Single-center orbital localization

The consequences of replacing the screening potential $-\rho V_a\rho$ in the modified Hartree-Fock equations $(F_a+V_a-\rho V_a\rho ){\varphi }_{\mathrm{ai}}={\epsilon }_{\mathrm{ai}}{\varphi }_{\mathrm{ai}}$ by a term $-{\rho }^{\mathrm{}\left(p\right)\mathrm{}}{V}_a{\rho }^{\mathrm{}\left(p\right)\mathrm{}}$ which screens the interaction potential only on a $p$-dimensional submanifold of the Hartree-Fock manifold are examined. It is shown that the orbitals are only semilocalized in the sense that the contributions from $p-1\mathrm{}$ neighboring atomic components are suppressed but beyond this distance the modified orbitals are not localized and resemble Bloch orbitals. One can, however, obtain approximate solutions with the proper local behavior by consistently truncating the orbital basis and neglecting the contribution of distant neighbors to the interaction potential $V_a$. The precautions which must be taken when using a one-center expansion in order to avoid multicenter integrals are pointed out.