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

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