  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.
