Atomic Self-Consistent-Field Calculations Using Statistical Approximations for Exchange and Correlation

The ${\rho }^{\frac{1}{3}}$ approximation introduced by Slater for the exchange terms in the Hartree-Fock (HF) equations, with or without Latter's cutoff in the tail of the potential function, leads to serious errors when applied to excited configurations of atoms. A more accurate approximation (HX) is suggested in which Hartree's method is used for the self-interaction correction, and a modification of the Slater ${\rho }^{\frac{1}{3}}$ term is used to approximate the remainder of the HF exchange terms. The HX scheme is fairly simple; iterative convergence to self-consistent-field solutions is quite stable; and the resulting one-electron radial wave functions produce acceptably accurate values of one-electron and total binding energies, interaction parameters $F^k$, $G^k$, and ${\zeta }_i$, and expectation values of ${\mathcal{r}}^n$. Perturbation-theory relativistic and correlation corrections are computed; the latter are based on the free-electron approximation, with empirical modifications to allow for differences between free and bound electrons. Results agree well with experimental ionization and excitation energies for a wide variety of configurations.