Differentiable but exact formulation of densityfunctional theoryTools Kvaal, Simen and Ekström, Ulf and Teale, Andrew M. and Helgaker, Trygve (2014) Differentiable but exact formulation of densityfunctional theory. Journal of Chemical Physics, 140 (18). 18A518 /118A518/14. ISSN 10897690
AbstractThe universal density functional F of densityfunctional theory is a complicated and illbehaved 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 groundstate 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 ε > 0, pairs of conjugate functionals (ε E, ε F) that converge to (E, F) pointwise everywhere as ε → 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 groundstate energy E(v) is exactly recoverable from the regularized groundstate 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 densityfunctional 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.
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