Discontinuity of the exchange-correlation potential from a density-functional viewpoint

An expression for the discontinuity of the exchange-correlation potential $v_{\mathrm{xc}}$ of an insulator is derived entirely within the framework of density-functional theory. The discontinuity is expressed in terms of changes of the exchange-correlation energy, $E_{\mathrm{xc}}$, of a perfect N-particle insulator when (a) a conduction electron is introduced, (b) a valence electron is removed, and (c) the external perturbation is applied to the perfect insulator (without changing N) such that the density change is equal to minus the sum of the density changes in (a) and (b).