Density-functional theory beyond the Hohenberg-Kohn theorem

A density-functional theory that treats all states of an electronic system on the same footing is introduced. The corresponding Kohn-Sham formalism can be applied to ground and excited states alike, does not suffer from a $v$-representability problem, and represents a rigorous formal basis for the common, but so far unjustified practice to treat excited states by Kohn-Sham methods. The presented density-functional theory emerges from a generalization of the constrained-search procedure. The new Kohn-Sham formalism is based on generalized adiabatic connections introduced here. The possible topologies of those generalized adiabatic connections are discussed. A density-based stationarity principle and a density theorem that represents a more general counterpart of the Hohenberg-Kohn theorem are presented. A method to take into account exactly exchange interactions in the presented Kohn-Sham formalism is introduced, implemented, and applied to atoms.