Koopmans's theorem in the restricted open-shell Hartree–Fock method. II. The second canonical set for orbitals and orbital energies

A treatment of the validity of Koopmans's theorem KT in the restricted open-shell Hartree–Fock ROHF method can be separated into two essentially different cases. The first of them involves the one-electron processes X→Xj in which the spin state of an ion Xj having a hole or an extra electron in the closed, open or virtual orbital j is correctly described by a one-determinant wave function. This case was analyzed using different methods by Plakhutin et al. J. Chem. Phys. 125, 204110 2006 and by Plakhutin and Davidson J. Phys. Chem. A 113, 12386 2009. In the present work we analyze more complex processes where the state of an ion cannot be described by a single determinant. An example of such processes is the removal of an alpha electron from the closed shell of a high-spin half-filled open-shell system X. For this case we give a slightly generalized formulation of KT in both the "frozen" orbital approximation i.e., within the canonical ROHF method and the limited configuration interaction approach for ionized systems. We also show that a simultaneous treatment of KT for all one-electron ionization processes possible leads to the necessity of introducing in the canonical ROHF method two different sets of orbitals and two respective sets of orbital energies. The theory developed is compared with the previous formulations of KT in the restricted ROHF and unrestricted Hartree-Fock methods, and in the unrestricted density functional theory. The practical applicability of the theory is verified by comparing the KT estimates of the vertical ionization potentials in molecules O2 and NO2 with the respective experimental data