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}$${\mathit{C}}_{\mathit{x}}$F${\mathcal{J}}_{\mathrm{\ensuremath{\sigma}}}$ ${\mathrm{\ensuremath{\rho}}}_{\mathrm{\ensuremath{\sigma}}}^{4/3}$(r)[1+\ensuremath{\beta}G(${\mathit{x}}_{\mathrm{\ensuremath{\sigma}}}$)]dr, where ${\mathit{C}}_{\mathit{x}}$ is the Dirac constant, \ensuremath{\beta} is a constant, G(x) is a function of the gradient-measuring variable ${\mathit{x}}_{\mathrm{\ensuremath{\sigma}}}$=\ensuremath{\Vert}\ensuremath{\nabla}${\mathrm{\ensuremath{\rho}}}_{\mathrm{\ensuremath{\sigma}}}$\ensuremath{\Vert}/${\mathrm{\ensuremath{\rho}}}^{4/3}$, and the summation is over spin densities ${\mathrm{\ensuremath{\rho}}}_{\mathrm{\ensuremath{\sigma}}}$. Becke recommends G(${\mathit{x}}_{\mathrm{\ensuremath{\sigma}}}$)=${\mathit{x}}_{\mathrm{\ensuremath{\sigma}}}^{2}$/[1+0.0253${\mathit{x}}_{\mathrm{\ensuremath{\sigma}}}$${\mathrm{sinh}}^{\mathrm{\ensuremath{-}}1}$(${\mathit{x}}_{\mathrm{\ensuremath{\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 ${\mathcal{J}}_{\mathrm{\ensuremath{\sigma}}}$ ${\mathrm{\ensuremath{\rho}}}_{\mathrm{\ensuremath{\sigma}}}^{5/3}$(r)[1+\ensuremath{\alpha}G(${\mathit{x}}_{\mathrm{\ensuremath{\sigma}}}$)]dr, where ${\mathit{C}}_{\mathit{F}}$ is the Thomas-Fermi constant, \ensuremath{\alpha} 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 \ensuremath{\alpha}=4.4188\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}3}$, \ensuremath{\beta}=4.5135\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}3}$. The best \ensuremath{\alpha}-to-\ensuremath{\beta} ratio, 0.979, is close to unity, and calculations with \ensuremath{\alpha}=\ensuremath{\beta}=4.3952\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}3}$ 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.