Differentiable but exact formulation of density-functional theory

Kvaal, Simen; Ekstr�m, Ulf; Teale, Andrew M.; Helgaker, Trygve

Abstract

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 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 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.

Citation

Kvaal, S., Ekström, U., Teale, A. M., & Helgaker, T. (2014). Differentiable but exact formulation of density-functional theory. Journal of Chemical Physics, 140(18), Article 18A518. https://doi.org/10.1063/1.4867005

• Journal Article Type: Article
• Acceptance Date: Feb 1, 2014
• Online Publication Date: Mar 11, 2014
• Publication Date: May 14, 2014
• Deposit Date: Dec 16, 2015
• Publicly Available Date: Dec 16, 2015
• Journal: The Journal of Chemical Physics
• Print ISSN: 0021-9606
• Electronic ISSN: 1089-7690
• Publisher: American Institute of Physics
• Peer Reviewed: Peer Reviewed
• Volume: 140
• Issue: 18
• Article Number: 18A518
• DOI: https://doi.org/10.1063/1.4867005
• Public URL: https://nottingham-repository.worktribe.com/output/725286
• Publisher URL: http://scitation.aip.org/content/aip/journal/jcp/140/18/10.1063/1.4867005
• Additional Information: Copyright (2014) AIP Publishing. This article may be downloaded for personal use only. Any other use requires prior permission of the author and AIP Publishing.; ; The following article appeared in S. Kvaal, U. Ekström, A.M. Teale and T. Helgaker, J. Chem. Phys. 140, 18A518 (2014) and may be found at http://scitation.aip.org/content/aip/journal/jcp/140/18/10.1063/1.4867005