Correlation energy, structure factor, radial distribution function, and ...

PHYSICAL REVIEW B

15 JULY 1994-I

VOLUME 50, NUMBER 3

Correlation energy, structure factor, radial distribution function, and momentum distribution of the spin-polarized uniform electron gas Department

G. Ortiz of Physics, University of Illinois at Urbana Ch-ampaign,

1110 West

Green Street, Urbana, Illinois 61801

P. Ballone Institut fur Festkorperforschung, Forschungszentrum Julich, D 5ggg-5 Jiilich, Germany (Received 21 January 1994)

The properties of the three-dimensional uniform electron gas in the Fermi liquid regime are analyzed using variational Monte Carlo (VMC) and fixed-node diffusion Monte Carlo methods. Our study extends those of Ceperley [Phys. Rev. B 18, 3126 (1978)] and Ceperley and Alder [Phys. Rev. Lett. 45, 566 (1980)] to larger system sizes with improved statistics and, more importantly, to partial spin polarization. The density range 0.8 & r, & 10, which is the most relevant for density functional computations, is studied in detail. We analyze the size dependence of the simulation results and present an extended set of correlation energies extrapolated to the thermodynamic limit. Using the VMC method we analyze the spin dependence of the correlation energy, and we compare our results to several interpolation formulas used in density functional calculations. We summarize our results by a simple interpolation formula. In addition, we present results for the radial distribution function, the structure factor, the momentum distribution, and triplet correlation functions, and we discuss the comparison with many-body semianalytic theories.

I. INTRODUCTION The uniform electron gas is the simplest yet still relevant many-fermion model. Its Hamiltonian combines a quantum kinetic energy, a purely Coulombic electronelectron repulsion, and a stabilizing contribution of a homogeneous and rigid background that neutralizes the system. At zero temperature and zero magnetic field, the uniform electron gas is characterized by two parameters only, i.e., its number density p and spin polarization (, defined as the ratio of spin and number densities: I

pt

- pt

pg+

I

pg

where ptl~) is the density of spin-up (-down) electrons and p = pt + p~. We shall represent the number density by the dimensionless parameter r, defined by r, = (3/4z'p) its/ao, where ao is the Bohr radius. Despite its simplicity, this model is of great interest &om at least three points of view: It provides a first approximation to describe the valence electrons in simple metals. 2 It is the basic ingredient of density functional (DF) approximations, 3 both at the local density level and beyond. Finally, beyond the range of r, relevant for these applications, it displays a complex and fascinating phase diagram that includes the Wigner crystal, a variety of magnetic structures, and possibly more complex and ex-

otic structures. 5 These reasons, together with its role as a prototype of a many-body system, have motivated eight decades of intensive research on its static and dynamical properties. A clear and comprehensive introduction Mahan, and more details are provided

0163-1829/94/50(3)/1391(15)/$06. 00

is provided by by Singwi and 50

Tosi. We now outline some of the key developments. The first recognition of the role of statistics in determining the electron gas properties was due to Pauli and Sommerfeld, r soon followed by Bloch's evaluation of exchange energy. s The pioneering years were closed with the paper by Wigner, 4 containing both a discussion of the crystal phase and the first accurate approximation for the correlation energy e, (r, ) that is still used in density functional theory (DFT) computations. During the 1950s, the development of perturbation theory led to the random phase approximation (RPA) of Pines and Bohm and exploited by Gell-Mann and Brueckner to compute the electron gas correlation energy in the limit of high densities (r, + 0 limit). A more modern approach, based on the "equation of motion formalism, " was introduced by Singwi, Tosi, Land, and Sjolander (STLS).ii This formalism was extended to finite temperatures and applied extensively to study the thermodynamics and dynamical properties of the electron gas from the extreme quantum mechanical to the classical regime. iz The latest development, and the most relevant for our study, has been the introduction of stochastic methods [quantum Monte Carlo (QMC)] to compute the equilibrium properties of many-particle systems, applied to the electron gas problem by Ceperley and Alder. In its three main zero temperature variants [variational Monte Carlo (VMC), fixed-node (FN) diffusion Monte Carlo (DMC), and release-node difFusion Monte Carlo (RN DMC)], QMC provides a description of increasing sophistication of the many-body properties of the system. The last variant (RN DMC) is limited in principle only by statistical uncertainties (that are, however, formidable in practice) and by the size of the system that can be simu1391

Qc

1994 The American Physical Society

6. ORTIZ

1392

AND P. BALLONE

lated. The FN DMC has a further limitation in accuracy since it requires as input the location of the nodes of the electron many-body wave function, information that is Comparisons carried out in only known approximately. Refs. 14 and 15, however, show that this limitation is not a major one and FN DMC is the most practical scheme to provide very accurate estimates of the correlation energy. The VMC method, the oldest of the QMC schemes, still has an important role in providing simple estimates for e as well as a transparent scheme to analyze the importance of the different contributions to the many-body wave function. It also has an important role in providing the input for the other two more accurate schemes. The main result of the Ceperley and Ceperley-Alder computations is a series of correlation energies for different system densities that has been used widely in DF computations. In addition, few structural properties have been computed [the structure factor S(k) and the radial distribution function g(r)] and only recently appeared in the literature. is The First QMC computation of the response properties of the two-dimensional (2D) uniform electron gas has also been reported recently by Moroni, Ceperley, and Senatore. Despite the importance of their data and the interest of the method, there have been few attempts to extend the original Ceperley-Alder (CA) study and to test its accuracy. In the present paper we exploit computer and algorithm advances to extend the CA computation to larger system sizes with improved statistics and to partially spin-polarized systems. We concentrate on the density range that is the most important one for valence electron DF computations (0.8 r, 10) and we determine total and correlation energies. We study the size dependence and extrapolate to the thermodynamic limit and summarize the results in a simple interpolation formula. We perform a detailed study of structural properties, including radial correlation functions, the static structure factor, the momentum distribution, and the triplet correlation function. A preliminary account of the present study has been published in Ref. 18. When this manuscript was = 0.41 complete, we learned of a VMC study of the and uniform electron gas by Pickett and Broughton, we compare their results and ours below. Our study analyzes a wider density range and assesses reliably the size dependence of the computed quantities. Most important, we supplement the VMC results by more accurate DMC computations.

( (

(

II. THE

VARIATIONAL MONTE CARLO METHOD

We start our computation at the VMC level in the A tutorial introduction variance minimization version. to the VMC method is contained in Ref. 21. Here, mainly to fix the notation, we note that the VMC method is based on an ansatz for the many-body wave function, i.e. , the so-called trial wave function 4~, containing some free parameters. Then the expectation value for a given operator A is given by

50

" ~~( ") [' [&~~( ")]/~~(

d

(A)

)

I

=

iv

(2) ~@ ( N)~2

which has the form of an average of A4z/4z over the (non-negative) probability distribution 4~ and can therefore be evaluated easily and efficiently by Monte Carlo (MC) techniques. Following Ref. 20, we vary the free parameters in order to optimize the Buctuation of the local energy: ~

2=

dr~ ~

C&(r~)

~'

[HC

~

&(r~)/4&(r~) —E&]'

N~@(

iv)~2

where H is the system Hamiltonian and Ez is a suitWe choose to minimize e2 instead able trial energy. of the energy expectation value because, as discussed in Ref. 20, this strategy provides a balanced optimization of the wave function since it does not emphasize the energy over other observable properties. This is particularly important for our study, in which we discuss a variety of energy, momentum, and structural properties. Although more precise results will be provided by the FN DMC computations described below, we have taken particular care in the VMC step for the following reasons. (i) The VMC method is simple, but nevertheless provides useful and reliable estimates for the correlation energy. In addition, it is based on simple analytic wave functions that can be stored easily and shared with other researchers. It is also easy to analyze the correlation energy in terms of different contributions (two- and manybody terms, static and dynamic correlation, 22 etc. ). This differs from the DMC method, which is based on a long sequence of electron configurations, in most cases difficult to interpret. (ii) An accurate VMC is the prerequisite for an effiDMC computation. Although cient importance-sampled the converged total energy does not depend on the trial state 4'z (apart from its nodes which act as hard-wall boundary conditions), the number of iterations required to reach convergence and the residual total energy Huctuations does depend upon 4'~. (iii) Semianalytic computations based on a variational approach have been performed (mainly in the Fermi hypernetted chain approach ) and the VMC method allows a direct comparison with these results. In our computation we consider N = N~ + N~ electrons in a rhombic dodecahedron cell (fcc Wigner-Seitz cell) with periodic boundary conditions (PBCs). Nt~~l is the number of spin-up (-down) electrons in the cell. The nearly spherical shape of this cell mimics the isotropic environment of the Buid phase better than the simple cubic cell and has been preferred despite the slight disadvantage in imposing the PBCs during the simulation. The electron charge is neutralized by a uniform background of positive charge whose number density p~ is equal to the average electron density p.

CORRELATION ENERGY, STRUCTURE FACTOR, RADIAL. . .

The Hamiltonian

1393

is

(4) where P is the linear momentum. Atomic hartree units are used throughout the paper. We consider trial wave functions of the Jastrow type, 4 with one determinant product and explicit two-body correlations only:

4'z(ri, ..., r~) = D+ D J(ri, ..., rN) D+ and D are the electron spin-up and -down Slater determinants of single particle orbitals. The orbitals are the Hartree-Fock (HF) solutions for the finite, periodically repeated uniform electron gas, i.e. , they are constructed Rom the Ng, N~ lowest kinetic energy plane waves whose momentum label G is a reciprocal lattice vector of the simulation cell. The corresponding HF total energy is used to compute the correlation energy e, for each system size. A translationally invariant ground state requires the inclusion in the determinants of all the G's related by symmetry operations of the lattice and thus restricts the choice of the number of particles in the cell N to a well-defined sequence, dependent on the cell symmetry. In the equation above, (the Jastrow factor~4) is a symmetrical and positive definite function. Considering only two-body correlations, the Jastrow factor reads J(r ) = g,.&. exp[v;~(r;j)]. The potentials v, j(r), which are different for like and unlike spin electrons, satisfy the well known Kato cusp condition for T m 0. For an extended system the optimal v;z(r) should decay as 1/r at large distances to account for the plasmon dispersion relation. For finite systems, however, the sum over periodic replicas produces an effective two-body potential v', which has cubic symmetry in our computation. If one of the two electrons is taken as the center of the simulation box, v' approaches the boundary of the cell with zero normal gradient and is rather Hat in the interstitial region2s (see Fig. 1). To mimic this behavior without resorting to expensive evaluations of v' by the Ewald sum, we consider a finite range v;~, vanishing outside a sphere of radius Rq tangential to the unit cell: v;~(r) = 0 if r Rq. As discussed below, we have verified that this truncation involves an insignificant loss of correlation en-

J

)

ergy. Inside the sphere, the two-body potential v;~ is of the Pade form, with the desired property of being linear at small r in order to satisfy the cusp condition vzj

(r):

vzj

(r)

+ Azj

exp (

cl'&jr

)

and

T&Rg 8;q

——0

)

g

Ic=O

Pg

(r —Rs)", r

) Rg.

(6)

FIG. 1. Coulomb potential due to an electron at the center of a rhombic dodecahedron cell periodically repeated in space, evaluated by the Ewald method.

j

The fa;j are determined by the electron-electron cusp condition (1/4 and 1/2 for like and unlike spins, respectively) and (b, j, s;j, A, j, n;j and Rs are variational parameters. The (Psj are chosen in such a way that 4'z and its two first derivatives are continuous everywhere. Two extensions of the present Jastrow form, devised to include many-body correlations, are discussed in Ref. 27. The optimization of the variance cr2 with respect to the variational parameters is carried out following closely the procedure of Ref. 20. We start with a set of trial parameters and we perform a short MC run (typically 1000 MC steps per electron), during which we select a small population of independent'and representative configurations (usually 1000 configurations). Keeping fixed this representative sample, we optimize the variational parameters using a standard minimization routine. This process is repeated several times until we achieve self-consistency between the input and output parameters. Usually four or five iterations are required. In this minimization process the o~ is typically reduced by a factor of 10 with respect to the HF value. The energy is not optimal, but test runs where we minimized the energy instead of o 2 do not change the results significantly. The variational parameters 8, 6, o. have dimensions of integer power of length and their optimal values display a clear scaling with r, that is, however, far from quantitative. All the variational parameters display only a weak dependence on size. Optimal wave functions for the simulated systems listed below are available &om the authors. After optimization, we equilibrate the system by generating more than 5000 steps per electron which we discard from the random walk. We then accumulate statistics during at least 10000 steps per electron. We have verified that the finite range of v;z does not compromise the quality of our variational wave function. First of all, we repeated the computation by Ceperley for a cubic box, N = 162 electrons at r, = 1 and = O, i in which he used two wave functions (i) potential v;j (r) from the RPA and (ii) v;j(r) of the Yukawa form], both with the long-range 1/r tail evaluated by an Ewald sum. According to Ref. 13, the total energies provided by these two forms are similar. Our energy is intermediate be-

j

(

G. ORTIZ AND P. BALLONE

1394

50

TABLE I. Variational Monte Carlo correlation energies (e, ) for finite systems obtained from trial wave functions of the form of Eq. (5). e, is in eV per electron N

178 226 274

338 458

169 181 229 259 283 226 226 232

0.21 0.42 0.56

0.8

1

2

3

4

5

8

10

-1.49(1) -1.41(1) -1.42(1) -1.39(1) -1.54(1)

-1.38(1) -1.32(1) -1.34(1) -1.31(1) -1.44(1)

-1.094(4) -1.042 (4) -1.056(4) -1.041(4) -1.116(3)

-0.914(2) -0.880(2) -0.894(4) -0.881(3) -0.939(2)

-O. 793(2) -O. 771(1)

-0.782(2) -O. 771(1) -0.814(1)

-0.7143(8) -0.689(1) -0.699(l) -0.6901(8) -0.726(1)

-0.5422(5) -0.5349(5) -0.5361(5) -o.53oo(9) -0.5538(5)

-0.4733(5) -0.4623(5) -0.4686(4) -0.4645 (5) -0.4831(5)

-0.63(1) -0.68(1) -0.75(1) -0.72(1) -0.71(1)

-0.59(1) -0.63(1) -0.681(5) -0.668(5) -0.659(5)

-0.492 (3) -0.511(3) -0.550(3) -0.531(3) -0.543(3)

-0.430(1) -0.454(2) -0.476(2) -0.468(2) -0.464(2)

-0.385 (1) -0.392(2) -0.418(l) -0.4144 (8) -0.4041(8)

-0.350(1) -0.3644(8) -0.3855(8) -0.3740(8) -0.3712(7)

-0.2754(8) -0.2889(3) -0.3010(3) -0.2961 (3) -0.2939(6)

-0.2472 (7) -0.2556(5) -0.2681(5) -0.2618(3) -0.2603(5)

-1.43(1) -1.26(1) -1.24(1)

-1.35(1) -1.20(1) -1.17(1)

-1.053(4)

-0.879(2) -O. 797(2) -0.781(2)

-0.768(1) -0.700(1) -0.680(1)

-0.685(1) -0.628(1) -0.606(l)

-0.529(5) -0.490(5) -0.4?1(5)

-0.4581(5) -0.4279(5) -0.4121(5)

-0.943(4) -0.920(4)

tween the two, slightly worse than that given by the RPA potential and slightly better than that given by the Yukawa form. We conclude that the truncation of v' to a finite range does not lead to a significant loss of correlation energy, the latter being mainly determined by short-range effects. We have also verified that this approximation does not modify significantly other characteristic properties of Coulomb systems. For instance, the value of S(k)/k [S(k) is the structure factor] extrapolated to k = 0 is always close to the exact I/2m~ value, 2s where V'4mp is the plasma frequency. Finally, we note that the VMC results are supplemented by DMC computations, for which the Jastrow part of the trial wave function O'T only affects the statistical fluctuation in the result, not the average value. For these reasons we believe that the short-range character of v, ~(r) does not limit the quality of the computation. The second and most difFicult problem we analyzed is that of the size dependence of the correlation energy. Up to the sizes considered in our study (N = 458 electrons), the density of states is still far from the ideal one in the N -+ oo limit. One-electron orbitals group in shells of G vectors, separated by energy gaps. Total, kinetic, and potential energy are oscillating functions of N and the amplitude of these oscillations is comparable to e, . The correlation energy itself is much less N dependent. As we show below, the oscillation amplitude for ~ is of the order of 0.05 eV per electron for sizes in excess of 200 electrons. Although small, this finite-size error is much larger than the statistical one and is enough to blur the spin dependence of e„particularly at low val-

u„=

TABLE II. Fit parameters

bo,

bq

(. To reduce this error significantly, we have used an extrapolation scheme closely related to that proposed We performed simulations for several sysby Ceperley. tem sizes, for spin unpolarized (( = 0), fully polarized (( = 1), and partially polarized ((=0.21,0.42, 0.56) systems. The results are reported in Table I. We used these results to fit the total energy E~(r„() with the Pade form of Ref. 13 (modified to take into account the spin dependence), interpolating between the low and high density size dependence of the total energy: ues of

E (r

&)

=Eiv(r

&)

—&~(r.

&N(rs,

() = &Tiv (rs () — ~

l

N b (

) —1 DUN(r„() )

Here ATN and AU~ are the difference in HF kinetic and potential energy between the finite and infinite system:

AT~(r„() = T~(r„() —T (r„()

(10)

with a similar expression for AUiv. The functions bo(r, ) are used as and bi(r, ), together with the energies fitting variables. For each r„we include in the fit all the = 0, = 1, and the intermediate sizes simulated with reported in Table I. The best fit is obtained for n = 4 and the r, dependent 6's listed in Table II. The ability

E,

(

(

used in the extrapolation

(

of the total energy to the

N:

oc

0.8

1

2

3

4

5

8

10

4.362 6.515

3.610 5.103

0.638

-0.043 0.963

-0.051 0.659

-0.019 0.305

-0.020 0. 108

-0.027 0.080

1.914

(8)

with

limit.

bo

&)

..

CORRELATION ENERGY, STRUCTURE FACTOR, RADIAL.

50

0.62

because of the larger number of free parameters. The difference between the two extrapolations ( 0.01 eV per electron) provides an estimate of the residual uncertainty in the extrapolated correlation energies.

0.61 cd

1395

0.6

III. THE DIFFUSION MONTE

K 059

CARLO METHOD

Starting from the optimized trial wave functions (@T ) described in Sec. II, we have performed fixed-node DMC for the N = 226, = 0 and N = 229, = 1 systems. We followed the method of Ref. 31, with the inclusion of some of the improvements discussed in Ref. 32. In order to define the quantities discussed below and to describe the specific choices made in our implementation, we outline the main features of the method. The computation starts from an initial population of N~ walkers (R) in configuration space (R = rN), which is then relaxed by propagation in imaginary time until when it approximates the stationary distribution f~~(R) = @T (R) 4'o(R) (up to a normalization constant), where 4'o is the many-body wave function with lowest energy that has the same nodes as 4T. This procedure determines a rigorous upper bound to the exact fermion ground state energy. At each time step and for every configuration, we attempt to move each of the electrons by sampling the approximate Green's function (short-time approximation):

0.58

(

(

0.57 1.17

1.16 (0

~ 1.15

Cn

14

1.13 1.12 100

I

200

500

400

300 N

FIG. 2. Total energy of the electron gas at r, = 1. Solid circles, VMC computations for 6nite systems periodically repeated in space; squares, E~(r» t) —Civ(r„() from Eq. (8) (see text); full line, extrapolated value E

G(R m R

)i)—

(r„().

x exp

correlation energy, reported in Table III and discussed extensively in Sec. III, are in fair agreement with the variational results of Ref. 13. We have explored other extrapolation schemes, mainly to assess the reliability of our results. In particular, we have used the form based on Fermi liquid considerations, used in Ref. 30 for the 2D electron gas:

=E

+. bi(r„()

bT~+

b2

27

—w i EL, (R) + El, (R') 2

(i2) describing a diffusive random-walk of width ~7 with a superimposed drift of velocity Fq. In the equation above ET is a trial energy and r is the time step and we discuss appropriate values for both quantities below. We define the local gradient as F = V'@T (R)/O'T (R) and the drift Fg is given by

(r„()

The results obtained by this extrapolation

[R' —R —~Fq (R) j'

x exp

of this extrapolation to reduce the uncertainty in e, can be seen in Fig. 2. The extrapolated values for the VMC

EN

1

are very close

to those given by Eq. (9) with, however, a slightly worse TABLE III. The negative of the VMC correlation energy (in eV per electron), extrapolated to the to the values of Ceperley (Ref. 13).

N;

oo limit.

C refers

0.8

1

2

3

4

5

8

10

1.73 1.57(2)

1.60 1.45(1)

1.203 1.146(7)

0.991 0.984(7)

0.853 0.850(6)

0.754 0.748(4)

0.569 0.571(4)

0.494 0.497(5)

0.85 0.785(9)

0.79 0.716(8)

0.621 0.569(6)

0.525 0.493(6)

0.460 0.430(5)

0.413 0.398(4)

0.322 0.313(4)

0.283 0.277(5)

1396

G. ORTIZ AND P. BALLONE

which reduces to F in the limit ~ -+ energy is defined by

ET+ El, ('R) =

0. Finally, the local

(HIT)/@T —ET

~ sgn

if ~(H@T)/CT —ET~ &

' —

, (H4'7 )/@T otherwise. The definitions of Fq and EI. aim at improving the behavior of the algorithm for walkers close to the nodal surfaces, where F and (H4'7 )/4T may diverge. This is the cause of large contributions to the time-step error. Detailed balance is enforced by accepting or rejecting the move with probability

P('R m 'R',

w)

( = 0) or N~ = 200 (for ( = 1) configurations

during a long VMC run. This initial population was propagated in imaginary time with a time step ~ = 10 r, a.u. during a relaxation time v„= 1.2r, a.u. , after which no drift was apparent in the quantities we used to monitor the system evolution (such as the total energy or the radial distribution function). The initial trial energy ET was equal to the variational total energy minus a small r, -dependent quantity. During the transient period of time w„{but not during the runs to generate statistics), the value of E~ was updated periodically to approach an asymptotically stationary population. Every A7. = 0.2r, a. u. the number Xgr of configurations added (or deleted) to keep the population constant was used to estimate the

6

correction to ET according to

= min[1;A(R — i 'R', 7.)]

exp [ GET — AT]

where

~ 'R; 7-) G(R ~ R', ~)

~4 T('R') ~' G('R'

]IT(R)~'

The ratio of accepted to attempted mean-square displacements defines an effective time step 7', : accepted

(p2$I attempted which accounts for the reduced diffusion rate. After attempting the movement of all the electrons in a configuration, the time is incremented by 7, and the resulting configuration undergoes a branching process determined by

El, 'R

+ El. 'R 2

(18)

( is

99%%u&'&

a random number uniformly

(0, 1). When M

Q) Emix 0

step. Fermi statistics are enforced by the rules that forbid node crossing: When 'R and R' belong to cell pockets of the configuration space where the trial wave function 4T has opposite signs, the move is rejected and the walker is restored to 'R. The process described above is carried out for all the N~ configurations, and the total size of the population is then rescaled to the initial value by random deletion or duplication of the appropriate number of configurations. The evolution step is performed for a sufIicient number of time steps (relaxation time) to approach the optimal fixed-node distribution fFN('R) Once we have . reached fFN(R), the time evolution is continued for an additional time ~, to accumulate statistics for the quantities of in-

terest. Our practical implementation proceeded as follows. We generated the initial population, distributed according to @T ~, by selecting at random N~ ——300 (for ~

=

The updating of E~ converges to a value slightly lower than (but indeed very close to) the average of the local energy El, ('R), and statistics were accumulated during a time v; = 2r, a. u. The acceptance probability determined by Eq. (15) was always in excess of and node crossing (causing the rejection of the move) occurred rarely. These two observations show that the time step was short enough to produce only a small "finite-step error. " In order to check the residual error, we repeated the simulation for r, = 1 with a time step 7 that was half of the one used in the main computation, and we found that the difference between the two local energy averages was smaller than the error bar. The favorable behavior of the algorithm with respect to the time-step error is a consequence of the softness of the repulsive 1/r potential, which is seldom sampled at close distances because of the exchange-correlation hole. The first and most important quantity we studied is the ground state total energy, evaluated as a time average over the stationary population:

distributed in is zero, the configuration is deleted from the population. When M & 0, M copies of the configuration are added to the population for the next time where

50

+s

M,

E, (Z)

i 1 &w

the standard terminology, we call Eo '" the "mixed estimator" for the total energy. The computed values are extrapolated to the thermodynamic limit by the Pade form Eq. (9), with the coefficients determined in Sec. II. The use of this extrapolation formula in the DMC case is justified by the fact that finite-size errors are mainly determined by the discrete density of states of the finite system and are therefore nearly the same at the VMC and the DMC level of approximation. The resulting correlation energies per electron are reported in Table IV and discussed in Sec. IV. The difference between the UMC and the DMC results is quantitatively more important at low r„but it is never very large. The moderate gain in total energy achieved by the DMC method is the result of the compensation between the decrease of potential energy and the increase of kinetic energy in goFollowing

CORRELATION ENERGY, STRUCTURE FACTOR, RADIAL.

50

..

1397

TABLE IV. The negative of the DMC correlation energy (in eV per electron), extrapolated to the to the values of Ceperley and Alder as quoted in Ref. 36.

0.8 1.75 1.65(2)

1.62

0.93 0.828(9)

2

3

1.013 1.018(7)

0.872

1.53(1)

1.227 1.193(7)

0.86 0.794(8)

0.655 0.606(6)

0.546 0.508(6)

1

ing &om the VMC to the DMC method. This is refiected in the structural properties discussed in Sec. V. The small differences between the present results and those of Ref. 18 are due mainly to the extrapolation to the oo limit, which now is based on a larger set of The differences between the finite size resizes. and r, sults (i.e. , before the extrapolation) are compatible with the estimated error bar for e, of 0.01 eV per electron. Even at the DMC level, our results differ &om those of CA by more than our estimated error bars. Again, we believe that the discrepancy is mainly due to the extrapolation procedure, as suggested also by the fact that the difFerences are larger at high density, where the finite size effects are also larger. Ground state expectation values of observables other than the total energy are, in general, more difBcult to evaluate. For a general operator A that does not commute with the Hamiltonian H, the "mixed estimator" analogous to Eq. (20) can be written:

N;

(A)

mix

) .) .

A4'T ('R;) O'T

(R;)

(A)

extr

—@T )

10

0.893(5)

0.505 0.5117(5)

0.474 0.448(5)

0.422 0.403(4)

0.326 0.321(4)

0.286 0.281(5)

),

e, (r„() = e, (r„0) + a, (r. f(&) [1 —( ]

+[e,(r„1) —e, (r„0)]f (I )( the second, whose energy, is

(22)

( dependence

is that of the exchange

() = ec(r. ) o) + [e.(r»1) —ec(ra) 0)]f((),

(24)

with

I

(1 + ()4/3 + (1 f(t,') = (2,/,

Whether this is a better estimate of the "exact" expec-

tation value (A)'" = (@p A tllp)/(tlip @p) depends, of course, on the relative sign and magnitude of the linear and higher order contributions in (4p —tIIT) to (A)'". From a computational point of view, it also depends on the accuracy by which we can estimate the difFerence between the variational and mixed estimator. This point is ~

8

0.583 0.583(4)

The correlation energies computed by the DMC method and extrapolated to the thermodynamic limit by Eq. (8) have been interpolated to provide a simple input to DFT computations. We have used two different forms, that of Perdew and Wang (PW), simple and physically motivated, and that of Perdew and Zunger (PZ), ss which has the advantage of being already implemented in many of the DFT computer codes. According to the first, which is based upon the spin-interpolation expression proposed by Vosko, Wilk, and Nusair (VWN):sr

by evaluating

(~~ ~~)

5

IV. THE CORRELATION ENERGY

ee(rs,

2 (A) mix

C refers

discussed in more detail below. In the following, under the heading DMC we always report the mixed estimator. At the same time, we provide also the corresponding VMC value, from which the extrapolated estimate can be evaluated. To make the extrapolation easier, most of the computed quantities have been fitted by simple analytic functions, and the parameters of the fit are listed in the tables.

It is possible to have an expression that is accurate to second order in the difference (4'p the "extrapolated estimator":

oo limit.

0.771 0.769(4)

4

(21)

Nw

N:

~

()4/3 2)

(25)

The function a, (r, ) = 8 e, (r„0)/8( in Eq. (23) is referred to as the spin stifFness. In the PW interpolation the functions — a, (r, ), e, (r„0), and e, (r„1) are fitted to the functional form

I

g(r, ) = —2A(1+. air, ) ln

1+

2A(Pir,

+ P2rg + Psrs

+ P4r

(26) )

I

As described in Ref. 35, the parameters A, Pi, and P2 are determined by the known high density expansion of e (r„t,'). ai, Ps, and P4 are free Parameters adjusted to fit our QMC results. The optimal parameters for

e, (r„0) and (r„e1) are reported in Table V, where we list those adjusted to both the DMC and the VMC results. Both interpolations have been obtained by minimizing the square deviation from the MC data with

G. ORTIZ AND P. BALLONE

1398

TABLE V. Optimal fit parameters for the correlation energy {in hartrees) according to the Perdew-Wang interpola-

50

spin polarization

function

(=0 A

0.'y

Pi Pz

Ps p4

0.031091 0.026481 7.5957 3.5876 -0.46647 0.13354

VMC

(=1

(=0

0.015545 0.022465 14.1189 6.1977 -0.56043 0.11313

0.031091 -0.002257 7.5957 3.5876 -0.52669 0.03755

(=I 0.015545 -0.009797 14.1189 6.1977 -0.91381 0.01538

(r„()

the constraint that e, is everywhere negative and monotonic with respect to r, . Although reasonable, the fit accomplished with the PW form is not very good, neither for e, (r„0) nor for e, (r„1), with the largest deviations occurring around r, = 1. Our data are not sufBcient to determine cr, (r, ) with confidence. As we describe below, however, our results are compatible with the PW parametrization for n, (r, ) and differ from the PZ spin dependence. The PZ interpolation for e, (r„0) and e, (r„ 1) assumes the following functional form: 'Y

g(r, ) =

r, &1

I+Pi ~r, +Pzr, &

A ln

r, + B + C r, ln r, + D r,

,

r,

( 1. (27)

Again, A and B are established by the asymptotic expanat high density. The other constants are sion for e, determined by fitting the MC data with a function that is continuous and has continuous derivative at r, = 1. The optimal fit parameters are reported in Table VI. The fit provided by the PZ form for e, (r„0) and e, (r„1) is significantly better than that given by the PW equation,

(r„()

Eq. (26).

„()

is The determination of the spin dependence of e, (r particularly important for moderate values of ( (( & 0.5), which are the most relevant for local spin density (LSD) calculations. This region of parameters is also the most challenging one, since energy differences between closely values are small and dominated by finite size spaced oo with the Pade effects. Only the extrapolation to X form Eq. (9) allows one to draw a smooth and monotonic

(

as

e, (r„() —e, (r„0) e, (r„1) —e, (r„0)

tion. DMC

T(r„(), defined

(28)

We notice that the correlation energy gain on going from the VMC to the DMC method is almost the same = 0 and = 1, suggesting that T(r„() is nearly for the same when computed at the VMC or the DMC level. For this reason, and in order to save the many expensive DMC computations needed to explore the range of interest, we discuss on the basis of the VMC results. Our results for in the 0.8 & r, & 10, 0 & & 0.56 region are reported in Fig. 3, where they are compared with the PW and PZ spin functions. From the figure it is apparent that the PW spin parametrization is systematically closer to the simulation data than the PZ one. Uncertainties in the simulation data do not allow us to draw any conclusion about the accuracy of the r, dependence in the PW On the scale of the figure it is diKcult to distinguish the PW spin polarization function from the VWN one, which is therefore equally accurate in reproducing the simulation results. On the basis of the above comparisons, we recommend the use of Eq. (26) with the PW (or, equivalently, VWN) The spin dependence for LSD computations. functions e, (r„0) and e, (r„1) should, however, be computed by the PZ form with the coefBcients of Table VI, which provides a very good fit to our simulation data. In the discussion below, we use this procedure to describe the equation of state of the system. Besides being useful for DF computations, the interpolations described above provide a simple test of the consistency of our computation via the virial theorem. For a purely Coulombic system the virial theorem states

(

(

(r„()

T(r„() T(r„()

(

T(r„().

T(r„()

that

—r, dE(r,' ) = rs

2

T(r, ) + U(r, )

(29)

.

From the above expression, the kinetic energy T(r, ) and potential energy U(r, ) can be computed from the total

0.3

~

TABLE VI. Optimal fit parameters for the correlation energy (in hartrees) according to the Perdew-Zunger interpolation.

(=0 pi Pz A

B C D

DMC

-0.103756 0.56371 0.27358 0.031091 -0.046644 -0.00419 -0.00983

(=1 -0.065951 1.11846 0.18797 0.015545 -0.025599 -0.00329 -0.00300

(=0

VMC

-0.093662 0.49453 0.25534 0.031091 -0.046644 -0.00884 -0.00688

(=1 -0.055331 0.93766 0.14829 0.015545 -0.025599 -0.00677 -0.00093

0.0

0.0

0.6

function T as a function of ( for 5. Full lines, PW interpolation; dashed line, PZ interpolation. The error bar is comparable to the size of the

FIG. 3. Spin-polarization

1 &

r,

symbols.

&

CORRELATION ENERGY, STRUCrURE FACTOR, RADIAL. . .

50

TABLE VII. Comparison of kinetic (T) and potential (U) energies computed by the mixed (TDMo), extrapolated (TnMc), and virial (Tn'Mo) estimators within the DMC method. TvMc and UvMg are the VMC results. Energies are in hartrees.

r, =l

+DMC

UvMc

UDMc

energy

~S

1.1362 1.1377 1.1392 1.1401 -0.5493 -0.5529 -0.5565 -0.5556

&VM C

E(r, ) = T(r, ) + U(r, ) r8

Ur,

d[r,

the partial structure factors ogy to a binary alloy, as 1

S„,„(k) =

1399

S„„(k)are

(p„(k)p„(—k)) —/N„N„bi, , p

S(k) = where

z„= gN„/N

) z„z„S„„(k), and N

= P„N„.

S„„.

V dN„, „(r')

N„(N —6„,„) 4nr' 'I

d[r.' &(r )]

(34)

The partial and total radial distribution functions can be defined simply in terms of Fourier transforms of the Here we give the real-space definition that is used in the actual evaluation of g„„(r) during the simulation:

as

dr.

(33)

and the structure factor is the weighted average

0.1391 0.1395 0.1399 0.1412 -0.2050 -0.2065 -0.2080 -0.2085

E(r, )]

defined, by anal-

(30)

(31)

It is easy to verify that our results satisfy the relations above with remarkable accuracy both for the VMC and the DMC computations. The virial theorem is also satisfied with very good accuracy by the finite size results, provided the kinetic and exchange energies are computed by their N dependent expressions. We exploit this observation to test the reliability of the extrapolated estimator defined in Sec. III. The discussion below is based on the N = 226 spinunpolarized results. For the VMC method the expectation values of kinetic and potential energy are directly comparable to the virial estimates. The two routes to evaluate T(r, ) and U(r, ) give very similar results and confirm once again the accuracy of the computation. The comparison is particularly interesting for the DMC method, since the virial estimator is, in principle, as accurate as the total energy and therefore provides a test for the extrapolated estimator described by Eq. (22). The comparison is carried out in Table VII. Despite the accumulated uncertainties implied by the fitting and derivation with respect to r„we conclude that the extrapolated estimator indeed provides a better evaluation of T(r, ) and U(r, ) and should also be preferred in computing other properties.

6)

dN„„(r') is the number of electron pairs whose r' distance r' satisfies r —6/2 r + 6/2 and the spin is (p, , v). The average is over all the electrons of appropriate spin and over configurations, either VMC or DMC. The global radial distribution function g(r) is defined by a relation analogous to Eq. (34) with, however, where

( (

z„= N„/¹

The first point we discuss is the size dependence of these functions, which we have analyzed using the VMC results. Figure 4 displays g(r) for r, = 1, = 0, and three diferent system sizes: N=178, 338,458. It is apparent that little size dependence is observed in g(r) and, therefore, in S(k). This is true for all the simulated r, and spin polarization and is consistent with the observation that the size dependence of the total energy is mainly contained in the kinetic energy and not in the potential energy, which can be expressed as an integral over g(r) [see Eq. (41) below]. The relative insensitivity of these functions on size is underlined by the comparison with the results of Ref. 19, whose g(r), computed for systems with only N = 114 electrons, are very close to ours. In the following, we present and discuss the results for N = 226, = 0, and N = 229, = 1 as representatives of the ideal extended systems.

(

(

(

(

V. THE STRUCTURE FACTOR AND RADIAL DISTRIBUTION FUNCTION

--

N=178

— N=338

The main structural property we have computed is the static structure factor S(k) and the related radial distribution function g(r). Starting from the Fourier transform of the density of spin-p electrons

p„(k) =

) exp(ik.

r~)

(32)

N=458

r/r, FIG. 4. Size dependence of the radial distribution function g(r) (( = 0), computed by the VMC method.

G. ORTIZ AND P. BALLONE The second point we analyze is the evolution in the shape of g(r ) as correlation is included by approximations of increasing accuracy. This is illustrated in Fig. 5, which shows the spin parallel and antiparallel radial distribution functions for r, = 5, computed in the HF, VMC, and DMC methods. It is apparent that, although the VMC method already provides the major part of the improvement with respect to the HF method [especially for gag(r)], the many-body correlations brought in by the DMC method play an important role in defining the shape of g(r): The short-range depletion hole is deeper in the DMC method than in the VMC method, the rise around r/r, = I is steeper, and finally, the DMC method enhances the peak in g~~(r), which starts to develop around r, = 3 and becomes more conspicuous at larger r, . All these features are consistent with the observation in Sec. III that the potential energy decreases and the kinetic energy increases on going from the VMC to the DMC method, resulting in a moderate difference of total energy. Additional information on the manybody correlations is contained in the three-body correlation function, brieBy discussed below. The r, dependence of g(r) and S(k), computed by the mixed estimator in the DMC method, is shown in Figs. 6 and 7. The main features apparent in the figure for g(r) are the good scaling with r, of the depletion hole diameter, its progressive deepening with increasing r„and the tendency to a nonmonotonic g(r) at lower densities. All these trends are also reQected in the r, dependence of

S(k). The g(r) or S(k) of the uniform electron gas provides the key input for a variety of functionals proposed in the framework of density functional theory to describe exchange and correlation in inhomogeneous systems. To make easier the use of our results in this context, we have fitted our g(r) to a simple analytic form closely related to that discussed in Ref. 38:

g„, (u)

=I+

A+&

xexp —g ~ where u

= r/r,

+u&

u+Du'+Eu'+Fu'

2

The coeffi. cients A,

(36)

B, . . . , rI

depend on

FIG. 5. Spin parallel (lower three curves) and antiparallel (upper curves) radial distribution function for r, = 5 and 0 computed in the Hartree-Fock-method (dash-dotted line), in the variational method (dashed line), and in the diffusion Monte Carlo method (full line).

50

ins

0

0

FIG. 6. Radial distribution functions g„, (r ) (t = 0) computed by the DMC method (mixed estimator): r, = I (dotted line), r, = 3 (dash-dotted line), r, = 5 (dashed line), and r, = 10 (full line).

r,

and on the relative value of the p, v spins. Sum rules and thermodynamic definitions establish several relations between the coefficients in Eq. (36). The conditions enforced in our fit are the following: the Pauli principle,

g„„(o)=o,

(37)

Kimball relations (from Ref. 39), dgP, -(o) dp

(0)

(38)

and from Ref. 40

d'g„ „(O)

2

(&

3

d'g„, „(0)

(39)

k rs factor S(k) computed by the DMC method (mixed estimator). The r, considered and the symbols are the same as those of Fig. 6.

FIG. 7. Structure

..

CORRELATION ENERGY, STRUCTURE FACTOR, RADIAL.

50

1401

TABLE IX. Optimal 6t parameters for the VMC radial functions according to Eq. (36) (( = 0).

perfect screening,

distribution

3 —" P o

[g„„(u) —1] = —b„„,

du u

potential energy de6nition,

3 2

0

du u [g(u)

—1] = r, (U(r, )) =

' E(r )]

d[&.

dr. (41)

and the long-wavelength lim

A:-+o

limit of

S(k),

k2

du u

2

24)p

[g(u)

—1] .

(42)

0

The remaining degrees of freedom are used to 6t the simulation results for g(r) and S(k), 4i which also enter in the determination of g(r) via the thermodynamics quantities [such as U(r, )] derived from interpolation formulas [Eq. (27)]. The resulting parameters are reported in Table VIII. In Table IX we report the corresponding parameters for the VMC results, needed for the evaluation of the "extrapolated estimator" results. From the 6t it is easy to extract several quantities discussed in the framework of approximate schemes or applications of the electron gas theory. The Brst contact is with theories for the dielectric response of the electron gas. Defining the static local-field correction G(k) via the well-known relation

4x yp(k) k~

y

4m

yp(k) G(k)

lim G(k) p

= pp

fk)' I

(44)

kp

TABLE VIII. Optimal fit parameters for the DMC radial functions according to Eq. (36) (I,' = 0).

r, =l

r. =3

r, =5

r, =10

A++ A+

-1 -0.4552

-1 -0.8156

-1 -0.9536

-1 -0.9952

B++ B+

0

0

0

0.5448 -0.1295 -0.1813 0.1942 0.0032 0.0898 0.0057 -0.0461 -0.0004 0.5180 0.2562

0.5532 -0.3961 0.4209 0.2298 -0.4340 0.2206 0.1149 -0.0841 -0.0089 0.5492 0.3363

0 0.2320 -0.4557 0.9293 0.2447 -0.5384 0.2507 0.0670

E+ F++ F+ f++ 9+

r, =10

-1 -0.8698

-1 -0.9887

B++ B+

0

0

0

0

0.6740

1.0167

1.0567

0.7512 -0.5959 -0.3827 0.1667 0.0711 -0.0220 -0.0047 0.0011 0.2177 0.0669

0.6509 0.6293 0.3035 -0.2161 -0.3731 0.0064 0.1047 0.0026 -0.0089 0.1587 0.3050

0.1129 0.5442 1.3241 -0.1142 -0.9365 -0.0313 0.2305 0.0068 -0.0188 0.1711 0.3372

C++ C+ D+ D+

F++ F+ f++ f+—

where kF

-0.6099 -0.8804 0.2775 0.3029 -0.0595 -0.0383 0.0046 0.3808 0.1228

= (37r2p) i~a

is the Fermi wave vector and

(4)

~

(9.

4

)~

,

(r„())

de,

2

d'e, (r. () (, 12(" ~

(45)

drs

By making the assumption

k2+

4x yp(k,

id)

4'

(u)

yp(k,

G(k)

(46)

and computing an approximate radial distribution function via the Huctuation-dissipation theorem, it is possible to obtain the additional approximate relation lim

G(k)

= [1 —g(0)]

(47)

The simulation values for these two limits are reported = 0. Moreover, in Table X as a function of r, for it is possible to use our g(r) [or S(k)] in order to compute G(k) according to approximate theories, such as the STLS scheme (see Ref. 42 for a similar manipulation of the CA simulation results)

(

distribution

C++ C+ D++ D+ @++

r, =5

-1 -0.6611

(43)

where c(k) is the static dielectric function of the uniform electron gas and yo is the density response function computed in the free electron approximation (Lindhard function), it is straightforward to verify that the simulation results constrain the long-wavelength limit (compressibility sum rule2s) of G(k):

s

r. =3

-1 -0.3260

E++

OO

S(k)

r, =1 A++ A+

(40)

-0.0941 0.0044 0.5536 0.4099

0.0475 -0.6803 -0.1314 -0.1175 1.2968 0.8157 -0.7958 -0.2499 0.1304 0.6568 0.5163

G(k)

=

-p

(27r)' q'

or the Utsumi-Ichimaru

[S(]&-q]) —1]

(48)

(UI) interpolation

TABLE X. Short [1 —g(0)] and long (po) wavelength limits of the static local-field correction G(k) estimated by the DMC method. Qo

1

—g(0)

r. =1

r. =3

r, =5

r, =10

0.2567 0.7276

0.2722 0.9078

0.2850 0.9768

0.3079 0.9976

G. ORTIZ AND P. BALLONE

50

s=& 0

(=021

( = 0.42

(=056 1

FIG. 9. Spin dependence of the radial distribution function g(r)

k/kF FIG. 8. Local-field correction G(k) of the spin-unpolarized uniform electron gas computed by (a) the STLS relation [Eq. (47)] and (b) the UI interpolation [Eq. (48)]. In order of increasing G(k), the solid lines display the results for r, = 1, 3, 5, and 10. The dashed line is the Toigo-Woodruff approximation and the dotted line is the LDA G(k) for r, = 3.

G(k)

=A

/

x

4kF2

F

k2

4kFk

2kF ln 2kF

(

/

t'k't — C (kF)

Sty(k)

+ zg

Sgg(k)

+ 2/ztz~ St~(k),

(51)

/

+k

SNS(k)

= ztzg [Sgt(k) —Sgt (k)] (52)

(49)

—k

—» —

— [1 —g(0)]with A = 0.029 (0 & r, 15), B = — — —", = --, ', A. ", C and —, —, A, [1 g(0)] The local-field corrections obtained according to these two recipes are displayed in Fig. 8, together with the LDA local-field correction [G(k) = »(k/kF)z] and the Toigo%oodruK function. 4 Of course, the strength of the STLS and UI theories comes mainly from their self-consistency requirements and the application of Eqs. (48) and (49) out of context is not justified. However, Fig. 8 provides useful information on the shape of G(k), which is compatible with simulation results, and the comparison between diferent recipes allows us to estimate the uncertainty associated with the approximations underlying semianalytical theories. for partially spinUsing the VMC computations polarized systems, we analyze the spin dependence of the radial distribution function g(r„(). The results of the simulation for r, = 1 are reported in Fig. 9, from which it is apparent that g(r„g) is relatively insensitive

'»+

= zg

ss(k) = ztz~[z„ St&(k) + zt St~(k) -2dztzt St~(k)]

~

(k) +(B+ ssA) A/ (kF )

SNN(k)

(50)

(k )' +C — +B~ (kF) (kF) ~

(

for moderate polarizations. Only for large spin polarizations the radial distribution function significantly deviates from g(r„( = 0), crossing over to the behavior of g(r„( = 1), characterized by zero contact value and a slightly wider depletion hole. The same behavior is apparent also from the results of Ref. 19. This observation is emphasized by the transformation from the (St~, Stt, Sgg) set of functions to the numbernumber (SNN, equal to the global structure factor), spinspin (Sss), and number-spin (SNs) structure functions, defined as

to

S

z Cf}

zz C/}

krs FIG. 10. Number-number SNN (full line), spin-spin Sss SNs (dotted line) partial (dashed line), and number-spin structure factors computed by the DMC method (mixed estimator).

CORRELATION ENERGY, STRUCTURE FACTOR, RADIAL.

50

..

1403

such as the Kirk-

on gs. Second, a simple approximation wood superposition (for = 0)

(

=

g3(rl r2 r3)

gpss(ri2) gZ't(ris) g&Z(r23)

+-,' g~~(r»)

g~i(r») gt~(r»)

(54)

provides a rather good representation of g3. This is surprising, when one considers that this approximation is based on weak coupling, classical statistical mechanics.

VI. MOMENTUM DISTRIBUTION FIG. 11. Three-body correlation function g3(ri, rs, rs) for three points describing an equilateral triangle (r, = 10 and ( = 0). Full line, DMC computation; dash line, VMC computation; dash-dotted line, Kirkwood approximation [Eq. (53)].

In our simulation we have computed the one-body density matrix p(r), defined by

..., r;+ r, .. ., rg) pr)= 4(ri, 4(ri, ... 1';, ... and its Fourier transform, tion p(q):

z„=

where N„/N. A related transformation can be defined for the radial distribution function. The transformation described above is the analog of the "numbercomposition" description defined in the theory of classical liquid alloys, 4s and is related to the "number-charge" transformation used for classical Coulomb liquids. 4s As in the classical case of systems with strong short-range order, the transformation shows that among the three correlation functions, only the first two (i.e. , SNN and Sss) are relevant, while the last one (SNs) is negligible for all but the largest values of (see Fig. 10). As a last point of this section, we discuss the threebody correlation function gs(ri, rz, rs), defined by

(

gs(ri, rz, rs)

= I'(ri, r2, rs) Vp

I'(ri, r2, rs) is the average number of three electron clusters whose geometry is (ri, r2, rs) and bV is the volume in the configurational space associated with this configuration. In Fig. 11 we display g3 computed by the VMC and DMC methods as a function of distance for three points (ri, r2, rs) describing an equilateral triangle and we note two features. First, the many-body correlations sampled by the DMC method have a sizable effect where

P(q) with the normalization

d«' 0

tp(q)

=

,

,

P(q

=

dr

e iq r

re)

i.e. , the momentum

distribu-

i' .. r;+ r, ... rN) "(r„. @(ri, ",ri~ "~ rN) ,

,

In this definition, an average over the central particle i is implied. These two quantities have been computed by selecting at random M independent configurations generated by the VMC or the DMC method, displacing each of the electrons in turn by the vector r, and averaging over the contributions so obtained. The sampling in r needed to define p(r) or to compute the integral to transform to p(q) has been computed by selecting at random Q intermediate positions r+r, within the simulation box. In our computation we took )4=300 and Q=300. The results for p(q) computed by the DMC method at four different values of r, are reported in Fig. 12. Again, we analyzed the size dependence by the VMC method and found that it is only weak, apart from the fact that with increasing N the sampling in q is more dense, and the discontinuity at the Fermi vector Ic~ is increasingly well defined. We also notice that there is little change in going from the VMC to the DMC method, the difference between the two sets of computations being comparable to the statistical error. The results of the simulation for p(q) have been fitted to the simple formula

A+ B (q/k+)'+ C (q/k&)'+ D (q/k+)',

q & k&

&

, D (kF/q)'

q) kF

+ E (kF/q)

constraint

—H(k~ —q)l = o,

where O(q) is the unit step function. The optimal fit parameters are reported in Table XI. Prom the fit several quantities can be evaluated, the most important of which is the quasiparticle renormalization factor at the Fermi surface Z~, also reported in Table XI.

(56)

(57)

VII. SUMMARY AND CONCLUSIONS and diffusion Monte Carlo methods, a systematic study of the properties of the homogeneous electron gas in the density range 0.8 & r, & 10 and for several values of the spin polarization ((=0, 0.21, 0.42, 0.56, 1). Our simulations cover the size range kom N = 169 to N = 458 electrons. We have analyzed in detail the size dependence

By variational

we have performed

(

G. ORTIZ AND P. BALLONE

50

TABLE XI. Optimal fit parameters for the DMC momentum distribution (( = 0) according to Eq. (56). Zs is the quasiparticle renormalization factor at the Fermi surface. The RPA value is from Ref. 1.

go gg

0

0.5

0.0

8NI~

1.0

~

1.5

F

FIG. 12. Momentum distribution of the spin-unpolarized uniform electron gas computed by the DMC method. Solid circles, r, = 1; empty circles, r, = 3; empty squares, r, = 5; solid squares, r, = 10.

N:

of the correlation energy and performed an extrapolation to the oo limit. Our results show that, although the VMC method with explicit two-body correlations provides a good description of the system energy, the remaining many-body contributions brought in by the DMC method are important and necessary to give an accurate evaluation of the correlation energy. The role of the DMC model is even more important for the structural properties such as the radial and three-body correlation function and the structure factor. The correlation energies estimated by the DMC method have been interpolated by two simple analytic formulas, that of Perdew and Wang and that of Perdew and Zunger. Our VMC data for partially polarized systems support the validity of two (equivalent) accurate spin functions T(r„(), i.e. , again that of Ref. 35 and the Vosko-Wilk-Nusair function 7 which, therefore, are recommended for local spin density computations. In Secs. V and VI of the paper we have discussed several structural properties (the structure factor, the radial distribution function, and the three-body correlation function) and the momentum distribution of the homogeneous electron gas. Besides having intrinsic interest, these functions provide a bridge to the response properties of the system, via sum rules and approximate relations [we have discussed in some detail the static localfield correction G(k)j. Moreover, they provide the input

G. D. Mahan, Many Particle Ph-ysics (Plenum, New York, 1991), Chap. 5. N. W. Ashcroft and N. D. Mermin, Solid State Physics (Holt-Saunders Editions, New York, 1976). R. O. Jones and O. Gunnarsson, Rev. Mod. Phys. 61, 689

(1989).

E. P. Wigner, Phys. Rev. 46, 1002 (1934); see Solid State Phys. 1, 367 (1955).

gF zRPA

q/kp

also D. Pines,

K. Moulopoulos and N. W. Ashcroft, Phys. Rev. Lett. 69,

r, =1

r. =3

r, =5

r, =10

1.0000 -0.0510 -0.0178 0.0364 -0.0017 0.0253

0.9806 -0.0725 -0.0041 -0.0171 0.3411 -0.2968

0.9828 -0.2146 0.0466 0.0040 0.4712 -0.3728

0.9125 -0.2799 -0.0338 0.0894 0.9063 -0.7955

0.9451 0.859

0.8425 0.700

0.7204 0.602

0.5774

for refined approximations to the exchange-correlation energy within density functional theory. To make easier the use of our data in this context, we have interpolated our results by simple expressions. Additional data (especially for partially or fully polarized systems) are available from the authors. The last information we should like to provide is the = 0, size of the computations. For lV' = 226 and 10 VMC steps require 5000 sec on the Cray- YMP One DMC simulation requires supercomputer. 50 h on the Cray-YMP. The total simulation, therefore, required about 1000 h of a supercomputer, which is comparable to the cost of large scale DFT or quantum chemistry computations.

(

ACKNOW'LEDGMENTS This work was supported by National Science Foundation Grant No. DMR-91-17822. The computations were performed using the CRAY-YMP at the National Center for Supercomputing Applications. We gratefully acknowledge several useful discussions with D. M. Ceperley and C. Umrigar. We are indebted to Gaetano Senatore for a useful clarification concerning the local-field corrections. We would like to thank R. O. Jones for his careful We also acknowledge kind reading of the manuscript. hospitality and several discussions with visiting and resident scientists at the Centre Europeen de Calcul Atomique et Moleculaire in Paris.

2555 (1992)) 70, 2356(E) (1993). K. S. Singwi and M. P. Tosi, in Solid State Physics, edited by H. Ehrenreich, F. Seitz, and D. Turnbull (Academic, New York, 1981), Vol. 36, p. 177. W. Pauli, Z. Phys. 41, 81 (1927); A. Sommerfeld, Naturwissenschaften 15, 825 (1927). F. Bloch, Z. Phys. 57, 545 (1929). D. Pines and D. Bohm, Phys. Rev. 85, 338 (1952). M. Gell-Mann and K. A. Brueckner, Phys. Rev. 106, 364

CORRELATION ENERGY, STRUCTURE FACTOR, RADIAL.

SO

(1957). K. S. Singwi, M. P. Tosi, R. H. Land, and A. Sjolander, Phys. Rev. 1'76, 589 (1968). S. Ichimaru, Rev. Mod. Phys. 54, 1017 (1982). D. M. Ceperley, Phys. Rev. B 18, 3126 (1978). D. M. Ceperley and B. J. Alder, Phys. Rev. Lett. 45, 566 (1980). D. M. Ceperley, in Recent Progress in Many-body Theories, edited by J. G. Zabolitsky (Springer, Berlin, 1981), p. 262. J. P. Perdew and Y. Wang, Phys. Rev. B 46, 12947 (1992). S. Moroni, D. M. Ceperley, and G. Senatore, Phys. Rev. Lett. 69, 1837 (1992). G. Ortiz and P. Ballone, Europhys. Lett. 23, 7 (1993). W. E. Pickett and J. Q. Broughton, Phys. Rev. B 48, 14859 (1993). C. J. Umrigar, K. G. Wilson, and J. W. Wilkins, Phys. Rev. Lett. 60, 1719 (1988).

J. W. Negele

and H. Orland, Quantum Many Partic-le Sys tems (Addison-Wesley, Redwood City, CA, 1988). R. E. Watson, Ann. Phys. (N. Y.) 13, 250 (1961); I. Shavitt, in Advanced Theories and Computational Approaches to the Electronic Structure of Molecules, edited by C. E. Dyk-

stra (Reidel, Dordrecht, 1984). L. J. Lantto, Phys. Rev. B 22, 1380 (1980). R. Jastrow, Phys. Rev. 98, 1479 (1955). T. Kato, Commun. Pure Appl. Math. 10, 151 (1957). J. P. Hansen, Phys. Rev. A 8, 3096 (1973). G. Ortiz and P. Ballone (unpublished). D. Pines and P. Nozieres, The Theory of Quantum Liquids (Addison-Wesley, Redwood City, CA, 1989). The amplitude of the oscillations for the kinetic energy is one order of magnitude larger than that for the potential energy.

B. Tanatar

and D. M. Ceperley,

(1989). P. J. Reynolds,

Phys. Rev. B

D. M. Ceperley,

B. J.

39, 5005

Alder,

and

..

1405

W. A. Lester, Jr. , J. Chem. Phys. 77, 5593 (1982). C. J. Umrigar, K. J. Runge, and M. P. Nightingale,

in Monte Carlo Methods in Theoretical Physics, edited by S. Caracciolo and A. Fabbrocini (ETS, Pisa, 1990), p. 161. C. J. Umrigar, M. P. Nightingale, and K. J. Runge, J. Chem. Phys. 99, 2865 (1993). D. M. Ceperley and M. H. Kalos, in Monte Carlo Methods in Statistical Physics, edited by K. Binder (SpringerVerlag, Berlin, 1979), p. 145; P. J. Reynolds et aL, Int. J. Quantum Chem. 29, 589 (1986). J. P. Perdew and Y. Wang, Phys. Rev. B 45, 13 244 (1992). J. P. Perdew and A. Zunger, Phys. Rev. B 23, 5048 (1981). S. H. Vosko, L. Wilk, and M. Nusair, Can. J. Phys. 58,

1200 (1980). S. Ichimaru, H. Iyetomi, and S. Tanaka, Phys. Rep. 149, 92 (198?). J. C. Kimball, Phys. Rev. A 7, 1648 (1973); J. Phys. A 8, 1513 (1973). A. K. Rajagopal, J. C. Kimball, and M. Banerjee, Phys. Rev. B 18, 2339 (1978).

The structure factor corresponding to the interpolation formula Eq. (36) is computed numerically via a fast Fourier transform since the odd power terms in g(r) do not have a closed form analytic transform. G. Senatore and G. Pastore, Phys. Rev. Lett. 64, 303

(1990). K. Utsumi and S. Ichimaru, Phys. Rev. B 22, 5203 (1980); S. Ichimaru and K. Utsumi, ibid. 24, 7385 (1981); K. Utsumi and S. Ichimaru, Phys. Rev. A 26, 603 (1982). F. Toigo and T. O. Woodruff, Phys. Rev. B 2, 3958 (1970). A. B. Bathia and D. E. Thornton, Phys. Rev. B 2, 3004 (1970). J. P. Hansen and I. R. McDonald, Theory of Simple Liquids (Academic, London, 1976). Our programs run on a workstation and are efficiently vectorized on the Cray YMP supercomputer.