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The non-uniform electron gas

Mihaly A. Csirik, Andre Laestadius

Abstract

The non-uniform (or inhomogeneous) electron gas has received much attention in many-body quantum mechanics and quantum chemistry in the early days of density functional theory, mainly as a theoretical device to construct gradient approximations via linear response theory. In this article, motivated by the recent works of Lewin, Lieb and Seiringer, we propose a definition of the quantum (resp. classical) non-uniform electron gas through the use of the grand-canonical Levy-Lieb functional (resp. the grand-canonical strictly correlated electrons functional), establish these systems as rigorous thermodynamic limits and analyze their basic properties. The non-uniformity of the gas comes from an arbitrary lattice-periodic background density.

The non-uniform electron gas

Abstract

The non-uniform (or inhomogeneous) electron gas has received much attention in many-body quantum mechanics and quantum chemistry in the early days of density functional theory, mainly as a theoretical device to construct gradient approximations via linear response theory. In this article, motivated by the recent works of Lewin, Lieb and Seiringer, we propose a definition of the quantum (resp. classical) non-uniform electron gas through the use of the grand-canonical Levy-Lieb functional (resp. the grand-canonical strictly correlated electrons functional), establish these systems as rigorous thermodynamic limits and analyze their basic properties. The non-uniformity of the gas comes from an arbitrary lattice-periodic background density.
Paper Structure (30 sections, 27 theorems, 225 equations)

This paper contains 30 sections, 27 theorems, 225 equations.

Table of Contents

  1. Introduction
  2. Preliminary notions
  3. Notations
  4. Lattice-periodic functions
  5. Classical density functional theory
  6. Grand-canonical Levy--Lieb functional
  7. Main results
  8. Classical non-uniform electron gas
  9. Quantum non-uniform electron gas
  10. Definition based smeared indicators
  11. Definition based on a transition region
  12. Thermodynamic limit and equivalence of definitions
  13. Further results
  14. Proofs for the classical case
  15. Known bounds
  16. ...and 15 more sections

Key Result

Theorem 2.1

For any $\mathcal{L}$-periodic complex-valued function $u\in L_{\mathrm{loc}}^1(\mathbb{R}^d)$, the mean value of $u$ over $\mathbb{R}^d$ exists and equals its mean value over a unit cell $\Lambda\subset\mathbb{R}^d$ of $\mathcal{L}$, i.e. independently and uniformly in $a\in\mathbb{R}^d$.

Theorems & Definitions (39)

  • Theorem 2.1
  • Theorem 3.1: Classical non-uniform electron gas
  • Remark 1
  • Theorem 3.2: Local density approximation
  • Theorem 3.3: Quantum non-uniform electron gas
  • Remark 2
  • Theorem 3.4: Local density approximation
  • Proposition 3.5: Semiclassical bound
  • Theorem 4.1: Lieb--Oxford inequality
  • Theorem 4.2: Graf--Schenker inequality
  • ...and 29 more