Hubbard $U$ through polaronic defect states

Since the preliminary work of Anisimov and co-workers, the Hubbard corrected DFT+$U$ functional has been used for predicting properties of correlated materials by applying on-site effective Coulomb interactions to specific orbitals. However, the determination of the Hubbard $U$ parameter has remained under intense discussion despite the multitude of approaches proposed. Here, we define a selection criterion based on the use of polaronic defect states for the enforcement of the piecewise linearity of the total energy upon electron occupation. A good agreement with results from piecewise linear hybrid functionals is found for the electronic and structural properties of polarons, including the formation energies. The values of $U$ determined in this way are found to give a robust description of the polaron energetics upon variation of the considered state. In particular, we also address a polaron hopping pathway, finding that the determined value of $U$ leads to accurate energetics without requiring a configurational-dependent $U$. It is emphasized that the selection of $U$ should be based on physical properties directly associated with the orbitals to which $U$ is applied, rather than on more global properties such as band gaps and band widths. For comparison, we also determine $U$ through a well-established linear-response scheme finding noticeably different values of $U$ and consequently different formation energies. Possible origins of these discrepancies are discussed. As case studies, we consider the self-trapped electron in BiVO$_4$, the self-trapped hole in MgO, the Li-trapped hole in MgO, and the Al-trapped hole in $\alpha$-SiO$_2$.