Koopmans' analysis of chemical hardness with spectral-like resolution.

Three approximation levels of Koopmans' theorem are explored and applied: the first referring to the inner quantum behavior of the orbitalic energies that depart from the genuine ones in Fock space when the wave-functions' Hilbert-Banach basis set is specified to solve the many-electronic spectra of spin-orbitals' eigenstates; it is the most subtle issue regarding Koopmans' theorem as it brings many critics and refutation in the last decades, yet it is shown here as an irrefutable "observational" effect through computation, specific to any in silico spectra of an eigenproblem; the second level assumes the "frozen spin-orbitals" approximation during the extracting or adding of electrons to the frontier of the chemical system through the ionization and affinity processes, respectively; this approximation is nevertheless workable for great deal of chemical compounds, especially organic systems, and is justified for chemical reactivity and aromaticity hierarchies in an homologue series; the third and the most severe approximation regards the extension of the second one to superior orders of ionization and affinities, here studied at the level of chemical hardness compact-finite expressions up to spectral-like resolution for a paradigmatic set of aromatic carbohydrates.