General embedded cluster protocol for accurate modeling of oxygen vacancies in metal-oxides

The O vacancy (Ov) formation energy, $E_\textrm{Ov}$, is an important property of a metal-oxide, governing its performance in applications such as fuel cells or heterogeneous catalysis. These defects are routinely studied with density functional theory (DFT). However, it is well-recognized that standard DFT formulations (e.g. the generalized gradient approximation) are insufficient for modeling the Ov, requiring higher levels of theory. The embedded cluster method offers a promising approach to compute $E_\textrm{Ov}$ accurately, giving access to all electronic structure methods. Central to this approach is the construction of quantum(-mechanically treated) clusters placed within suitable embedding environments. Unfortunately, current approaches to constructing the quantum clusters either require large system sizes, preventing application of high-level methods, or require significant manual input, preventing investigations of multiple systems simultaneously. In this work, we present a systematic and general quantum cluster design protocol that can determine small converged quantum clusters for studying the Ov in metal-oxides with accurate methods such as local coupled cluster with single, double and perturbative triple excitations [CCSD(T)]. We apply this protocol to study the Ov in the bulk and surface planes of rutile TiO2 and rocksalt MgO, producing the first accurate and well-converged determinations of $E_\textrm{Ov}$ with this method. These reference values are used to benchmark exchange-correlation functionals in DFT and we find that all studied functionals underestimate $E_\textrm{Ov}$, with the average error decreasing along the rungs of Jacob's ladder. This protocol is automatable for high-throughput calculations and can be generalized to study other point defects or adsorbates.