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^{1/3}$${\mathit{C}}_{\mathit{x}}$F${\mathcal{J}}_{\mathrm{\sigma }}$ ${\mathrm{\rho }}_{\mathrm{\sigma }}^{4/3}$(r)[1+βG(${\mathit{x}}_{\mathrm{\sigma }}$)]dr, where ${\mathit{C}}_{\mathit{x}}$ is the Dirac constant, β is a constant, G(x) is a function of the gradient-measuring variable ${\mathit{x}}_{\mathrm{\sigma }}$=‖∇${\mathrm{\rho }}_{\mathrm{\sigma }}$‖/${\mathrm{\rho }}^{4/3}$, and the summation is over spin densities ${\mathrm{\rho }}_{\mathrm{\sigma }}$. Becke recommends G(${\mathit{x}}_{\mathrm{\sigma }}$)=${\mathit{x}}_{\mathrm{\sigma }}^2$/[1+0.0253${\mathit{x}}_{\mathrm{\sigma }}$${\sinh }^{\mathrm{-}1}$(${\mathit{x}}_{\mathrm{\sigma }}$)]. It is demonstrated that the kinetic energy can be represented with comparable accuracy by the formula T=$2^{2/3}$${\mathit{C}}_{\mathit{F}}$F ${\mathcal{J}}_{\mathrm{\sigma }}$ ${\mathrm{\rho }}_{\mathrm{\sigma }}^{5/3}$(r)[1+αG(${\mathit{x}}_{\mathrm{\sigma }}$)]dr, where ${\mathit{C}}_{\mathit{F}}$ 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×${10}^{\mathrm{-}3}$, β=4.5135×${10}^{\mathrm{-}3}$. The best α-to-β ratio, 0.979, is close to unity, and calculations with α=β=4.3952×${10}^{\mathrm{-}3}$ 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.