Tuning LDA+U for electron localization and structure at oxygen vacancies in ceria
We examine the real space structure and the electronic structure (particularly Ce4f electron localization) of oxygen vacancies in CeO2 (ceria) as a function of U in density functional theory studies with the rotationally invariant forms of the LDA+U and GGA+U functionals. The four nearest neighbor Ce ions always relax outwards, with those not carrying localized Ce4f charge moving furthest. Several quantification schemes show that the charge starts to become localized at U≈3eV and that the degree of localization reaches a maximum at ∼6eV for LDA+U or at ∼5.5eV for GGA+U. For higher U it decreases rapidly as charge is transferred onto second neighbor O ions and beyond. The localization is never into atomic corelike states; at maximum localization about 80–90% of the Ce4f charge is located on the two nearest neighboring Ce ions. However, if we look at the total atomic charge we find that the two ions only make a net gain of (0.2–0.4)e each, so localization is actually very incomplete, with localization of Ce4f electrons coming at the expense of moving other electrons off the Ce ions. We have also revisited some properties of defect-free ceria and find that with LDA+U the crystal structure is actually best described with U=3–4eV, while the experimental band structure is obtained with U=7–8eV. (For GGA+U the lattice parameters worsen for U>0eV, but the band structure is similar to LDA+U.) The best overall choice is U≈6eV with LDA+U and ≈5.5eV for GGA+U, since the localization is most important, but a consistent choice for both CeO2 and Ce2O3, with and without vacancies, is hard to find.