First Ionization Energy as the Asymptotic Limit of the Average Local Electron Energy

The
first vertical ionization energy of an atom or molecule is
encoded in the rate of exponential decay of the exact natural orbitals.
For natural orbitals represented in terms of Gaussian basis functions,
this property does not hold even approximately. We show that it is
nevertheless possible to deduce the first ionization energy from the
long-range behavior of Gaussian-basis-set wave functions by evaluating
the asymptotic limit of a quantity called the average local electron
energy (ALEE), provided that the most diffuse functions of the basis
set have a suitable shape and location. The ALEE method exposes subtle
qualitative differences between seemingly analogous Gaussian basis
sets and complements the extended Koopmans theorem by being robust
in situations where the one-electron reduced density matrix is ill-conditioned.