Proc. Indian Acad. Sci. (Chem. Sci.), Vol. 106, No. 2, April 1994, pp. 91-102. Printed in India. Density approximation to the average Hartree-Fock exchange potential for atoms O V GRITSENKO, A RUBIO, L C BALBAS and J A ALONSO* Departamento de Ffsica Te6rica, Universidad de Valladolid, E-47011 Valladolid, Spain Abstract. A non-local density-based approximation to the average Hartree-Fock (HF) exchange potential is developed. The new potential is formulated within the spin-dependent version of the weighted density approximation, and is based on a novel form of the (exchange- only) pair-correlation function for electrons in finite systems. The results for total energies and one-electron orbital energies of atoms are reasonably accurate in comparison with those obtained using the exact average HF potential, or the exact orbital-dependent HF potential. Keywords. Density approximation; Hartree-Fock exchange potential; weighted-density approximation. 1. Introduction The Hartree-Fock (HF) method is often used for the study of atoms and molecules (Froese-Fischer 1977). In this method one has to solve self-consistently one set of one-electron equations of the form (Hartree atomic units will be used through the paper). - + Vn(r) + V,(r) + v~i~(r) ~i~(r) = ~iOi~(r). (1) I 1-v2 1 In these equations V~ is the attractive nuclear potential, Vc is the classical electrostatic potential of the electronic cloud V~(rx)= Idr2 p(r2), (2) J r12 and vxi~ is the spin- and orbital-dependent HF exchange potential for an electron in the spin-orbital ~i: vx, trl) = 1 Idr2 % re)p (rl, re). (I)i~(r 1 ) ,J r12 On the other hand, p is the electron density, built from the occupied one-electron * For correspondence 91 92 0 V Gritsenko et al spin-orbitals, p(r) = ~ &~(r)Ot~(r), (4) i,r and p~(rx, r2) is the first-order density matrix component with spin index a: p~(r,, r2) ---- ~ &~(rx )~,(r2). (5) i The fact that the HF exchange potential vxi~ is orbital-dependent led first Slater (1951, 1974) and then other authors (Karsch and Rennert 1972; Alonso and Girifalco 1978) to propose a simplified alternative to the HF method where equations (1) are replaced by others, [- ~ V2 § Vn(r) + V.(r) + W~.~(r)30,~(r) f ,,~,~(r ), (6) containing an orbital-independent exchange potential W~. This Slater (S) potential is obtained by averaging the orbital-dependent HF exchange potentials, namely 14~,~(r) = ~162r162 ,., (7) Incidentally, it is also worth recalling that equations like (6), with an orbital-independent exchange potential V,o, also arise in the standard formulation of the density functional (DF) formalism (Kohn and Sham 1965). In this case the exchange potential is the functional derivative of the exchange energy functional E~[p]: DF OE,~] v (r)-- ap~(r) " (8) The Slater potential of (7) can also be written in an alternative form which is more convenient for some purposes as W~(rl) = f dr2P2(r2) C"(rl' r2), (9) r12 where Cr is the ~ component of the Fermi (or exchange) pair-correlation function Cr = p2(rl'r2) (10) p~(rl) p~(r2)" More recently, working within the framework of the optimized effective potential (OEP) method (Sharp and Horton 1953; Talman 1989; Wang et al 1990), Krieger et al (1992) have set up a method leading to one-electron equations similar to (6), although containing an optimized exchange potential. The Krieger-Li-Iafrate (KLI) optimal potential is given by the expression ~i0" (r) O/q(r)( Vix~tcr-- Vxia)int VXL](r~xr . = WS~(r) + p~(r) ' (I1) Density approximation to the average HF exchange potential 93 where pint i~ and V,+~i"t are evaluated from the following integrals vi".XlO't = fdr**(r) vx,` (r)~,(r) ) (12) vintx,o= f dr ~*(r) vgLn'r'(l),io t ) i~tr),- , (13) The optimal potential from (11) yields one-electron orbitals whose Slater determinant minimizes the expectation value of the HF Hamiltonian. The feature we want to stress is that w..S is a component of the KLI potential. Furthermore, it has been shown (Krieger et al 1992) that the exchange potential of Harbola and Sahni (HS; 1989) and Li et al (1989) can be obtained from (11) by making a plausible approxima- tion for the second term of the r.h.s, of this equation. Calculations for closed-shell atoms (Krieger et al 1992) have shown that I4/~, is the dominant term of the KLI and HS potentials at all electron-nucleus distances. The values of various electronic properties calculated with the Slater potential W,as were found to be close to those obtained from the KLI and HS potentials. Moreover, WS(r) provides the correct asymptotic behavior for the KLI and HS potentials (Ortiz and Ballone 1991; Engel et al 1992). For instance, for a neutral atom WS(r)_~ _ 1, for r ~ ~, (14) r and this induces the same behavior in VTM and Vus At the nucleus, W s has a finite Xr Xd " X~ value, specific for each atom. The features presented above generate new interest in the problem of calculating the average HF exchange potential WS. Equations (9) and (10) show that Vr o can be evaluated from the first-order density matrix of the system. To our knowledge, this has only been made for atoms. But for the purpose of applications to more complex systems, it should be desirable to derive accurate approximations to WS from the knowledge of the electron density only (that is, the diagonal part of the first-order density matrix). In this paper we propose an approximation to W~, developed within the weighted-density approximation (WDA) scheme (Alonso and Girifalco 1978; Gunnarsson et al 1979). Our approach is based on a novel form of the pair-correlation function C,(rl,r2) recently developed for finite systems (Gritsenko et al 1993). In section 2 we introduce the WDA approximation to the average HF exchange potential and we compare the exact and approximate potentials for Ne and Cu § A comparison is then made of orbital and total energies in w1673 and 4, this time for all atoms from He to Ar and for selected heavier atoms. 2. Density-based approximations to the Slater potential Various density-based approximations to the Slater potential WS have been proposed in the literature (Slater 1951, 1974; Karsch and Rennert 1972; Gopinathan et al 1976; Alonso and Girifalco 1978; Malkina et al 1990), mainly in the context of the local density (LDA) (Kohn and Sham 1965) or local spin-density (LSDA) (Hedin and 94 0 V Gritsenko et al Lundqvist 1971; Von Barth and Hedin 1972) approximations, namely wLSDAIr ~ ~ 1,=pr 3 ~ dr 2 Cr162 (15) 3 r12 Here C~(rl2;Pr is the HF pair-correlation function for a spin-polarized homo- geneous electron gas with constant density precisely equal to pr the true spin- density of the inhomogeneous system at r~. Inserting the HF pair-correlation function, calculated by Dirac (1930), we arrive at the LSDA approximation to wSxr (Slater 1951): wLSDA(r)xa.. -- - 3 pl/a (r). (16) First of all we compare WX~LsDA to the exact Ws Xff" It is well known that the local density approximation is unable to reproduce the correct long-range asymptotics. This can be observed in figures 1 and 2, for Ne and Cu + respectively, which show that the LSDA underestimates electron exchange, compared to the exact Slater potential, at low densities (large r). The LSDA potential decays, in fact, exponentially. On the contrary, the LSDA substantially overestimates exchange at high densities (small r), while at intermediate densities it provides a fair approximation to wS~. The LSDA potential plotted in these figures was constructed from the HF wave functions of Clementi and Roetti (1974). On the other hand the average HF potential wS has been taken directly from Kim and Gordon (1974) and Hartree (1958) for Ne and Cu § respectively. One obvious comment about the potentials plotted in figures 1 and 2 is that the exchange potentials are the same for both spin components (a = T, ~) since Ne and Cu § have dosed subshells. Some years ago, an approximation to WSx,the non-spin-polarized analogue of wS, was proposed by Alonso and Girifalco within the context of WDA (Alonso and Girifalco 1978; Gunnarsson et al 1979). Contrary to the LDA, the true density p(r2) is retained in the integrand of (9) in this scheme, and only the pair-correlation function is approximated, namely wWDA(rl) = fdr 2p(r2) cWDA(r12;P(rl)). (17) r12 This equation shows that the pair-correlation function C(rl,r2), which is a symmetric function to the interchange of the electron coordinates rl,r 2, and in general is not spherically symmetric with respect to the exchange-hole center rl, has been replaced by cWDA(r12;~(rl) ), which, in contrast, is spherically symmetric around rl and is not symmctric with respect to the interchange of r~ and r 2. cWDA(r12; ~(r~)) is the Dirac (or HF) pair-correlation function for a homogencous electron gas (Dirac 1930) with constant density equal to the weighted density ~(r 1). This weighted-density parameter is determined at every point rl from the sum rule f dr2p(r2)CWVA(rl2;~(rl))= -- 1, (18) which sets the normalization of the exchange-hole charge. The WDA approximation to the Slater potential was tested by Alonso and Girifalco (1978) by performing Density approximation to the average HF exchange potential 95 | | -- Sloter --- LDA r -- ~VDA -2.o Slater 6 -7.0 LDA -..I WDA -1.5 Z ILl rlB Ne Z -1.0 ,,,\\ Z X \\ W \ \\\N~ \x\ -0.5 o o.1 0.2 oJ 0.4 as 0.6 0.6 1.o ts .o 2.s o 35 ~.o r(a.u.) r (a.u.) Figure 1.