Comparing Hubbard parameters from linear-response theory and Hartree-Fock-based approach

Density-functional theory with on-site $U$ and inter-site $V$ Hubbard corrections (DFT+$U$+$V$) is a powerful and accurate method for predicting various properties of transition-metal compounds. However, its accuracy depends critically on the values of these Hubbard parameters. Although they can be determined empirically, first-principles methods provide a more consistent and reliable approach; yet, their results can vary, and a comprehensive comparison between methods is still lacking. Here, we present a systematic comparison of two widely used approaches for computing $U$ and $V$, namely linear-response theory (LRT) and the Hartree-Fock-based pseudohybrid functional formalism, applied to a representative set of oxides (MnO, NiO, CoO, FeO, BaTiO$_3$, ZnO, and ZrO$_2$). We find that for partially occupied transition-metal $d$ states, these two methods yield consistent $U$ values, but they differ for nearly empty or fully filled $d$ shells. For O-$2p$ states, LRT always predicts large $U$ values ($\sim$10 eV), whereas the pseudohybrid formalism produces system-dependent values depending on the level of localization and hybridization for the electronic states. Even larger differences are found for the inter-site $V$: the former predicts consistently small values ($<1$ eV), while the latter produces larger values ($\sim3$ eV), reflecting its explicit dependence on relative charge redistribution. Our results show that while parallels between these two methods exist, they rely on distinct assumptions for determining $U$ and $V$, leading to variations in predictions of material properties.