Conjoint gradient correction to the Hartree-Fock kinetic- and exchange-energy density functionals

Becke [J. Chem. Phys. 84, 4524 (1986); Phys. Rev. A 38, 3098 (1988)] has shown that the Hartree-Fock exchange energy for atoms (and molecules) can be excellently represented by a formula K=2^1/3C_xFJ_σ ρ_σ^4/3(r)[1+βG(x_σ)]dr, where C_x is the Dirac constant, β is a constant, G(x) is a function of the gradient-measuring variable x_σ=‖∇ρ_σ‖/ρ^4/3, and the summation is over spin densities ρ_σ. Becke recommends G(x_σ)=x_σ²/[1+0.0253x_σsinh^-1(x_σ)]. It is demonstrated that the kinetic energy can be represented with comparable accuracy by the formula T=2^2/3C_FF J_σ ρ_σ^5/3(r)[1+αG(x_σ)]dr, where C_F is the Thomas-Fermi constant, α is a constant, and G(x) is just the same function that appears in the formula for K. Recommended values, obtained by fitting data on rare-gas atoms, are α=4.4188×10^-3, β=4.5135×10^-3. The best α-to-β ratio, 0.979, is close to unity, and calculations with α=β=4.3952×10^-3 are shown to give remarkably accurate values for both T and K. It is briefly discussed how the above-noted equations for K and T can both result from scaling arguments and a simple assumption about the first-order density matrix.