Vertical Ionization Energies, Generalized Kohn-Sham Orbital Energies, and the Curious Case of the Copper Oxide Anions

Are the vertical ionization energies from a bound electronic system, initially in its ground state, equal to minus the corresponding exact Kohn-Sham orbital energies of density functional theory (DFT)? This is known to be true for the first or lowest vertical ionization energy. We show that the correction from time-dependent DFT arises from the continuum and need not vanish. Recent work compared the experimental photoemission thresholds of the molecules Cu₂O^-, CuO^-, CuO₂^-, and CuO₃^- with minus the corresponding orbital energies from a generalized gradient approximation (GGA) and its global and range-separated hybrids with exact exchange, finding striking differences which were attributed to self-interaction error, strong correlation, or both. Here, we extend that work to include the local spin density approximation (LSDA), its Perdew-Zunger self-interaction correction with Fermi-Löwdin localized orbitals (LSDA-SIC), a quasi-self-consistent locally scaled-down version of LSDA-SIC (QLSIC), and the Quantum Theory Project QTP02 range-separated hybrid functional, all but LSDA implemented in a generalized Kohn-Sham approach. QTP02 impressively yields a near equality for many sp-bonded molecules. However, for the copper oxide anions studied here, none of the tested methods reproduces the experimental photoemission thresholds.