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 ${\epsilon }_k$ to the higher VIEs $I_k$. Chong, Gritsenko and Baerends (CGB) established empirically that $I_k+{\epsilon }_k\propto \sqrt{{\epsilon }_{\mathrm{HOMO}}-{\epsilon }_k}$. We ask whether a similar relation holds for the exact generalized Kohn-Sham (GKS) system, which incorporates a fraction $\alpha$ 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 $\alpha =0,0.25,0.5,0.75,1.0$ 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 $\alpha$, $c(\alpha )=c(0)-s\alpha$, where $c(0)=1.185\pm 0.017$ eV${}^{1/2}$ and $s=1.987\pm 0.041$ eV${}^{1/2}$, with $R^2\ge 0.9996$ 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${}^{1/2}$, which we conjecture is its origin. The coefficient vanishes at ${\alpha }^{\ast }=0.597\pm 0.008$. At ${\alpha }^{\ast }$ the HOMO-referenced GKS occupied levels satisfy ${\epsilon }_k\approx -I_k$ 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 $\alpha =0$.