Atomic Energy Levels for the Thomas-Fermi and Thomas-Fermi-Dirac Potential

The eigenvalues of the Schr\"odinger equation have been obtained for the Thomas-Fermi and Thomas-Fermi-Dirac atomic potentials. Electron self-interactions were taken into account by modifying the potentials to give asymptotically the field of a unit charge. All levels were treated from $1s$ to $7d$ for a range of Z-values sufficient to permit easy interpolation. It was found that the energies, for either the Thomas-Fermi or Thomas-Fermi-Dirac potentials, agree in general as well with experimental ionization energies as the Hartree or Hartree-Fock approximations. Applications of the statistical potential to other atomic problems are indicated.