CONJOINT GRADIENT CORRECTION TO THE HARTREE-FOCK KINETIC-ENERGY AND EXCHANGE-ENERGY DENSITY FUNCTIONALS

Becke [J. Chem. Phys. 84, 4524 (1986); Phys. Rev. A 38, 3098 (1988)] has shown that the Hatree-Fock exchange energy for atoms (and molecules) can be excellently represented by a formula K = 2(1/3)C(x) integral SIGMA-sigma-rho-sigma-4/3(r)[1+BG(x-sigma)]dr, where C(x) is the Dirac constant, beta is a constant, G(x) is a function of the gradient-measuring variable x-sigma = \NABLA-rho-sigma\/rho-4/3, and the summation is over spin densities rho-sigma. Becke recommends G(x-sigma) = x-sigma-2/[1 + 0.0253x-sigma-sinh-1(x-sigma)]. It is demonstrated that the kinetic energy can be represented with comparable accuracy by the formula T = 2(2/3) C(F) integral SIGMA-sigma-rho-sigma-5/3(r)[1 + alpha-G(x-sigma)]dr, where C(F) is the Thomas-Fermi constant, 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 alpha = 4.4188 x 10(-3), beta = 4.5135 x 10(-3). The best alpha-to-beta ratio, 0.979, is close to unity, and calculations with alpha = beta = 4.3952 x 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.