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.

Using the Heyd–Scyseria–Ernzerhof (HSE) hybrid functional in the framework of the density functional theory (DFT), we probe the insight into the characteristic gap level and n-type conductivity of the intrinsic rutile TiO2. Thermodynamic and kinetic investigations have been conducted to elaborate the favorability for the formation of the possible n-type defects and unintentional impurities in rutile TiO2. Results show that oxygen vacancy is clearly identified to induce a deep localized state inside the forbidden energy region through localizing two excess electrons at two Ti4+ ions along the [001] direction and reducing them into Ti3+ ions, accounting for the characteristic gap level observed experimentally. The eg orbital composition of this gap level offers an accountable explanation of the experimentally measured ferromagnetism in TiO2–x , while the electron transition from this characteristic level is contributable to the photocatalytic behaviors and visible photoluminescence of slightly reduced TiO2. Also, unintentional incorporation of hydrogen substitution for oxygen acts as a shallow donor, providing a consistent explanation of the n-type conductivity in TiO2. The fundamental understanding of these characteristic properties and the associated functionalities would be essential to improving and expanding the practical applications of TiO2-based materials and devices.

R ecently, we reported a Heyd−Scuseria−Enrzerhof (HSE) 1 hybrid functional study of oxygen vacancy ordering and electron localization in bulk CeO 2 , with the aim to obtain an atomic-level insight into the multiple oxygen vacancies in CeO 2 and the associated effects on the photocatalytic, photovoltaic, and magnetic properties properties. 2 Our main conclusions are that oxygen vacancies energetically prefer to order in the <111> direction and that such vacancy ordering and the caused electron localization may profoundly influence the photocatalytic, photovoltaic, and magnetic properties. In a Comment, 3 Ganduglia-Pirovano et al. questioned three points in our paper: (i) the localization of two excess electrons left behind by oxygen removal, (ii) the associated lattice relaxation, and (iii) the missing of the citations to some previous works. In this Reply we try to address these comments and explain why there are evident discrepancies between our results and theirs.Regarding the excess charge localization, Ganduglia-Pirovano et al. think two excess charges should localize on two secondnearest Ce ions to the oxygen vacancy and reduce them into Ce 3+ ions, and disagree with our results (where the two excess charges were found to localize on two Ce ions nearest to the oxygen vacancy). Theoretically, this pair of Ce 3+ ions can be the first neighbor (1N), second neighbor (2N), or third neighbor (3N) to the vacancy and so on. To identify the most stable configuration of Ce 3+ ions, we use the approach of predetermining the Ce 3+ sites (as described in ref 4) to inspect all possible configurations of Ce 3+ ions in the case of a single vacancy, and their respective energies are listed in Table 1. (Note that the configurations with 4N and beyond are strongly disfavored because of their much higher energies.) Clearly, the 1N−1N configuration is the most stable, with 17 meV less than the 2N−2N case. It is consistent with several recent reports 5−11 but different from other ones (i.e., the 2N−2N configuration is the most stable). 4,12−15 Apparently, no general agreement on the excess charge localization has been achieved, considering that related experimental results are yet to be desired.In our opinion, the first factor responsible for the discrepancy is attributed to the different lattice relaxation schemes used. In our case, a full relaxation scheme has been used, that is, the relaxation on the shape, volume, and atomic positions. The different spin configurations of Ce 3+ ions have also been taken into account and compared. As expected, there is little difference between the optimized structures, with a single vacancy using the limited relaxation (fixed volume and shape) and full relaxation. However, the influence of the full relaxation on the systems with divacancy and four vacancies is not

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