Hartree-Fock Calculations for Atoms with Inner-Shell Vacancies

For configuration states such as $2p^{5}3p^5{}_{}{}^1{}_{}{}^{}S$, ${}_{}{}^3{}_{}{}^{}P$, ${}_{}{}^1{}_{}{}^{}D$ the Hartree-Fock method produces total wave functions with an incorrect form as the atomic number $Z\to \infty$. It is shown how a correct form can be obtained, with similar accuracy as the Hartree-Fock wave function, for a statistically weighted average of $2p^{5}3p^5{}_{}{}^3{}_{}{}^{}S$, ${}_{}{}^1{}_{}{}^{}P$, ${}_{}{}^3{}_{}{}^{}D$.