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Differentiable but exact formulation of density-functional theory

Kvaal, Simen; Ekström, Ulf; Teale, Andrew M.; Helgaker, Trygve


Simen Kvaal

Ulf Ekström

Trygve Helgaker


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.


Kvaal, S., Ekström, U., Teale, A. M., & Helgaker, T. (2014). Differentiable but exact formulation of density-functional theory. Journal of Chemical Physics, 140(18),

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 AIP Publishing
Peer Reviewed Peer Reviewed
Volume 140
Issue 18
Article Number 18A518
Public URL
Publisher URL
Copyright Statement Copyright information regarding this work can be found at the following address:
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


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Copyright Statement
Copyright information regarding this work can be found at the following address:

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