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=21/3{C}ₓF{J}_{{{σ}}}$ ${{{ρ}}}_{{{σ}}}4/3(r)[1+βG({x}_{{{σ}}}$)]dr, where ${C}ₓis the Dirac constant, β is a constant, G(x) is a function of the gradient-measuring variable{x}_{{{σ}}}=∇{{{ρ}}}_{{{σ}}}/{{{ρ}}}4/3$, and the summation is over spin densities ${{{ρ}}}_{{{σ}}}$. Becke recommends G(${x}_{{{σ}}}$)=${x}_{{{σ}}}²$/[1+0.0253${x}_{{{σ}}}sinh^-1(x_σ)]. It is demonstrated that the kinetic energy can be represented with comparable accuracy by the formula T=22/3{C}FF{J}_{{{σ}}}$ ${{{ρ}}}_{{{σ}}}5/3(r)[1+αG({x}_{{{σ}}}$)]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.