# The density-functional theory of systems with noninteger particle numbers and the relevance of the gradient expansion to atoms and molecules

The first Hohenberg-Kohn theorem is extended to noninteger particle numbers. As a result of this extension; one can show that there exists a noncrossing theorem for ground ensembles of different particle numbers. This theorem excludes some densities as possible ground-state densities of a given system, provided the ground-state densities of this and other systems are known at a neighboring particle number. This theorem produces inequalities that functionals modelling the exchange-correlation energy and the noninteracting kinetic energy must fulfill It was proved in the early 1980's that the exchange-correlation potential undergoes a discontinuous positive jump as the particle number crosses an integer. This explained significant discrepancies in Local Density Approximation calculations of the fundamental band gap. The proof relies on a physically reasonable assumption that the shape of the exchange-correlation potential of the interacting system changes continuously with respect to the particle number. We prove that the noninteracting kinetic energy varies continuously with respect to the particle number and argue that this strongly indicates that the assumption is correct. In the special case where the particle number crosses 1, a rigorous proof is presented for the existence and size of the discontinuity of the exchange-correlation potential. The ensemble-search noninteracting kinetic energy of a 1- or 2-particle system is shown to be given by the von Weizsacker functional A new density scaling is developed under which the first terms of the gradient expansion for the noninteracting kinetic energy and the exchange energy become exact as the scaling constant goes to infinity. This is a limit that should be fulfilled by approximate functionals for these quantities. The successful Generalized Gradient Approximations for exchange do not, and we explain why