Investigation of cusps and steps in the non-additive kinetic potential functional V^nadfrom analytic inversion

The non-additive kinetic potential functional V^nad is a key issue in density-dependent embedding methods, such as Frozen Density Embedding Theory and Partition-DFT. V^nad is a bifunctional of pair of electron densities ρ_A and ρ_B. We previously reported the exact analytical inversion procedure to generate reference V^nad for weakly overlapping ρ_A and ρ_B(M. Banafsheh, T. A. Wesolowski, Int. J. Quant. Chem. 118 (2018): e25410). The behaviour of V^nad at the vicinity of the nuclei has been questioned since the beginning. Available computational tools and methods in the past led to a cusp at nuclei in V^nad calculations. We analyse existence and non-existence properties of the cusp in V^nad analytically, and compare against nuclear cusps condition for the ground-state density and resulting cusp in the Kohn-Sham potential. We show the agreement of numerical calculations with this fact for various diatomic model systems of two and four electrons. The results are compared to the von Weizsäcker functional (exact for one orbital) and other kinetic energy functionals. 

In addition, we find that the well-known step structure of V_xc associated with molecular dissociation also appears in V^nad, even in local and semi-local functionals in the region where the two subsystems' densities weakly overlap.