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

Abstract

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.

Figures (15)

• Figure 1: Protocol for computing Hubbard parameters within the "self-consistent" scheme using (a) LRT (DFPT) Timrov:2021 and (b) eACBN0 Lee:2020. In the LRT workflow, Hubbard parameters are evaluated self-consistently after the electronic charge density has been fully converged. In contrast, the eACBN0 scheme updates the charge density and the Hubbard parameters simultaneously until convergence is reached. In both workflows, atomic positions are fixed and no structural relaxations are performed.
• Figure 2: DFT+$U$ and DFT+$U$+$V$ self-consistent calculations for TMOs without on-site corrections on O-2$p$ states ($U_O$), using the workflow described in Fig. \ref{['Fig:flowchart']}. (a) On-site Hubbard $U_{TM}$ parameter for 3$d$ orbitals of TM ions and inter-site Hubbard $V_{TM, O}$ parameter between 3$d$ orbitals of TM ions and 2$p$ orbitals of oxygen, (b) band gap, and (c) magnetic moment. LRT and eACBN0 denote the method used to obtain the Hubbard parameters. The experimental band gaps and magnetic moments for MnO Kurmaev2008prbElp15301991prbCheetham1983prbFender1968JCP, FeO Bowen1975JSSCRoth1958prbBattle1979JPCSSP, CoO Elp60901991prbKhan1970prbJauch2001prb, and NiO Kurmaev2008prbSawatzky1984prlCheetham1983prbFender1968JCP are indicated by the dashed blue lines in panels (b) and (c), respectively.
• Figure 3: DFT+$U$ and DFT+$U$+$V$ self-consistent calculations for TMOs with on-site $U_{O}$ corrections on O-2$p$ states. The remainder of the caption is analogous to Fig. \ref{['Fig:TMO_wo_O']}.
• Figure 4: DFT+$U$ and DFT+$U$+$V$ self-consistent calculations for ZrO$_{2}$ without on-site corrections on O-2$p$ states ($U_O$). (a) On-site Hubbard $U_{Zr}$ parameter for 4$d$ orbitals of Zr ions and inter-site Hubbard $V_{Zr, O}$ parameter between 4$d$ orbitals of Zr ions and 2$p$ orbitals of oxygen, and (b) band gap. The experimental band gap Bersch:2008 is shown by the dashed blue line in panel (b).
• Figure 5: DFT+$U$ and DFT+$U$+$V$ self-consistent calculations for ZrO$_{2}$ with on-site $U_{O}$ corrections on O-2$p$ states. The remainder of the caption is analogous to Fig. \ref{['Fig:ZrO2_wo_O']}.