  The electronic structure of cerium oxide is investigated here using a
combination of ab initio one-electron theory and many-body physics, with
emphasis on the nature of the 4f electron shell of cerium ions. We propose to
use the hybridization function as a convenient measure for the degree of
localization of the 4f shell of this material, and observe that changing the
oxidation state is related to distinct changes in the hybridization between the
4f shell and ligand states. The theory reveals that CeO$_2$ has essentially
itinerant 4f states, and that in the least oxidized form of ceria, Ce$_2$O$_3$,
the 4f states are almost (but not fully) localized. Most importantly, our model
points to that diffusion of oxygen vacancies in cerium oxide may be seen as
polaron hopping, involving a correlated 4f electron cloud, which is located
primarily on Ce ions of atomic shells surrounding the vacancy.
