Discontinuity of the exchange-correlation potential: Support for assumptions used to find it

It was shown by J. P. Perdew, R. G. Parr, M. Levy, and J. L. Balduz [Phys. Rev. Lett. 49, 1691 (1982)]; J. P. Perdew and M. Levy [Phys. Rev. Lett. 51, 1884 (1983)]; and L. J. Sham and M. Schl\"uter [Phys. Rev. Lett. 51, 1888 (1983)] that the exact Kohn-Sham exchange-correlation potential ${v}_{\text{xc}}(\stackrel{P\vec}{r})$ of an open system may jump discontinuously as the particle number crosses an integer, with important physical consequences. The original derivations of the size and nature of the discontinuity rely on an implicit assumption that, when $N\ensuremath{\rightarrow}J$ from above and below, the effective potential of the interacting system with particle number $N$ becomes identical (modulo a constant) to the effective potential at an integer particle number $J$ so that the Kohn-Sham orbitals change continuously as $N$ passes through $J$ at fixed external potential. We prove under mild assumptions that the noninteracting kinetic energy of the interacting system is a continuous function of the particle number (as is the density itself), and hence argue that the original assumption is likely correct; if so, then all energy components are continuous. A rigorous proof of the existence and size of the ${v}_{\text{xc}}$ discontinuity is presented in the special case where the particle number crosses 1, by showing that, even when the Levy-Lieb constrained search for a minimum kinetic energy expectation value is taken to go over ensembles instead of wave functions, the noninteracting kinetic energy is still given by the von Weizs\"acker functional for all densities with real particle number $N$ between 0 and 2 inclusive. We also prove that the von Weizs\"acker functional is a lower bound on the ensemble-search noninteracting kinetic energy for all real $N>2$. To illustrate the ${v}_{\text{xc}}$ discontinuity and related results, we construct an analytic model density for the ${\text{H}}^{\ensuremath{-}}$ ion and plot the exchange-correlation potential as the particle number approaches 1 from above and below, and 2 from below. Finally, we discuss the behavior of the free-electron continuum as the particle number crosses an integer and the case where the particle number of a hydrogen or helium atom crosses $J=2$, the maximum number of electrons they can bind.