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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 $({}^ε\!E,{}^ε\!F)$ that converge to $(E,F)$ pointwise everywhere as $ε\rightarrow 0^+$, and such that ${}^ε\!F$ 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 ${}^ε\!E(v)$ in a simple way. All concepts and results pertaining to the original $(E,F)$ pair have direct counterparts in results for $({}^ε\! E, {}^ε\!F)$. The Moreau-Yosida regularization therefore allows for an exact, differentiable formulation of density-functional theory. In particular, taking advantage of the differentiability of ${}^ε\!F$, a rigorous formulation of Kohn-Sham theory is presented that does not suffer from the noninteracting representability problem in standard Kohn-Sham theory.