  We present a first-principles study of the low-temperature rhombohedral phase
of BaTiO$_3$ using Hubbard-corrected density-functional theory. By employing
density-functional perturbation theory, we compute the onsite Hubbard $U$ for
Ti($3d$) states and the intersite Hubbard $V$ between Ti($3d$) and O($2p$)
states. We show that applying the onsite Hubbard $U$ correction alone to
Ti($3d$) states proves detrimental, as it suppresses the Ti($3d$)-O($2p$)
hybridization and drives the system towards a cubic phase. Conversely, when
both onsite $U$ and intersite $V$ are considered, the localized character of
the Ti($3d$) states is maintained, while also preserving the Ti($3d$)-O($2p$)
hybridization, restoring the rhombohedral phase of BaTiO$_3$. The generalized
PBEsol+$U$+$V$ functional yields good agreement with experimental results for
the band gap and dielectric constant, while the optimized geometry is slightly
less accurate compared to PBEsol. Zone-center phonon frequencies and Raman
spectra are found to be significantly influenced by the underlying geometry.
PBEsol and PBEsol+$U$+$V$ provide satisfactory agreement with the experimental
Raman spectrum when the PBEsol geometry is used, while PBEsol+$U$ Raman
spectrum diverges strongly from experimental data highlighting the adverse
impact of the $U$ correction alone in BaTiO$_3$. Our findings underscore the
promise of the extended Hubbard PBEsol+$U$+$V$ functional with first-principles
$U$ and $V$ for the investigation of other ferroelectric perovskites with mixed
ionic-covalent interactions.
