## 3 Results
### Finite Size Correction Schemes to Energy for Charged Supercell
#### 3.1.2 Alignment Correction  
Among the various procedures [39; 62; 66; 62] outlined in the literature to align the average electrostatic potential in the defect supercell with that in the bulk supercell, we use the scheme provided by [PERSON] et al. [39]. The key advantage of this scheme (referred to in the literature as the FNV scheme [49]) is that the long-range 1/\(r\) potential is removed before the alignment is determined for the remaining short-range potential. If the size of the simulation cell is big enough to fully localize the  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Defect** & \(\mathbf{E_{corr}^{image}}\) & \(\mathbf{\Delta P_{corr}^{image}}\) & \(\mathbf{\Delta\Omega_{corr}^{image}}\) & \(\mathbf{\Delta\Omega^{\prime}}\) \\ \hline \(V_{\text{f}}^{4-}\) & \(1.07\) & \(-0.03\) & \(0.28\) & \(40.2\) \\ \(V_{\text{f}}^{5-a}\) & \(0.65\) & \(-0.01\) & \(0.39\) & \(31.2\) \\ \(V_{Si}^{5-}\) & \(0.57\) & \(-0.02\) & \(0.34\) & \(1 defect charge density within a sub-region of the supercell, the short range potential reaches a plateau (Figure 2b) far from the defect, which yields the alignment as a well-defined quantity [52]. The short-range potential away from the defect site is obtained by taking the difference between the electrostatic potential from a model charge density \(V^{\text{model}}\) and the DFT electrostatic potential from the charged defect \(V^{DFT}_{q}\) and the perfect bulk \(V^{DFT}_{b}\):  
\[\Delta V=\Delta V_{q/b}=\left(V^{\text{DFT}}_{q}-V^{\text{DFT}}_{b}\right) \big{|}_{\text{far}}-V^{\text{model}}\big{|}_{\text{far}} \tag{11}\]  
The model charge density is the charge distribution used to capture the localized defect charge (as shown in Figure 2a). In the FNV alignment correction scheme, we use a Gaussian charge distribution for the model charge density, with width equal to 1 Bohr. The DFT electrostatic potential (blue curve), and the potential due to model charge density (red curve) are shown in Figure 2b. The difference is the short-range electrostatic potential (green curve) which reaches a plateau away from the defect site (centered at origin), thereby giving the alignment correction \(\Delta V\) (highlighted as a dashed line in the plots). The computed value of \(\Delta V\) for \(V^{\text{+}}_{\text{U}}\) is \(-0.17\) eV, and it is computed in a similar manner for all other charged defect studied. For defects with large atomic relaxations the charge density and potential can show strong variation close to the defect site, resulting in uncertainty in extracting the exact value of the alignment correction. Recent reviews [38; 49] of charged defect correction schemes highlight similar issues with relaxed geometries. While this source of error in potential alignment correction needs to be quantified, the results from Reference [46] suggest that the changes are relatively small, i.e., within 40-100 meV, and unlikely to make a significant change to the defect's volume dependence. To obtain the corrections to pressure and volume due to potential alignment correction, we compute the defect formation energy with and without the alignment correction for a set of isotropically-scaled volumes. Figure 3 shows the defect formation energy as function of volume; from the fit to the energy-volume data we compute the volume corresponding to the energy minima, calculated with two approaches. In approach (1), the potential alignment is computed between defect and bulk, with volume change (isotropic-scaling) applied to both the defect and the bulk supercell, such that the defect and bulk have the same volume. In approach (2), the potential alignment is computed such that volume change is applied only to the defect supercell and bulk is kept fixed to its pristine volume. For approach (2) we have employed the LZ scheme [62] because the FNV scheme cannot be applied when the defect and bulk have different volume. That is, the difference in the two approaches is the volume at which the bulk is kept when computing the potential alignment correction of the defective cell.