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 vs-representable. It is shown that this assumption does not hold for pairs of densities: ρtot being the Kohn-Sham density in some system and ρB obtained from such partitioning of ρtot that the difference ρtot - ρB vanishes on a Lebesgue measurable volume element. The inversion procedure is still applicable for ρtot - ρB 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.