224 A . L E O N A R D 175

See, e. g., K. Huang, Statistical Mechanics (John Wiley & Sons, Inc., New York, 1963).

2Ref. 1, p. 226. 3See, e.g., E. T. Copson, Theory of Functions of a

Complex Variable (Oxford University Press, London, 1935).

I. INTRODUCTION

The problem of obtaining the ground-state energy of a quantum electron gas has been approached in a number of ways. The original work on this sub­ject was done by Wigner,x who calculated the "cor­relation energy" of an electron gas for very high densities ( r s < l ) and then extended the resul ts in such a way as to obtain the correc t value at very low densities (rs» 1). He estimated his resul ts to be correct everywhere to within 20%. Later calculations employing perturbation-expansion techniques2""5 gave improved resul ts for high den­sit ies while improved calculations for low densi­t ies 6 were carr ied out using Wigner's lattice mod­el. In addition to Wigner's t reatment, the region of metallic densities (1 <rs <. 5) has been treated7

by truncating the Martin-Schwinger equations and using a random-phase approximation.5 Finally, a recent paper8 has shown how the problem may be treated by approximating, the Slater sum for the system by an effective Boltzmann factor. Tjie aim of this paper is to improve upon these calculations for the metallic density region, and also, to find a method which will give good agreement with the above-mentioned resul ts in their respective regio gions of validity. It is further desirable that the method used for the electron gas be of sufficient flexibility so that it may later be generalized to include multicomponent sys tems. In accordance with these ideas, we shall present a parametr ic Rayleigh-Ritz variational method for calculating the ground-state energy, using a t r ia l wave func­tion of the form

$ = Dex9(-i £ u(r.)), ( i . i )

N. I. Muskhelishvili, Singular Integral Equations (P. Noordhoff, Groningen, The Netherlands, 1953).

5F. London, Superfluids (Dover Publications, Inc., New York, 1954), Vol. II.

where u{r) is an unknown function to be determined by variation and D is the wave function for an ideal

rgy gas of s p i n - ! par t ic les . i

II. DERIVATION OF THE BASIC EQUATION or-i The Hamiltonian operator H for an electron gas n can be written as j

» #=-(KV2m)A2 + y( rV ' - ry , (2.1)

N i- where A2 = 2 1 v2 (2.2)

a- 1 a

in and F(rV . . rjyO represents the Coulomb potential. j 7 The expectation value of the energy (E) 9 is then id

a (E)=Q-lfip*[-(t?/2m)A2 + V]il)dT , (2.3) e

where Q^ J^tydr , (2.4) dm >ns and dr sd r ,d rV • • dvAT . (2.5) 1 N

[e We must now reduce this expression to a more io tractable form in order to apply the variational 3 principle. > F i r s t , we will t reat the potential-energy te rm

given by

{PE)=Q-1j^Vipdr . (2.6)

'- Since VGv • *Yn) represents the Coulomb potential, we may define a two-body te rm v{rfj) in the follow­ing manner:

*> 7 ( r V - - r )= TtvirJ. (2.7)

i<j J

PHYSICAL REVIEW VOLUME 175, NUMBER 1 5 NOVEMBER 1968

A Parametric Approach to the Ground-State Energy of an Electron Gas

M. S. Becker, A. A. Broyles, and Tucson Dunn

Physics Department, University of Florida, Gainesville, Florida (Received 11 March 1968)

A parametric form is chosen for the ground-state wave function of an electron gas. The expression for the energy of the system is derived and then minimized with respect to the unknown parameters. The energies thus obtained agree well with the results of other au­thors .

175 G R O U N D - S T A T E E N E R G Y O F E L E C T R O N GAS

Thus we may write

<PE)= j>Npfv(r)g(r)dr,

225

(2.8)

where the radial distribution function g(r) is given by

g(r) = N(N-i)p-2Q-1ftp*ipdr3 *"<&#. (2. 9)

The function v(r) can be expressed as 5

/ x o - i T * *£•?• (2.10)

where the pr ime indicates that the k equal to zero is not included in the summation. This occurs be­cause of the presence of the uniform positive back­ground. Thus

performing some manipulations, we find that the kinetic energy may be written as

/v(r12)rfr1^r2 = S g i^6(£)6( - £ )= 0.

Therefore,

^pfv(r)g(r)dr = \pJv(r)[g(r) - l]dr .

(2.11)

(2.12)

For the case under consideration, v(r) may be ex­panded as 9

v(r)= e2{r~l -aQL~l- axr-2L~z •• •). (2.13)

Since the factor g(r) - 1 approaches zero rapidly as r becomes large, we find that, in the limit as L approaches <*>, we have

(VE)=^Npfv(r)g(r)dr

= ^Npe2J[g{r)-l]r~1dr. (2.14)

In order to reduce the kinetic-energy term in Eq. (2. 3), we f i rs t note that integration by par t s yields the following relationship:

Ji/j V2ipdr^ fVip - V ^ r , (2.15)

where the surface t e rms cancel as a resul t of the periodic boundary conditions imposed upon ip. Substituting our chosen form for the wave function given bysEq. (1.1) into the above equation, and

(KE)=(n2/2m)Q'1fVip -Vipdr =fNeF+J (2.16)

where J={W/2m)Q-1J^Iipdr; (2.17)

and /= S V.u(r..)-V.u{r.u). (2.18) i,j,kl « l lk

In obtaining the above result , use was made of the relationship

(K2/2m)(D*V2D)= ( l ? / 2 m ) S . k2(J0*D)

=fNeF(D*D), ( 2 - 1 Q )

where Cp denotes the Fermi energy. The last t e rm in Eq. (2.16) can be written in t e rms of the radial distribution function g(r) and the three body distribution function given by

£-3(?i, r2, r3)

= N(N~ l)(N~2)p-3Q~1fip*il)c{r4 • • -ctr . (2. 20)

Doing this, and adding the expression obtained for the potential energy in Eq. (2„ 14), we find that

(E)/N=feF+±pe2f[gI(r)~l]r-idr

+ (ti2p/Sm)f[du(r)/drfg(r)dr + (K2p2/8m)l,

+ l>pe2f[g(r)-gI(r)]r-1dr, (2.21)

where

£ = fvMri2) •vMr13)g3{r1, r2, r3)dr12dr13, (2. 22)

and where we have added and subtracted a t e rm involving the ideal-gas radial distribution function gj(r) in order that the f i rs t two t e rms of the above equation constitute the familiar Hatree-Fock solu­tion2 to the problem. The last three t e rms con­tain the unknown quantity u(r), and their sum is called the correlation energy. It is this quantity that we a re pr imari ly concerned with he re .

III. APPLICATION OF APPROXIMATIONS

In order to proceed further, we now resor t to some approximations.

(A)

F i r s t , the energy expression is simplified by the application of either a random-phase approximation or a superposition approximation,.10 The random-phase approximation is used for the high-density cal­culations while the superposition approximation is used for the metallic region.

1. Random-Phase Approximation

In Eq. (2.16) we saw that the kinetic energy contained a term labeled J which we defined in Eq. (2.17). We want to apply the random-phase approximation to this t e rm. F i r s t , we assume that the unknown func­tion u(r) may be written as

«(r)=0-1EfV(fe)^ff#?. (3-D

Using Eq. (3. l ) in Eq. (2.18) yields

7=flT2S j C ^ _ (k. m)cj>(k)<t)(m)e lJ m,k

zk-r . ~zm«r . iZ

i(k + m)> r (3.2)

226 BECKER, BROYLES, AND DUNN HI

The random-phase approximation (R.P.A.) assumes that the r a , which appears in the above equation, is distributed at random so that the factor £ ;exp[z(k + m ) - r j may be replaced by N^ _ ;£. Performing

this substitution in Eq. (3. 2) and summing over m gives

zk* r

/-Mr2EEWte)e li. (3-3)

i, jk

Thus Eq. (2.17) may be written as r ' , ik-r..l

J^(t?p/Bm)Q'1j ij)*ij}\,pE fe^tej+sr1 JS.^J k2<p2(k)e tJ\dT . (3.4)

Using the definition of g(r) as given by Eq. (2. 9), converting the summations to integrals, and adopting the same procedure used to obtain the factor g(r) - 1 in Eq. (2.12), we finally obtain

J « f o W p / 8 m ) ( 2 i r ) - ^ ^ ^ . (3.5)

Substituting this resul t into Eq. (2.16) and combining with Eq. (2.14), we find that

E R P A = 3 _ r i rr te)_ x l <2R+_J_ fX262(x)d^

r

^ ^ J R ^ ) - ^ (3.6)

where y2 = Se2KF/37reF , (3.7)

and where the Fe rmi units have been used (see Appendix). As before in Eq. (2. 21), the last three te rms represent correlation energy.

2. Superposition Approximation

Equation (20 21) contains an integral involving the three-body distribution function g3(r1} r2, r3) . This in­tegral is labeled L and is defined by Eq0 (2. 22). In order to simplify L, *we will apply the Kirkwood super­position approximation (S.A.) given by replacing the function g3(r19 r2, r3) by the product g(r12)g(r13)g(r23). Performing this substitution in L yields

L =* /V i t tO^) . ^Mri3k(r12)g(r13)g(r23)dr12dr13 . (3.8)

A brief consideration of the above shows that

L<* $Vxu(r12)- V\u(r13)g(r12)g(r13)[g(r23) - 2]dr12dr13 . (3.9)

In order to uncouple the variables in the above, we now introduce a 6 function as follows:

L&f \u{r12y ^Mnskir^gir^lgir^) - 1] x 5(r23 - r13 + r12)dr12df13dr23 . (3.10)

Noting that 6 ( r ) = ( 2 7 r ) - 3 / / - k ' r d k , (3.11)

we see that, after some manipulation, Eq. (3010) can be written as

Substituting this resul t into Eq. (2. 21) and changing to Fe rmi units (see Appendix), we obtain

(3.12)

£S.A, ;_3 _ y

where, once again, the last three t e rms constitute the correlation energy.

175 GROUND-STATE ENERGY OF ELECTRON GAS 227

(B)

Next, we must calculate the radial distribution function g(r) as a functional of the unknown quantity u(r), To do this, we note that the product D*D can be approximated by writing

D*D<*ex9.(-gje(r.j)) , (3.14)

where 9(r) has been determined from the ideal-gas calculations of Lado. n This enables us to write g(r) as

g(r) *N(N- l ) p - 2 / e x p { - ^[eir^+uir^jdr^ • • drN/fexp{~ £j[e(rij)+u(rij)'ftdr1- • • dvN . (3.15)

Radial distribution functions of this form may be readily calculated by several methods. We have chosen to use the Percus-Yevick integral equation12 for the metallic densities and a perturbation formula13 for the high-density points. Let us now define the structure factor S(x) in the usual manner by writing

S(x) = pG(x)-l (3.16)

where G(x) is the Fourier transform of [g(R)- l ] . The perturbation approximation referred to above tells us that

Six) «S7(*)/[1 + pQ&ySjM] , (3.17)

where 0(#) is the Four ier transform of the effective potential as shown in Eq. (3.1), and the ideal-gas s t ructure factor Sj(x) is given by

SI(x) = pGI(x)+l=*x-(x3/l6), x^2; or 1, x ^ 2 . (3.18)

Substituting this result into Eq. (3. 6) and simplifying, we find that for high densities the correlation energy may be written as

. « (l/24<ir4)fx*<j>Hx)S(x)dx+ (3y2/S)f[S(x)-SI(x)]dx.

(C)

(3.19)

Finally, although the function u(r) is unknown, some information about i ts form is already available. Specifically, as r becomes infinitely large, u(r) must vanish, and the work of Dunn and Broyles indicates that w(0) is finite and nonzero. This can be expected since the electrons can tunnel into the classical Coulomb b a r r i e r . The work of Dunn and Broyles and others1 4 further indicates that the function u(r) falls off as (r)-1 in the l a r g e - r limit. A simple functional form of u(r) which obeys these conditions is given by

br\ u2(r) = ar~1(l- e~ ) , (3.20)

where a and b are unknown parameters to be determined. If this form of u(r) i s adopted, one can express the wave function, and consequently, also the energy for the system, as a function of the two parameters alone. Thus the ground-state energy may be calculated simply by computing the energy for various val­ues of the two parameters and determining which values give the lowest energy. The accuracy of the minimum energy obtained in this manner depends rather heavily on how well the parametr ic form chosen for u(r) agrees with the true form of u(r). In the hopes of obtaining better numerical resul ts , the calcu­lations were repeated with a form more flexible than that given by Eq. (3. 20). It is given by

u4(r) = ar"1(l - e~ )+ aifP+r2)'1 ,

where a, b, a, and /3 are all unknown parameters to be determined by variation.

(3.21)

IV. RESULTS

The minimum correlation energies, for various densities, using Eq. (3. 20) and Eq. (3.21), are given in the tables below. The notation used in Table I is the following: rs is the ion sphere r a ­dius in Bohr units, y*= 1.13r s , E2 is the minimum correlation energy found by using Eq. (3. 20), E4

is the minimum correlation energy found by using Eq. (3. 21). The resul ts are presented in fermi units and, as mentioned before, the high-density

points (y2= 0. 01 and 0.1) were calculated using Eq. (3.19), while the lower density points (y2

= 1.0, 3.0, and 5. 0) were done using the last three t e rms of Eq. (3.13) and the Percus-Yevick inte­gral equation.

From a consideration of the Table I, it i s ap­parent that, for metallic densit ies, the two-pa­rameter effective potential given by Eq. (3.20) yields essentially the same resul ts as the four-parameter form given by Eq. (3.21). It is possible, therefore, that for the metallic density region, the

228 BECKER, BROYLES, AND DUNN 175

TABLE I.

r s

0.0113 0.113 1.13 3.39 5 65

The ground-state correlation energies in Fermi Units.

y2

0.01 0.1 1.0 3.0 5.0

See Appendix.

E2

-0.000 0121 -0.000 85 -0.041 -0.31 -0.69

# 4

-0.0000127 -0.000 88 -0.041 -0.31 -0.69

TABLE III.

r2

0.01 0.1 1.0 3.0 5.0

Ground-

a

0.15 0.75 1.5 4.2 5.3

See Appendix.

-state wave functions u n i t s . a

b

0.20 0.33 1.25 1.4 1.5

a

0.4 2.5

in Fermi

P

10.0 8.0

two-parameter function is quite a good approxima­tion to the optimum two-body effective potential. A comparison with the resul ts of other authors which seems to substantiate this hypothesis is given in Table II. Thus we conclude that the effective potential given byEq. (3.20) may be rather good guess. The val­ues obtained for the parameters a, b, a, and p are given in Table III.

In conclusion, let us discuss one final point. It may appear strange that the superposition approx­imation has been used for g3(r19 r2, r3) while the Percus-Yevick integral equation was used instead of the Born-Green-Yvonne (BGY) integral equa­tion15?16 for the purpose of computing the radial distribution function g{r). The reason for this is

TABLE II. Comparison of ground-state correlation energies in Fermi units. a

r2

0.01 0.1 1.0 3.0 5.0

Variational method

-0.0000127 -0.00088 -0.041 -0.31 -0.69

Carr

-0.000 013 -0.00082 -0.044

Wigner

-0.034 -0.24 -0.57

Dunn and Broyles

-0.051 -0.30 -0.62

See Appendix.

that the work of Broyles, Chung, and Sahlin17 indi­cates that the Percus-Yevick equation is superior to the BGY equation for computing g(r), and a prop­er form of the Percus-Yevick equation for calcu­lating £-3 was not readily available.

APPENDIX

Fermi Units

Fermi units are based on the definition of the Fe rmi energy given by

€F=t?KF2/2m ,

where Kp= (3?r2p)l/3 .

The unit of length in these units is

aF=l/Kp .

In order that the variables of integration in this paper will be dimensionless, the following t rans ­formations are made in the energy expressions:

and K = KFx ,

where R and x a re dimensionless.

*E. Wigner, Trans. Faraday Soc. 34, 678 (1938). 2M. Gell-Mann and K. A. Brueckner, Phys. Rev. 106,

364 (1957). 3D. R. DuBois, Ann. Phys. (New York) 7, 174 (1959). 4W. J. Carr , J r . , and A. A. Maradudin, Phys. Rev.

133, A371, (1964). "~^D. Bohm and D. Pines, Phys. Rev. 92, 609 (1953).

6W. J. Carr , J r . , Phys. Rev. 122, 1437 (1961). 7L. Hedin, Phys. Rev. 139, A796 (1965). 8Tucson Dunn and A. A. Broyles, Phys. Rev. 157, 156

(1967). 9S. G. Brush, H. L. Sahlin, and E. Teller, J . Chem.

Phys. 45, 2102 (1966). 10 J. G. Kirkwood, J . Chem. Phys. 3, 300 (1935).

n F . Lado, J. Chem. Phys. 47, 5369 (1967). 12J. K. Percus and G. J . Yevick, Phys. Rev. 110, 1

(1958). 13 A. A. Broyles, H. L. Sahlin, and D. D. Car ley, Phys.

Rev. Letters 10, 319 (1963). 14T. Gaskell, Proc . Phys. Soc. (London) 7_, 1182

(1966). 15M. Born and H. S. Green, A General Kinetic Theory

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Chem. Phys. 37, 2462 (1962).