Koopmans’ theorem and selection rules for one-electron ionization processes in orbitally degenerate systems

Abstract

One-electron ionization processes [Formula: see text] in orbitally degenerate systems, such as atoms with the open-shell configuration p N, can be divided into two groups. The first group involves the processes that are allowed in photoelectron spectra. The processes of this group in atoms obey the familiar selection rules (SRs) formulated within the Russell–Saunders L, S coupling. All other ionization processes, for which SRs are not obeyed, belong to the second group. Here, we analyze the validity of Koopmans’ theorem (KT) for the processes of the second group forbidden by SRs. We show that the general formulation of KT in the Hartree–Fock method [Plakhutin, J. Chem. Phys. 148, 094101 (2018)] is implicitly based on the assumption that a [Formula: see text] process is allowed by SRs, and this presents a limitation of KT. To overcome the latter, we develop an extension of KT that enables estimating the energies of SR-forbidden processes. We prove that the variational condition underlying KT gives different results for SR-allowed and SR-forbidden processes. For the former processes, this condition gives the familiar KT relationship I i = − ɛ i, while for SR-forbidden processes, the respective relationship between I i and ɛ i takes a more complex form. The practical applicability of the extension of KT is verified by applying it to the totality of ionization processes in the valence 2 s and 2 p shells of atoms C, N, and O in their ground and excited states, which involves a total of 29 SR-allowed and 34 SR-forbidden processes. For all of these processes, we compare KT estimates of ionization energies (IEs) with the relevant experimental data. For comparison, we also present the respective estimates of IEs derived with a ΔSCF approach. Particular attention is paid to the analysis of the validity of KT in the specific cases of violation of Hund’s rules for cation states.