Differentiable but exact formulation of density-functional theory
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 ({sup ε}E, {sup ε}F) that converge to (E, F) pointwise everywhere as ε → 0{sup +}, and such that {sup 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 {sup ε}E(v) in a simple way. All concepts and results pertaining to the original (E, F) pair have direct counterparts in results for ({sup ε}E, {sup ε}F). The Moreau–Yosida regularization therefore allows for an exact, differentiable formulation ofmore »
- Authors:
-
- Centre for Theoretical and Computational Chemistry, Department of Chemistry, University of Oslo, P.O. Box 1033 Blindern, N-0315 Oslo (Norway)
- Publication Date:
- OSTI Identifier:
- 22253466
- Resource Type:
- Journal Article
- Journal Name:
- Journal of Chemical Physics
- Additional Journal Information:
- Journal Volume: 140; Journal Issue: 18; Other Information: (c) 2014 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0021-9606
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 74 ATOMIC AND MOLECULAR PHYSICS; 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; DENSITY FUNCTIONAL METHOD; FUNCTIONALS; GROUND STATES
Citation Formats
Kvaal, Simen, Ekström, Ulf, Helgaker, Trygve, Teale, Andrew M., and School of Chemistry, University of Nottingham, University Park, Nottingham NG7 2RD. Differentiable but exact formulation of density-functional theory. United States: N. p., 2014.
Web. doi:10.1063/1.4867005.
Kvaal, Simen, Ekström, Ulf, Helgaker, Trygve, Teale, Andrew M., & School of Chemistry, University of Nottingham, University Park, Nottingham NG7 2RD. Differentiable but exact formulation of density-functional theory. United States. https://doi.org/10.1063/1.4867005
Kvaal, Simen, Ekström, Ulf, Helgaker, Trygve, Teale, Andrew M., and School of Chemistry, University of Nottingham, University Park, Nottingham NG7 2RD. 2014.
"Differentiable but exact formulation of density-functional theory". United States. https://doi.org/10.1063/1.4867005.
@article{osti_22253466,
title = {Differentiable but exact formulation of density-functional theory},
author = {Kvaal, Simen and Ekström, Ulf and Helgaker, Trygve and Teale, Andrew M. and School of Chemistry, University of Nottingham, University Park, Nottingham NG7 2RD},
abstractNote = {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 ({sup ε}E, {sup ε}F) that converge to (E, F) pointwise everywhere as ε → 0{sup +}, and such that {sup 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 {sup ε}E(v) in a simple way. All concepts and results pertaining to the original (E, F) pair have direct counterparts in results for ({sup ε}E, {sup 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 {sup 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.},
doi = {10.1063/1.4867005},
url = {https://www.osti.gov/biblio/22253466},
journal = {Journal of Chemical Physics},
issn = {0021-9606},
number = 18,
volume = 140,
place = {United States},
year = {Wed May 14 00:00:00 EDT 2014},
month = {Wed May 14 00:00:00 EDT 2014}
}