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. While 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) subset of its domain. In this article, we apply a tool from convex analysis, Moreau-Yosida regularization, to construct, for any epsilon > 0, pairs of conjugate functionals (E-epsilon, F-epsilon) that converge to (E, F) pointwise everywhere as epsilon -> 0(+), and such that F-epsilon is (Frechet) 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 E-epsilon(v) in a simple way. All concepts and results pertaining to the original (E, F) pair have direct counterparts in results for (E-epsilon, F-epsilon). The Moreau-Yosida regularization therefore allows for an exact, differentiable formulation of density-functional theory. In particular, taking advantage of the differentiability of F-epsilon, a rigorous formulation of Kohn-Sham theory is presented that does not suffer from the noninteracting representability problem in standard Kohn-Sham theory. (C) 2014 AIP Publishing LLC.