## 3 Results
### Structure of Charged Defects
#### 3.2.1 Lattice Relaxation around an Electron and Hole  
An electron or hole is simulated in DFT calculation by changing the total number of electrons by +1 or -1, in the optimized bulk structures. The relaxation volumes of electrons and holes in bulk Si, GaAs, and UO\({}_{2}\), obtained after relaxing (\(p=0\)) the charged bulk structures, are summarized in Table 5. Addition and removal of an electron results in expansion and contraction of the lattice, respectively. These volume changes (-12 to 22 A\({}^{3}\) or 0.3 to 0.7\(\Omega_{0}\), depending on the system) are large and result in about 1% change in their respective lattice parameters for the current supercell sizes. There is not much data in the literature on the relaxation volumes of electrons and holes to compare our results to. However, a few DFT studies [12; 25] on defects in Si report the relaxation volumes of an electron of \(0.68\Omega_{\text{Si}}\) to \(0.72\Omega_{\text{Si}}\) and a hole of \(-0.78\Omega_{\text{Si}}\), where \(\Omega_{\text{Si}}\) is the volume per Si atom in a perfect supercell. Our computed values for the relaxation volume of an electron and hole in Si are in a similar range: \(0.74\Omega_{\text{Si}}\) and \(-0.84\Omega_{\text{Si}}\), respectively. A DFT calculation of an electron or hole generally yields a delocalized electron, as it represents the exchange of electron or hole with the bulk conduction band or the valence band edge, unless the added (or removed) extra charge has a preference to localize on a specific atomic site. To verify this, we performed Bader charge analysis [67; 68] and looked at the difference in charge density (planar averaged) between the electron (or hole) and bulk for both UO\({}_{2}\) and GaAs. For GaAs, the difference in charge density (Figure 5) between the charged and uncharged systems is periodic across the As and Ga atomic planes normal to the \(z\)-axis, with no evidence of charge localization. In contrast, for UO\({}_{2}\), the difference in charge density is not uniform (Figure 5), indicating localization of the additional electrons within the supercell. A 2D slice (Figure 6) of the difference in charge density between the excess electron and bulk shows large variations in charge density across specific U and O sites. Furthermore, a Bader charge analysis on the structure with an extra electron in UO\({}_{2}\) confirms a U site having an effective Bader charge of +2.18, less than that of +2.54 on a U site in the bulk (or perfect) lattice, and neighboring O sites with Bader charge of -1.30, larger than of -1.27 on O site in bulk. A similar Bader charge analysis of the hole suggests partial localization at a U site with an effective Bader charge of +2.77. Such partial localization of an extra electron (\(e^{\prime}\) or U\({}_{\text{U}}\)\({}^{\otimes-}\)) or hole (\(h\). or U\({}_{\text{U}}\)\({}^{\otimes+}\)) has previously been seen in DFT + \(U\) studies of UO\({}_{2}\)[69]. To gauge the local atomic relaxations due to the presence of an excess electron or hole, we analyze the change in distances of the neighboring atoms for all the sites in the perfect and defective (electron or hole) structure. Figure 7 shows the relative change in nearest neighbor distances \(\left(\frac{\Delta d}{d_{0}}\right)\) for all the sites in the defect structure, plotted as the function of distance (\(d_{0}\)) of the given site from the defect. The change in the neighbor distance (\(\Delta d=d-d_{0}\)) is calculated by taking the average difference of distances for each set of neighbors at a given distance from the defect site between the bulk and defect structures. Figure 5: Difference in charge density (\(xy\)-planar averaged) between supercells containing an extra electron and bulk along the \(z\)-axis in UO\({}_{2}\) (top) and GaAs (bottom). The difference in charge density is normalized by the maximum value of the difference.