Excited-State Density-Functional Theory Revisited: on the Uniqueness, Existence, and Construction of the Density-to-Potential Mapping

The generalized constrained search formalism is used to address the issues concerning density-to-potential mapping for excited states in time-independent density-functional theory. The multiplicity of potentials for any given density and the uniqueness in density-to-potential mapping are explained within the framework of unified constrained search formalism for excited-states due to G\"orling, Levy-Nagy, Samal-Harbola and Ayers-Levy. The extensions of Samal-Harbola criteria and it's link to the generalized constrained search formalism are revealed in the context of existence and unique construction of the density-to-potential mapping. The close connections between the proposed criteria and the generalized adiabatic connection are further elaborated so as to keep the desired mapping intact at the strictly correlated regime. Exemplification of the unified constrained search formalism is done through model systems in order to demonstrate that the seemingly contradictory results reported so far are neither the true confirmation of lack of Hohenberg-Kohn theorem nor valid representation of violation of Gunnarsson-Lundqvist theorem for excited states. Hence the misleading interpretation of subtle differences between the ground and excited state density functional formalism are exemplified.