Orbital energies and negative electron affinities from density functional theory: Insight from the integer discontinuity

Orbital energies in Kohn–Sham density functional theory (DFT) are investigated, paying attention to the role of the integer discontinuity in the exact exchange-correlation potential. A series of closed-shell molecules are considered, comprising some that vertically bind an excess electron and others that do not. High-level ab initio electron densities are used to calculate accurate orbital energy differences, $Δϵ$⁠, between the lowest unoccupied molecular orbital (LUMO) and the highest occupied molecular orbital (HOMO), using the same potential for both. They are combined with accurate vertical ionization potentials, $I0$⁠, and electron affinities, $A0$⁠, to determine accurate “average” orbital energies. These are the orbital energies associated with an exchange-correlation potential that averages over a constant jump in the accurate potential, of magnitude $ΔXC=(I0−A0)−Δϵ$⁠, as given by the discontinuity analysis. Local functional HOMO energies are shown to be almost an order of magnitude closer to these average values than to $−I0$⁠, with typical discrepancies of just 0.02 a.u. For systems that do not bind an excess electron, this level of agreement is only achieved when $A0$ is set equal to the negative experimental affinity from electron transmission spectroscopy (ETS); it degrades notably when the zero ground state affinity is instead used. Analogous observations are made for the local functional LUMO energies, although the need to use the ETS affinities is less pronounced for systems where the ETS values are very negative. The application of an asymptotic correction recovers the preference, leading to positive LUMO energies (but bound orbitals) for these systems, consistent with the behavior of the average energies. The asymptotically corrected LUMO energies typically agree with the average values to within 0.02 a.u., comparable to that observed with the HOMOs. The study provides numerical support for the view that local functionals exhibit a near-average behavior based on a constant jump of magnitude $ΔXC$⁠. It illustrates why a recently proposed DFT expression involving local functional frontier orbital energies and ionization potential yields reasonable estimates of negative ETS affinities and is consistent with earlier work on the failure of DFT for charge-transfer excited states. The near-average behavior of the exchange-correlation potential is explicitly illustrated for selected systems. The nature of hybrid functional orbital energies is also mentioned, and the results of the study are discussed in terms of the variation in electronic energy as a function of electron number. The nature of DFT orbital energies is of great importance in chemistry; this study contributes to the understanding of these quantities.