screening moments obtained under the static LFCF. Dynamic Static 1 .0 -0.9955281 -0.994870 1.5 -1 .007725 -1 .00955 2.0 -0.9878318 -0.985772 2.5 -0.9933104 -0.992360 3.0 -0.9705490 -0.966804

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-205Table 9.3 Values of (minus one times) h(r) = g(r) 1 at the radial origin at zero temperature. Comparison is made with results obtained from the Quantal Hartree equations under the static and dynamic local field correction factor approximations. The values corresponding to the analogue pair distribution are also presented. Dynami c Analogue Static 1.0 .81565 .80388 .77287 1.5 .87668 .89783 .80273 2.0 .92277 .94465 .81942 2.5 .96570 .96961 .82914 3.0 1 .01480 .98359 .84664

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-206Table 9.4 Results for the dynamic local field correction factor with the inclusion of an excess kinetic energy value of 0.01 ryd at zero temperature. -Pot -screening -h(r = 0) 1.0 1.1374 1.5 .79253 2.0 .61189 2.5 .50291 .9955540 .88120 1.007454 1.06873 .9881874 1.33991 .9929164 2.38550

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-207I/asyiYiptote-l/stat i c lindhard rs2.00 tau0.20 tau0.00 q [1/bohrl Figure 9.1 Difference in the reciprocals of the static temperature Lindhard function and its leading asymptotic form.

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-208lindhard memory function rs2.00 taue.ee hw3.B0 Iryd) riw2.00 Iryd) hw-l.a0 Irydl Figure 9.2 Lindhard memory function versus wave vector at zero temperature varying imaginary frequencies.

PAGE 214

-209Figure 9.3 Lindhard memory function versus imaginary frequency at zero temperature for varying wave vectors.

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-210lindhard memory function [1/bohr] Figure 9.4 Lindhard memory function versus wave vector at finite temperature

PAGE 216

-211Figure 9.5 Lindhard memory function versus imaginary frequency at finite temperature tau = 0.2 for varying wave vectors.

PAGE 217

-212physical/analogue h(r) Figure 9.6 Plot of the zero temperature physical /analogue pair distribution functions for rg = 1.0. The physical PDF was computed under a dynamic LFCF approximation neglecting excess kinetic energy contributions.

PAGE 218

-213physical/analogue h(r) • • • s — • (Nj oo -ir' IS) Figure 9.7 Plot of the zero temperature physical /analogue pair distribution functions for = 1.5. The physical PDF was computed under a dynamic LFCF approximation neglecting excess kinetic energy contributions.

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-214phys i cal/anal ogue h(r) J — \ — I — I — I — I — \ I I I I I I I I I I I I I I I I I Figure 9.8 Plot of the zero temperature physi cal /analogue pair distribution functions for rg = 2.0. The physical PDF was computed under a dynamic LFCF approximation neglecting excess kinetic energy contributions.

PAGE 220

-215Figure 9.9 Plot of the zero temperature physi cal /analogue pair distribution functions for = 2.5. The physical PDF was computed under a dynamic LFCF approximation neglecting excess kinetic energy contributions.

PAGE 221

-216Figure 9.10 Plot of the zero temperature physi cal /analogue pair distribution functions for = 3.0. The physical PDF was computed under a dynamic LFCF approximation neglecting excess kinetic energy contributions.

PAGE 222

-217physical/analogue h(r) csi m zf in Figure 9.11 Plot of the zero temperature physi cal /analogue pair distribution functions for = 2.0. The physical PDF was computed under a dynamic LFCF approximation including an excess kinetic energy contribution of 0.01 ryd.

PAGE 223

-218Figure 9.12 Plot of the zero temperature physi cal /analogue pair distribution functions for = 1.5. The physical PDF was computed under a dynamic LFCF approximation including an excess kinetic energy contribution of 0.01 ryd.

PAGE 224

CHAPTER X CONCLUSIONS In this dissertation we have examined the non-zero temperature quantal electron gas. Except for the approach espoused by Pokrant^^ and his antecedents, few numerical results of thermodynamic properties at intermediate degeneracy have been previously published, despite the overwhelming amount of work that has been conducted on the fully degenerate or completely classical electron gas systems. Of late this has been changing: temperature-dependent effects have been published 20 7fi by Dandrea et al and Tanaka et al using a dielectric formalism of the many-body problem by employing the non-zero temperature Lindhard ideal fermion density response functions. However the effects of electron-electron interactions have been introduced through static local field corrections that (1) have employed the STLS approximation which is known to give worse results than the classical HNC equation in the non-degenerate regime, or (2) are ad hoc in nature and fitted to reproduce zero temperature and classical results. -219-

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-220This dissertation has also employed the dielectric formalism, but the electron-electron interactions are incorporated by a non-perturbati ve local field correction factor which is obtained by a quantal extension of classical fluid integral equations. The formalism asymptotically reduces to standard statistical mechanical approaches in the classical limit, while in the completely degenerate limit it reduces to a nonlocal exchange-correlation functional approximation within the Kohn-Sham (density functional) theory. The validity of this approximation is reflected in the good agreement we have obtained at zero temperature with results obtained from the 3Q_42 Green's Function Monte Carlo method. The various approaches can be contrasted on several points. First there is a fundamental difference between those theories utilizing the dielectric formalism (including that of the present dissertation) and the nonzero-temperature variational principle utilized by Pokrant. The dielectric formalism must use a coupling-constant integration to obtain values for the excess kinetic energy. This is usually done implicitly when computing the free energy. In the method of Pokrant the excess kinetic energy can be computed explicitly. Although it is not given accurately, it is a relatively unimportant contribution to the internal energy, and does not affect values of the free energy computed by integrating the internal energy over inverse temperature at constant density. When computing free energies directly, the cruder approximations in the dielectric formalism have a machine time advantage over the method presented in this dissertation. This is because an additional set of

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-221equations (the Kohn-Sham equations) appears within the self-consistency calculation. This relative speed of numerical calculation is responsible for the fact that the free energy obtained from the cruder dielectric methods have been calculated over wide regions of temperature and density. These have been presented in Pade approximate form, where the sole dependance upon the classical coupling parameter, "gamma," is modified by temperature dependent coefficients. This method has been criticized for giving inaccurate thermodynamic derivative quantities; this may be attributed to parameterizing the coupling in terms of gamma, which is clearly inappropriate. (At zero temperature gamma diverges, while it is known that the free energy depends solely on the 43 parameter r^.) A more apt description would be in terms of the parameter *3 N where p = (^ ira^)"^ ^ P (2vy 2>m which reduces to gamma in the classical limit, and is proportional to r^ in the fully degenerate limit. Then the free energy would be independent of a second parameter (needed to specify the temperature and density) in both extremes, and by inference, only loosely

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-222dependent on it in the intermediate degeneracy regime. Also this description would be more amendable to an analytic fit. A second point of contrast concerns the peaked structure of the interaction energy per particle near absolute zero. This feature is not present in the method of Pokrant. The relevant data are not directly reported in the calculations of Tanaka et al or Dandrea et al However Dandrea et al do report a feature that we found to be directly associated with the hump. Starting at absolute zero and increasing temperature, we find that g(r = 0), after initially decreasing to a minimum value (located at the temperature corresponding to the maximum of the interaction energy), then increases for more weakly coupled plasmas. Thirdly, the dielectric method presented in this dissertation admits modifications of the form of the fluid equation obtained in the classical limit. In classical statistical mechanics, a blending of fluid equations was found to be an efficient algorithm for satisfying the compressibility sum rule. Satisfying this constraint generally improved the accuracy of the thermodynamic quantities with respect to experimental values. In this research we found that the utility of such a procedure was limited to the near classical regime, but is totally absent in other dielectric formalisms. Lastly it should be noted that the method of Pokrant allows a reformulation of the electron gas into a multispecies description of spin-up and spin-down particles. It has been shown^^^ that in statistical mechanical descriptions of real plasmas (i.e. consisting of positive ions and electrons) the resolution of the constituents into spindifferentiated species, while not appreciably affecting the

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-223charge structure factors, modifies the calculated internal energies, chemical potentials, and compressibilities. The extension of the dielectric formalism presented in this dissertation to resolve spindifferentiated species remains an open avenue of research.

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REFERENCES 1. N. W. Ashcroft and D. Stroud in Solid State Physics Vol. 33 (Academic press, New York, 1978). 2. N. W. Ashcroft in The Liquid State of Matter: Fluids Simple and Complex (North-Holland, Amsterdam, 1982). 3. P. A. Egel staff, An Introduction to the Liquid State (Academic Press, New York, 1967). 4. S. Tanaka and S. Ichimaru, J. Phys. Soc. Japan 53, 2039 (1984). 5. S. Ichimaru, Rev. Mod. Phys. 54. 1017 (1982). 6. M. Baus and J. P. Hansen, Physics Reports 59, 1 (1980). 7. N. H. March and M. P. Tosi Coulomb Liquids (Academic Press, London, 1984). 8. K. S. Singwi and M. P. Tosi in Solid State Physics Vol. 36 (Academic Press, New York, 1981). 9. D. Pines and P. Nozieres, The Theory of Quantum Liquids (Benjamin, New York, 1966). 10. K. A. Dawson and N. H. March, Phys. Chem. Liq. M, 131 (1984). 11. P. Eisenberger, P. M. Platzman, K. C. Pandey, Phys. Rev. Lett. 31. 311 (1973). 12. P. M. Platzman and P. Eisenberger, Phys. Rev. Lett. 33, 152 (1974); Solid State Commun. 14, 1 (1974). 13. P. Eisenberger, P. M. Platzman, P. Schmidt, Phys. Rev. Lett. 34 18 (1975). ~' 14. S. Rahman and G. Vignale, Phys. Rev. B30, 6951 (1984). G. Mukhopadhyay, R. K. Kalia, K. S. Singwi, Phys. Rev. Lett. 34 950 (1975). ^ J G. Mukhopadhyay and A. Sjolander, Phys. Rev. BH, 3589 (1978). -224-

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BIOGRAPHICAL SKETCH Brian Gregory Wilson was born in Chicago, Illinois, on September 24, 1956. He is the son of Francis G. and Jeanne M. Wilson. He received his undergraduate instruction in physics at the Illinois Institute of Technology and was awarded his B.S. degree in 1978. He attended Stanford University under an Air Force Thermionics Emission Research Program where he received his M.S. degree in electrical engineering in 1979 and an Engineer's degree in electrical engineering in 1980. He entered the University of Florida in the fall of 1980 to pursue graduate studies in the field of physics. On passing the doctoral qualifying examination in 1981 he began research under the direction of then department chairman Professor Charles Hooper. In 1982 he married Katherine A. Nadworny, also a graduate student in physics at the University of Florida. From 1983 onward his research was conducted at the Physics Department of the Lawrence Livermore National Laboratory, where he came under the direction of Dr. Forest Rogers. On graduation he will take up a postdoctoral position at the laboratory. -235-

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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. / I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, of Doctor of Philosophy. dissertation for the degree I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Kwan Chen Professor of Astronomy

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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Neil Sullivan Professor of Physics This dissertation was submitted to the Graduate Faculty of the Department of Physics in the College of Liberal Arts and Sciences and to the Graduate School and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. May 1987 Dean, Graduate School


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