Exact relations between the electron density and external potential for systems of interacting and noninteracting electrons



The differential virial theorem (DVT) is an explicit relation between the electron density ρ(r), the external potential, kinetic energy density tensor, and (for interacting electrons) the pair function. The time-dependent generalization of this relation also involves the paramagnetic current density. We present a detailed unified derivation of all known variants of the DVT starting from a modified equation of motion for the current density. To emphasize the practical significance of the theorem for noninteracting electrons, we cast it in a form best suited for recovering the Kohn–Sham effective potential vs(r) from a given electron density. The resulting expression contains only ρ(r), vs(r), kinetic energy density, and a new orbital-dependent ingredient containing only occupied Kohn–Sham orbitals. Other possible applications of the theorem are also briefly discussed. © 2012 Wiley Periodicals, Inc.