Oxygen vacancy clusters on ceria: Decisive role of cerium f electrons

Defects such as oxygen vacancies dominate the electronic and chemical properties of ceria. However, fundamental understanding of such defects, especially clusters of vacancies, is sparse. In this work, we use density-functional theory with the addition of the Hubbard $U$ term to investigate various oxygen vacancies, including the vacancy monomer, dimer, trimer, and tetramer, in which subsurface vacancies can also be involved. We show that the individual surface and subsurface vacancies have very similar stabilities; the vacancy dimer consisting of two surface vacancies, which is not reported experimentally, is stable in theory; between the two vacancy trimers observed in experiments, the triangular surface vacancy cluster is more stable than the double linear surface vacancy cluster containing a subsurface vacancy, which agrees with some experiments but disagrees with some others; and the linear vacancy tetramer emerges as the most stable among the possible tetramers containing subsurface vacancies, although it is less stable than those containing no subsurface vacancies. These findings are rationalized in terms of the electronic change upon the removal of oxygen, namely, the localization of resulting excess electrons on $\text{Ce}\text{ }f$ orbitals. We identify a correlation between the energy levels of the occupied $f$ states of reduced Ce ions and their coordination numbers, which proves pivotal in interpreting formation energy and stability of various vacancies. Comparisons are made with experiments and apparent discrepancies are discussed. Results for gold adsorption on the vacancy clusters are presented, and the implications these have in catalysis are briefly discussed.