Universal Functionals in Density Functional Theory

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. An appendix is devoted to the Hohenberg-Kohn theorem and the role of many-body unique continuation in its proof.

Final version of a chapter to appear in the book "Density Functional Theory - Modeling, Mathematical Analysis, Computational Methods, and Applications", edited by Eric Cancès and Gero Friesecke, Springer

References in corpus (17)

• The Pauli principle revisited
• Geometric methods for nonlinear many-body quantum systems
• Diffusion quantum Monte Carlo study of three-dimensional Wigner crystals
• Coulomb and Riesz gases: The known and the unknown
• How tight is the Lieb-Oxford bound?
• Improved Lieb-Oxford exchange-correlation inequality with gradient correction
• Kirzhnits gradient expansion for a D-dimensional Fermi gas
• Lower Bounds on the Exchange-Correlation Energy in Reduced Dimensions
• Low-density Phase Diagram of the Three-Dimensional Electron Gas
• Itinerant-electron magnetism: the importance of many-body correlations
• Energy Cost to Make a Hole in the Fermi Sea
• Improved Lieb-Oxford bound on the indirect and exchange energies
• The strictly-correlated electron functional for spherically symmetric systems revisited
• Building Kohn-Sham potentials for ground and excited states
• Floating Wigner Crystal and Periodic Jellium Configurations
• Some properties of the potential-to-ground state map in quantum mechanics
• First order expansion in the semiclassical limit of the Levy-Lieb functional

Cited by in corpus (9)

• Coulomb and Riesz gases: The known and the unknown
• Analyticity of the one-particle density matrix
• Ground state energy is not always convex in the number of electrons
• Classical Density Functional Theory: The Local Density Approximation
• Exchange-only virial relation from the adiabatic connection
• The Validity of the Local Density Approximation for Smooth Short Range Interaction Potentials
• Grand-Canonical Optimal Transport
• Optimizers for the finite-rank Lieb-Thirring inequality
• Mixed state representability of entropy-density pairs