Electron densities in search of Hamiltonians

By utilizing the knowledge that a Hamiltonian is a unique functional of its ground-state density, the following fundamental connections between densities and Hamiltonians are revealed: Given that ρ_α, ρ_β,,ρ_ω are ground-level densities for interacting or noninteracting Hamiltonians H₁, H₂,,HM (M arbitrarily large) with local potentials v₁,v₂,,vM, but given that we do not know which ρ belongs with which H, the correct mapping is possible and is obtained by minimizing ${∫}d{{→}}{r} [{v}₁({{→}}{r}){{ρ}}_{{α}}({{→}}{r})+{v}₂({{→}}{r}){{ρ}}_{{β}}({{→}}{r})+{⋯}{v}M({{→}}{r}){{ρ}}_{{ω}}({{→}}{r})]$ with respect to optimum permutations of the ${ρ}$'s among the $v$'s. A tight rigorous bound connects a density to its interacting ground-state energy via the one-body potential of the interacting system and the Kohn-Sham effective one-body potential of the auxiliary noninteracting system. A modified Kohn-Sham effective potential is defined such that its sum of lowest orbital energies equals the true interacting ground-state energy. Moreover, of all those effective potentials which differ by additive constants and which yield the true interacting ground-state density, this modified effective potential is the most invariant with respect to changes in the one-body potential of the true Hamiltonian. With the exception of the occurrence of certain linear dependencies, a density will not generally be associated with any ground-state wave function (is not wave function v representable) if that density can be generated by a special linear combination of three or more densities that arise from a common set of degenerate ground-state wave functions. Applicability of the "constrained search" approach to density-functional theory is emphasized for non-v-representable as well as for v-representable densities. In fact, a particular constrained ensemble search is revealed which provides a general sufficient condition for non-v representability by a wave function. The possible appearance of noninteger occupation numbers is discussed in connection with the existence of non-v representability for some Kohn-Sham noninteracting systems.