Validity of the Extended Koopmans' Theorem

The generalized Koopmans' theorem (EKT) yields an estimate of ionization potentials (IPs) of an N-electron system. This estimate (IP(EKT)) is obtained as an eigenvalue of a generalized eigenvalue problem. Katriel and Davidson provided a proof [Katriel, J.; Davidson, E. R. Proc. Natl. Acad. Sci. U.S.A., 1980, 77, 1403.] that the EKT predicts the exact lowest IP for ground states of Coulomb systems. However, subsequently, several articles have been published challenging the exactness of the EKT and providing disproofs. This apparent contradiction is resolved by demonstrating that the lowest eigenvalue of the generalized Koopmans' procedure does, in general, not exist. This explains why contradictory results are obtained about the lowest IP(EKT) since its existence has implicitly been assumed. Nonetheless, it will also be shown here that the generalized Koopmans' approach gives IPs that are arbitrarily close to the exact lowest ionization energy. The eigenvalues obtained according to the EKT have an accumulation point given by the exact lowest IP.