VOLUME 87, NUMBER 3 PHYSICAL REVIEW LETTERS 16JULY 2001

Quantum Monte Carlo Analysis of Exchange and Correlation in the Strongly Inhomogeneous Electron Gas

Maziar Nekovee,1,* W. M. C. Foulkes,1 and R. J. Needs2 1The Blackett Laboratory, Imperial College, Prince Consort Road, London SW7 2BZ, United Kingdom 2Cavendish Laboratory, Madingley Road, Cambridge CB3 0HE, United Kingdom (Received 31 July 2000; published 25 June 2001) We use the variational quantum Monte Carlo method to calculate the density-functional exchange- correlation hole nxc, the exchange-correlation energy density exc, and the total exchange-correlation energy Exc of several strongly inhomogeneous electron gas systems. We compare our results with the local density approximation and the generalized gradient approximation. It is found that the nonlocal contributions to exc contain an energetically significant component, the magnitude, shape, and sign of which are controlled by the Laplacian of the electron density.

DOI: 10.1103/PhysRevLett.87.036401 PACS numbers: 71.15.Mb, 71.10.–w, 71.45.Gm

The Kohn-Sham density-functional theory (DFT) [1] rs ෇ 2a0 (approximately the same as for Al). The strong shows that it is possible to calculate the ground-state prop- density modulations were one dimensional and periodic, erties of interacting many-electron systems by solving only with a roughly sinusoidal profile. Three different modula- 0 0 one-electron Schrödinger-like equations. The results are tion wave vectors q # 2.17kF were investigated, where kF exact in principle, but in practice it is necessary to ap- is the Fermi wave vector corresponding to n0. The strong proximate the unknown exchange-correlation (XC) energy variation of n͑r͒ on the scale of the inverse local Fermi ͓ ͔ 21 2 21͞3 functional, Exc n , which expresses the many-body effects wave vector kF͑r͒ ෇ ͓3p n͑r͔͒ results in a strik- ͑ ͒ in terms of the electron density n r . The current popu- ingly nonlocal behavior of nxc that cannot be described by larity of density-functional methods in condensed matter semilocal corrections to the LDA. We show, however, that physics, quantum chemistry, and materials science reflects the LDA errors in exc have a dominant and energetically the remarkable success of fairly simple approximate XC significant component, the magnitude, shape, and sign of energy functionals. which are controlled by the semilocal quantity =2n͑r͒. Be- In the local density approximation (LDA), the XC hole cause it depends only on n and j=nj, the GGA is unable ͑ 0͒ nxc r, r about an electron at r is approximated by the XC to correct the LDA errors in Exc resulting from this com- hole of a uniform electron gas of density n ෇ n͑r͒. This ponent adequately, and worsens the LDA in two of our works surprisingly well, but not well enough for many three systems. The relevance of Laplacian terms has been chemical and biological applications. The most widely pointed out previously [6], but our calculations provide the used correction to the LDA is the generalized gradient first quantitative evidence of their importance in strongly approximation (GGA) [2–4], in which the effects of in- inhomogeneous systems and for exc. homogeneityR are modeled using the semilocal approxima- Our starting point is the adiabatic connection formula ഠ ͑ j j͒ tion Exc dr f n, =n , where f is some parametrized [7], which expresses Exc͓n͔ as the volume integral of an nonlinear function of n and =n. A common feature of all XC energy density exc defined by [8] current GGAs is that their construction is guided by limit- Z ͑ ͒ ͑ 0͒ ͑ ͓ ͔͒ ෇ 1 0 n r nxc r, r ing behaviors and sum rules; they are designed to fit vari- exc r, n dr 0 , (1) ous integrated quantities such as total exchange energies of 2 jr 2 r j atoms or ionization energies of molecules, but incorporate where nxc is the coupling-constant-averaged XC hole. The little or no information about the behavior of local quanti- definition of exc used in the construction of the GGA dif- 0 fers from ours by an integration by parts that does not affect ties such as nxc͑r, r ͒ and the exchange-correlation energy density e ͑r͒ in strongly inhomogeneous systems. This the integrated Exc. We favor Eq. (1), however, because of xc its clear physical interpretation and because it aids compar- may explain why current GGAs, although better than the LDA in many situations, are not consistently able to de- ison with the LDA XC energy density, which is constructed liver the very high accuracy of ϳ0.1 eV required to study, from an approximate hole. e.g., most chemical reactions. To open a new direction in The XC hole nxc is obtained via a coupling-constant the search for improved functionals, we use the variational integration, quantum Monte Carlo (VMC) method [5] to calculate n n͑r͒n͑r0͒ 1 n͑r͒n ͑r, r0͒ xc Z xc and e for several strongly inhomogeneous electron gases 1 X X xc ෇ ͗ l ͑ ͒ ͑ 0 ͒ l͘ and analyze the performance of the LDA and the GGA in dl C j d r 2 ri d r 2 rj jC , 0 i j͑fii͒ these systems in detail. (2) The inhomogeneous electron gases considered all had where Cl is the antisymmetric ground state of the Ham- 0 ෇ ͑͞ 3͒ ˆ l ˆ ˆ ˆ l the same average electron density n 3 4prs , where iltonian H ෇ T 1lVee 1 V associated with coupling

036401-1 0031-9007͞01͞87(3)͞036401(4)$15.00 © 2001 The American Physical Society 036401-1 VOLUME 87, NUMBER 3 PHYSICAL REVIEW LETTERS 16JULY 2001 constant l. Here Tˆ and Vˆ ee are the operators for the ki- we performed additional VMC calculations of the XC en- netic and electron-electron interaction energies, and Vˆ l ෇ ergies of finite homogeneous electron gases with N ෇ 64 l SV ͑ri͒ is the one-electron potential needed to hold the and rs ෇ 0.8, 1, 2, 3, 4, 5, 8, and 10. This enabled us electron density nl͑r͒ associated with Cl equal to n͑r͒ to construct a Perdew-Zunger parametrization [14] of the for all values of l between 0 and 1. Our VMC method VMC XC energy per electron of a finite uniform electron [9] for calculating nxc and exc from Eqs. (1) and (2) is gas with N ෇ 64. By using this parameterization to cal- LDA a generalization of the scheme used by Hood et al. [10]. culate exc in all systems studied, we largely eliminated It amounts to treating both Cl and V l variationally and the systematic errors in the calculated differences between VMC LDA determining the variational parameters by simultaneously exc and exc . The same parametrization was also used GGA minimizing the variance of the local energy [11] and the as input for the evaluation of Exc [2], a procedure that l GGA VMC deviation of n from n [9]. we assume mitigates the errors in Exc 2 Exc . The VMC simulations were performed for a finite spin- In Fig. 1 we show snapshots of the deformation of VMC ෇ 0 unpolarized electron gas in a face-centered cubic simu- nxc around an electron moving in the q 1.11kF sys- lation cell subject to periodic boundary conditions. The tem along a line parallel to q, the direction of maximum exact interacting ground-state density n͑r͒ was chosen inhomogeneity, from a density maximum towards the tail a priori, and the constrained minimization scheme was of n͑r͒. The XC hole is plotted as a function of r0 around then used to find, at each l, the exact (within VMC) wave a fixed electron at r, with r0 ranging in a plane parallel l l LDA function C and external potential V corresponding to to q. Also shown is the corresponding LDA hole nxc ෇ l͑ ͒ VMC that density [12]. At full coupling (l 1), V r is the [7]. At the density maximum (not shown), both nxc LDA exact external potential of the many-electron system with and nxc are centered at the electron. However, unlike ͑ ͒ ͑ ͒ LDA VMC ground-state density n r . The input density n r was gen- nxc , which is always spherically symmetric, nxc erated by solving, within the LDA, the Kohn-Sham equa- is contracted in the direction of the inhomogeneity. As tions for an external potential of the form Vq cos͑q ? r͒, the electron moves away from the density maximum to 0 0 where Vq was fixed at 2.08eF and eF is the Fermi en- a point on the slope (top panel), the nonlocal nature of 0 VMC LDA ergy corresponding to n . The advantages of this proce- nxc becomes manifest. While nxc is still centered VMC dure are that the input electron density is guaranteed to be at the electron and is rather diffuse, nxc lags behind noninteracting v-representable and that the Slater determi- near the density maximum and is much more compact. VMC nant of single-particle orbitals is by construction the exact The nonlocal behavior of nxc becomes remarkable at VMC many-body wave function corresponding to l ෇ 0. The the density minimum. Here nxc has two large nonlocal ϳ (density-functional) exchange contributions to exc, nxc, minima, each centered at a density maximum 2.80 a.u. and Exc, obtained from this Slater determinant, are there- fore also exact [9]. 0 0 We consider potentials with q ෇ 1.11kF , 1.55kF , and 0 2.17kF, and cells containing 64, 78, and 69 electrons, respectively. The adiabatic calculations were performed using six equidistant values of l in the range ͓0, 1͔ and with the Slater-Jastrow ansatz given in [9] as our many-body wave function. At each l, we used a total of 20 variational parameters in Cl and up to seven coefficients in the plane-wave expansions of nl and V l. The optimization of the parameters in Cl and V l was performed using 96 000 statistically uncorre- lated electron configurations. This was sufficient to reduce the root mean square deviation of nl͑r͒ from n͑r͒ to less than 0.5% of n͑r͒ for all values of l and all systems. The expectation values were calcu- lated [5] using 106 independent configurations of all electrons. Throughout, we used the modified electron- electron interaction described in [13], which virtually eliminates the finite-size errors arising from the long range of the Coulomb potential. The statistical errors were negligible except in nxc in low-density regions, where 0 FIG. 1 (color). The VMC and LDA nxc͑r, r ͒ for the q ෇ they were much smaller than the differences between nxc 0 LDA 1.11kF system plotted around an electron fixed at r (indicated and nxc . The largest systematic errors are caused by the 0 l in the figure), with r ranging in a plane parallel to q: (top) finite size of the system and the approximate nature of C . electron on the slope; (bottom) electron at a density minimum. These errors combine such that, even in a homogeneous The solid line shows the direction parallel to q (see Fig. 2 for VMC LDA electron gas, exc fi exc . To circumvent this problem, the profile of the electron density along this line). 036401-2 036401-2 VOLUME 87, NUMBER 3 PHYSICAL REVIEW LETTERS 16JULY 2001

VMC ෇ 0 away from the electron (the small fluctuations of nxc q 1.11kF system) to negative (for the two other sys- at this point are statistical errors). The LDA hole, by tems) as q increases and the negative contributions to 2 contrast, is spread over theR whole system in order to Dexc, which occur where = n͑r͒ , 0, become dominant. 0 LDA͑ 0͒ ෇ satisfy the sum rule [7]: dr nxc r, r 21. This The GGA corrections are by construction always negative VMC LDA ෇ striking nonlocality of nxc also occurs in the other two [2–4]; they improve Exc for the q 1.11kF system but systems we considered, and similar nonlocal behavior worsen it for the two other systems. We note that GGA has previously been observed for the exchange plus corrections also worsen the LDA in quasi-two-dimensional Coulomb hole of the hydrogen molecule [15]. Clearly, electron gases [17], a shortcoming shared by the recently semilocal corrections are unable to significantly improve proposed meta-GGA (MGGA) [18]. Although our sytems the LDA description of the XC hole in our systems, and are not quasi two dimensional [19], we believe that a simi- fully nonlocal approximations are required. We found, lar semilocal effect lies behind the failure of the GGA in VMC however, that, despite the strong nonlocality of nxc , both cases, namely, an increasing negative contribution to VMC the LDA errors in exc can be described in terms of a the LDA errors in exc originating from the strongly nega- semilocal quantity, the Laplacian of the electron density. tive Laplacian around the peak of n͑r͒. Similar errors are LDA VMC In Fig. 2 we show exc 2 exc for two of the strongly known to occur near the nuclei in molecules, but are over- LDA inhomogeneous systems studied, where exc is calculated compensated by positive errors in the bonding and outer using the exact ground-state density n͑r͒. (The very simi- regions [20]. We note that, since =2n͑r͒ integrates to zero, lar graph for the third system has been omitted to save an energy-density correction proportional to =2n͑r͒ would LDA space.) The results are plotted along a line parallel to not improve Exc . As shown below, however, the depen- q (we call this direction y͒ . Also shown are n͑r͒ and dence on =2n observed here is nonlinear. =2n͑r͒ plotted along the same line. It is apparent that The exchange-correlation energy density is a unique the shape, magnitude, and sign of the LDA errors in exc functional of the electron density. Provided that the closely follow the shape, magnitude, and sign of =2n͑r͒. electron density has a convergent Taylor expansion about The LDA errors in exc are large and negative in regions the point r, it can therefore be written as exc͑r, ͓n͔͒ ෇ 2 where = n͑r͒ is large and negative (around density maxi- exc͓r, n͑r͒, =i n͑r͒, =i=j n͑r͒,...͔. The GGA may be ma), and large and positive in regions where =2n͑r͒ is viewed as an attempt to approximate this mapping by large and positive. The GGA XC energy density is not a nonlinear function of n͑r͒ and =in͑r͒ only. In the defined via Eq. (1) and is not shown here [16]. The case of the exchange energy (although not, unfortunately, VMC values of the integrated Exc are shown in Table I, the correlation energy), a uniform scaling argument [21] LDA ෇ LDA VMC ෇ ͑ ͒ LDA along with the differences DExc Exc 2 Exc and shows that ex Fx s, l,... ex , where ex is the exact GGA ෇ GGA VMC DExc Exc 2 Exc (the version of the GGA used (density-functional) exchange-energy density obtained l෇0 LDA here is due to Perdew, Burke, and Ernzerhof [2]). The from C [9], Fx is an enhancement factor, ex LDA errors in Exc reflect the profound effect of the Lapla- is the exchange-energy density within the LDA, and 2 2 cian errors in exc and change sign from positive (for the s ෇ j=nj͓͞2kF ͑r͒n͑r͔͒ and l ෇ = n͓͞4kF ͑r͒n͑r͔͒ are a dimensionless gradient and a dimensionless Laplacian,

0 0 respectively. Figures 3a and 3b are scatter plots compar- q=1.11kF q=1.55kF 0.0010 ing the values of Fx at different points in space with the values of s and l at those points. Because n͑r͒ appears in 0.0005 the denominator of l and s, the maximum absolute values 0.0000 of l and s occur where the electron density is smallest and −0.0005 (a.u.)

xc are largest for the q ෇ 1.11k system.

e F −0.0010 ∆ The two-valued nature of Fig. 3a arises because the −0.0015 mapping from s to position is two-valued in our systems. −0.0020 For example, s is zero at both the minimum and the maxi- mum of the density, where the required corrections to the 0.10 LDA exchange hole are completely different, as can be in-

0.00 ferred from Fig. 1 [22]. In sharp contrast, Fig. 3b shows

−0.10 TABLE I. Exchange-correlation energies (Hartrees per elec- density (Laplacian) density tron) and the LDA and GGA errors in this quantity for different Laplacian −0.20 values of the wave vector q. The statistical errors in EVMC are 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 xc λ 0 λ 0 indicated. y/ F y/ F ͞ 0 VMC LDA GGA LDA VMC q kF Exc DExc DExc FIG. 2. The upper graphs show exc 2 exc along a direc- tion parallel to q for two different strongly inhomogeneous 1.11 20.3289 6 0.001 10.0042 10.0001 systems. The lower graphs show the corresponding electron 1.55 20.3127 6 0.001 20.0005 20.0074 densities (light lines) and Laplacians (heavy lines). Distances 0 0 2.17 20.2882 6 0.001 20.0066 20.0140 are in units of the Fermi wavelength lF ෇ 2p͞kF . 036401-3 036401-3 VOLUME 87, NUMBER 3 PHYSICAL REVIEW LETTERS 16JULY 2001

[1] P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964); 2.2 W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965). 2.0 [2] J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996); J. P. Perdew and Y. Wang, Phys. Rev. B 1.8 33, 8800 (1986). [3] A. D. Becke, Phys. Rev. A 38, 3098 (1988); A. D. Becke, 1.6 x J. Chem. Phys. 98 5648 (1993).

F , 1.4 [4] C. Lee, W. Yang, and R. G. Parr, Phys. Rev. B 37, 785 (1988).

1.2 q=1.11kF [5] B. L. Hammond and W.A. Lester, Jr., and P. J. Reynolds, q=1.55kF Monte Carlo Methods in Ab Initio Quantum Chemistry q=2.17k 1.0 F (World Scientific, Singapore, 1994). (a) (b) [6] E. Engel and S. H. Vosko, Phys. Rev. B 50, 10 498 (1994); 0.8 0.0 1.0 2.0 3.0 4.0 −2 4 10 16 22 28 C. J. Umrigar and X. Gonze, Phys. Rev. A 50, 3827 (1994); s l A. D. Becke, J. Chem. Phys. 109, 2092 (1998). FIG. 3. The left panel (a) plots values of the exact exchange [7] R. G. Parr and W. Yang, Density Functional Theory of Atoms and Molecules (Oxford University Press, Oxford, enhancement factor Fx against values of the reduced density gradient for three strongly inhomogeneous systems. The right 1988). panel (b) plots values of the exact exchange enhancement against [8] Hartree atomic units with e ෇ m ෇ 4pe0 ෇ h¯ ෇ 1 are values of the reduced density Laplacian. used throughout. [9] M. Nekovee et al., Adv. Quantum Chem. 33, 189 (1999). that Fx is a simple and almost unique function of l, with the [10] R. Q. Hood et al., Phys. Rev. Lett. 78, 3350 (1997). physically different regions around the maximum and the [11] C. J. Umrigar, K. G. Wilson, and J. W. Wilkins, Phys. Rev. minimum of the density corresponding to regions of nega- Lett. 60, 1719 (1988); A. J. Williamson et al., Phys. Rev. B 53, 9640 (1996). tive and positive Laplacian, respectively. This suggests [12] At l ෇ 1 this procedure might be viewed as a VMC re- that the inclusion of Laplacian terms may allow the con- alization of Hohenberg and Kohn’s first theorem: given struction of simpler and more accurate approximate func- the exact ground-state density, we find the corresponding tionals. Springborg and Dahl recently reached a similar ground-state many-body wave function. conclusion [23] based on their studies of the exchange- [13] L. M. Fraser et al., Phys. Rev. B 53, 1814 (1996); A. J. energy densities of closed shell atoms, although their defi- Williamson et al., Phys. Rev. B 55, R4851 (1997). nition of ex differs from ours. In most gradient expansions, [14] J. P. Perdew and A. Zunger, Phys. Rev. B 23, 5048 (1981). the Laplacian terms allowed by symmetry are transformed [15] M. A. Buijse and E. J. Baerends, in Density Functional into j=n͑r͒j2 terms via an integration by parts. This is pos- Theory of Molecules, Clusters, and Solids, edited by D. E. sible only when the dependence on l is linear, however, Ellis (Kluwer Academic Publishers, Dordrecht, 1995). which is not the case here, as can be seen from Fig. 3b. [16] See, however, Fig. 4 of Ref. [9], where preliminary results containing relatively large finite-size errors Furthermore, the integration by parts destroys the physi- ϳ VMC LDA,GGA ( 20.0003 a.u.) are shown for ex 2 ex and cal interpretation in terms of the XC hole and so hinders VMC LDA,GGA ec 2 ec . further progress. We note that there is a one-to-one rela- [17] Y.-H. Kim et al., Phys. Rev. B 61, 5202 (2000); L. Pollack tionship between position and l in each of our systems, and and J. P. Perdew, J. Phys. Condens. Matter 7, 1239 (1999). so an enhancement factor of the form Fx͑l͒ might not be [18] J. P. Perdew, S. Kurth, A. Zupan, and P. Blaha, Phys. Rev. as universal as Fig. 3b suggests. However, the consistent Lett. 82, 2544 (1999). results obtained for all three systems, the energetic signifi- [19] In fact, the GGA corrections have the wrong sign at large cance of the Laplacian terms, and the strong similarities q, when the fixed-height potential barriers are narrow. In between the form of the Laplacian and the LDA errors in this limit, the electrons tunnel easily from well to well and our systems become more three dimensional. exc (see Fig. 2) give us confidence in the physical impor- tance of the Laplacian in describing inhomogeneity correc- [20] P.R. T. Schipper, O. V. Gritsenko, and E. J. Baerends, Phys. tions to the LDA. Rev. A 57, 1729 (1998). [21] See Levy and Perdew [Phys. Rev. A 32, 2010 (1985)] for We thank R. Q. Hood for useful discussions, G. Ra- a derivation for the case F ෇ F ͑s͒. The generalization jagopal and A. J. Williamson for help with the VMC x x to the case Fx ෇ Fx ͑s, l͒ is straightforward. codes, and L. Smith for help with visualization. M. N. [22] For s ෇ 0, the value of Fx is ,1 on the lower branch of was supported by an EU Marie Curie Fellowship under this scatter plot and .1 on the upper branch, in contrast to GGA Grant No. ERBFMBICT961736. Our calculations were the GGA enhancement factor, which satisfies Fx # 1 as performed on the CRAY-T3E at EPCC. s ! 0. The GGA enhancement factor is constructed such that it approaches the uniform electron gas value of 1 in the limit as s ! 0; the exact Fx , however, is a function of all density gradients and need not approach 1 when s ෇ 0 *Present address: Complexity Research Group, Admin2 but l is nonzero. PP5, British Telecom Laboratories, Martlesham Heath, [23] M. Springborg and J. P. Dahl, J. Chem. Phys. 110, 9360 Suffolk IP5 3RE, UK. (1999). 036401-4 036401-4