TDDFT+$U$: Hubbard corrected approximate density-functional theory in the excited-state regime

We develop a generalization of the Kohn-Sham density functional theory (KS-DFT) + Hubbard $U$ (DFT+$U$) method to the excited-state regime. This has the form of Hubbard $U$ corrected linear-response time-dependent DFT, or `TDDFT+$U$'. Combined with calculated linear-response Hubbard $U$ parameters, it may provide a computationally light, first-principles method for the simulation of tightly-bound excitons on transition-metal ions. Our presented implementation combines linear-scaling DFT+$U$ and linear-scaling TDDFT, but the approach is broadly applicable. In detailed benchmark tests on two Ni-centred diamagnetic coordination complexes with variable $U$ values, it is shown that the Hubbard $U$ correction to an approximate adiabatic semi-local exchange-correlation interaction kernel lowers the excitation energies of transitions exclusively within the targeted localized subspace, by increasing the exciton binding of the corresponding electron-hole pairs. This partially counteracts the Hubbard $U$ correction to the exchange-correlation potential in KS-DFT, which increases excitation energies into, out of, and within the targeted localised subspace by modifying the underlying KS-DFT eigenspectrum. This compensating effect is most pronounced for optically dark transitions between localized orbitals of the same angular momentum, for which experimental observation may be challenging and theoretical approaches are at their most necessary. Overall, our results point to shortcomings in the contemporary DFT+$U$ corrective potential, either in its functional form, or when applied to transition-metal orbitals but not to ligand ones, or both.