Vertical ionization energies from inverse generalized Kohn-Sham calculations at fixed electron density

Abstract: The exact Kohn--Sham (KS) system must have its highest occupied orbital energy equal to the negative of the lowest vertical ionization energy (VIE). No comparable theorem connects the deeper occupied levels $\varepsilon_k$ to the higher VIEs $I_k$. Chong, Gritsenko and Baerends (CGB) established empirically that $$I_k + \varepsilon_k\propto\sqrt{\varepsilon_{\mathrm{HOMO}}-\varepsilon_k}$$. We ask whether a similar relation holds for the exact generalized Kohn-Sham (GKS) system, which incorporates a fraction $α$ of Hartree-Fock exchange. We invert near-exact CCSD/aug-cc-pVQZ ground-state densities of LiH, LiF, H$_2$O, HF, CO, N$_2$ and F$_2$ at $α= 0, 0.25, 0.5, 0.75, 1.0$ and obtain the exact GKS systems at each admixture. As the density is held fixed and only the operator is varied, the resulting occupied spectra separate the effect of exact-exchange admixture on orbital energies from its effect on the density, two contributions inseparable in a self-consistent hybrid calculation. The CGB relation holds broadly across the family, with a proportionality coefficient linear in $α$, $c(α)=c(0)-sα$, where $c(0)=1.185\pm0.017$ eV$<sup>{1/2}$ and $s=1.987\pm0.041$ eV$<sup>{1/2}$, with $R<sup>{2}\ge0.9996$ for H$_2$O, CO, HF and N$_2$. Using the response-potential matrix elements tabulated by Gritsenko, Braïda and Baerends, we decompose the CGB coefficient into response and spectroscopic parts, and find $s$ to agree with the homogeneous-electron-gas step-height coefficient $K_g = 1.9932$ eV$<sup>{1/2}$, which we conjecture is its origin. The coefficient vanishes at $α<sup>{*}=0.597\pm0.008$. At $α<sup>{*}$ the HOMO-referenced GKS occupied levels satisfy $\varepsilon_k\approx-I_k$ across the whole occupied range. The zero crossing is fixed by the core orbitals; in the outer valence the CGB coefficient is already indistinguishable from zero at $α=0$.