Physical interpretation and evaluation of the Kohn–Sham and Dyson components of the ε–I relations between the Kohn–Sham orbital energies and the ionization potentials

Theoretical and numerical insight is gained into the ε–I relations between the Kohn–Sham orbital energies εi and relaxed vertical ionization potentials (VIPs) Ij, which provide an analog of Koopmans’ theorem for density functional theory. The Kohn–Sham orbital energy εi has as leading term −niIi−∑j∈Ωs(i)njIj, where Ii is the primary VIP for ionization (φi)−1 with spectroscopic factor (proportional to the intensity in the photoelectron spectrum) ni close to 1, and the set Ωs(i) contains the VIPs Ij that are satellites to the (φi)−1 ionization, with small but non-negligible nj. In addition to this “average spectroscopic structure” of the εi there is an electron-shell step structure in εi from the contribution of the response potential vresp. Accurate KS calculations for prototype second- and third-row closed-shell molecules yield valence orbital energies −εi, which correspond closely to the experimental VIPs, with an average deviation of 0.08 eV. The theoretical relations are numerically investigated in calculations of the components of the ε–I relations for the H2 molecule, and for the molecules CO, HF, H2O, HCN. The derivation of the ε–I relations employs the Dyson orbitals (the ni are their norms). A connection is made between the KS and Dyson orbital theories, allowing the spin-unrestricted KS xc potential to be expressed with a statistical average of individual xc potentials for the Dyson spin–orbitals as leading term. Additional terms are the correction vc,kin,σ due to the correlation kinetic effect, and the “response” vresp,σ, related to the correction to the energy of (N−1) electrons due to the correlation with the reference electron.