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 p...
| Main Authors: | , , , |
|---|---|
| Format: | Article |
| Published: |
American Institute of Physics
2014
|
| Online Access: | https://eprints.nottingham.ac.uk/31103/ |
| _version_ | 1848794128228286464 |
|---|---|
| author | Kvaal, Simen Ekström, Ulf Teale, Andrew M. Helgaker, Trygve |
| author_facet | Kvaal, Simen Ekström, Ulf Teale, Andrew M. Helgaker, Trygve |
| author_sort | Kvaal, Simen |
| building | Nottingham Research Data Repository |
| collection | Online Access |
| description | 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. |
| first_indexed | 2025-11-14T19:11:16Z |
| format | Article |
| id | nottingham-31103 |
| institution | University of Nottingham Malaysia Campus |
| institution_category | Local University |
| last_indexed | 2025-11-14T19:11:16Z |
| publishDate | 2014 |
| publisher | American Institute of Physics |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | nottingham-311032020-05-04T16:45:03Z https://eprints.nottingham.ac.uk/31103/ Differentiable but exact formulation of density-functional theory Kvaal, Simen Ekström, Ulf Teale, Andrew M. Helgaker, Trygve 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. American Institute of Physics 2014-03-11 Article PeerReviewed Kvaal, Simen, Ekström, Ulf, Teale, Andrew M. and Helgaker, Trygve (2014) Differentiable but exact formulation of density-functional theory. Journal of Chemical Physics, 140 (18). 18A518 /1-18A518/14. ISSN 1089-7690 http://scitation.aip.org/content/aip/journal/jcp/140/18/10.1063/1.4867005 doi:10.1063/1.4867005 doi:10.1063/1.4867005 |
| spellingShingle | Kvaal, Simen Ekström, Ulf Teale, Andrew M. Helgaker, Trygve Differentiable but exact formulation of density-functional theory |
| title | Differentiable but exact formulation of density-functional theory |
| title_full | Differentiable but exact formulation of density-functional theory |
| title_fullStr | Differentiable but exact formulation of density-functional theory |
| title_full_unstemmed | Differentiable but exact formulation of density-functional theory |
| title_short | Differentiable but exact formulation of density-functional theory |
| title_sort | differentiable but exact formulation of density-functional theory |
| url | https://eprints.nottingham.ac.uk/31103/ https://eprints.nottingham.ac.uk/31103/ https://eprints.nottingham.ac.uk/31103/ |