The Abdus Salam International Centre for Theoretical Physics Non-local Energy Density Functionals: Models plus Some Exact General Results

Holas and March (Phys. Rev. A51, 2040, 1995) gave a formally exact expression for the force @Vxc(r ~ ) @r ~ associated with the exchange-correlation potential Vxc(r ~ ) of density functional theory. This forged a precise link between rstand second-order density matrices and Vxc(r ~ ). Here models are presented in which these low-order matrices can be related to the ground-state electron density. This allows non-local energy density functionals to be constructed within the framework of such models. Finally, results emerging from these models have led to the derivation of some exact `nuclear cusp' relations for exchange and correlation energy densities in molecules, clusters and condensed phases. MIRAMARE { TRIESTE February 2001 1 Background and outline In their early attempts to transcend the Thomas-Fermi (TF) approximation, March and Young [MY:1] recognized that electron density theory should be regarded as a branch of density matrix theory. Their early studies led to a single-particle kinetic energy functional Ts[ ] of the form Ts[ ] = TW [ ] + (N) TTF [ ] (1.1) where TTF [ ] and TW [ ] were the TF kinetic energy and the von Weizsacker inhomogeneity kinetic energy respectively. In particular, the formula of MY for (N) led to this TF contribution correctly vanishing in the case of a single level occupancy only. Subsequently Herring and coworkers [2,3] gave stronger arguments as to why in Ts[ ] it was physically appropriate to subtract out the full von Weizs acker contribution TW [ ] in constructing the single-particle kinetic energy functional Ts[ ]. Of course, the most common route to implement DFT has been since Slater's work [4] to return to orbitals from which to obtain the single-particle kinetic energy by standard quantum mechanics. The only question remaining then concerns the one-body potential V (r ~ ), now including exchange and correlation interactions, to insert into the Schrodinger equation (the so-called Slater-Kohn-Sham (SKS) equations [4,5]). However, within the last decade, major e ort has been focussed on orbital free DFT [6-10]. This search for (presently approximations to) the single-particle kinetic energy functional Ts[ ] is then, of course, additional to the construction of V (r ~ ), given formally by V (r ~ ) = VHartree(r ~ ) + Vxc(r ~ ) : (1.2) One direction of progress on the force @Vxc=@r ~ associated with Vxc(r ~ ) was pointed out by Holas and March [11,12], who showed, via a many-electron form of the di erential virial theorem [13] that the force @Vxc(r ~ )=@r ~ could be written, quite explicitly, in terms of rstand second-order density matrices (1 and 2DMs) of the many-electron system under discussion, plus, of course, information contained in the Slater-Kohn-Sham determinant of the `reference' system with the same ground-state density as the correlated electron assembly under discussion. The problem of nding Vxc(r ~ ) is central, whether one works with SKS orbitals to obtain Ts, or with the single-particle kinetic energy functional Ts[ ] in orbital-free theory since the Euler-equation of DFT reads [14] = ÆTs Æ (r ~ ) + VHartree(r ~ ) + Vxc(r ~ ) : (1.3) where is the chemical potential (constant throughout the entire charge cloud) of the molecule, cluster or condensed phase being considered. The outline of the present work is then as follows. In section 2 below, a one-dimensional model of harmonic con nement of independent fermions will be utilized to expose a non-local