Improving virtual Kohn–Sham orbitals and eigenvalues: Application to excitation energies and static polarizabilities

Conventional continuum exchange-correlation functionals (e.g., local density approximation, generalized gradient approximation) offer a poor description of many response properties, such as static polarizabilities and single photon vertical excitation energies to Rydberg states. These deficiencies are related to errors in the virtual Kohn–Sham orbitals and eigenvalues, which arise due to a fundamental deficiency in the potentials of conventional continuum functionals. Namely, although these potentials approximately average over the exact integer discontinuity in energetically important regions, they fail to do so asymptotically, because they vanish. Our recent functional HCTH [J. Chem. Phys. 109, 6264 (1998)] was designed with this deficiency in mind, although its potential still fails to exhibit the appropriate asymptotic form. In this paper, we present a new procedure that explicitly corrects this asymptotic deficiency for any continuum functional. Self-consistent Kohn–Sham calculations are performed using a corrected potential, which equals the conventional potential ${\mathrm{\delta E}}_{\mathrm{XC}}{\mathrm{[\rho }}_{\alpha }{\mathrm{,\rho }}_{\beta }{\mathrm{]/\delta \rho }}_{\sigma }({r})$ in energetically important regions, but which asymptotically behaves in the required average manner ${\text{−(1/r)+I}}_{\sigma }{\mathrm{+\epsilon }}_{\mathrm{HOMO}\mathrm{,\sigma }}.$ The quantity $\text{−(1/r)}$ is determined using a nonlocal expression; $I_{\sigma }$ is an approximate $\sigma$ spin ionization potential; and ${\epsilon }_{\mathrm{HOMO}\mathrm{,\sigma }}$ is the highest occupied $\sigma$ spin eigenvalue. By applying this correction to the HCTH potential, we accurately reproduce the hydrogen atom eigenvalue spectrum, without significantly changing the total energy. We determine corrected orbitals and eigenvalues for a variety of molecules, and use them to compute excitation energies and static polarizabilities. We compare the results with those from a variety of other exchange-correlation functionals. Excitations to Rydberg states are determined as accurately as those to valence states; for CO, ${\mathrm{N}}_2,$ ${\mathrm{H}}_{\mathrm{2}}\mathrm{CO},$ and ${\mathrm{C}}_{\mathrm{2}}{\mathrm{H}}_{\mathrm{4}},$ mean absolute errors are less than 0.35 eV. The static isotropic polarizabilities of 14 small molecules are of MP2 quality.