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 semilocal and hybrid approximations. Excitation energies are similar in quality to time-dependent density functional theory, especially at high fractions of exact exchange. The approach therefore offers an easy-to-implement and low-cost route to robust prediction of molecular excitation energies