Differentiable but exact formulation of density-functional theory

The universal density functional F of density-functional theory is a complicated and ill-behaved function of the density-in particular, F is not differentiable, making many formal manipulations more complicated. Whilst F has been well characterized in terms of convex analysis as forming a conjugate pair (E,F) with the ground-state energy E via the Hohenberg-Kohn and Lieb variation principles, F is nondifferentiable and subdifferentiable only on a small (but dense) set of its domain. In this article, we apply a tool from convex analysis, Moreau-Yosida regularization, to construct, for any ε>0, pairs of conjugate functionals \({}^εE,{}^εF\) that converge to (E,F) pointwise everywhere as ε→ 0^+, and such that {}^εF is (Fréchet) differentiable. For technical reasons, we limit our attention to molecular electronic systems in a finite but large box. It is noteworthy that no information is lost in the Moreau-Yosida regularization: the physical ground-state energy E(v) is exactly recoverable from the regularized ground-state energy {}^εE(v) in a simple way. All concepts and results pertaining to the original (E,F) pair have direct counterparts in results for \({}^ε E, {}^εF\). The Moreau-Yosida regularization therefore allows for an exact, differentiable formulation of density-functional theory. In particular, taking advantage of the differentiability of {}^εF, a rigorous formulation of Kohn-Sham theory is presented that does not suffer from the noninteracting representability problem in standard Kohn-Sham theory.