Differentiable but exact formulation of density-functional theory
Kvaal, Simen; Ekström, Ulf; Teale, Andrew M.; Helgaker, Trygve
Andrew M. Teale
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.
|Journal Article Type||Article|
|Publication Date||May 14, 2014|
|Journal||The Journal of Chemical Physics|
|Peer Reviewed||Peer Reviewed|
|APA6 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), https://doi.org/10.1063/1.4867005|
|Copyright Statement||Copyright information regarding this work can be found at the following address: http://eprints.nottingh.../end_user_agreement.pdf|
|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.or...40/18/10.1063/1.4867005
Copyright information regarding this work can be found at the following address: http://eprints.nottingham.ac.uk/end_user_agreement.pdf
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