Optimized effective atomic central potential

A self-consistent set of equations is derived for an atomic central potential such that the energy given by the orbitals for the potential is minimized. It is shown that this effective potential behaves like $\frac{-e^2}{r}$ for large $r$ values. The equations have been solved for carbon, neon, and aluminum, and the resulting total energies exceed the Hartree-Fock total energies by less than 0.005%. The theory leads to an effective, local, central exchange potential analogous to the $X\alpha$ potential.