Single excitation energies obtained from the ensemble HOMO-LUMO gap: exact results and approximations

In calculations based on density functional theory, the "HOMO-LUMO gap" (difference between the highest occupied and lowest unoccupied molecular orbital energies) is often used as a low-cost, ad hoc approximation for the lowest excitation energy. Here we show that a simple correction based on rigorous ensemble density functional theory makes the HOMO-LUMO gap exact, in principle, and significantly more accurate, in practice. The introduced perturbative ensemble density functional theory approach predicts different and useful values for singlet-singlet and singlet-triplet excitations, using semi-local and hybrid approximations. Excitation energies are of similar quality to time-dependent density functional theory, especially at high fractions of exact exchange. It therefore offers an easy-to-implement and low-cost route to robust prediction of molecular excitation energies.