  Semi-local 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 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) DFT/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 1,708 unique
reactions that were derived from the formation energies by systematic
enumeration. 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 50% (70%) to 0.08 (0.04)
eV/atom. A simplified correction scheme that does not involve SCAN/SCAN+U
compatibility terms still achieves an error reduction of 35% (50%). The method
and tools demonstrated here can be applied to other classes of materials or to
parameterize the corrections to optimize DFT+U performance for other target
physical properties.
