Modified Statistical Treatment of Kinetic Energy in the Thomas−Fermi Model

We try to improve the Thomas−Fermi model for the total energy and electron density of atoms and molecules by directly modifying the Euler equation for the electron density, which we argue is less affected by nonlocal corrections. Here we consider the simplest such modification by adding a linear gradient term to the Euler equation. For atoms, the coefficient of the gradient term can be chosen so that the correct exponential decay constant far from the nucleus is obtained. This model then gives a much improved description of the electron density at smaller distances, yielding in particular a finite density at the nucleus that is in good qualitative agreement with exact results. The cusp condition differs from the exact value by a factor of 2. Values for the total energy of atomic systems, obtained by coupling parameter integration of the densities given by the Euler equation, are about as accurate as those given by the very best Thomas−Fermi−Weizsäcker models, and the density is much more accurate. Possible connections to orbital-free methods for the kinetic-energy functional in density functional theory are discussed.