Is the non-additive kinetic potential always equal to the difference of effective potentials from inverting the Kohn–Sham equation?

Abstract

The relation used frequently in the literature according to which the non-additive kinetic potential which is a functional depending on a pair of electron densities is equal (up to a constant) to the difference of two potentials obtained from inverting two Kohn–Sham equations, is examined. The relation is based on a silent assumption that the two densities can be obtained from two independent Kohn–Sham equations, i.e., are v s-representable. It is shown that this assumption does not hold for pairs of densities: ρ tot being the Kohn–Sham density in some system and ρ B obtained from such partitioning of ρ tot that the difference ρ tot − ρ B vanishes on a Lebesgue measurable volume element. The inversion procedure is still applicable for ρ tot − ρ B but cannot be interpreted as the inversion of the Kohn–Sham equation. It is rather the inversion of a Kohn–Sham-like equation. The effective potential in the latter equation comprises a “contaminant” that might even not be unique. It is shown that the construction of the non-additive kinetic potential based on the examined relation is not applicable for such pairs.