On the v-representability of ensemble densities of electron systems

Analogously to the case at zero temperature, where the density of the ground state of an interacting many-particle system determines uniquely (within an arbitrary additive constant) the external potential acting on the system, the thermal average of the density over an ensemble defined by the Boltzmann distribution at the minimum of the thermodynamic potential, or the free energy, determines the external potential uniquely (and not just modulo a constant) acting on a system described by this thermodynamic potential or free energy. The paper describes a formal procedure that generates the domain of a constrained search over general ensembles (at zero or elevated temperatures) that lead to a given density, including as a special case a density thermally averaged at a given temperature, and in the case of a v-representable density determines the external potential leading to the ensemble density. As an immediate consequence of the general formalism, the concept of v-representability is extended beyond the hitherto discussed case of ground state densities to encompass excited states as well. Specific application to thermally averaged densities solves the v-representability problem in connection with the Mermin functional in a manner analogous to that in which this problem was recently settled with respect to the Hohenberg and Kohn functional. The main formalism is illustrated with numerical results for ensembles of one-dimensional, non-interacting systems of particles under a harmonic potential.