Accurate and simple density functional for the electronic exchange energy: Generalized gradient approximation

TLDR

The authors propose a generalized gradient approximation for the exchange energy, expressed as \(E_x[n]=A_x\int d^3r\,n^{4/3}F(s)\) with \(F(s)=(1+1.296s^2+14s^4+0.2s^6)^{1/15}\), derived from a gradient‑expanded exchange hole constrained to be negative and to represent a deficit of one electron. The resulting functional is simple enough for routine self‑consistent calculations and yields exchange energies within 1 % of Hartree‑Fock values, while markedly improving valence‑shell removal energies and jellium surface energies compared to other simple functionals.

Abstract

The electronic exchange energy as a functional of the density may be approximated as ${E}_{x}[n]={A}_{x}\ensuremath{\int}{d}^{3}r{n}^{\frac{4}{3}}F(s)$, where $s=\frac{|\ensuremath{\nabla}n|}{2{k}_{F}n}$, ${k}_{F}={(3{\ensuremath{\pi}}^{2}n)}^{\frac{1}{3}}$, and $F(s)={(1+1.296{s}^{2}+14{s}^{4}+0.2{s}^{6})}^{\frac{1}{15}}$. The basis for this approximation is the gradient expansion of the exchange hole, with real-space cutoffs chosen to guarantee that the hole is negative everywhere and represents a deficit of one electron. Unlike the previously publsihed version of it, this functional is simple enough to be applied routinely in self-consistent calculations for atoms, molecules, and solids. Calculated exchange energies for atoms fall within 1% of Hartree-Fock values. Significant improvements over other simple functionals are also found in the exchange contributions to the valence-shell removal energy of an atom and to the surface energy of jellium within the infinite barrier model.