Noncrossing theorem for the ground ensembles of systems with noninteger particle numbers

Abstract

That the external potential ${v}_{\mathit{ext}}(\stackrel{P\vec}{r}\phantom{\rule{0.1em}{0ex}})$ of a system of electrons is determined uniquely by the ground-state density is one of the central statements of the first Hohenberg-Kohn theorem. It is known that the validity of this statement extends to densities $n(\stackrel{P\vec}{r}\phantom{\rule{0.1em}{0ex}})$ with noninteger particle number [i.e., $n(\stackrel{P\vec}{r}\phantom{\rule{0.1em}{0ex}})$ integrates to a number that is not an integer] if the functional derivative of ${T}_{s}[n(\stackrel{P\vec}{r}\phantom{\rule{0.1em}{0ex}})]+U[n(\stackrel{P\vec}{r}\phantom{\rule{0.1em}{0ex}})]+{E}_{xc}[n(\stackrel{P\vec}{r}\phantom{\rule{0.1em}{0ex}})]$ exists or (without relying on the existence of functional derivatives) if the ground-state energy is a strictly convex function of the particle number. In the present article, a proof that relies neither on the existence of the above functional derivative nor on the strict convexity of the ground-state energy is presented. The fact that the density determines the external potential leads to a noncrossing theorem for ground-state densities. The noncrossing theorem produces knowledge as to what the integer-particle-number ground-state densities of a system cannot be. The noncrossing theorem produces inequalities that the functional derivatives of the exchange-correlation energy functional ${E}_{xc}[n(\stackrel{P\vec}{r}\phantom{\rule{0.1em}{0ex}})]$ and the noninteracting kinetic energy functional ${T}_{s}[n(\stackrel{P\vec}{r}\phantom{\rule{0.1em}{0ex}})]$ must fulfill.