## 3 Results  
### Density Functional Theory Simulations  
There have been some theoretical papers on the topic of Cu-doped anatase TiO\({}_{2}\)[57; 58; 59; 60]. A DFT + U study [59] of anatase doped with Cu at a concentration of 6.25%, applied U = 8 eV to the Ti 3d states; this recovers the bulk TiO\({}_{2}\) energy gap but is not to be recommended for other properties. No +U correction is applied to the Cu 3d states, despite issues with describing these electronic states. A reduction in the band gap was reported in all studies and this is attributed to a combination of Cu 3d and O 2p states above the valence band maximum (VBM). [PERSON] et al. provided a comprehensive study [58] of Cu-doping in anatase which combined both experiment and theory including DFT + U calculations with the +U correction on Ti 3d orbitals only. However, the inclusion of a +U correction for Cu 3d states is particularly important in correctly describing the Cu\({}^{2+}\) oxidation state [48; 49; 61]. They reported greater band gap reduction with increases in dopant concentration due to the covalent character of the Cu-O interaction leading to new states at the VBM. DFT studies have examined the impact of Cu-doping of the anatase (101) surface [62; 63] with a focus on surface mediated phenomena in catalysis. A generalized gradient approximation (GGA) study showed that Cu dopants at the anatase (101) surface inhibit the dissociation of adsorbed water to hydroxyls [62]. The authors attributed enhancements in the photocatalytic activity to arise instead from the electronic properties and inter-bandgap states, rather than the promotion of water dissociation. A GGA + U study found little difference in the formation energies for substitutional doping of Cu at surface and sub-surface Ti sites [63] suggesting both are equally favorable. The emergence of impurity levels in the TiO\({}_{2}\) bandgap led to a predicted red shift in light absorption compared to unmodified TiO\({}_{2}\). A theoretical account of Cu-doping in rutile, in the context of the effect of oxygen vacancies on the magnetic moment of Cu-doped TiO\({}_{2}\)[64], reported that oxygen vacancies are most stable near the Cu impurity. Figure 1a shows the local atomic structure in the vicinity of the Cu dopant. The geometry around the dopant is symmetric in which apical Cu-O and equatorial Cu-O distances are 2.01 A and 1.99 A, these can be compared with corresponding Ti-O distances of 2.02 A and 2.00 A for undoped anatase. Figure 1b displays the excess spin density plot and shows that the hole states are spread over the  
oxygen sites neighboring the dopant. The computed spin magnetizations are 0.13 \(\upmu\)B for both apical sites and 0.2 \(\upmu\)B for the four equatorial sites. The Bader charge of the Cu dopant is 9.6 electrons and the spin magnetization is 0.97 \(\upmu\)B, consistent with the d\({}^{9}\) configuration of the Cu\({}^{2+}\) oxidation state. For the solution with the +U correction applied to the O 2p states, the local atomic structure near the dopant site is distorted asymmetrically; this set up results in localization of the hole states predominantly on the apical oxygen sites and the resulting Cu-O\({}_{\text{Ap}}\) distances are 2.27 A and 2.42 A. These elongated bonds are consistent with formation of an oxygen polaron as has been observed in other materials [46; 47; 65; 66; 67; 68; 69]. The computed spin magnetizations for these O sites are 0.78 \(\upmu\)B and 0.73 \(\upmu\)B and the Bader charges are reduced by 0.5 electrons after hole localization, from 7.3 to 6.8 electrons. The four equatorial Cu-O\({}_{\text{Eq}}\) distances are 1.99-2.00 A. For the remainder of the calculations the differences arising from U(O) = 0 eV vs. U(O) = 7 eV are quantitative in nature and the general characteristics of Cu-doped anatase are consistent, in particular charge compensation. Thus, for brevity we include only the results arising from calculations for which U(O) = 0 eV.