Properties of the multi-electron densities “between” the Hohenberg-Kohn theorems and variational principle

The properties of multi-electron densities are analyzed along with their behavior with respect to the two Hohenberg-Kohn theorems and the fundamental extension by Ziesche, Ayers and Levy in this manner. This analysis is continued with the form of density functionals and density differential and/or integral operators on different levels of dimensions between the variational principle (4N-dimension) and Hohenberg-Kohn theorems (3-dimension). The trend in ionization potentials is commented upon. The exact density functional operator of H-like atoms and one-electron systems is also discussed with the two-electron systems, not only as simple “forever prototypes”, but as a certain projection of one-electron density formalism of N ⩾ 1 electron systems to N = 1 and 2. The review part of this work is focusing primarily on functional analytical properties.