Semistatistical Model for Electrons in Atoms

A simple modification of the Thomas-Fermi statistical model for electrons in atoms and ions is presented, which corrects one of the model's major deficiencies, namely, its overestimation of the electron density near the nucleus. Exploiting the fact that near the nucleus the potential is nearly Coulombic, boundary conditions are derived from wave mechanics and incorporated into the model equations. The resulting equations are expressed in integral form and solved by a straightforward iterative technique. Results for total atomic energies show marked improvement over the corresponding results obtained from the Thomas-Fermi model, with only a slight increase in the required computational effort. The procedure is extended to obtain a relativistic statistical model of the atom. Numerical results show good agreement with wave mechanics in total energy calculations and in the form of the density near the nucleus. The relativistic effects are of greater magnitude than exchange or inhomogeneity corrections to the Thomas-Fermi model for large $Z$ atoms.