Exact density-functional theory as parallel ensemble variational hierarchies: from Lieb's formulation to Kohn-Sham theory

Abstract

Exact density-functional theory is recast here as two parallel exact ensemble variational hierarchies: an interacting hierarchy rooted in Lieb's ensemble formulation and a noninteracting hierarchy rooted in exact noninteracting ensemble theory. The Kohn-Sham construction links the two on a common admissible density class. In this recasting, Levy-Lieb, Hohenberg-Kohn, and Kohn-Sham formulations appear as constrained specializations of broader ensemble variational structures. Fractional particle number and fractional occupations enter naturally in the same ensemble setting, while piecewise linearity, one-sided chemical potentials, derivative discontinuity, and Janak-type relations emerge as consequences of the underlying variational geometry. We also clarify several distinctions that are often compressed together in standard expositions, including functional domain versus representability class, density reproduction versus spectral interpretation, and the relation between state-space realization and density-level supporting-potential structure.