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

The electronic exchange energy as a functional of the density may be approximated as $E_x\mathrm{}\left[n\right]=A_x\int d^{3}r{n}^{\frac{4}{3}}F\left(s\right)\mathrm{}$, where $s=\frac{\left|\nabla n\right|}{2k_{F}n}$, $k_F={\left(3\mathrm{}{\pi }^{2}n\right)}^{\frac{1}{3}}$, and $F\left(s\right)={(1+1.296\mathrm{}{s}^2+14\mathrm{}{s}^4+0.2\mathrm{}{s}^6)}^{\frac{1}{15}}$. 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.