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Oxygen vacancies in bulk CeO 2 have been investigated using the Heyd−Scuseria−Ernzerhof (HSE) hybrid functional method. Results show that oxygen vacancies tend to linearly order in the ⟨111⟩ direction of CeO 2 , yielding much more dispersive gap states with the weakened electron localization compared to the case of a single vacancy. Such vacancy ordering and electron localization give rise to a profound influence on material properties. First, the dispersive gap states are expected to act as stepping stones to facilitate the electron excitation from valence band to conduction band, contributing to extended optical absorption in the longer wavelengths and thus enhancing photovoltaic and photocatalytic functionalities. Also, the linear ordering of oxygen vacancies leads to the electron localization on Ce ions and oxygen vacancy sites, inducing the polarization of electrons on vacancy sites which effectively enhances stability of ferromagnetism. The fundamental understanding of these functional mechanisms is presented in detail. Additionally, the kinetic analysis of the oxygen-vacancy cluster has also been performed, and its high kinetic stability suggests its physical existence in bulk CeO 2 . The outcome of this work offers great promise for practical application of CeO 2 in visible-light photocatalysis and photovoltaics as well as magneto-optic and spintronic devices.

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It is rather difficult to model the electronic properties of wide-gap oxides with moderate electronic correlation using first principles methods in the framework of the density functional theory (DFT) together with the Hubbard correction to account for the nonlocal effect in the exchange-correlation (XC) functionals. In this contribution we present a case study of the Nb-doped anatase TiO 2 , to demonstrate such limitation in the GGA + U formalism. It is found that overcorrection owing to a big effective Hubbard parameter U 0 leads to an erroneous band structure, but an inadequate U 0 value results in significantly undervalued prediction of the energy band gap. Fortunately, such limitation in U 0 correction can be addressed satisfactorily through a physically justifiable extrapolation scheme, with electronic structures thus corrected being in excellent agreement with the experimentally observed transparent conductivity of Nb-doped anatase. High resolution X-ray photoelectron spectroscopy (XPS) offers solid support to the theoretical work that Nb doping lowers the valence band without introduction of gap states, and the measured valency of Nb in TiO 2 is in excellent agreement with the calculated value. Also, the current method has been successfully applied to other well-known and yet electronically rather different TCO systems, the Al-doped ZnO and F-doped SnO 2 .

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