## 3 Results
### Density Functional Theory Simulations
In response to the charge imbalance which arises when replacing Ti\({}^{4+}\) with lower valent Cu\({}^{2+}\), the system compensates through formation of an oxygen vacancy. The most stable site for this compensating oxygen vacancy is the O site neighboring the Cu dopant in the equatorial position and this has a computed formation energy of \(-\)0.35 eV. The top panels of Figure 2 show the geometry of the charge compensated system in the vicinity of the Cu dopant; the oxygen vacancy site is indicated by a black circle. After formation of the charge compensating oxygen vacancy, the Cu and Ti ions to which the removed O atom was bound have five-fold coordination. The Ti ions move off their lattice sites and outwards from the vacancy site and this distortion leads to the shortening of metal-oxygen bonds opposite the vacancy by 0.13-0.16 A. After oxygen vacancy formation, four of the Cu-O bonds contract by 0.01-0.06 A. The fifth Cu-O distance, involving the O\({}_{\text{Eq}}\) atom opposite the vacancy site, increases by 0.07 A. In this configuration the Bader charge on the dopant is 9.7 electrons and the spin magnetization is 0.75 \(\upmu\)B, indicating a Cu\({}^{2+}\) oxidation state. For the formation of a second, reducing oxygen vacancy, the most stable site is a second equatorial site neighboring the dopant and adjacent to the first vacancy. This oxygen vacancy forms with an energy cost of +3.27 eV and the local geometry about the dopant is shown in the bottom panels of Figure 2, with the oxygen vacancies indicated by black circles. This energy cost is similar to undoped bulk anatase so that Cu doping has no significant effect on the reducibility of anatase. Figure 1: (**a**) Relaxed atomic structure of bulk anatase TiO\({}_{2}\) in the vicinity of the Cu dopant and (**b**) excess spin density plot for Cu-doped anatase TiO\({}_{2}\). Spin isosurfaces enclose densities of up to 0.02 eV/Å\({}^{3}\). In this and subsequent figures Ti is represented by a grey sphere, O by a red sphere and Cu by an orange sphere. Both Cu-O\({}_{\text{Ap}}\) distances contract to 1.93 A, while the two remaining Cu-O\({}_{\text{Eq}}\) distances are 2.08 A and 2.10 A. After formation of the second oxygen vacancy two electrons are released and we analyze computed Bader charges and spin magnetizations to determine where these electrons are localized. For the Cu dopant the Bader charge increases from 9.7 to 10.3 electrons and the spin magnetization is 0 \(\upmu_{\text{B}}\), consistent with reduction from Cu\({}^{2+}\) to Cu\({}^{+}\). The second electron localizes at a Ti site which is five-fold-coordinated due to formation of the second oxygen vacancy. This site has a Bader charge of 1.7 electrons and a spin magnetization of 0.95 \(\upmu_{\text{B}}\), indicating reduction to Ti\({}^{3+}\). For comparison, the Ti\({}^{4+}\) ions in TiO\({}_{2}\) have computed Bader charges of 1.29-1.34 electrons. The projected electronic density of states (PEDOS) plots for pure TiO\({}_{2}\) anatase is shown in Figure 3a, highlighting the typical O 2p dominated valance band region and the Ti 3d dominated conduction band region. Figure 3b shows the PEDOS for Cu doping where a clear Cu\({}^{2+}\) derived state is present in the TiO\({}_{2}\) energy gap. For the ground state of Cu-doped anatase, with a single, compensating oxygen vacancy, and the reduced system, with two oxygen vacancies, the PEDOS are shown in the bottom panels of Figure 3. For the ground state system (Figure 3c) the empty 3d state of the Cu\({}^{2+}\) ion lies above the conduction band minimum (CBM) of the TiO\({}_{2}\) host, so that the dopant has no impact on the magnitude of the band gap. However, states emerge in the band gap after formation of the second oxygen vacancy, as shown in Figure 3d.