Exchange-correlation potential in terms of the idempotent Dirac density matrix of DFT

A formal proof has recently been proposed [I.A. Howard, N.H. March, Chem. Phys. Lett. 402 (2005) 1.] to show that the exchange-correlation potential Vxc(r) of density functional theory (DFT) is, in principle, determined by the idempotent Dirac density matrix γ(r, r′) generated by the one-body potential V(r) = Vext(r) + VHartree(r) + Vxc(r). Here we turn this 'existence' theorem into a concrete result by expressing the Laplacian ∇2Vxc(r) solely in terms of γ(r, r′), which, although idempotent and therefore not many-electron in character, has the exact ground state density ρ(r) on its diagonal γ(r, r). As a specific illustration, we calculate ρ(r) by diffusion QMC and hence γ(r, r′) for the Be atom ground-state. Vxc(r) is thereby obtained with satisfactory accuracy.