## 3 Results
### Finite Size Correction Schemes to Energy for Charged Supercell
#### 3.1.2 Alignment Correction
Figure 2: **(a)** Planar-averaged charge density difference across the supercell for \(V^{\text{4-}}_{\text{U}}\) as obtained from static (non-relaxed) DFT calculations. The difference is taken between the charge densities of the charged defect and bulk. **(b)** Planar-averaged potential difference \(\left(V^{\text{DFT}}_{q}-V^{\text{DFT}}_{b}\right)\) across the supercell for \(V^{\text{4-}}_{\text{U}}\), as obtained from DFT calculations (blue curve). The model potential (\(V^{\text{model}}\)) from the Gaussian charge is shown in red, and the difference potential \(\Delta V_{\sharp}\) is shown in green. The \(z\)-axis distance is measured from the location of the defect. The potential falls off again at a distance of \(\sim\)16 Å as that is the size of the cell and where the periodic image of the defect is located. Results are shown for a 144 atom UO\({}_{2}\) supercell with volume of 1936.76 Å\({}^{3}\). Comparing the volumes corresponding to the minimum defect formation energies with and without the alignment correction gives an estimate of the correction to the relaxation volume due to the alignment correction. Based on approach (1), top panels in Figure 3, we find that this correction is small. For \(V_{\text{U}}^{4-}\) in a 144 atom supercell, the volume correction is 1.09 A\({}^{3}\) (0.03 \(\Omega_{\text{UO2}}\)), while that for \(V_{\text{Ga}}^{3-}\) in a 216 atom supercell it is about 0.81 A\({}^{3}\) (0.02 \(\Omega_{\text{GaAs}}\)). However, based on approach (2), bottom panels in Figure 3, the correction is very significant; for \(V_{\text{U}}^{4-}\) the volume correction is \(-\)40.33 A\({}^{3}\) (\(-\)1.001 \(\Omega_{\text{UO2}}\)), while that for \(V_{\text{Ga}}^{3-}\) it is \(-\)51.2 A\({}^{3}\) (\(-\)1.2 \(\Omega_{\text{GaAs}}\)). This gives final relaxation volumes of \(V_{\text{U}}^{4-}\) to be 0.05 A\({}^{3}\) and that of \(V_{\text{Ga}}^{3-}\) to be \(-\)19.9 A\({}^{3}\). Approach (1) computes the alignment correction by aligning the common reference (average electrostatic potential) in the defect and bulk supercells, as volumes are kept same, as in References [36; 59]. Based on the analytical expression derived using FNV scheme by [PERSON] et al. [49], the potential alignment follows a \(1/\Omega\) (where \(\Omega\) is supercell volume) dependence, and converges much faster (\(\Omega^{-2}\)) than the image charge correction (\(\Omega^{-4/3}\)) with increasing cell size. This suggests that the potential alignment has a weak volume dependence, which falls in line with the results obtained using approach (1), as shown in Figure 3a. The large volume change as calculated from approach (2) can be broken down into two independent components. As shown in Figure 4, the first component (very small) is actually the same as approach (1), while the second (much larger) component aligns strained bulk (in the same charge state as the defect) with the pristine bulk. Clearly, the volume change of the second component is as large as  
Figure 3: Formation energy of strained charged defects (**a**) \(\left(V_{\text{U}}^{4-}\right)\) and (**b**) \(\left(V_{\text{Ga}}^{3-}\right)\) with and without potential alignment correction. (**top panels**) The potential alignment is computed with respect to the strained bulk of same volume, and (**bottom panels**) the potential alignment is computed with respect to the pristine unstrained bulk. The dashed lines indicate the corresponding relaxation volumes extracted from these calculations. that of the strained defect with respect to the pristine bulk, as shown for both charged uranium and gallium vacancies. This suggests that one probably does not need to introduce a defect to calculate the volume dependence of the potential alignment correction, instead it can, to a good approximation, be computed by just using the strained bulk (in same charge state as the defect).