# Universal Functionals in Density Functional Theory

Lewin M, Lieb EH, Seiringer R. 2023.Universal Functionals in Density Functional Theory. In: Density Functional Theory. Mathematics and Molecular Modeling, , 115–182.

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https://doi.org/10.48550/arXiv.1912.10424
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*Book Chapter*|

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*English*

Author

Lewin, Mathieu;
Lieb, Elliott H.;
Seiringer, Robert

^{ISTA}^{}Book Editor

Cances, Eric;
Friesecke, Gero

Department

Series Title

Mathematics and Molecular Modeling

Abstract

In this chapter we first review the Levy–Lieb functional, which gives the lowest kinetic and interaction energy that can be reached with all possible quantum states having a given density. We discuss two possible convex generalizations of this functional, corresponding to using mixed canonical and grand-canonical states, respectively. We present some recent works about the local density approximation, in which the functionals get replaced by purely local functionals constructed using the uniform electron gas energy per unit volume. We then review the known upper and lower bounds on the Levy–Lieb functionals. We start with the kinetic energy alone, then turn to the classical interaction alone, before we are able to put everything together. A later section is devoted to the Hohenberg–Kohn theorem and the role of many-body unique continuation in its proof.

Publishing Year

Date Published

2023-07-19

Book Title

Density Functional Theory

Page

115-182

ISBN

ISSN

IST-REx-ID

### Cite this

Lewin M, Lieb EH, Seiringer R. Universal Functionals in Density Functional Theory. In: Cances E, Friesecke G, eds.

*Density Functional Theory*. 1st ed. MAMOMO. Springer; 2023:115-182. doi:10.1007/978-3-031-22340-2_3Lewin, M., Lieb, E. H., & Seiringer, R. (2023). Universal Functionals in Density Functional Theory. In E. Cances & G. Friesecke (Eds.),

*Density Functional Theory*(1st ed., pp. 115–182). Springer. https://doi.org/10.1007/978-3-031-22340-2_3Lewin, Mathieu, Elliott H. Lieb, and Robert Seiringer. “Universal Functionals in Density Functional Theory.” In

*Density Functional Theory*, edited by Eric Cances and Gero Friesecke, 1st ed., 115–82. MAMOMO. Springer, 2023. https://doi.org/10.1007/978-3-031-22340-2_3.M. Lewin, E. H. Lieb, and R. Seiringer, “Universal Functionals in Density Functional Theory,” in

*Density Functional Theory*, 1st ed., E. Cances and G. Friesecke, Eds. Springer, 2023, pp. 115–182.Lewin, Mathieu, et al. “Universal Functionals in Density Functional Theory.”

*Density Functional Theory*, edited by Eric Cances and Gero Friesecke, 1st ed., Springer, 2023, pp. 115–82, doi:10.1007/978-3-031-22340-2_3.**All files available under the following license(s):**

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arXiv 1912.10424