Uniform electron gas from the Colle-Salvetti functional: Missing long-range correlations

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Uniform electron gas from the Colle-Salvetti functional: Missing long-range correlations

Jianmin Tao,¹Paola Gori-Giorgi,^1,2John P. Perdew,¹and Roy McWeeny³

1Department of Physics and Quantum Theory Group, Tulane University, New Orleans, Louisiana 70118

2Unita` INFM di Perugia, Via A. Pascoli 1, Perugia 06123, Italy

3Dipartimento di Chimica e Chimica Industriale, Universita` di Pisa, Via Risorgimento 35, Pisa 56100, Italy 共Received 7 September 2000; published 12 February 2001兲

Colle and Salvetti关Theor. Chim. Acta 37, 329 共1975兲兴 approximated the correlation energy of a many- electron system as a functional of the Hartree-Fock one-particle density matrix. The most fundamental and least approximate version of this functional关their Eq. 共9兲兴 is found to yield only 25% of the true correlation energy of a uniform electron gas, and not 100% as previously believed. While short-range correlations are described surprisingly well by this approach, important long-range correlations are missing. Such correlations are energetically negligible in atoms, but cannot be ignored in more extended systems, including solids as well as molecules.

DOI: 10.1103/PhysRevA.63.032513 PACS number共s兲: 31.15.Ew, 71.10.Ca, 71.15.⫺m, 31.25.Eb I. INTRODUCTION

The correlation energy is the correction to the Hartree- Fock energy of a many-electron system. Starting from a pair- correlated many-electron wave function, Colle and Salvetti 关1,2兴 approximated the correlation energy as a functional of the one-particle Hartree-Fock density matrix ␳1

HF(r,r⬘^).

Later Lee, Yang, and Parr关3兴 transcribed the Colle-Salvetti 共CS兲 approximation into a widely used gradient-corrected functional of the electron density n(r). The CS work also provided a motivation for more recent density functionals that employ the orbital kinetic energy density 关4兴.

An early study by McWeeny关5兴 found that the most fundamental and least approximate form of the CS approach, Eq. 共9兲 of Ref. 关1兴, gave an accurate correlation energy per electron for the electron gas of uniform density, supporting the underlying CS approach. Encouraged by this result, Ra- jagopal, Kimball, and Banerjee 共RKB兲关6兴 made a CS-like ansatz for the pair-distribution function of a magnetic electron gas; long-range correlations were later built into the RKB model by Contini, Mazzone, and Sacchetti关7兴.

In the present work, however, we have not been able to reproduce the original result of Ref.关5兴. We find instead that Eq. 共9兲 of Ref. 关1兴 underestimates the magnitude of the correlation energy by about a factor of 4. Our result is conso- nant with a recent critique by Singh, Massa, and Sahni关8兴 of the CS wave function for the two-electron atom, which in particular points out that it is not normalized.

It is widely known that the CS correlation energy functional is accurate for smaller atoms关9,10兴, but not for molecules where it misses important long-range correlations 关2,11,12兴 which must be described by other approaches such as the configuration interaction method. We show here that the electron gas also has energetically important long-range correlations which are missed by all variants of the CS correlation energy functional. In both the electron gas 关13,14兴 and molecules关15–17兴, the exact correlation hole has a long- range part which is approximately cancelled by the long- range part of the exact exchange hole, an effect which could have been but was not included in the CS approach.

II. ELECTRON PAIR DENSITIES, ETC.

We begin with a review of definitions 关18兴. Given an N-electron wave function ⌿(r1␴1,...,r_N␴N), we define the pair density

␳2共r,r⬘兲⫽N共N⫺1兲

⫻_␴

兺

i⫽1

N

冕

j

^兿

⫽3 N

dr_j兩⌿共r␴1,r⬘␴2,...r_N␴N兲兩², 共1兲 the one-particle density matrix

␳1共r,r⬘兲⫽N_␴

兺

i⫽1

N

冕

^⌿^*^共r^␴¹^,r²^␴²^,...,r^N^␴^N^兲

⫻⌿共r⬘␴1,r₂␴2,...r_N␴N兲dr2...dr_N, 共2兲 and the electron density

n共r兲⫽␳1共r,r兲⫽ 1

N⫺1

冕

^dr^⬘^␳²^共r,r^⬘^兲. ^共3兲

While n(r)dr is the probability of finding an electron in dr,

␳2(r,r⬘^{)dr dr}⬘ is the probability of finding one electron in dr and another in dr⬘. We also define the pair-distribution function g(r,r⬘^):

␳2共r,r⬘兲⫽n共r兲n共r⬘兲g共r,r⬘兲. 共4兲 By integrating Eq.共1兲 over r⬘, we find that

冕

^dr^⬘ⁿ^共r^⬘^{兲关g共r,r}^⬘^{兲⫺1兴⫽⫺1.} ^共5兲

In other words, the density n(r⬘⁾关g(r,r⬘⁾⫺1兴 of the exchange-correlation hole around an electron at r represents a deficit of one electron.

The Hartree-Fock wave function ⌿^HF(r₁␴1,...,r_N␴N) is the energy-optimized single-determinant approximation to

⌿, and the correlation energy is

E_c⫽具⌿兩Hˆ兩⌿典⫺具⌿^HF兩Hˆ兩⌿^HF典^, 共6兲

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where Hˆ is the Hamiltonian. The Hartree-Fock density is often close to the true density n(r), and for the uniform electron gas these densities are identical. For simplicity, we shall assume n(r)⫽n^HF(r) in the equations that follow.

Then, in Hartree atomic units,

E_c⫽

冕

^dr

冕

^dr^⬘

冉

^⫺¹²^ⵜ²^r^⬘

冊

^关^␳¹^共r,r^⬘^兲⫺^␳¹^HF^共r,r^⬘^兲兴

⫹1

2

冕

^dr

冕

^dr^⬘兩r⫺r¹ ⬘兩关␳2共r,r⬘兲⫺␳2

HF共r,r⬘兲兴, 共7兲

where the first term is the kinetic energy of correlation T_c and the second is the potential energy of correlation V_c. Moreover, for a spin-unpolarized or closed-shell system

␳2

HF共r,r⬘兲⫽n共r兲n共r⬘兲⫺1 2兩␳1

HF共r,r⬘兲兩², 共8兲

␳1

HF共r,r⬘兲⫽2

兺

_␣ ^␺^␣^HF^*^共r^⬘^兲^␺^␣^HF^共r兲, ^共9兲

where the␺_␣^HF(r) are the occupied Hartree-Fock orbitals.

Now we introduce the correlation contribution to the pair- distribution function

g_c共r,r⬘兲⫽g共r,r⬘兲⫺g^HF共r,r⬘兲. 共10兲 Clearly

V_c⫽1

2

冕

^dr

冕

^dr^⬘ⁿ^共r兲n共r兩r⫺r^⬘^兲g⬘^c兩^共r,r^⬘^兲. 共11兲 At least for the uniform gas of fixed density n, we can also write关19兴

E_c⫽1

2

冕

^dr

冕

^dr^⬘ⁿ^共r兲n共r兩r⫺r^⬘^兲g¯⬘^c^兩^共r,r^⬘^兲^, ^共12兲 g

¯_c共r,r⬘兲⫽

冕

0 1

d␭ gc␭共r,r⬘兲, 共13兲

where g_c^␭ is the correlation contribution to the pair- distribution function for electron-electron interaction ␭/

兩r⫺r⬘兩. The coupling constant␭ varies from 0 共the Hartree- Fock wave function兲 to 1 共the physical wave function兲. Since Eq.共5兲 must hold for every ␭ 共including the ␭⫽0 or Hartree- Fock limit兲

冕

^dr^⬘ⁿ^共r^⬘^兲g^c^␭^共r,r^⬘^兲⫽0. ^共14兲

Typically g_c^␭(r,r⬘) is negative for small u⫽兩r⬘⫺r兩, and positive for large u. Comparison of Eqs. 共11兲 and 共12兲 with Eq.共14兲 then suggests that, as gc␭becomes more long ranged in u, it must also become deeper at small u, and E_c and V_c must become more negative. The Coulomb cusp condition, which follows from the dominance of␭/兩r⬘^{⫺r兩 as r}⬘^{→r, is}

d

dug^␭共r,r⫹u兲兩u⫽0⫽␭g^␭共r,r兲. 共15兲

III. COLLE-SALVETTI APPROACH

With the concepts of the previous section in mind, consider the antisymmetric CS wave-function 关1兴 for a spin- unpolarized system

⌿^CS共r1␴1,...,r_N␴N兲⫽⌿^HF共r1␴1,...r_N␴N兲

⫻

兿

_i_{⬎ j} ^关1⫺^␸^共rⁱ^,r^j^兲兴, ^共16兲

where

␸共r,r⬘兲⫽exp关⫺␤²共R兲u²兴

再

¹^{⫺⌽共R兲}

冉

¹^⫹^u²

冊冎

^共17兲

is a Jastrow correlation factor. Here u⫽兩r⬘⫺r兩 and R⫽(r

⫹r⬘)/2. Equation共17兲 has a cusp at u⫽0, and vanishes as u→⬁. The inverse radius of the correlation hole is chosen to be

␤共R兲⫽qn^1/3共R兲, 共18兲

where q⫽2.29 from a fit of Eq. 共19兲 of Ref. 关1兴 to the correlation energy of the He atom. For atoms, this fit partially compensates 关8兴 for the CS neglect of Tcand other approximations.

Having defined the CS approximation to the wave function, we must discuss the CS approximation to the correlation energy. First, Colle and Salvetti 关1兴 assumed that

␳1(r,r⬘⁾⫽␳1

HF(r,r⬘), making T_c⫽0; this is a much more doubtful approximation than the assumption n(r)⫽n^HF(r), as we shall see. Second, they argued that␸is small because of strong damping by the Gaussian function in Eq. 共17兲, so that

␳2

CS共r,r⬘兲⫽␳2

HF共r,r⬘兲关1⫺␸共r,r⬘兲兴², 共19兲

which is properly positive and satisfies the cusp condition of Eq.共15兲 for ␭⫽1. The corrections to Eq. 共19兲 that arise when

␸ is not small, but which vanish in two-electron systems, have been discussed by Soirat, Flocco, and Massa关20兴. Thus we arrive at Eq. 共9兲 of Ref. 关1兴,

E_c^CS⫽1

2

冕

^dr

冕

^dr^⬘^␳²^HF^共r,r^⬘^兲^关^␸²^共r,r^⬘兩r^兲⫺2⬘^⫺r兩^␸^共r,r^⬘^兲兴^, 共20兲 which via Eq.共8兲 expresses the correlation energy as a functional of the Hartree-Fock one-particle density matrix.

Equation 共20兲 is the most fundamental level of Colle- Salvetti theory, and the one tested for the uniform gas by McWeeny 关5兴. We note that several further approximations are made to arrive at Eq. 共19兲 of Ref. 关1兴, which has been used to estimate the dynamical correlation energy for mol- ecules关2兴 and is the basis of the Lee-Yang-Parr density functional 关3兴.

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To complete Eq. 共20兲, Colle and Salvetti had to find an expression for⌽(R) in Eq. 共17兲. To do so, they wrote down the equation n(r)⫺n^HF(r)⫽0, with n(r) constructed from the right-hand side of Eq. 共3兲. The result, Eq. 共10兲 of Ref.

关1兴, imposes the constraint of Eq. 共14兲 for ␭⫽1, as pointed out in Ref.关8兴. However, the resulting integral equation for

⌽(R) cannot always be solved explicitly without further ap- proximation. Colle and Salvetti assumed that the density n varies little over the range of ␸, an assumption which is error-free for the uniform gas. Thus they could have obtained an algebraic quadratic equation for ⌽(R) 共the ‘‘full qua- dratic equation’’兲, which could have been solved exactly.

Instead they made a further approximation to simplify the coefficients of this quadratic equation, solved it in the high- and low-density limits, and interpolated between these limits with the simple formula

⌽共R兲⫽

冑

␲␤共R兲/关1⫹

冑

␲␤共R兲兴. 共21兲

IV. UNIFORM ELECTRON GAS USING EQ.„20…

Now let us consider an electron gas with a uniform density

n⫽3/共4␲^rs 3兲⫽kF

3/共3␲²兲. 共22兲

The Hartree-Fock pair-distribution function is

g^HF共y兲⫽1⫺1

2关3共sin y⫺y cos y兲/y³兴², 共23兲 where y⫽kFu. The exact correlation energy per electron is

⑀c共rs兲⫽tc共rs兲⫹vc共rs兲⫽

冕

0

⬁

du 2␲^nug^¯c共rs,u兲, 共24兲 where the potential energy of correlation is

vc共rs兲⫽

冕

0

⬁

du 2␲^nugc共rs,u兲. 共25兲

The functions⑀candv_care accurately known关21兴, as are the functions g¯_cand g_c关14,22,23兴, from a combination of theo- retical constraints and diffusion Monte Carlo simulations.

The CS approximation of Eq.共20兲 for the uniform gas is

⑀c

CS共rs兲⫽vc

CS共rs兲⫽

冕

0

⬁

du 2␲^nugc

CS共rs,u兲, 共26兲

g_c^CS共rs,u兲⫽g^HF共kFu兲关␸²共rs,u兲⫺2␸共rs,u兲兴, 共27兲

where␸is given by Eqs.共17兲, 共18兲, and 共21兲.

Figure 1 shows that⑀c

CSis only about¹₄of the true⑀c, and only about ¹₆ of the truev_c, for the range of valence electron densities 0.5ⱗrsⱗ10. As a check of our evaluation of the integral of Eq. 共26兲, two of us wrote independent computer programs which gave the same answer. Figure 1 also shows an analytic parametrization of⑀c

CSby McWeeny关5兴, which is close to ⑀cbut not to⑀c

CS.

The real-space analysis of the correlation energy is the integrand of Eq.共24兲 or Eq. 共25兲, since⑀cis the area under the curve of 2␲^nug^¯c(r_s,u) vs u. For r_s⫽3, Fig. 2 compares the CS and ‘‘exact’’ real-space analyses. At small u,g_c^CSimi- tates g_c, but g_c^CSis of much shorter range in u than either g_c or g¯_c.

Figure 3 shows the correlation contribution to the on-top (u⫽0) pair-distribution function, plotted as a function of rs.

FIG. 1. Correlation energy per particle of the uniform electron gas in three dimensions, as a function of the density parameter rsof Eq. 共22兲. ⑀c is the total correlation energy and v_c the potential energy of correlation. We compare Eq. 共9兲 of CS 共Ref. 关1兴兲 with McWeeny’s关5兴 parametrization thereof and with ‘‘exact’’ PW values from Ref.关21兴.

FIG. 2. Real-space analysis of the correlation energy of Fig. 1 for r_s⫽3. The area under each curve is the corresponding correlation energy, according to Eqs.共24兲–共26兲. The ‘‘exact’’ PW values are from Ref.关14兴.

FIG. 3. On-top pair-distribution function g_c(r_s,u⫽0) for the uniform electron gas, as a function of rs.

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We see that g_c^CS(r_s,u⫽0)⫽2¹关⌽²(R)⫺1兴 imitates gc(r_s,u

⫽0) in the high density (rs→0) limit. Why this should be so, in view of the incorrect long-range behavior of g_c^CS(r_s,u) for the uniform electron gas, is something we do not under- stand.关Since in an atom gc(r,r) is determined mainly by the local spin densities at r 关24兴, a roughly correct gc

CS(r_s,u

⫽0) might be inherited from the fit to the He atom.兴 Also of interest is the Fourier transform

S_c共rs,k兲⫽

冕

0

⬁

du 4␲^u²^ngc共rs,u兲sin ku

ku . 共28兲 We find

v_c共rs兲⫽ 1 共2␲兲³

冕

0

⬁

dk 4␲^k²⁴␲ k²

1

2S_c共rs,k兲, 共29兲

which decomposes the correlation energy into contributions from dynamic density fluctuations of various wave vectors k.

Figure 4 shows the wave-vector-space analysis of the corre- lation energy for r_s⫽3. The large-u error of gc

CS(r_s,u) mani- fests here as a small-k error.

By Eq.共14兲 we should find

lim

k→0

S_c共rs,k兲⫽

冕

0

⬁

du 4␲^u²^ngc共rs,u兲⫽0. 共30兲

The inset to Fig. 4 shows that the CS correlation does not exactly satisfy this condition, except in the limit r_s→0. In fact, the CS correlation hole contains a few hundredths of an electron. We can easily fix this problem, at least for the uniform gas, by replacing Eq.共21兲 by the exact solution of the full quadratic equation for⌽; this makes the CS correlation energy more negative than the CS curve in Fig. 1共by 45% at r_s⫽0, and by 75% at rs⫽10), but still far from the exact⑀c. The most serious error in Eq. 共20兲 for the uniform electron gas clearly arises from the strong Gaussian damping of

␸ ^{in Eqs.}共17兲 and 共18兲. The correct long-range behavior of g_c(r_s,u) in the electron gas is ⬃u^⫺4 关14,22兴. To achieve this from the wave function of Eq. 共16兲 requires 关25兴 that

␸⬀u^⫺1at large u. When␸decays this slowly, the justifica- tion for Eq. 共19兲 is also lost. In fact, if ␸⬀u^⫺1 at large u,

then Eq. 共19兲 would wrongly predict gc⬀u^⫺1 at large u, a behavior which is not only wrong but manifestly inconsistent with Eq.共14兲.

In summary, key ingredients of the Colle-Salvetti functional 关Eq. 共9兲 of Ref. 关1兴, or Eq. 共20兲 of this article兴 are inappropriate to the uniform electron gas: the strong Gauss- ian damping in Eqs.共17兲 and 共18兲, and the form of the pair density in Eq. 共19兲. For the He atom, on the other hand, Soirat et al. 关20兴 evaluated the same functional and found E_c^CS⫽⫺0.038 hartree, close to the exact ⫺0.042 hartree.

Without a long-range part, the CS real-space analysis for the correlation energy共Fig. 2兲 cannot describe the uniform electron gas, although it describes the He atom better 共Fig. 5 of Ref. 关26兴兲. While the long-range part of the CS gc is also seriously wrong in an atom 关8兴, this error is energetically important for the electron gas, and not for the atom 共where the probability of finding two electrons far apart is very low兲.

V. UNIFORM ELECTRON GAS USING OTHER COLLE-SALVETTI EXPRESSIONS

Thus far, we have only been concerned with our Eq.共20兲关Eq. 共9兲 of Ref. 关1兴 or Eq. 共6.10兲 of Ref. 关5兴兴. The double integral over r and r⬘ makes evaluation of this expression cumbersome for any nonuniform density. Colle and Salvetti 关1兴 therefore changed variables to R⫽(r⫹r⬘^{)/2 and} u⫽r⫺r⬘^, ^expanded ␳2

HF(R⫺u/2,R⫹u/2)⬇¹2 n²(R)关1

⫹K(R)u²/6兴, and performed the u integration analytically to obtain Eqs.共15兲 and 共16兲 of Ref. 关1兴, which express Ecas an integral over R alone of a function of n(R) and K(R).

For the uniform electron gas, where K⫽6kF

2/5, Eqs.共15兲 and 共16兲 of Ref. 关1兴 yield a correlation energy per electron

⑀c⫽⫺ 0.02209⫹0.00642rs

1⫹0.79431rs⫹0.1577rs

2, 共31兲

which is very close to the ‘‘CS Eq.共9兲’’ curve in our Fig. 1.

Amaral and McWeeny 关11兴 corrected Eq. 共16兲 of Ref. 关1兴, but the correction to Eq.共31兲 (0.00642→0.00432) is small.

Although Eqs.共15兲 and 共16兲 of Ref. 关1兴 seem to produce the desired simplification of Eq. 共9兲 of Ref. 关1兴, Colle and Salvetti replaced the function of n(R) and K(R) in their Eq.

共16兲 by another function of the same variables in Eq. 共19兲 of Ref. 关1兴, the final form of the CS functional. The new function was chosen phenomenologically to yield a good approximation to the correlation energy density ⑀c(R) of the He atom constructed from Eq. 共9兲 of Ref. 关1兴. For the uniform electron gas, Eq. 共19兲 of Ref. 关1兴 yields a correlation energy per electron

⑀c⫽⫺0.04918⫹0.01863e^⫺0.40828r^s 1⫹0.56314rs

, 共32兲

which surprisingly is about twice the ‘‘CS Eq.共9兲’’ curve of our Fig. 1. The Lee-Yang-Parr 共LYP 关3兴兲 density functional for the correlation energy is found from Eq.共19兲 of Ref. 关1兴 by making a density-gradient expansion of K(R). Since this expansion is exact for a uniform density, Eq.共32兲 is also the FIG. 4. Wave-vector-space analysis of the correlation energy of

Fig. 1 for r_s⫽3. The area under each curve, multiplied by 2kF/␲, gives the corresponding energy in hartree. The inset shows the CS violation of the sum rule of Eq.共30兲.

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LYP correlation energy for the electron gas. It is higher than the exact correlation energy by about 0.02 hartrees共Fig. 5 of Ref. 关4兴兲.

Table I compares the various CS expressions with one another and with the ‘‘exact’’ correlation energy for the elec- tron gas with r_s⫽3. Important long-range correlations are missing in all variants of CS theory.

A correct description of the uniform electron gas is re- quired not only for an accurate description of the solid met- als关4兴, but also to make a correlation energy functional more

transferable to a variety of systems. Some more-recent correlation energy functionals 关4兴 that make use of n(r) and its gradients, and sometimes of K(r), are correct by construc- tion for the uniform gas, accurate for atoms, and共when com- bined with similar exchange functionals兲 useful for molecu- lar atomization energies.

The description of long-range correlation by density- functional approximations, for use with exact exchange, is a key problem for the future. Perhaps this problem will be solved by constructing the correlation energy from the Hartree-Fock共or Kohn-Sham兲 one-particle density matrix, in the same spirit but not in the same way as in the Colle- Salvetti method.

Note added in proof. A very recent critical analysis of the Colle-Salvetti functional for two-electron atoms is presented in Ref.关27兴.

ACKNOWLEDGMENTS

This work was supported in part by the U.S. National Science Foundation under Grant No. DMR98-10620, and in part by the Fondazione Angelo Della Riccia共Firenze, Italy兲.

We thank V. Sahni and L. Massa for helpful comments.

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TABLE I. Correlation energy per electron for the uniform elec- tron gas with density parameter r_s⫽3, exactly and in various ver- sions of Colle-Salvetti theory.

Expression ⑀c(rs⫽3) 共hartree兲

‘‘Exact’’共Ref. 关21兴兲 ⫺0.0370

Eq.共9兲 of Ref. 关1兴 ⫺0.0098

Eq.共15兲 of Ref. 关1兴 ⫺0.0086

Eq.共2.6兲 of Ref. 关11兴 ⫺0.0073

Eq.共19兲 of Ref. 关1兴; Ref. 关3兴 ⫺0.0203