## 3 Results
### Structure of Charged Defects
#### 3.2.1 Lattice Relaxation around an Electron and Hole
Figure 6: (**a**) Charge density (slice along a (110) plane containing both U and O atoms) for bulk UO\({}_{2}\); (**b**) difference in charge density (within the same (110) plane) between the supercell with the extra electron and bulk UO\({}_{2}\), and (**c**) between the supercell containing a hole and bulk UO\({}_{2}\). Figure 7: \(Cont.\)  
In UO\({}_{2}\) with a partially localized extra electron on a U site (\(e^{\prime}\) or U\({}_{\text{U}}^{\phi-}\)) (Figure 7a), the local defect structure is complex. Neighboring atoms for any given site in the structure show outward relaxation, independent of the position of the site from the defect. For those sites at a distance of \(\sim\)5 A and beyond from the defect, the relative change in neighboring distance saturates to a constant value. This constant value is approximately equal to the fractional volume change \(\left(\frac{\Delta GY}{\Omega(bulk)}=\frac{11,52}{1936.76}\right)\) due to the addition of an electron in bulk UO\({}_{2}\), which is about \(\sim\)0.006 (or 0.6%). That is, it is what would be expected from an isotropic expansion of the cell. However, for the sites that are close to the defect (distance \(<\) 5 A) the change in neighboring distances is much larger than 0.6%. Similarly, for a partially localized hole on U site (\(h\). or U\({}_{\text{U}}\)\({}^{\phi+}\)) (Figure 7b), the structure exhibits a large inward relaxation of the lattice close to the defect site, and a constant inward relaxation far away from the defect site, equal to the fractional volume change (\(-\)0.6%) due to removal of an electron from bulk UO\({}_{2}\). This is expected in UO\({}_{2}\) since charges are known to form small polarons (known both experimentally and using higher levels of theory like DFT+\(Ul\) or hybrid-DFT), which by definition carry local lattice distortions [70]. By contrast, for GaAs and Si the electron (or hole) is completely delocalized and does not reside on any specific atomic site; therefore, the change in the neighboring distance is independent of the site considered: an electron (or a hole) induces a constant expansion in the lattice that is uniform over all atoms. Structure analysis of an electron or hole in GaAs (Figure 7c,d) shows that the change in neighboring distances between the defect and bulk structures is approximately a constant value. The structure with an extra electron shows a homogeneous expansion of \(\sim\)0.49% (Figure 7c) while that with an extra hole shows a homogeneous contraction of \(\sim\)0.40% of the lattice (Figure 7d), resulting in roughly \(\pm\)1% change respectively, in the lattice parameter. These calculations show that the addition or removal of an electron to or from a bulk lattice can result in delocalized or localized charge depending on the system. It is intriguing that independent of the behavior (localized or delocalized) of the charge, there is an expansion (or contraction) of the bulk lattice, resulting in significant relaxation volumes for electrons and holes determined within the DFT framework. Whether such an effect is real or a behavior particular to DFT calculations is not clear at present. However, a detailed examination like this of electrons and holes in bulk structures helps us to establish the dependence of relaxation volume and corresponding elastic dipole tensor purely on the behavior of the charged bulk system. Figure 7: Relative change (\(\Delta d/d_{0}\)) in average nearest neighbor distances (in %) between the structures with an extra electron (or hole) and the bulk: (**a**) electron, and (**b**) hole in UO\({}_{2}\); (**c**) electron, and (**d**) hole in GaAs. The red dashed line marks the % fraction volume change in the entire supercell (which is supercell size dependent), to show the overall expansion or contraction of the lattice, relative to a neutral system, in the respective cases. The black dashed line indicates zero change, for reference. In both materials, addition of an extra electron leads to a net expansion of the cell while a hole induces a contraction.