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Two-term asymptotics of the exchange energy of the electron gas on symmetric polytopes in the high-density limit

Thiago Carvalho Corso

TL;DR

This work derives two-term asymptotics for the exchange energy of the free electron gas at high density on strictly tessellating polytopes and lattice fundamental domains, revealing a bulk Dirac term and a boundary-layer surface correction. By constructing an exact wave kernel using domain symmetries and establishing uniform and $L^2$ estimates for the spectral function, the authors justify a continuum approximation $S_\lambda^{\text{ctm}}$ in the thermodynamic limit and extract precise interior and boundary contributions. They obtain explicit two-term expansions for the exchange energy $E_x(\lambda)$ and for a broad class of semi-local functionals $F(\lambda)$, including an integral constraint that links GGAs to the exact exchange energy, thereby informing functional design. The results illuminate finite-size and boundary-layer effects in the high-density limit and provide a rigorous bridge between spectral geometry and density-functional theory, with explicit constants in the Coulomb 3D case and implications for semi-local approximations in DFT.

Abstract

We derive a two-term asymptotic expansion for the exchange energy of the free electron gas on strictly tessellating polytopes and fundamental domains of lattices in the thermodynamic limit. This expansion comprises a bulk (volume-dependent) term, the celebrated Dirac exchange, and a novel surface correction stemming from a boundary layer and finite-size effects. Furthermore, we derive analogous two-term asymptotic expansions for semi-local density functionals. By matching the coefficients of these asymptotic expansions, we obtain an integral constraint for semi-local approximations of the exchange energy used in density functional theory.

Two-term asymptotics of the exchange energy of the electron gas on symmetric polytopes in the high-density limit

TL;DR

This work derives two-term asymptotics for the exchange energy of the free electron gas at high density on strictly tessellating polytopes and lattice fundamental domains, revealing a bulk Dirac term and a boundary-layer surface correction. By constructing an exact wave kernel using domain symmetries and establishing uniform and estimates for the spectral function, the authors justify a continuum approximation in the thermodynamic limit and extract precise interior and boundary contributions. They obtain explicit two-term expansions for the exchange energy and for a broad class of semi-local functionals , including an integral constraint that links GGAs to the exact exchange energy, thereby informing functional design. The results illuminate finite-size and boundary-layer effects in the high-density limit and provide a rigorous bridge between spectral geometry and density-functional theory, with explicit constants in the Coulomb 3D case and implications for semi-local approximations in DFT.

Abstract

We derive a two-term asymptotic expansion for the exchange energy of the free electron gas on strictly tessellating polytopes and fundamental domains of lattices in the thermodynamic limit. This expansion comprises a bulk (volume-dependent) term, the celebrated Dirac exchange, and a novel surface correction stemming from a boundary layer and finite-size effects. Furthermore, we derive analogous two-term asymptotic expansions for semi-local density functionals. By matching the coefficients of these asymptotic expansions, we obtain an integral constraint for semi-local approximations of the exchange energy used in density functional theory.
Paper Structure (10 sections, 17 theorems, 118 equations, 2 figures)

This paper contains 10 sections, 17 theorems, 118 equations, 2 figures.

Table of Contents

  1. Introduction and main results
  2. Wave kernel on symmetric polytopes
  3. Asymptotics of the spectral function
  4. Uniform estimates
  5. $L^2$ estimate
  6. Asymptotics of functionals
  7. Proof of Theorem \ref{['semilocalthm']}
  8. Proof of Theorem \ref{['exchangethm']}
  9. Strictly tessellating polytopes
  10. The wave kernel method

Key Result

Theorem 1.1

Let $\Omega \subset \mathbb R^n$ be a strictly tessellating polytope (see Definition def:kaleidoscopic) or the fundamental domain of a lattice. Let $E_x(\lambda)$ be the exchange energy defined in exchangespec and suppose that $n\geq 2$ and $\frac{n-1}{2}-\frac{n-1}{n+1}<s<n$. Then, for any $\epsilo where The leading exchange, the finite size, and the boundary layer constants are given by where

Figures (2)

  • Figure 1: Kaleidoscopic polytopes in $\mathbb R^3$. From left upper corner: rectangular parallelepiped, equilateral prism, 30-60-90 prism, isosceles (45-45-90) prism, quadrirectangular tetrahedron, trirectangular tetrahedron, and tetragonal disphenoid.
  • Figure 2: Example of strict tessellations of the space by some solids of Figure \ref{['figure:kaleidoscopicpolytopes']}.

Theorems & Definitions (38)

  • Theorem 1.1: Asymptotics of exchange energy
  • Theorem 1.2: Asymptotics of semi-local functionals
  • Remark : On the constants
  • Remark : Periodic case
  • Remark
  • Corollary 1.3: Integral constraint for GGAs
  • Remark : Kinetic energy approximations
  • Theorem 1.4: Asymptotics of the spectral function
  • Theorem 1.5: $L^2$ estimate of spectral function
  • Definition 2.1: Strictly tesselating polytopes
  • ...and 28 more