A Variational approach to the ground state energy of the electron gas

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A VARIATIONAL APPROACH TO THE

GROUND STATE ENERGY OF THE
ELECTRON GAS









By
MICHAEL SMITH BECKER














A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF TIE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY











UNIVERSITY OF FLORIDA
August, 1967











ACKNOWLEDGMENTS


The author wishes to express particular gratitude to Professor

Arthur A. Broyles for suggesting the topic for this dissertation,

and,also, for his continued assistance and encouragement during

the course of its development. Special thanks are also due to Dr.

Tucson Dunn whose help has been invaluable throughout every phase

of this work. The author also wishes to thank Dr. Earl W. Smith,

Dr. C. F. Hooper, Jr.,and Dr. Joon Lee for many helpful and en-

lightening discussions.

Finally, the author thanks Mrs. Margaret Dunn for undertaking

the typing of this manuscript.


















TABLE OF CONTENTS


ACKNOWLEDGMENTS . . . . . . .


LIST OF TABLES . . . . . . . .


LIST OF FIGURES . . . . . . .


Chapter


I. INTRODUCTION. . . . . . .


II. DERIVATION OF THE BASIC EQUATION. .


III. APPLICATION OF APPROXIMATIONS . .


The Random Phase Approximation .


The Superposition Approximation.


Variational Techniques . . .


IV. DISCUSSION AND RESULTS. . . . .


APPENDICES


A. THE ELECTRON GAS . . . . .


B. FERMI UNITS . . . . . . .


C. IDEAL GAS WAVE FUNCTION . . . .


D. RANDOM PHASE APPROXIMATION. . . .


E. NUMERICAL RESULTS FOR THE VARIATIONAL


LIST OF REFERENCES . . . . . . .


BIOGRAPHICAL SKETCH. . . . . . .


Page


. . . . . . ii


. . . . . . iv


. . . . . . v





. . . . . . 1


. . . . . . 4


. . . . . . 10


. . . . . . 10


. . . . . . 14


. . . . . . 17


. . . . . . 21





. . . . . . 26


. . . . . . 28


. . . . . . 29


. . . . . . 30
.~30


CALCULATIONS.


. . .


. . . .














LIST OF TABLES


Table "Page

1. The Ground State Correlation Energies. . . . . ... 22

2. Cancellation in the Intermediate Region. . . . . ... 23

3. Comparison of Ground State Energies. . . . . . ... 24

E.1. The Ground State Radial Distribution Functions . . .. .33

E.2. The Variational Parameters for Which Minimum
Energies Were Found. . . . . . . . ... .. .34



r1













LIST OF FIGURES


Figure Page

1. Variation of the First Parameter . . . . . ... 19

2. Variation of the Second Parameter. . . . . . ... 20

E.1. Comparison of Ground State Energies for Intermediate
Densities. . . . .. . . . . . . . . 35

E.2. Logarithmic Comparison of Ground State Energies. . . ... 36













CHAPTER I


INTRODUCTION


A standard procedure for obtaining an upper bound on the ground

state energy and an approximation to the ground state wave function

Y for a system is to write the energy in terms of the wave function

and then minimize it with respect to variation of the wave function.

This is the Rayleigh-Ritz variational method. We shall present a

means of using this procedure for an electron gas, using a trial
A/2
wave function of the form De where D is the wave function for

an ideal gas of spin 1/2 particles. We shall specialize to the form

where A is real and is given by


A = u(r..). (1.1)
i
That this is a reasonable assumption is indicated by the fact that

the system under consideration differs from an ideal gas only by

virtue of the presence of Coulomb forces. Such two-body forces

depend only on the distance of separation between the particles.

Some information about the form of u(r) is already available since

the wave function must obey certain boundary conditions in order to

properly describe the system under consideration. Specifically, as

r becomes infinitely large, u(r) must vanish, and, the work of Dunn112

indicates that u(0) is a constant, which allows tunneling effects to

be observed. The work of Dunn1,12 and others,2 further indicates
be observed. The work of Dunn, and others, further indicates





2



that the function u(r) galls off as (r)-1 in the large r limit. A

simple functional form for u(r) which obeys these conditions is given

by,


u(r) = -(l-e-br (1.2)
r

where a and b are unknown parameters to be determined. If this form

of u(r) is adopted, one can express the wave function, and,conse-

quently, also the energy for the system, as a function of the two

constants a and b. This fact leads to the simple variational pro-

cedure later employed in this thesis.

The Hamiltonian operator W for an electron gas can be written as

_'2
Y = +V(r . .rN) (1.3)


where

N N 3 -2
A2 = V2 = 1 (1.4)
a=l a a=l =l axj

and V(rl...rN) represents the Coulomb potential. The expectation

value of the energy, , is then


S...f*[- M A 2+V]Ydrl...drN
= (1.5)
!.. dr ...drN



Strictly speaking, u(r) is defined by

ik r
u(r) E 4Aab2 k k2(k+b2)'
k
where the prime on the summation sign indicates that the k = 0 term
is to be excluded from the summation. However, for our purposes,
Eq.(1.2) is valid as long as the quantity g(r)-l is used in the kinetic
energy expressions. This matter is discussed in detail in Chapter III.








In Chapters II and III it is shown that Eq.(1.5) can be written

in terms of integrals involving the radial distribution function g(r).

This function is defined as

02f... f Ydr3...drN
g(r12) = --, (1.6)
f.. .f Y' dri .. drN

where Q is the volume of the system. Noting that the product D D is

symmetric, we approximate it by writing


DD D exp(- ((r )) (1.7)
i
where the function i(r) has already been determined from the ideal gas
3 *
calculations of Lado and Dunn. Therefore Y Y may be written as

Y Y D Dexp(- u(ri.)) = exp(-j 6(r..)), (1.8)
i
where


e(r) E ((r)+u(r).


Since the product Y ? can be expressed in the form exp(-j 6(r.ij)), we
i are able to employ several powerful techniques for evaluating g(r).
4 11
Among the most useful of these are Lado's pertubation formula ', and

the Percus-Yevick integral equation. In this work the latter method

is employed in the region of metallic densities, while the former is

used for high densities.













CHAPTER II


DERIVATION OF THE BASIC EQUATION


In the introduction the expectation value of the energy was

defined in Eq.(1.5) as


-Y A2dT+fV' YdT
= (2.1)
IY* dr

where
-+ -
dT = drl...drN. (2.2)


We must now reduce this expression to a more tractable form in order

to apply the variational principle.

First, we will treat the potential energy term given by


= VdT (2.3)
ff YdT

Since V(rl...rN) represents the Coulomb potential, we may define a

two-body term v(rij) in the following manner:


V(rl...r ) = v(rij). (2.4)
i
Using the definition of the radial distribution function g(r), given

in Eq.(1.6), we write the potential energy term as

= -v(r12)g(r12)drldr2
(2.5)
= -v(r)g(r)dr.
2P









The function U(r) can be expressed as


- 6
v(r) = 1 V e ,k
k
k


where the prime indicates that

cluded in the summation. Thus


v(rl2)drldr2


the k equal to zero term is not in-




1 ik'+r -+ -
=- vke drldr2
k
(2.7)

= S vk6(k)6(-k) = 0.
k


Therefore


R-v(r)g(r)dr = P-fv(r)[g(r)-l]dr.


For the case under consideration, v(r) may be expanded as


(r) = 1 a0 alr2
v(r) = e2 -- + L.


Thus


fjv(r)g(r)dr = 1 -


ao alr2
L + L + [g(r)-l]dr.


(2.10)


In the limit as L approaches -, we have


fv(r)g(r)dr = [g(r)-l]- dr;


and

= N 2 [g(r)-l]dr.
2 N

In order to reduce the kinetic energy term in Eq.(2.1), we first

note that integration by parts yields the following relationship:


T A2ydT = + A *A\ddT,


(2.11)



(2.12)


(2.13)


(2.6)


(2.8)


(2.9)








where the surface terms cancel, due to the boundary conditions im-

posed on Y (see Appendix A). Substituting our form for the wave

function, given by


Y = DeA2, (2.14)


into Eq.(2.13) we find that


D*2D+D AD*AA+ + e dT


l* DAD D *D AD AA+ D(A)2 A
2 4
AD AD+ 2--- --- AA+ 4-"- d


Taking the complex conjugate of Eq.(2.15) and adding shows that


S(AD *AD)eAdT = [DA2D+A(D D).AA

(2.16)

+ DDA2A+ D D(AA)2]e dT.
2 2

In arriving at the above result, use was made of the fact that


D A2D = DA2D*. (2.17)


To show this, we note (as shown in Appendix C) that


A2D = (-k )D, (2.18)


where the k.'s form the complete set of propagation vectors making

up D. For the ground state of the system under consideration we will

have half the spins "up" and half the spins "down." The summation

sign employed in Eq.(2.18) therefore means, that for every value of

j, there will be two k.'s, one for spin "up" and one for spin "down."

Since the k.'s are real quantities, it is apparent from Eq.(2.18) that
J








D 2D = [-k. DD (2.19)
J

and


DA2D = (- ~DD (2.20)
J

Equation (2.19) along with Eq.(2.20) shows that Eq.(2.17) is valid.

Substitution of Eq.(2.16) into the right hand side of Eq.(2.15) yields


Ai AYdTr = -D A2D-

S(2.21)
D DA2A D D(AA)2 A
2 4 eT

Integrating the second term of Eq.(2.21) by parts gives


S[A(D D)*AA]eAdt = D*D[(AA)2+A2A]eAdT. (2.22)


Substituting Eq.(2.22) into Eq.(2.21) we find that


IAT .AdT = -D 2D+ A e dcr. (2.23)


To further simplify Eq.(2.23) we note that Eq.(2.17) may be rewritten

as


(D *2D) kD D, (2.24)
2m 2m j

where the expression h2 k2 is just the ideal gas ground state
where the expression 2m j
2m j 3 h2k2
energy, which we shall write as N, where E 2 (see Appendix
5 F' F 2m (see Appendix
B). Therefore, using Eq.(2.24) in Eq.(2.23), we see that the kinetic

energy may be written as

<2m.EY =dr
= -- -
fJT*d-u







f2
-f(AA)2y Yd2
5 NEF+ (2.25)
fT *dT

The expression [AA]2 is defined as

N
[AA]2 = V [ u(r i)].V [ a u(r )]. (2.26)
a=1 i
A brief manipulation shows that

N
[AA]2 = V u(r ).V u(r ), (2.27)
a,i,j

where the terms involving u(0) do not contribute, since u(0) is a

constant. Substituting Eq.(2.27) into Eq.(2.25) and using the re-

sult obtained in Eq.(2.12), we find that the expectation value of

the energy may be written as follows:

J Ja V u(r .).V u(r .) P*dc
32 a ai a aj
E = 3 Ne + E a -i- -
5 F 8m *Yd-T
(2.28)

+ Npe2 [g(r)-l]dr.


Equation (2.28) may be written in terms of two-and three-body dis-

tribution functions alone, where we define the three-body distribu-

tion function as
23f 'Y dr *.. .drN
g3(rl,r2,r3) = (2.29)
f* YdT
To this end we can use Eq.(1.8), along with symmetry considerations,

to show that
V u(r ai)V u(r .) 'Ydr
JI .Ia ai a aj
fT*TdT (2.30)

N(N-l)fV1u(r2)(r2 -Vu(rl2) dT N(N-l)(N-2)fV1u(rl2) Vlu(r 3)' *dr
=*T + f*T+
f Y"dr J'PPdT




9

Using the definition of g(r) as given by Eq.(1.6) along with Eq.(2.29)
we see that the right hand side of Eq.(2.30) may be written as

Np (Vu(r))2g(r)dr+Np2 Vlu(r2) Vlu(rl)g3(r,r 2,r3)drl2dr3. (2.31)

Using the above result in Eq.(2.28), we find that

NE +N n (7Vu(r) 2g(r)dr
+5 F 8m j[g -


+N8m jVlu(r2)'Vlu(rl3)3((lr2'r3)drl2drl3 (2.32)


+N pe2 [g(r)-ldr.















CHAPTER III


APPLICATION OF APPROXIMATIONS


In Chapter II the energy expression was reduced as far as pos-

sible without resorting to some form of approximation. To proceed
1,13
further, we will employ the random phase approximation13 in the
8 9
high density region, and the superposition approximation for the

region of metallic densities.


The Random Phase Approximation


It was shown in Eq.(2.25) that the kinetic energy involved a

term containing the expression (AA)2. We wish to apply the random

phase approximation to this function in order to reduce it, and con-

sequently, also the expression containing it, to a more easily handled

form. In the introduction we saw that the function u(r) could be

written as

-* --*
u(r) = (k)eikr. (3.1)
k

Substituting Eq.(3.1) into Eq.(2.27) yields

-* ->* .-* -*-
1 3ikr Imr
(AA)2 = X (ik (k)e ai)*(2mW(m)e aj). (3.2)
a i,j -
m,k

Collecting terms in the above equation gives

-> -> -f
1 ikr -im'rj i(k+m)r .
(AA)2= -(k-(m)e e a. (3.3)
i,j -m a
m,k








We are now in a position to apply the random phase approximation (see
-* ->* ->

Appendix D) to Eq.(3.3). This approximation replaces e i(km) ra by
a
N6- with the result that
k,-m
-> ->* ->- -
(AA)2 -ikr -im*rj
(AA)2 = e -(km)(k)_(m)e e), (3.4)
i,j k m
k,m

where 6- is the Kronecker delta symbol. Summing over m gives
k,-m

(AA)2 N 'k22(k)e rij. (3.5)
,j k

In order to simplify the notation in the following development,

let us now define the term in Eq.(2.25) which contains (AA)2 in the

following manner:


I --m (AA) 2Y dT, (3.6)
whemQ f

where Q = jY*dT. Combining Eq.(3.5) with Eq.(3.6) leads to the

conclusion that

-,,3 '2p 2 r
I = k2 2(k)+k k22(k)eirij Ydr. (3.7)
k k

From the footnote to Eq.(1.2) we see that

4 nab2
(k) = k2(k2 ) (3.8)


Substituting Eq.(3.8) into Eq.(3.7) gives


I = (4Tab2)2 k2(k b2)2 2 )2 P YdT. (3.9)
k iij >
k k

The first term is easily evaluated. Replacing the summation sign by

its corresponding integral and performing the integration yields









'___(4ab2) 1 dk a2b
ab k2(k+b) = N (2)-3 ( N c .(3.10)
k

From symmetry and the definition of g(r), as given by Eq.(1.6), we

see that the second term of Eq.(3.9) may be written as


S'2p(4ab2)2 i#
8m




- m(4ab2)2 2
k


1 ei '-ij T *k
Q kZ(k +b Z)2 d
k
*
f Yd



eikr12 -
k (kz+bz) g(r12)drdr2.


We now treat the right hand side of Eq.(3.11) in the same manner as

the potential energy term of Chapter II. To this end, consider the

following development:


J '


ikdrl2 dr2
e kr12 -d
k2(k2+b2)2- drdr2


= k2+b )2 eikrl2ddrld2
k

= 2 k2(k+b2) k)6(-k = 0.
k


i '
k


f Ix
k


(3.12)


ik'r
e 12 -12
k2(k2+b2)2 g(rl2)drldr2


(3.13)


-+ -+
ikr1
eik'rl2 -I -
k (k2+b2)2 [g(rl2)-l]drldr2"


Integrating over the summation in Eq.(3.13) yields


(3.11)


Therefore


!







- - -> ->
ik'r ik'r
1 r12 )3 e r12
n k2(k2+b2)2 = (2)- (k+b2) dk
k (3.14)

1 -br
= 7-r[2- (2+br)e-br].


Applying the results obtained in Eq.(3.13) and Eq.(3.14) to the right

hand side of Eq.(3.11),we find


-p3(4Tab2)2j 1 k (k2+b2) g(rl2)drldr2

-k (3.15)
-br
12p 2a2 f[2-(2+br1 )e 12]
2m ( JT r =2 [g(r12)-l]drldr2.

Integrating over the coordinates of particle 1 and changing to Fermi

units (see Appendix B), shows that the right hand side of this equa-

tion equals

2a2 br
NE (~- [2-(2+br)e-br][g(r)-1]rdr. (3.16)


Using the above result, along with that obtained in Eq.(3.10), in

Eq.(3.9) we obtain the desired expression for I:

a2b 2a2 -br
NF 6-- + 9-- [2-(2+br)e ][g(r)-l]rdr (3.17)


Substituting Eq.(3.17) into Eq.(2.28), and converting to Fermi units,

gives

-+-
NEF 5 6 T


2a2 ibr
+ ( -) [2-(2+br)ebr][g(r)-l]rdr (3.18)



+ -fI[g(r)-l1rdr,








where

8e2kF
y2 (3.19)
F

For purposes which will become apparent in the next chapter, we will

wish to cast Eq.(3.18) in a form which separates out a special term

involving the ideal gas radial distribution function. This special

term, symbolized by ex, is called the exchange energy, and is defined

by


ex = I[g(r)-l]rdr, (3.20)


where g (r) represents the radial distribution function of the ideal

gas whose wave function is D. Performing the separation we find that

Eq.(3.18) may be rewritten as

R.P. = 3 + j a2b
Nc 5 4 j 6g


2a2 f e-br
+ 2J[2-(2+br)e J3[g(r)-l]rdr (3.21)



+ Jf[g(r)-gi(r)]rdr.



The Superposition Approximation


The expression for the energy given in Eq.(2.32) involves a term

containing the function g3(rl,r2,r3). For convenience, we shall label

the integral expression in this term by


L = Vu(rl lu(rl)3(rl2,r3)drdrl3 (3.22)
The superposition approximation merely replaces the function g3(r,r2,r3
The superposition approximation merely replaces the function 93(r1)r2'r3)








by the product g(r12)g(r13)g(r23). Performing this substitution in

Eq.(3.22) yields


L = Vu(r12)Vlu(r13)g(r12)g(r13)g(r23)drl2dri3. (3.23)


By integrating over angles first,it is easy to see that


Vlu(r12) Vlu(rl3)g(r12)g(rl3)drl2drl3 = 0. (3.24)


Thus, Eq.(3.23) may be written as
-t -
L Vlu(r12).Vlu(rl3)g(rl2)g(rl3)[g(r23)-l]dr12drl3. (3.25)


In order to uncouple the variables in Eq.(3.25), we now introduce a

delta function as follows:


L = Viu(rl2)'Vlu(rl3)g(r12)g(r13)[g(r23)-l]

(3.26)

6(r23-'13+r12)dr12dr13dr23.


Noting that


6(r) = (2T)-3j e dk,


we see that Eq.(3.26) can be written as

-ik3 "rl2
L = (2T)-3 dk{ V1u(rl2)g(r 2)e 2dr12
k


(3.27)







(3.28)


J V1u(rl3)g(rl3)ikrl3drl3 1(g(23)-l)eik*r23dr23 *


The integrals involving the gradients are evaluated as follows:
Se r du(rl2) -"
Vlu(rl2)g(rl2)eik*rl2drl2 = rl2 dg(r2)e krl2drl2, (3.29)








where r12 is a unit vector in the direction of r12. Taking k in the

azimuthal direction and carrying out the integration over angles

yields


r12 dr2 g(r12)e dr12


Sdu(rl2) ikr cose 2
= ik(2) drl g(rl2)e 12 r2sin6cosjd dri2 (3.30)


Sdu(rl2) coskrl2 sinkrl2
= ik i dr1 2 kr12 k2r2 dl2


where i is a unit vector in the k direction. The integral involving
k
r13 is just the complex conjugate of this expression with r13 re-

placing r12-

Substituting the above results into Eq.(3.28) shows that

2 ik s
L dk (g(s)-l)ei ds
k (3.31)

(coskr sinkr du( g(r)r2dr 2
kr 721j6 dr

-t 'L-
->. ,-. ,-. ,- -r13
where r23 = s, and r r12 = r13

Noting the fact that

d sinkr coskr sinkr
dk lr kr k-r7'


and taking the derivative of u(r), we find that Eq.(3.31) may be

written as


L = dk g(s)-l)e ds { be-br
k (3.32)

(1-e-br) sinkr g(r r2dr2
r kr r




17

Finally, from Eq.(3.32) it is apparent that the expression for L may

be written in terms of Fourier transforms as follows:

3 + ik s +d fb-br (l-e ) ( ik.rd2
L(2)-3 dk (g(s)- e ds { be r r dr} (3.33)
J-2 {dkd r r
k

Using this equation, along with the derivative of u(r), in Eq.(2.32)

we find that, for the superposition approximation
3 a2l2e-br e-2br -br -2br
s 3 a2 l-2e +e 2b(e -e ) .2 -2br
= + -- +b2e g(r)dr
NEF 5 3r2 r
F

1 f iks d df -br (1-ebr
+ -87- dk (g(s)-l)eik ds dk be r (3.34)
k


er- t e dr + f[g(r)-l]rdr,
r

where Fermi units are again employed. Following the procedure adopted

in the first section of this chapter, we now rewrite Eq.(3.34) in terms

of the exchange energy. Doing this yields

2
s 3 2
NEF 5 4 g(r)-l]rdr

a2 l2e-b+e-br 2b-(e-br -2br
+ 3- r- r +b2e 2bg(r)dr


-br
1 k (g(s)l) iks b -br_ (1-e (3.35)
+ T887 dk g(s)- e ds ddk be r(3.35)
k
S -ik r 12 +
r e dr + [g(r)-g (r)]rdr.



Variational Techniques


So far, it has been the purpose of this work to reduce the

energy expression, given by Eq.(1.5), to a suitable form depending

on the two unknown parameters a and b. This having been done








in Eq.(3.21) and Eq.(3.35), we now proceed to the actual variational

techniques employed in calculating the ground state energies, and

their corresponding wave functions, for various densities. It is

apparent, upon consideration of Eq.(3.21) and Eq.(3.35), that the

last three terms of each of these equations are the only terms con-

taining the unknowns a and b. The sum of these terms is called the

correlation energy and is symbolized by the expression e Since

the first two terms of the final energy equations do not depend on

a and b, the minimum value of the energy with respect to the two

parameters can be found by minimizing e alone. The variational

principle tells us that, if we arbitrarily vary a and b and find a

combination of these two numbers which gives us the lowest value for

ec, then this value is the best approximation to the ground state

correlation energy for the originally adopted form of u(r). The

variation of a and b is carried out as follows:

(1) Fix b at some convenient value, and compute ec for several

values of a. This procedure will yield one particular value of a,

which we shall call al, corresponding to a local minimum in the

value of e An example of this, for a particular density, is given

in Fig. 1.

(2) Having found al, we now compute ec for various different

b values, holding a fixed at the value obtained in step 1. This

gives a value for b, called bl, at which ec is again found to be

a minimum (see Fig. 2).

(3) Fixing b at the value b1 we now vary a again,searching

for a still lower minimum in the neighborhood of al.

This process is carried on until reasonable accuracy is obtained.




19


b

0 .05 .10 .15 .20 .25
0- I I



a = 0.11

p 1030cm-3
b .17
b1 .17


-6










-7-



00
x
to



-8










-9










-10


Fig. 1. Variation of the First Parameter.





20


a

0 .05 .10 .15 .20 .25
0-


b = 0.15

p 1030cm-3

al .13
-6 i










-7






0


S-8-










-9









-10


Fig. 2. Variation of the Second Parameter.













CHAPTER IV


DISCUSSION AND RESULTS


Before presenting the results of the calculations described in

Chapter III, a brief summary of the approximations used to obtain

those results is in order. The approximations are:

(1) It is assumed that the wave function T can be expressed as


V = Dexp[- u(r )],
i
where D is the wave function for an ideal gas of spin 1/2 particles.


(2) ID12 = exp[- p(r )].3
i
(3) It is assumed that the effective potential u(r) can be

written as


a -br
u(r) = (l-e ),
r


where a and b are parameters to be determined.

(4) The energy expressions are simplified by the application
1,13
of either a random phase approximation13 or a superposition ap-

proximation.9,10

(5) The radial distribution function g(r) is computed by use

of either the Percus-Yevick integral equation or the Lado pertur-

bation formula. '








The numerical results of the energy calculations are given in the

table below. Graphs, and other pertinent information, can be found in

Appendix E. The notation employed in Table 1 is the following:

r = the ion sphere radius in Bohr units.

y2 = 1.13r
s

ER.P.A./L. = the minimum correlation energy found by using the

random phase approximation and the Lado perturbation

formula.

S.A./P.-Y. = the minimum correlation energy found by using the

superposition approximation and the Percus-Yevick

integral equation.

ES.A./L. = the minimum correlation energy found by using the

superposition approximation and the Lado perturbation

formula.



TABLE 1

The Ground State Correlation Energies




r Y2 ER.P.A./L. ES.A./L. ES.A./P.-Y.


.00113 .001 -.00000012

.0113 .01 -.000010

.113 .1 -.00053 -.00047

1.13 1.0 -.019 -.018 -.039

3.39 3.0 -.29

5.65 5.0 -.65








The accurate calculation of the correlation energies listed in

Table 1 is a somewhat delicate task. This is particularly true in

the region of intermediate densities. The difficulties encountered

are largely the consequence of cancellation among the various terms

which make up the correlation energy expression. Using the defini-

tion of ec given in the last section of Chapter III, along with Eq.

(3.35), we see that the correlation energy may be written as


Ec = cl+c2+3,


where

Sr -br -br -2br
a2 (1-e )- (e -e ) 2 2br
S= -e% 2b --- +b2e2bg(r)dr,
3 rj) r r


1 j ik s d -br (-e ) g(r) ikr 1-2
E2 7 dk [g(s)-1e ds1 k be r r e dr


and


3 = 2f[g(r)-g (r)]rdr.


Adopting this notation in Table 3 we are able to observe the

cancellation referred to above.



TABLE 2

Cancellation in the Intermediate Region



Y2 E1 E2 3 Eg


1.0 +.118 -.078 -.079 -.039
3.0 +.897 -.690 -.498 -.291
5.0 +1.34 -1.05 -.940 -.653








A comparison between the results of the variational method used

in this work, and the results of some other approaches is given below

in Table 4.


Comparison


TABLE 3

of Ground State Energies


Variational High
y2 method Density Wigner Dunn
(Parametric) Expansion


.001 -.00000012 -.00000018 -.00000057

.01 -.000010 -.000013 -.000026

.1 -.00053 -.00084 -.0011

1.0 -.039 -.043 -.034 -.051

3.0 -.29 -.24 -.30

5.0 -.65 -.568 -.624


W. J. Carr, Jr., Phys. Rev. 133, A371 (1964).

F. Wigner, Trans. Faraday Soc. 34, 678 (1938).

T. Dunn, dissertation, University of Florida (1966); Tucson Dunn
and A. A. Broyles, Phys. Rev. 157, 156 (1967).



The above table seems to imply that the parametric method is

giving reasonably accurate results over a wide range of densities,

including the region of metallic densities. This was the hoped-for

result. Another suggestive feature of these calculations concerns

the nature of the radial distribution function. Several methods of

calculating this function, for a given effective potential, utilize








an approximation which considers the contribution of the "bridge dia-

grams"4'14 in a cluster expansion of g(r) to be negligible. The cal-

culations done in this dissertation, along with those of T. Dunn,

contain that approximation, while those of Carr and Wigner do not.

The observed agreement between the results obtained by these different

approaches seems to indicate that the above assumption concerning the

"bridge diagrams" is a good approximation.

Finally, it must be noted that the parametric form chosen for

the effective potential in this work can only be expected to yield

a good approximation to the true answer. In order to determine the

exact form for the effective potential, one must apply a rigorous

variational procedure to the expression for the energy given by

Eq.(2.32). This process would yield an integro-differential equa-

tion involving the effective potential and the radial distribution

function. The derivation of such an equation is now in progress, and

it is hoped that the results of this dissertation will provide the

initial information necessary to solve it.













APPENDIX A


THE ELECTRON GAS


The electron gas is defined as a system of N electrons in a

cube of edge length L, and volume Q = L3. It is assumed that when

N and Q are large enough, the surface effects are negligible, even

though periodic boundary conditions are imposed upon the wave func-

tion T. The number of particles and the volume are assumed to be
N
infinite in such a way that the density p = -, remains finite.

In order that the total charge of the system remains neutral,

we assume that the electrons move in a uniform positive background

whose total charge just cancels that of the electrons. Each elec-

tron has associated with it an "ion sphere" whose radius is given by

1/3
r = 13. (A.I)

1
We now write the density p(r) as

p(r) = pe(r)+p, (A.2)

where p (r) is the electron density, and p is the density of the uni-

form positive background. Using Eq.(A.2), we see that the potential

energy for the system may be written as


e2 )p(r') +d d'
U -d P drdr' J
2 J I
Ir-r! |

S(r)p (r') p (r) + '
e2 fPe e (' e drdr'
Sdrd'r'-2p1 drdr'+p2- (A.3)
2rJ fr- r'i J r-r'|








Since the uniform positive background assured us of charge neutrality,

the average electron charge density must be equal to that of the posi-

tive background. Using this fact in Eq.(A.3), we see that partial

cancellation occurs in the last two terms to yield

-+ -+
2 fPe(r)p (r')
U -drdr'
r-r|

(A.4)

e2 -_Pe(
It can further be shown that Eq.(A.) may be wrdrd'.

It can further be shown6 that Eq.(A.4) may be written as


1 ik'rij, (A.5)
i,j k

where


47Te2
Vk k2-'














APPENDIX B


FERMI UNITS


Fermi units are based on the definition of the Fermi energy

given by

42k2
F
SF 2m '


where


kF = (3m2p)1/3


The unit of length in these units is


1
F k


There are two more important quantities used in this work. Expressed

in Fermi units they are:
1
1. The density p = .8e

2. The density parameter y2 F F
3F
In order that the variables of integration in this paper be di-

mensionless, the following transformations are made:


R
r-
kF


and


k = kFX.













APPENDIX C


IDEAL GAS WAVE FUNCTION


We define the ideal gas wave function as follows:

N-
ik *r.
D = e J P JS (C.I)
P

where S accounts for spin, p is the permutation operator, and e is
given by
given by


P


+1 for even permutations

-1 for odd permutations.


From Eq.(1.4) and the above definition of D we see that

N -,
N iEk .-r.
A2D = I V2 Xe j J P S
j=1 i p

Taking the gradients gives


(C.2)


(C.3)


N-
N iEk .*r.
A2D = (- 2.)e j p'S.
j=1 p p p


N-
ik .pj rj
J ( 2 P3
P kpj)e J


Therefore,


N-,
N iEk .*r.
= I-kfi e j Pi
J P
j p


N i "k *r
A2D -k e J
j p


N
= D.
jJ


But,


(C.4)


(C.5)














APPENDIX D


RANDOM PHASE APPROXIMATION


Consider the expression


M= eik'ri-im*rj ei(k+)-ra (D.1)
i,j a

Averaging over the position of particle a gives

-+ i 3i(k+m).r *
M eik.rie-im.rj p3fe aY YdT'
M = Xe 1e j (D.2)
i,j fv/ dT

where

->-t -> -> ->
dT' = dr ...dr. -dri ...drj. dr ...dr .
1 i-1 i+l j-l j+1 N

-f+-
From the definition of g3(rl,r2,r3) given by Eq.(2.29) we see that

Eq.(D.2) may be written as


Sik-r. -im-rj i(k+m) ra - -
M = e -re j(p) e a (r i,r .)dr (D.3)
i,j f a

The random phase approximation assumes that the particles are uni-

formly distributed at random. We can apply this hypothesis to Eq.(D.3)
-,+-
by setting g3(r ,r.,r.) equal to 1. This yields


i-kri -im rp i(k+m)
M i eik re-irjpei dra (D.4)
i,j a

Noting that


i(k+m) r -
0-1 e adr = 6-> (D.5)
J a k, -m'








we see that Eq.(D.4) becomes

N -> + +- +
M eikr ie-ir (N65, ). (D.6)
k,-m
i,j

Taking a different view of the problem,we might have considered
N -)> -)
the term .ei(k+m) ra alone. The theory of "Random Walk" tells us
a
that, if (k+m) is not equal to zero, then the summation is equal to
- ->
YN. If, however,(r+m) is equal to zero, then the summation equals N.

Since our model allows the number of particles to approach infinity,

we see that the (k+m) = 0 term will dominate. Thus

N - -- )
.ei(k+m)ra = N6r a. (D.7)
k,-m














APPENDIX E


NUMERICAL RESULTS FOR THE
VARIATIONAL CALCULATIONS


The following are a series of graphs and tables summarizing the

results of the calculations done in this work. The high density

curve in Fig. (E.2) comes from the results of Bohm and Pines.








TABLE E.1

The Ground State Radial Distribution Functions


g(r)

R y2 = .001 y2 = .01 y2 = .1 y2 = 1.0 y2 = 3.0 y2 = 5.0


.2

.4

.6

.8

1.0

1.4

1.8

2.2

2.6

3.0

3.4

3.8

4.2

5.2

6.2

7.2

8.2

9.2

10.2


.5039

.5157

.5349

.5606

.5918

.6661

.7470

.8244

.8903

.9402

.9732

.9914

.9988

.9975

.9969

.9996

.9998

.9994

.9998


.5029

.5148

.5341

.5598

.5911

.6655

.7465

.8240

.8900

.9399

.9730

.9912

.9986

.9974

.9968

.9995

.9997

.9993

.9998


.4921

.5051

.5254

.5521

.5841

.6598

.7420

.8205

.8873

.9378

.9713

.9900

.9977

.9969

.9966

.9994

.9997

.9993

.9998


.2130

.2531

.2989

.3487

.4030

.5209

.6432

.7579

.8545

.9270

.9740

.9992

1.009

1.004

.9989

1.000

1.000

.9996

.9999


.0239

.0427

.0709

.1094

.1594

.2926

.4597

.6381

.8011

.9278

1.009

1.047

1.053

1.014

.9908

.9950

1.001

1.001

1.000


.00656

.01528

.03134

.05735

.09598

.1486

.3885

.5894

.7829

.9271

1.035

1.077

1.078

1.011

.9809

.9938

1.005

1.003

.9991








TABLE E.2

The Variational Parameters for Which Minimum
Energies Were Found


2 R.P.A./L. aS.A./L. aS.A./P.-Y. bR.P.A./L. S.A./L. bS.A./P.-Y.


.001 .025 .08 -

.01 .13 .17 -

.1 .48 .45 .28 .3

1.0 1.3 1.5 .4 1.25

3.0 4.2 1.4

5.0 5.3 1.5


The notation used in the table above is the following:

R.P.A. means random phase approximation.113
4,11
L. means Lado perturbation formula.4
9,10
S.A. means superposition approximation.90

P.-Y. means Percus-Yevick integral equation.











0 Variational method
- Dunn

- Carr

Wigner


0



- 0
- -


0 I 2 3 4 5

r.

Fig. E.1. Comparison of Ground State Energies for Intermediate
Densities.










[] Variational method
Bohm and Pines
- -Dunn
- Wigner


-I
I -













'I
I _
10-


Fig. E.2. Logarithmic Comparison of Ground State Energies.
Fig. E.2. Logarithmic Comparison of Ground State Energies.


/
/


-4
10













LIST OF REFERENCES


1. Tucson Dunn, dissertation, University of Florida (1966).

2. T. Gaskell, Proc. Phys. Soc. (London) 7, 1182 (1961).

3. Fred Lado and Tucson Dunn, private communication, University
of Florida (1965).

4. Fred Lado, dissertation, University of Florida (1964).

5. J. K. Percus and G. J. Yevick, Phys. Rev. 110, 1 (1958).

6. D. Bohm and D. Pines, Phys. Rev. 92, 609 (1953).

7. David D. Carley, dissertation, University of Florida (1963).

8. Terrel L. Hill, Statistical Mechanics (McGraw-Hill Book Company,
Inc., New York, 1956).

9. J. G. Kirkwood, J. Chem. Phys. 3, 300 (1935).

10. W. L. McMillan, Phys. Rev. 138, A442 (1965).

11. F. Lado, Phys. Rev. 135, A1013 (1965).

12. Tucson Dunn and A. A. Broyles, Phys. Rev. 157, 156 (1967).

13. D. Bohm and D. Pines, Phys. Rev. 82, 625 (1951).

14. J. E. Mayer and E. Montroll, J. Chem. Phys. 9, 2 (1941).














BIOGRAPHICAL SKETCH


Michael Smith Becker was born April 4, 1938, in New York City,

New York. In June, 1957, he graduated from Georgetown Preparatory

School. He attended Duke University and the University of Florida,

and received a Bachelor of Science degree from the latter in De-

cember, 1963. In January, 1964, he enrolled in the Graduate School

at the University of Florida, with the financial assistance of a

National Science Foundation traineeship.








This dissertation was prepared under the direction of the chair-

man of the candidate's supervisory committee and has been approved by

all members of that committee. It was submitted to the Dean of the

College of Arts and Sciences and to the Graduate Council, and was ap-

proved as partial fulfillment of the requirements for the degree of

Doctor of Philosophy.



August, 1967


Dean, College of Arts and Sciences


Dean, Graduate School


Supervisory Committee:




Chairman


-^--
t U


________J ___ ff_

^^J'.

PAGE 1

A VARIATIONAL APPROACH TO THE GROUND STATE ENERGY OF THE ELECTRON GAS By MICHAEL SMITH BECKER A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PAKTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA August, 1967

PAGE 2

iiMiVFRSITY OF FLORIDA mnmim 3 1262 08552 621/

PAGE 3

ACKNOWLEDGMENTS The author wishes to express particular gratitude to Professor Arthur A. Broyles for suggesting the topic for this dissertation, and, also, for his continued assistance and encouragement during the course of its development. Special thanks are also due to Dr. Tucson Dunn whose help has been invaluable throughout every phase of this work. The author also wishes to thank Dr. Earl W. Smith, Dr. C. F. Hooper, Jr., and Dr. Joon Lee for many helpful and enlightening discussions. Finally, the author thanks Mrs. Margaret Dunn for undertaking the typing of this manuscript. IX

PAGE 4

TABLE OF CONTENTS Page ACKNOWLEDGMENTS ii LIST OF TABLES iv LIST OF FIGURES v Chapter I. INTRODUCTION 1 II. DERIVATION OF THE BASIC EQUATION 4 III. APPLICATION OF APPROXIMATIONS 10 The Random Phase Approximation 10 The Superposition Approximation 14 Variational Techniques 17 IV. DISCUSSION AND RESULTS 21 APPENDICES A. THE ELECTRON GAS 26 B. FERMI UNITS 28 C. IDEAL GAS WAVE FUNCTION 29 D . RANDOM PHASE APPROXIMATION 30 E. NUMERICAL RESULTS FOR THE VARIATIONAL CALCULATIONS 32 LIST OF REFERENCES 37 BIOGRAPHICAL SKETCH 38 111

PAGE 5

LIST OF TABLES Table ^^Se 1. The Ground State Correlation Energies 22 2. Cancellation in the Intermediate Region 23 3. Comparison of Ground State Energies 24 E.'l. The Ground State Radial Distribution Functions 33 E.2, The Variational Parameters for Which Minimum Energies Were Found 34 IV

PAGE 6

LIST OF FIGURES Figure Page 1. Variation of the First Parameter 19 2. Variation of the Second Parameter 20 E.l. Comparison of Ground State Energies for Intermediate Densities • 35 E.2. Logarithmic Comparison of Ground State Energies 36

PAGE 7

CHAPTER I INTRODUCTION A standard procedure for obtaining an upper bound on the ground state energy and an approximation to the ground state wave function • for a system is to write the energy in terms of the wave function and then minimize it with respect to variation of the wave function. This is the Rayleigh-Ritz variational method. We shall present a means of using this procedure for an electron gas, using a trial A/ 2 wave function of the form De , where D is the wave function for an ideal gas of spin 1/2 particles. We shall specialize to the form where A is real and is given by A -I u(r..). (1.1) That this is a reasonable assumption is indicated by the fact that the system under consideration differs from an ideal gas only by virtue of the presence of Coulomb forces. Such two-body forces depend only on the distance of separation between the particles. Some information about the form of u(r) is already available since the wave function must obey certain boundary conditions in order to properly describe the system under consideration. Specifically, as 1 12 r becomes infinitely large, u(r) must vanish, and, the work of Dunn ' indicates that u(0) is a constant, which allows tunneling effects to 1 12 2 be observed. The work of Dunn, ' and others, further indicates

PAGE 8

that the function u(r) galls off as (r) ^ in the large r limit. A simple functional form for u(r) which obeys these conditions is given by/ u(r) = ^(l-e"^""), (1.2) where a and b are unknown parameters to be determined. If this form of u(r) is adopted, one can express the wave function, and, consequently, also the energy for the system, as a function of the two constants a and b. This fact leads to the simple variational procedure later employed in this thesis. The Hamiltonian operator V for an electron gas can be written as ^= -1^ a2+V (?,... ?^) (1.3) 2m i N where N N 3 -2 a=l a=l il=l 8xj and V(r,...r ) represents the Coulomb potential. The expectation value of the energy, , is then = . (1.5) /.../y ^i-dr^.-.dr Strictly speaking, u(r) is defined by .7^ -iik-r e u(r) E Airab^^ ^^TJ^^^:^^ k where the prime on the summation sign indicates that the k = term is to be excluded from the summation. However, for our purposes, Eq.(1.2) is valid as long as the quantity g(r)-l is used in the kinetic energy expressions. This matter is discussed in detail in Chapter III.

PAGE 9

In Chapters II and III it is shown that Eq.(1.5) can be written in terms of integrals involving the radial distribution function g(r) . This function is defined as fi2/... U ^dr,...dr j'-'h ^dr^-.-dr^ where ^ is the volume of the system. Noting that the product D D is symmetric, we approximate it by writing D*D = exp[-I ^(r )], (1-7) i
PAGE 10

CHAPTER II DERIVATION OF THE BASIC EQUATION In the introduction the expectation value of the energy was defined in Eq.(1.5) as ^JH' A^ydx+JW YdT — 1 , (2.1) /'i' VdT where dT = dr^. . .dr . (2.2) We must now reduce this expression to a more tractable form in order to apply the variational principle. First, we will treat the potential energy term given by =l4^. (2.3) /f I'd T Since V(ri...r^) represents the Coulomb potential, we may define a two-body term v(r..) in the following manner: V(?^...?^) = I v(r ). (2.4) i = |N 2 v(ri2)8(ri2)dridr2 (2.5) v(r)g(r)dr.

PAGE 11

The function U(r) can be expressed as 1 ikT k where the prime indicates that the k equal to zero term is not included in the summation. Thus (2.6) Therefore \ "* "* 1 v' ik'ri 9 ->• -* v(ri2)dridr2 = '^ L \^ >'^dridr2 k = ^l V 6(k)6(-k) = 0. k ^ (2.7) 2 v(r)g(r)d? = £-|v(r)[g(r)-l]dr. (2.8) For the case under consideration, v(r) may be expanded as' v(r) = e2 1 _ _ r L '^1' + — 5— + (2.9) Thus 2 v(r)g(r)dr = _ Ml \k a^r^ 2 r L + [g(r)-l]dr. (2.10) In the limit as L approaches ~, we have .2 2 v(r)g(r)dr = peand = N pe^ [g(r)-l]i d?; :[g(r)-l]dr. (2.11) (2.12) In order to reduce the kinetic energy term in Eq.(2.1), we first note that integration by parts yields the following relationship: H'*A2>l'dT = + Al* •A'i'dT, (2.13)

PAGE 12

where the surface terras cancel, due to the boundary conditions imposed on V (see Appendix A) . Substituting our form for the wave function, given by 'F = De A/2 (2.14) into Eq.(2.13) we find that 2. 4 e dx AD •AD+(^ DAD +D AD> „ , , D D(AA)^ )-AA+ )—^ ^A e di, (2.15) Taking the complex conjugate of Eq.(2.15) and adding shows that (AD •AD)e^dT = [D a2d+A(D D) "AA 1*1 1 * o A + |D DA^AHD D(AA)2]e dx, In arriving at the above result, use was made of the fact that (2.16) D A^D = DA^D . (2.17) To show this, we note (as shown in Appendix C) that a2d = (-Ik2)D, (2.18) where the k.'s form the complete set of propagation vectors making up D. For the ground state of the system under consideration we will have half the spins "up" and half the spins "down." The summation sign employed in Eq.(2.18) therefore means, that for every value of j, there will be two k.'s, one for spin "up" and one for spin "down." Since the it 's are real quantities, it is apparent from Eq.(2.18) that J

PAGE 13

D A^D = (-);k?D d] (2.19) and DA^D = [-j;k2)DD . (2.20) Equation (2.19) along with Eq.(2.20) shows that Eq.(2.17) is valid. Substitution of Eq.(2.16) into the right hand side of Eq.(2.15) yields AV •Al'dx = ^* . o^ A(D D) -AA -D A'^D— ^^ ^ D DA^A D D(AA)2 e dx. (2.21) Integrating the second term of Eq.(2.21) by parts gives [A(D*D) -AAle^dT = D D[(AA)2+A2A]e dx. (2.22) Substituting Eq.(2.22) into Eq.(2.21) we find that A'F -AH-dx = .d%2d+ ^4M1: e dx. (2.23) To further simplify Eq.(2.23) we note that Eq.(2.17) may be rewritten as 3 (2.24) h v"*"9 where the expression -r— )k. is just the ideal gas ground state ^ 2m . 1 „ , J 3 h2k| energy, which we shall write as — Ne , where e^ = 2^ ^^^^ Appendix B) . Therefore, using Eq.(2.24) in Eq.(2,23), we see that the kinetic energy may be written as = fi2 r * -^JAH' •A'l'dx zm-' ic /H' Vdx

PAGE 14

= J Nsp+ — The expression [AA]^ is defined as (2.25) [AA]2= I Vj;_u(r )].VJ Iu(r^J]. a=l i ^ -V fi3K^d?,...d?^ 83(^1, ra.rg) = — . Jf "i'dT (2.29) To this end we can use Eq.(1.8), along with symmetry considerations, to show that y V uCr .) -V u() .. y V u(r .) -V u(r .) r . a ai a ai a.i. .1 /y Vdx (2.30) !(N-1)/V^u(r^2) •^1^(^12^'^*'^^^ N(N-l) (N-2) /v^u(ri2) •^I^^^IS^'*' '^^'^ : + T . J Y ydi

PAGE 15

Using the definition of g(r) as given by Eq.(1.6) along with Eq.(2.29) we see that the right hand side of Eq.(2.30) may be written as Np (Vu(r))2g(r)dr+Np2 '^l^(r^2)'V^niv^^)g^(v^,r^,r^)dr^2'^^13(2-31) Using the above result in Eq.(2.28), we find that 3 ., ,„ -52£ = -rNe +N 5 F Sm (;Vu(r))2g(r)d? +N •n^p^ 8m Viu(ri2) •Viu(r^3)g3(ri,r2,r3)dri2dri3 (2.32) +N pej ^[g(r)-l]d?.

PAGE 16

CHAPTER III APPLICATION OF APPROXIMATIONS In Chapter II the energy expression was reduced as far as possible without resorting to some form of approximation. To proceed 1 13 further, we will employ the random phase approximation ' in the 8 9 high density region, and the superposition approximation ' for the region of metallic densities. The Random Phase Approximation It was shown in Eq.(2.25) that the kinetic energy involved a term containing the expression (AA) . We wish to apply the random phase approximation to this function in order to reduce it, and consequently, also the expression containing it, to a more easily handled form. In the introduction we saw that the function u(r) could be written as u(r) = \ I ^{V.)e^^'''. (3.1) k Substituting Eq.(3.1) into Eq.(2.27) yields (AA)2=^j; I [ (ik.^(k)e^'''''ai).(2m<},(ni)e^'"*''aj). (3.2) a i,j ^ T* '^ m,k Collecting terms in the above equation gives ,0 1 r r ',,->• ->-^ /, V / V ik'r^ -im^r-iv i(k+m) 'r^ ,^ o> (AA)2 = -^ I I -(k-m)(j)(k)(}>(m)e ^e J ),e ^. (3.3) -> -> a m,k 10

PAGE 17

11 We are now in a position to apply the random phase approximation (see r. i(k+m) Ta , Appendix D) to Eq.(3.3). This approximation replaces ^e by a N5-+ -^ with the result that k,-m -y -* .-y (AA)2=-^ I ):'-(k.m)0)(k)Km)e-^^*^i'^-^°'*^ f^^ >. e"^™*^J(N6--), (3.4) k, -m '^ k,m where 6^-> is the Kronecker delta symbol. Summing over m gives k,-m 'v. N r r'. o , o „ ^ ik*r,-^ (AA)2^fr I I kV(k)e^'^-^i3 (3.5) i,J In order to simplify the notation in the following development, let us now define the term in Eq.(2.25) which contains (AA)2 in the following manner: 8mQ (AA)2'1' YdT, (3.6) where Q = /v H-dx conclusion that Combining Eq.(3.5) with Eq.(3.6) leads to the 8mQj pl'k2<|>2(k)+-| I l'k2^2(k)ei^-^ij it '^^ k f 'I'dT, (3.7) From the footnote to Eq.(1.2) we see that . . 47iab^ '^^^> = k2(k^+b'^) (3.8) Substituting Eq.(3.8) into Eq.(3.7) gives l=|^(4.ab2)^ omQ P4 k2(k2+b2)2 " n .^. it ^ -^ y y' 9. .H. ^ k 2(k2+b2)2 Y >i'dT. (3.9) The first term is easily evaluated. Replacing the summation sign by its corresponding integral and performing the integration yields

PAGE 18

12 5|^(4,ab^)^r ^:T^di^ » I^U.)y^Si^ N f^ C^.O.IO) From symmetry and the definition of g(r) , as given by Eq.(1.6), we see that the second term of Eq.(3.9) may be written as 1 v' e^^'^ij .J.J Q ^ k^(k^+b^)^ -f^^P,, , 2^2 ^ -" k g^(4Trab2)2 -^ •i* ydx 2^3 -ft^p 8m -(Airab^) 2\2 /y ^Fdi 1 Y e '^ (3.11) k g(ri2^'^^l'^^2" We now treat the right hand side of Eq.(3.11) in the same manner as the potential energy term of Chapter II. To this end, consider the following development : ik'r. dr^dr2 P7k^+b^ i ^^dr^dr-2 (3.12) = ^^l i,2(i,2^b^)2 6(k)6(-k) = 0. Therefore -> -) e^^'^12 L 1,2 r\/^a-u'^\ 2 k'^Ck'^+b'^)' g(r^2)'^^1^^2 Y e ^^ ^ k^Tk^+b^ [g(rj2)-l]dr^dr2. (3.13) Integrating over the sunmiation in Eq.(3.13) yields

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13 -> -> ik.ri2 1 r' e ^12 ik«r^2 (3.14) V[2-(2+br)e ^^]. 87Tb 'r Applying the results obtained in Eq.(3.13) and Eq.(3.14) to the right hand side of Eq. (3.11), we find -n2p3 8m (4TTab2)2 1 _i 1 y' 1 Q ^ k'^(k'^+b2)^ g(rj2)d^^l'^^2 (3.15) 2m ^671^''^ [2-(2+br^2)e ^""^^J 12 -[g(r^2)~l^d^l'^^2Integrating over the coordinates of particle 1 and changing to Fermi units (see Appendix B) , shows that the right hand side of this equation equals 2 ^^A^) [2-(2+br)e ^^] [g(r)-l]rdr. (3.16) Using the above result, along with that obtained in Eq.(3.10), in Eq.(3.9) we obtain the desired expression for I: ^ '^ ^, a^b ^ 2a2 'F 611 9tt [2-(2+br)e ^""j [g(r)-l] rdr (3.17) Substituting Eq.(3.17) into Eq.(2.28), and converting to Fermi units, gives ^^^R.P. ^ 3 ^ a2b Ne 5 6tt 2a' MI!t) 9tt' [2-(2+br)e ''''] [g(r)-l]rdr (3.18) n [g(r)-l]rdr,

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14 where Y 8e2k 2 = F 3Tiep • (3.19) For purposes which will become apparent in the next chapter, we will wish to cast Eq.(3.18) in a form which separates out a special term involving the ideal gas radial distribution function. This special term, symbolized by e , is called the exchange energy, and is defined by ^x 4 [gj(r)-l]rdr. (3.20) where g-|-(r) represents the radial distribution function of the ideal gas whose wave function is D. Performing the separation we find that Eq.(3.18) may be rewritten as R.P. Ne, 5 4 a^b [g^(r)-l]rdr+^ ^2a2 [2-(2+br)e ^""j [g(r)-l]rdr (3.21) H [g(r)-g^(r)]rdr. The Superposition Approximation The expression for the energy given in Eq.(2.32) involves a term containing the function g3(r^ ,r2 .rj) . For converfience, we shall label the integral expression in this term by L = V^u(r^2>-^l"(^13^83^^1'^2'^3)'^^12'^^13^^'^^^ .^ -> -> . The superposition approximation merely replaces the function 83(^1 »'^2'^3'^

PAGE 21

15 by the product g(r^2)g(^i3^S(r23) • Performing this substitution in Eq. (3.22) yields L = '^^uir^^)'V^uir^^)g(r^^)g{v^^)g(r^^)dr^^dr^^. (3.23) By integrating over angles first, it is easy to see that Viu(ri2) •Viu(ri3)g(ri2)g(ri3)dri2dri3 = 0. (3.24) Thus, Eq.(3.23) may be written as L = Viu(ri2)-Viu(ri3)g(ri2)g(ri3)[g(r2 3)-l]dri2dri3. (3.25) In order to uncouple the variables in Eq.(3.25), we now introduce a delta function as follows: L = Viu(ri2)'Viu(ri3)g(ri2)g(ri3)[g(r2 3)-l] ,-y -y 6(r2 3-ri3+ri2)dri2dri3dr2 3. (3.26) Noting that 6(r) = (2^)-3 e'-^'^dk, (3.27) we see that Eq.(3.26) can be written as L = (2u) -3 dk{ Viu(ri2)g(ri2)e ^''dri; Viu(ri3)g(ri3)e"^ '^^^dris (g(r23)-l)e''^-"23d,23 (3.28) The integrals involving the gradients are evaluated as follows: du(ri2) Viu(ri2)g(ri2)e ^^dri2 = ri2 dr 12 ;(ri2)e^^*^12dr^2, (3-29)

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16 where ri2 is a unit vector in the direction of ri2' Taking k in the azimuthal direction and carrying out the integration over angles yields f du(ri2) ri2 dr 12 . , ik'rio, ;(ri2)e ^-^dri; i^(2.) du(ri2) dr 12 , , ikr cose . njin^ g(ri2)e 12 r]^sinecos9d6dri2 (3.30) = ij^(4Tii) du(ri2) dr 12 g(ri2)r?2 coskri2 sinkri2 kr 12 K r^2 dri2, where i is a unit vector in the k direction. The integral involving k ri3 is just the complex conjugate of this expression with rjs replacing ri2' Substituting the above results into Eq.(3.28) shows that dk _ ->-> (g(s)-l)e ds fcoskr _ sinkr^ du^rl (r)r2dr ^ kr k^r^ ^ dr ^^^ (3.31) where r2 3 = s, and r ri2 ^^13' Noting the fact that d j -sinkri _ coskr sinkr dk ^ kr^ ^ " kr " ~k^' and taking the derivative of u(r) , we find that Eq.(3.31) may be written as TT dit (g(s)-l]e^^-^d^ ^dk be -br (3.32) r sinkr ^^ ^2dr}' kr'^ r

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17 Finally, from Eq.(3.32) it is apparent that the expression for L may be written in terms of Fourier transforms as follows .,• • , , /-, -br ik's L=(2^)-^ dk .g (s)-l)^ ds {Mk be -br (1-e ) g(r) ik'r -»i^ . . ° 2 6 dr| . (3.33) Using this equation, along with the derivative of u(r), in Eq.(2.32) we find that, for the superposition approximation „ 2 5 3ti Ne, -br. -2br -br -2br. l-2e +e 2b(e -e ) ^^2 -2br +b'^e g(r)dr 288tt dk ik's ,-> (g(s)-l]e^'^-^ds { ^dk be -br (1-e-^^) (3.3A) ^ei^-^d?}%J^[[g(r)-l]rdr, r A J where Fermi units are again employed. Following the procedure adopted in the first section of this chapter, we now rewrite Eq.(3.34) in terms of the exchange energy. Doing this yields c = 1+ ^[[g-^(r)-l]rdr 3tt -br, -2br -br -2br, l-2e +e 2b(e -e ) ^^2 -2br :t +h-^e g(r)dr 288iT' dk (g(s)-l)e^^'^d^ { ik's ,^| r d dk be -br d-e'^^) (3.35) ^ e^^'^d?}% ^j[g(r)-g^(r)]rdr. Variational Techniques So far, it has been the purpose of this work to reduce the energy expression, given by Eq.(1.5), to a suitable form depending on the two unknown parameters a and b. This having been done

PAGE 24

18 in Eq.(3.21) and Eq.(3.35), we now proceed to the actual variational techniques employed in calculating the ground state energies, and their corresponding wave functions, for various densities. It is apparent, upon consideration of Eq.(3.21) and Eq.(3.35), that the last three terms of each of these equations are the only terms containing the unknowns a and b. The sum of these terms is called the correlation energy and is symbolized by the expression e . Since the first two terms of the final energy equations do not depend on a and b, the minimum value of the energy with respect to the two parameters can be found by minimizing e alone. The variational principle tells us that, if we arbitrarily vary a and b and find a combination of these two numbers which gives us the lowest value for e , then this value is the best approximation to the ground state correlation energy for the originally adopted form of u(r). The variation of a and b is carried out as follows: (1) Fix b at some convenient value, and compute e for several values of a. This procedure will yield one particular value of a, which we shall call a^ , corresponding to a local minimum in the value of e . An example of this, for a particular density, is given in Fig. 1. (2) Having found a^ , we now compute e for various different b values, holding a fixed at the value obtained in step 1. This gives a value for b, called bj, at which e is again found to be a minimum (see Fig. 2). (3) Fixing b at the value b^ we now vary a again^ searching for a still lower minimum in the neighborhood of aj . This process is carried on until reasonable accuracy is obtained.

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19 .05 .15 I .20 I .25 L_ ^^ a = 0.11 30 -3 p = 10 ^^cm' b^ ^ .17 -6-7X -8-9-10 Fig. 1. Variation of the First Parameter.

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20 -6-7o X u -8 -9-10.0 5 I .10 I b = 0.15 p = 103"cin-3 a^ = .13 .20 I .2 5 L_ Fig. 2. Variation of the Second Parameter.

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CHAPTER IV DISCUSSION AND RESULTS Before presenting the results of the calculations described in Chapter III, a brief sununary of the approximations used to obtain those results is in order. The approximations are: (1) It is assumed that the wave function "i" can be expressed as --, y = Dexp[i I u(r )], where D is the wave function for an ideal gas of spin 1/2 particles. (2) 1d|2 = exp[I *(r )].^ i u(r) = -d-e^ ), where a and b are parameters to be determined. (4) The energy expressions are simplified by the application r . t. . ^ • . 1,13 . . of either a random phase approximation or a superposition ap9,10 proximation. (5) The radial distribution function g(r) is computed by use of either the Percus-Yevick integral equation or the Lado perturK V ^ 1 ^.11 bation formula. 21

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22 The numerical results of the energy calculations are given in the table below. Graphs, and other pertinent information, can be found in Appendix E. The notation employed in Table 1 is the following: r = the ion sphere radius in Bohr units. s '^ Y^ = 1.13r . s ^Tj T) A /T ~ '-^^ minimum correlation energy found by using the random phase approximation and the Lado perturbation formula. S.A./P.-Y. = the minimum correlation energy found by using the superposition approximation and the Percus-Yevick integral equation. E , = the minimum correlation energy found by using the O . A. / Li t superposition approximation and the Lado perturbation formula. TABLE 1 The Ground State Correlation Energies r s

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23 The accurate calculation of the correlation energies listed in Table 1 is a somewhat delicate task. This is particularly true in the region of intermediate densities. The difficulties encountered are largely the consequence of cancellation among the various terms which make up the correlation energy expression. Using the definition of e given in the last section of Chapter III, along with Eq. (3.35), we see that the correlation energy may be written as e^ = Ci+C2+e3: where C2 = 288TT' a 3tt dk{ (1-e ) , -br -2br. „, i 2b ^^—^ ^ +b2e-2^^ g(r)dr, r [g(s)-l]e^^-^d^}4 -br. be -br_ (l-e-"'^) ^ e^^-^d?} , and ^3 = f [g(r)-g^(r)]rdr. Adopting this notation in Table 3 we are able to observe the cancellation referred to above. TABLE 2 Cancellation in the Intermediate Region Y^

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24 A comparison between the results of the variational method used in this work, and the results of some other approaches is given below in Table 4. TABLE 3 Comparison of Ground State Energies

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25 an approximation which considers the contribution of the "bridge dia4 lA grams" ' in a cluster expansion of g(r) to be negligible. The calculations done in this dissertation, along with those of T. Dunn, contain that approximation, while those of Carr and Wigner do not. The observed agreement between the results obtained by these different approaches seems to indicate that the above assumption concerning the "bridge diagrams" is a good approximation. Finally, it must be noted that the parametric form chosen for the effective potential in this work can only be expected to yield a good approximation to the true answer. In order to determine the exact form for the effective potential, one must apply a rigorous variational procedure to the expression for the energy given by Eq.(2.32). This process would yield an integro-dif ferential equation involving the effective potential and the radial distribution function. The derivation of such an equation is now in progress, and it is hoped that the results of this dissertation will provide the initial information necessary to solve it.

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APPENDIX A THE ELECTRON GAS The electron gas is defined as a system of N electrons in a cube of edge length L, and volume fi = L^. It is assumed that when N and Q are large enough, the surface effects are negligible, even though periodic boundary conditions are imposed upon the wave function Y. The number of particles and the volume are assumed to be infinite in such a way that the density p = — , remains finite. In order that the total charge of the system remains neutral, we assume that the electrons move in a uniform positive background whose total charge just cancels that of the electrons. Each electron has associated with it an "ion sphere" whose radius is given by 1/3 r = (t^] . (A.l) We now write the density p(r) as p(r) = p^(?)+p, (A. 2) where p (r) is the electron density, and p is the density of the uniform positive background. Using Eq.(A.2), we see that the potential energy for the system may be written as U=^ P^^)P< ^' ) d?d?' r-r! e2 pJr)pAr') ^ _^ fpj?) — drdr'-2p 26 drdr'+p2 ^^^} . (A.3) r-r

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27 Since the uniform positive background assured us of charge neutrality, the average electron charge density must be equal to that of the positive background. Using this fact in Eq.(A.3), we see that partial cancellation occurs in the last two terms to yield ^2 U = 2 J Pg(r)p^(r') drdr' r-r' (A.4) e2 Pe(-) J C.p drdr'. It can further be shown that Eq.(A.4) may be written as 1 V v' i^T-;i,j kwhere (A. 5) Aire'

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APPENDIX B FERMI UNITS Fermi units are based on the definition of the Fermi energy given by '^ ^F 2m ' where kp = (3tt^p) The unit of length in these units is 2.n1/3 There are two more important quantities used in this work. Expressed in Fermi units they are: 1. The density p = „ ? . JIT 2. The density parameter y = Se^k 3^ep In order that the variables of integration in this paper be dimensionless, the following transformations are made: -> -> R and -> -> k. = k X. r 28

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APPENDIX C IDEAL GAS WAVE FUNCTION We define the ideal gas wave function as follows; iZk .T. D = ):e e J PJ ^S (C.l) where S accounts for spin, p is the permutation operator, and e is given by +1 for even permutations e = -1 for odd permutations. From Eq.(1.4) and the above definition of D we see that 3=1 ^ P P iEk . -r . A^D = I V2 le^e J P^ ^S Taking the gradients gives (C.2) N-> -> N iZk^.'r. a2d = I h {-P.)e J P^ ^S, But, iZk . T . Therefore, N^ ^ N iZk^.-r. N y Te (-k2.)e j pj ^ = r-yk2]ye e j p^ ^ . , P PJ ^ . J'^ P N^ iXk .'r. a2d = [-Ik^j^e e J PJ JS • J P J P N J 29 (C.3) (C.4) (C.5)

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APPENDIX D RANDOM PHASE APPROXIMATION Consider the expression H = I ei^*^i-i"''^j^e'-^^"^^*^a^ Averaging over the position of particle a gives M J4' Ydx i.J where (D.l) (D.2) ,->->->-> -> ^ QT = drT...dr. Tdr.,T...dr. , dr . , ^ . • .dr„. 1 1-1 1+1 j-1 j+1 N From the definition of g3(r^,r2,r3) given by Eq.(2.29) we see that Eq.(D.2) may be written as „ r ik-r-i -im"r4 , . K = I e ^e J (p) -> -> -> i(k+m) 'r^ g3(r3,r..rj)dr^. (D.3) The random phase approximation assumes that the particles are uniformly distributed at random. We can apply this hypothesis to Eq.(D.3) ->->•->• by setting gq(r ,r.,r.) equal to 1. This yields 3 a 1 J -> ^ -> -> ., V ik-ri -im-r^ H I e ^e Jp i(k+m) ^^ e dr . a r e ^dr = 6^->, a k,-m (D.5) 30

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31 we see that Eq.(D.A) becomes ikT-; -iniT^ (D.6) M = I e"^-^ie-^"^-^J(N6-. -.) Taking a different view of the problem, we might have considered N . /r>->, -^ ^1, ^ r i.(k+m) 'r-, the term ^e ^ alone. The theory of "Random Walk" tells us a that, if (k+m) is not equal to zero, then the summation is equal to 04. If, however, (r+m) is equal to zero, then the summation equals N Since our model allows the number of particles to approach infinity, -> -» we see that the (k+m) = term will dominate. Thus N .,;>->. -> i(k+m)*r_ <=> a ~ M,j , k,-m V i(k+m)«r_ ^,le a . N6j> _->. (d.7)

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APPENDIX E NUMERICAL RESULTS FOR THE VARIATIONAL CALCULATIONS The following are a series of graphs and tables summarizing the results of the calculations done in this work. The high density 6 curve in Fig. (E.2) comes from the results of Bohm and Pines. 32

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33 TABLE E.l The Ground State Radial Distribution Functions

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34 TABLE E.2 The Variational Parameters for Which Minimum Energies Were Found ^ ^R.P.A./L. ^S.A./L. ^S.A./P.-Y. '^R.P.A./L. ^S.A./L. ^S.A./P.-Y. .001

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35 Fig. E.l. Comparison of Ground State Energies for Intermediate Densities.

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36 Fig. E.2. Logarithmic Comparison of Ground State Energies,

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LIST OF REFERENCES 1. Tucson Dunn, dissertation, University of Florida (1966). 2. T. Gaskell, Proc. Phys . Soc. (London) 1_, 1182 (1961). 3. Fred Lado and Tucson Dunn, private communication, University of Florida (1965). 4. Fred Lado, dissertation, University of Florida (1964). 5. J. K. Percus and G. J. Yevick, Phys. Rev. 110 , 1 (1958). 6. D. Bohm and D. Pines, Phys. Rev. _22, 609 (1953). 7. David D. Carley, dissertation. University of Florida (1963). 8. Terrel L. Hill, Statistical Mechanics (McGraw-Hill Book Company, Inc., New York, 1956). 9. J. G. Kirkwood, J. Chem. Phys. 3, 300 (1935). 10. W. L. McMillan, Phys. Rev. 138, A442 (1965). 11. F. Lado, Phys. Rev. 135, A1013 (1965). 12. Tucson Dunn and A. A. Broyles, Phys. Rev. 157' , 156 (1967). 13. D. Bohm and D. Pines, Phys. Rev. 82, 625 (1951). 14. J. E. Mayer and E. Montroll, J. Chem. Phys. 9_. 2 (1941). 37

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BIOGRAPHICAL SKETCH Michael Smith Becker was born April 4, 1938, in New York City, New York. In June, 1957, he graduated from Georgetown Preparatory School. He attended Duke University and the University of Florida, and received a Bachelor of Science degree from the latter in December, 1963. In January, 1964, he enrolled in the Graduate School at the University of Florida, with the financial assistance of a National Science Foundation traineeship. 38

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This dissertation was prepared under the direction of the chairman of the candidate's supervisory committee and has been approved by all members of that committee. It was submitted to the Dean of the College of Arts and Sciences and to the Graduate Council, and was approved as partial fulfillment of the requirements for the degree of Doctor of Philosophy. August, 1967 c<^ jC M ^-n^ Dean, College of Arts and Sciences Dean, Graduate School Supervisory Committee: id. /. M^

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/.I -(I 5 3 >~