Differential virial theorem for the fractional electron number: Derivative discontinuity of the Kohn-Sham exchange-correlation potential

We extend the differential virial theorem put forth by Holas and March [Phys. Rev. A 51, 2040 (1995)] to a statistical mixture of densities corresponding to different numbers of electrons. This allows us to derive an expression for the gradient of the exchange-correlation potential for densities that lead to noninteger numbers of electrons. We apply this to study the jump in the exchange-correlation potential as the electron number increases through an integer. We show that the difference between the gradient of the exchange-correlation potential for N electrons, where N is an integer, and that for N+f electrons, where f is a small positive number, is similar to the electric field in and around a charged metal sphere: It is zero inside, peaks at the surface, and then decays. Consequently, the corresponding potential is constant in the interior of the system and decays sharply at the surface.