Analytic inversion procedure for the exact non-additive kinetic potential functional V^nad

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 pairs of specific electron densities ρ_A and ρ_B. We report here an inversion procedure to generate reference V^nad for weakly overlapping ρ_A and ρ_B. To obtain the exact V^nad we used an analytical inversion procedure that we proposed (M. Banafsheh, T.A. Wesolowski, Int. J. Quant. Chem. 118 (2018): e25410). We discuss the constraints on the choice of electron densities to assure their admissibility. Mathematical challenges of satisfying these constraints will be presented in detail. The potential at small overlap is constructed for various diatomic systems of four electrons at different interatomic distances. These results are compared with the potential obtained using common kinetic functional approximations. V^nad is also presented for some diatomic systems including more than 4 electrons in which two electrons are localized with high precision in space and the accuracy of V^nad is assured. We are now studying the forward Kohn-Sham problem for some small diatomic systems using the analytically inverted potential and comparing with the standard Kohn-Sham approach.