Reformulation of the $\mathrm{LDA}+U$ method for a local-orbital basis

We present a local orbital approach to the evaluation of the on-site repulsion energy $U$ for use in the $\mathrm{LDA}+U$ method of Anisimov and co-workers. Our objectives are to make the method more firmly based, to concentrate primarily on ground-state properties rather than spectra, and to test the method in cases where only modest changes in orbital occupations are expected, as well as for highly correlated materials. Because of these objectives, we employ a differential definition of $U$. We also define a matrix $\mathbf{U}$, which we find is very dependent on the environment of the atom in question. The formulation is applied to evaluate $U$ for transition-metal monoxides from VO to NiO using a local-orbital basis set. The resulting values of $U$ are typically only 40–65 % as large as values currently in use. We evaluate the $\mathbf{U}$ matrix for the $e_g$ and $t_{2g}$ subshells in paramagnetic FeO, and illustrate the very different charge responses of the $e_g$ and $t_{2g}$ states. The sensitivity of the method to the choice of the $d$ orbitals, and to the basis set in general, is discussed.