Simple modification of Thomas-Fermi theory

A simple modification of the Thomas-Fermi (TF) theory is suggested by replacing rho5/3 with alpha (N)r-1 rho4/3 in the TF atomic kinetic-energy density where alpha (N) obviously varies linearly with N1/3, N being the total number of electrons. This modification gives rise to a previously unsolved, non-linear differential equation which has been solved numerically subject to the boundary conditions. Although the calculated electronic density and radial density for atoms have rather long tails, the atomic energy values are in satisfactory agreement with the corresponding Hartree-Fock values. Both at the nuclear site and in the intermediate region, the electron density is considerably improved over the original TF values. The present work is likely to lead to a systematic improvement of algebraic and differential single equations for the direct calculation of electron density in many-electron systems and may also be useful in other areas of non-linear sciences.