Data-driven approach to parameterize SCAN+U for an accurate description of 3d transition metal oxide thermochemistry

Abstract

Semilocal density-functional theory (DFT) methods exhibit significant errors for the phase diagrams of transition-metal oxides that are caused by an incorrect description of molecular oxygen and the large self-interaction error in materials with strongly localized electronic orbitals. Empirical and semiempirical corrections based on the $\mathrm{DFT}+U$ method can reduce these errors, but the parameterization and validation of the correction terms remains an on-going challenge. We develop a systematic methodology to determine the parameters and to statistically assess the results by considering interlinked thermochemical data across a set of transition metal compounds. We consider three interconnected levels of correction terms: (1) a constant oxygen binding correction, (2) Hubbard-$U$ correction, and (3) $\mathrm{DFT}/\mathrm{DFT}+U$ compatibility correction. The parameterization is expressed as a unified optimization problem. We demonstrate this approach for $3d$ transition metal oxides, considering a target set of binary and ternary oxides. With a total of 37 measured formation enthalpies taken from the literature, the dataset is augmented by the reaction energies of 1710 unique reactions that were derived from the formation energies by systematic enumeration. To ensure a balanced dataset across the available data, the reactions were grouped by their similarity using clustering and suitably weighted. The parameterization is validated using leave-one-out cross validation (CV), a standard technique for the validation of statistical models. We apply the methodology to the strongly constrained and appropriately normed (SCAN) density functional. Based on the CV score, the error of binary (ternary) oxide formation energies is reduced by 40% (75%) to 0.10 (0.03) eV/atom. A simplified correction scheme that does not involve $\mathrm{SCAN}/\mathrm{SCAN}+U$ compatibility terms still achieves an error reduction of 30% (25%). The method and tools demonstrated here can be applied to other classes of materials or to parameterize the corrections to optimize $\mathrm{DFT}+U$ performance for other target physical properties.