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

TL;DR

This paper reconstructs exact density-functional theory using convex variational structures, revealing parallel ensemble hierarchies and clarifying foundational distinctions in the theory.

Contribution

It introduces a novel parallel ensemble hierarchy framework connecting Lieb's formulation and Kohn-Sham theory, with insights into fractional occupations and variational geometry.

Findings

Reveals two parallel exact ensemble hierarchies in DFT.

Shows fractional occupations naturally arise in the ensemble setting.

Clarifies distinctions like functional domain versus representability class.

Abstract

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