The analog of Koopmans' theorem for virtual Kohn-Sham orbital energies

An analog of Koopmans' theorem is formulated for the energies, epsilon a, of virtual Kohn-Sham (KS) molecular orbitals (MOs) from the requirement that the KS theory provides, in principle, not only the exact electron density, but also its exact response. The starting point is the Kohn-Sham analog of Koopmans' theorem, relating the vertical ionization energies, I-i, to the energies, epsilon(i), of the occupied MOs (Chong, D. P.Gritsenko, O.V.Baerends, E.J. J. Chem. Phys. 2002, 116, 1760). Combining this with the coupled-perturbed equations of time-dependent density functional theory (TDDFT), exact relations between the energies, epsilon(a), of virtual KS MOs and the excitation energies, omega(ia), and vertical ionization energies (VIPs), I-i, are obtained. In the small matrix approximation for the coupling matrix K of TDDFT, two limiting cases of these relations are considered. In the limit of a negligible matrix element, K-ia,K- ia, the energy, epsilon(a), can be interpreted as (minus) the energy of ionization from the phi(i) -> phi(a) excited state, epsilon(a) approximate to -I-a, where -I-a is defined from the relation I-i = omega(ia) + I-a. This relation breaks down in special cases, such as charge-transfer transitions and the HOMO-LUMO (highest occupied molecular orbital - lowest unoccupied molecular orbital) transition of a dissociating electron-pair bond (also of charge-transfer character). The present results highlight the important difference between virtual orbital energies in the Kohn-Sham model (epsilon(a) approximate to -I-a) and in the Hartree-Fock model (epsilon(a) approximate to -A(a)). Kohn-Sham differences epsilon(a) - epsilon(i) approximate the excitation energy, omega(ia), while Hartree-Fock differences epsilon(HF)(a) approximate to epsilon(HF)(i) do not approximate excitation energies but approximate the difference of an ionization energy and an electron affinity, I-i - A(a).