Decoupling many-body interactions in the CeO₂(111) oxygen vacancy structure with statistical learning and cluster expansion

Oxygen vacancies (V_O's) are of paramount importance in influencing the properties and applications of ceria (CeO₂). Yet, comprehending the distribution and nature of V_O's poses a significant challenge due to the vast number of electronic configurations and intricate many-body interactions among V_O's and polarons (Ce³⁺ ions). In this study, we established a cluster expansion model based on first-principles calculations and statistical learning to decouple the interactions among the Ce³⁺ ions and V_O's, thereby circumventing the limitations associated with sampling electronic configurations. By separating these interactions, we identified specific electronic configurations characterized by the most favorable V_O-Ce³⁺ attractions and the least favorable Ce³⁺-Ce³⁺/V_O-V_O repulsions, which are crucial in determining the stability of vacancy structures. Through more than 10⁸ Metropolis Monte Carlo samplings of V_O's and Ce³⁺ ions in the near surface of CeO₂(111), we explored potential configurations within an 8 × 8 supercell. Our findings revealed that oxygen vacancies tend to aggregate and are abundant in the third oxygen layer with an elevated V_O concentration primarily due to extensive geometric relaxation, an aspect previously overlooked. This work introduces a novel theoretical framework for unraveling the complex vacancy structures in metal oxides, with potential applications in redox and catalytic chemistry.