# Water on Actinide Dioxide Surfaces: A Review of Recent Progress
## 2 Oxygen Vacancies on AnO\({}_{2}\) Surfaces
### Theoretical Calculations  
Several DFT simulations have been reported on the study of oxygen vacancies on the top-surface and the subsurface of AnO\({}_{2}\), providing insights in terms of the electronic structure of the reduced surfaces, the oxidation states of actinide atoms, the formation energies of vacancies, and the electronic structure of excess electrons owing to oxygen vacancies. Figure 3: XPS spectra of reduced UO\({}_{2}\) (111) surface [49]. The broad shoulder of U4_f_\({}^{7/2}\) peak can be better fitted by considering different oxidation states of uranium from +4 to 0. Table 4 shows the calculated formation energies of oxygen vacancies on (111) surface of ThO\({}_{2}\), UO\({}_{2}\) and PuO\({}_{2}\). The formation energy is defined as \(E_{v}=E_{slab}-\left(E_{v,slab}+\frac{1}{2}E_{O_{2}}\right)\), where \(E_{v,slab}\) and \(E_{slab}\) are the total energy of the slab with and without oxygen vacancy, respectively. \(E_{O_{2}}\) is the total energy of one isolated O\({}_{2}\) molecule in its ground state. From these calculations, it is seen that the oxygen vacancy formation energies decrease from ThO\({}_{2}\) to PuO\({}_{2}\)[44, 53, 54, 55]. Recent studies found that the oxygen vacancy formation energies are correlated with the reduction potential of An\({}^{4+}\), in accordance with the \(f\)-band energy drops as going down the actinide series [53, 56]. Interestingly, the fundamental question regarding the relative stability of oxygen vacancy in the top-surface and subsurface is still debatable, although similar DFT + \(U\) calculations were used [53, 54]. The conclusions for PuO\({}_{2}\) (111) surface are consistent that the oxygen vacancy formation energy is slightly lower in the top-surface than in the subsurface in previous studies [44, 53], and the energy difference is within 0.05 eV. However, it has been demonstrated that oxygen vacancy in the subsurface is more stable than in the top-surface on CeO\({}_{2}\) (111) surface, which has an identical structure to AnO\({}_{2}\)[47]. Besides the debate of the relative stability of oxygen vacancy in the top-surface and subsurface of AnO\({}_{2}\), inconsistent results about the distribution of the excess electrons were also reported. On the UO\({}_{2}\) (110) surface, some theoretical calculations based on Perdew-Burke-Ernzerhof (PBE) functional and an effective Hubbard \(U\) value of 4.0 eV found that the excess electrons are localized on two uranium atoms, resulting in two U\({}^{3+}\) next to the vacancy site [54]. Other theoretical studies reported the two electrons to be delocalized over three uranium atoms by natural population analysis for the embedded cluster calculations [53]. The inclusion of a Hubbard-like on-site Coulomb interaction in DFT + \(U\) yields a more accurate treatment of the correlated electrons, which overcomes deficiencies of the pure LDA/GGA functionals and has achieved a wide range of successes in treating AnO\({}_{2}\)[7, 8]. However, the inclusion of Hubbard term introduces multiple energy local minima, which complicates the search for the ground state [17, 18, 19, 20]. Several methods, such as \(U\)-ramping [19] and occupation matrix control (OMC) [17, 18], have been proposed to overcome this difficulty presented by DFT + \(U\). These methods have been shown to locate low-energy solutions for a variety of systems containing strongly correlated \(d\) and \(f\) electrons [17, 18, 19]. Recently, these approaches have been applied to the study of reduced AnO\({}_{2}\) (111) surfaces [55]. It was found for ThO\({}_{2}\), UO\({}_{2}\), and PuO\({}_{2}\) (111) surfaces that the oxygen formation energy is 0.23-0.38 eV lower for the subsurface vacancy. This is consistent with the calculations for CeO\({}_{2}\) (111) surface, where explicit searching of the low-energy solutions has been carried out by assigning excess electrons to different sites near the vacancy [47, 57].