Vertical Ionization Energies from the Average Local Electron Energy Function
Abstract
It is a non-intuitive but well-established fact that the first and higher vertical ionization energies (VIE) of any N-electron system are encoded in the system's ground-state electronic wave function. This makes it possible to compute VIEs of any atom or molecule from its ground-state wave function directly, without performing calculations on the (N-1)-electron states. In practice, VIEs can be extracted from the wave function by using the (extended) Koopmans' theorem or by taking the asymptotic limit of certain wave-function-based quantities such as the ratio of kinetic energy density to the electron density. However, when the wave function is expanded in a Gaussian basis set, the latter method fails because the ratio diverges in the asymptotic limit. We show that, in such cases, the first VIE of any finite system can still be estimated by taking the asymptotic limit of the average local electron energy function. This function is constructed from an exact or approximate ground-state wave function of the system and approaches a system- and method-dependent constant in the asymptotic limit. For Hartree--Fock and density-functional theory, this limit reduces to the eigenvalue of the highest-occupied molecular orbital and hence the first VIE according to Koopmans' theorem. We also show that, in the finite-basis approximations of these theories, this constant will generally be more negative than the eigenvalue of the highest-occupied molecular orbital. The results are generalized to finite-basis-set post-Hartree--Fock theory.