Exact exchange-correlation potential and approximate exchange potential in terms of density matrices

An exact expression in terms of density matrices (DM) is derived for δF[n]/δn(r), the functional derivative of the Hohenberg-Kohn functional. The derivation starts from the differential form of the virial theorem, obtained here for an electron system with arbitrary interactions, and leads to an expression taking the form of an integral over a path that can be chosen arbitrarily. After applying this approach to the equivalent system of noninteracting electrons (Slater-Kohn-Sham scheme) and combining the corresponding result with the previous one, an exact expression for the exchange-correlation potential ${\mathit{v}}_{\mathrm{xc}}$(r) is obtained which is analogous in character to that for δF[n]/δn(r), but involving, besides the interacting-system DMs, also the noninteracitng DMs. Equating the former DMs to the latter ones, we reduce the result for the exact ${\mathit{v}}_{\mathrm{xc}}$(r) to that for an approximate exchange-only potential ${\mathit{v}}_{\mathit{x}}$(r). This leads naturally to the Harbola-Sahni exchange-only potential.