Atomic and molecular exchange-correlation charges in Kohn–Sham theory Giuseppina Menconi,a David J. Tozer*a and Shubin Liub a Department of Chemistry, University of Durham, South Road, Durham, UK DH1 3L E b Department of Chemistry, Duke University, Durham, NC 27768-0349, USA Received 17th April 2000, Accepted 30th June 2000 Published on the Web 7th August 2000 A new deÐnition of the exchange-correlation charge has recently been proposed (S. Liu, P. W. Ayers and R. G. Parr, J. Chem. Phys., 1999, 111, 6197). This charge, which is related to the exchange-correlation potential by the Poisson equation, is attractive due to its dependence on a single electronic coordinate. Using the method of Zhao, Morrison and Parr (Phys. Rev. A., 1994, 50, 2138), accurate charges have previously been calculated for the spherical systems He, Ne and Ar. By using a gaussian basis set implementation of this method, we demonstrate that the charge can be routinely determined for both atoms and small molecules ; we consider He, Ne, HF, CO and We highlight the deÐciencies of Ðnite gaussian basis sets, and graphically examine the N 2 . charges. The molecular charges show the same characteristics as those observed in the atomic charges, with structure that is closely related to that of the associated exchange-correlation potentials. The determination of accurate exchange-correlation charges may aid the development of improved exchange-correlation functionals and model potentials. 1 Introduction and background In KohnÈSham density functional theory (DFT) the exchange- correlation charge r@) has conventionally been related to o XC (r, the exchange-correlation energy by the expression1,2 E XC [o] E XC [o] \ 1 2 PP o(r)o XC (r, r@) o r [ r@ o dr dr@ (1) where o(r) is the electron density. The dependence of r@) o XC (r, on two electronic co-ordinates makes it a difficult quantity to visualise. Furthermore, it has no simple relationship to the exchange-correlation potential, which is the central quantity in the KohnÈSham equations, determining molecular struc- tures, response properties, etc. In light of this, Liu et al.3 have proposed an alternative deÐnition of the exchange-correlation charge. Their new charge, denoted depends only on a q XC (r), single electronic coordinate and is related to the asymp- totically vanishing exchange-correlation potential through the Poisson equation +2v XC (r) \[ 4pq XC (r) (2) or equivalently v XC (r) \ P q XC (r@) o r [ r@ o dr@ (3) i.e. the new exchange-correlation charge is the charge distribu- tion whose classical Hartree potential is the exchange- correlation potential. Eqn. (2) and (3) have also been considered in the recent work of Go rling.4 An important achievement of modern DFT research is the determination of exchange-correlation potentials from elec- tron densities.5 h11 One such approach is due to Zhao, Morri- son and Parr (ZMP).11 If the exact electron density were known then the ZMP method would yield the exact exchange- correlation potential. Furthermore, because the ZMP formal- ism naturally expresses the potential in the form (3), it would also yield the exact exchange-correlation charge of Liu et al. The ZMP method is based on the constrained search for- mulation of Levy and Perdew : 12 the orbitals that minimise the non-interacting kinetic energy, subject to the constraint that the density is exact, are the KohnÈSham orbitals. In the ZMP method the non-interacting kinetic energy is minimised subject to the constraint [sufficient to make o(r) \ o 0 (r)] 1 2 PP [o(r) [ o 0 (r)][o(r@) [ o 0 (r@)] o r [ r@ o dr dr@ \ 0 (4) where o(r) and are the trial and exact densities respec- o 0 (r) tively ; this leads to a set of equations for the KohnÈSham orbitals. By adding appropriate explicit density functionals to the minimisation procedure, the equations can be cast in the form of the conventional KohnÈSham equations, with the exchange-correlation potential identiÐed as v XC (r) \ lim j?= A [ 1 N P oj(r@) o r [ r@ o dr@ ] j P oj(r@) [ o 0 (r@) o r [ r@ o dr@ B (5) where N is the number of electrons and j is the Lagrange multiplier associated with the constraint (4). At the solution point j ] O, the trial density oj(r) equals the exact density and the potential in eqn. (5) is the exact exchange- o 0 (r) correlation potential. This ZMP potential vanishes asymp- totically and so corresponds to the exact potential on the electron deÐcient side of the integer.13,14 The ZMP method expresses the potential as the classical Hartree potential of a charge distribution, and so is intrinsi- cally linked to the new deÐnition of the exchange-correlation charge. Comparing eqn. (5) with eqn. (3) gives the expression for the exact charge within the ZMP formalism q XC (r) \ lim j?= A [ oj(r) N ] j[oj(r) [ o 0 (r)] B (6) Given that oj(r) and both integrate to N electrons, inte- o 0 (r) gration of this expression o†ers a simple derivation of the sum rule P q XC (r)dr \[1 (7) DOI : 10.1039/b003049j Phys. Chem. Chem. Phys., 2000, 2, 3739È3742 3739 This journal is The Owner Societies 2000 (