The first-principles calculation of correlated materials within density functional theory remains challenging, but the inclusion of a Hubbard-type effective on-site Coulomb term $\left(U_{\mathrm{eff}}\right)$ often provides a computationally tractable and physically reasonable approach. However, the reported values of $U_{\mathrm{eff}}$ vary widely, even for the same ionic state and the same material. Since the final physical results can depend critically on the choice of parameter and the computational details, there is a need to have a consistent procedure to choose an appropriate one. We revisit this issue from constraint density functional theory, using the full-potential linearized augmented plane wave method. The calculated $U_{\mathrm{eff}}$ parameters for the prototypical transition-metal monoxides—MnO, FeO, CoO, and NiO—are found to depend significantly on the muffin-tin radius $R_{\mathrm{MT}}$, with variations of more than 2–3 eV as $R_{\mathrm{MT}}$ changes from 2.0 to 2.$7a_B$. Despite this large variation in $U_{\mathrm{eff}}$, the calculated valence bands differ only slightly. Moreover, we find an approximately linear relationship between $U_{\mathrm{eff}}\left(R_{\mathrm{MT}}\right)$ and the number of occupied localized electrons within the sphere, and give a simple scaling argument for $U_{\mathrm{eff}}$; these results provide a rationalization for the large variation in reported values. Although our results imply that $U_{\mathrm{eff}}$ values are not directly transferable among different calculation methods (or even the same one with different input parameters such as $R_{\mathrm{MT}})$, use of this scaling relationship should help simplify the choice of $U_{\mathrm{eff}}$.

• Received 19 June 2017
• Revised 25 December 2017

DOI:https://doi.org/10.1103/PhysRevB.97.035117