How to Address Nonlinearity & Estimate Uncertainty w/ Polynomial Fitting in Hubbard U Calculations for DFT+U

Abstract. Finite-difference linear response calculations for the Hubbard U and Hund’s J parameters in DFT+U—a corrective method aiming to improve the accuracy of DFT calculations of strongly correlated electron systems like Mott insulators—typically involve linear regression of small response datasets. Many subspaces of material systems, however, are found to exhibit nonlinear response (e.g., curvature) that may disrupt the extraction of the required linear component. For these systems, low-order polynomial regressions can capture curvature in linear response datasets, permitting the quantification of uncertainty on the Hubbard parameters through the unbiased root-mean-square error. What remains unclear in this context is the optimal strategy practitioners can employ to avoid the overfitting that occurs at higher polynomial degrees. In this work, we determine uncertainties on the Hubbard parameters and investigate polynomial degree selection strategies based on leave-one-out cross validation (LOOCV) and repeated K-fold cross validation (RKFCV). We show that LOOCV is a practical technique, not least due to its simple closed-form formula, given the small perturbation-occupancy datasets typically used in DFT+U. For larger datasets, LOOCV exhibits high variance in its estimated errors, whereas RKFCV provides more stable estimates. In practical calculations, we have found that fitting to a quadratic or cubic polynomial before taking the slope at zero perturbation of the resulting fit satisfactorily addresses the issue of nonlinear response.