DFT$+U(\omega)$: Frequency-dependent Hubbard $U$ correction

In contemporary first-principles atomistic simulation based on DFT, the augmentation of approximate exchange-correlation functionals with spatially or energetically localized corrections, such as DFT$+U$, is a successful approach for improving its applicability to strongly interacting systems. Electronic screening is a dynamical process, and since the Hubbard $U$ parameter, in particular, is a measure of the screened Coulomb interaction, its frequency-dependent generalisation for the dynamical regime is possible. We introduce a conceptually pragmatic and computationally straightforward method, named DFT$+U(\omega)$, for calculating and incorporating strong dynamical screening effects in spectroscopic calculations based on Kohn-Sham DFT. Our method is designed to be a minimal dynamical extension of DFT$+U$, one in which computing approximate dynamical Hubbard $U$ functions only requires functionality that is widely available. We demonstrate our effective plasmon fitting and self-energy approximation scheme for DFT$+U(\omega)$, which enables the resulting low-energy dynamical model to be solved at the $G_0 W_0$ level, and beyond, efficiently and effectively.