## 3 Results
### Finite Size Correction Schemes to Energy for Charged Supercell
#### 3.1.1 Image Charge Correction  
The image charge correction is needed because, in periodic supercell calculations, a charged defect (in the presence of the homogeneous background charge density) has long-range electrostatic interactions with its periodic images. If the dimensions of the unit cell were infinite, this electrostatic interaction would be zero. However, in periodic DFT calculations the supercell dimensions are finite; therefore we need to correct the total energy of the charged system to account for this spurious Coulomb interaction. This correction is referred to as the monopole correction by [PERSON] and [PERSON] [63], and is designed to transform the electrostatic energy of a periodic lattice of point charges in a neutralizing background into the electrostatic energy of single point charge. As such, it is not unique to DFT, but occurs whenever the energetics of a system with net charge is considered. The correction term is given as [39; 63]:  
\[\Delta E_{corr}^{Image}=E_{isolated}-E_{periodic}=\frac{\alpha q^{2}}{2\varepsilon \Omega^{1/3}} \tag{8}\]  
where \(\alpha\) is the Madelung constant, \(\Omega\) is the volume of the bulk supercell in which the defect is created, and \(\varepsilon\) represents the electronic and ionic low frequency dielectric constant of the material, computed from DFT calculations (Table 1). In our calculations, we use the average value [38; 64], \(\frac{1}{3}tr\big{(}\varepsilon_{ij}\big{)}\) of the dielectric tensor as the effective value for dielectric constant in Equation (8). The energy correction given by Equation (8) is computed directly from VASP, and the corresponding correction to the pressure \(\Delta P_{corr}^{image}\) can either be calculated directly from the slope of the image-charge correction (\(\Delta E_{corr}^{Image}\)) as a function of volume or via the relation as derived in Reference [13]:  
\[\Delta P_{corr}^{image}=-\frac{\partial E_{corr}^{image}}{\partial\Omega}=- \frac{1}{3}\bigg{(}\frac{\alpha q^{2}}{2\epsilon\Omega^{\frac{4}{3}}}\bigg{)} \tag{9}\]  
with the corresponding volume correction given as,  
\[\Delta\Omega_{corr}^{image}=-\Omega_{Bulk}\bigg{(}\frac{\Delta P_{corr}^{ image}}{B}\bigg{)} \tag{10}\]  
The computed image charge corrections to energy, pressure and volume are summarized in Table 4 for charged defects in UO\({}_{2}\), GaAs and Si. The corrections to volume are very small compared to the corrections in the total energy and to the corresponding defect relaxation volumes: \(\Delta\Omega_{corr}^{image}\) -0.01-0.02 \(\Delta\Omega^{\prime}\), for the three systems studied here. Thus, the correction to the relaxation volumes due to the image charges is only a 1%-2% effect. Our computed value of \(\Delta\Omega_{corr}^{image}\) for the charged (2+) Si vacancy is 0.02 \(\Omega_{\text{Si}}\), which is similar to that reported by [PERSON] and co-workers [14]. We note that the correction to the relaxation volume and pressure (or stresses) decreases with the increasing size of the simulation supercell and is independent of the sign of the charge on the defect, depending only on the magnitude of the charge. For example, for the U vacancy, the energy and volume corrections are 0.82 eV, and 0.2 A\({}^{3}\) in a \(3\times 3\times 3\) supercell, compared to 1.23 eV, and 0.32 A\({}^{3}\) in \(2\times 2\times 2\) supercell. The above changes in the pressure and volume are computed based on two assumptions. The first assumption is that the value of the bulk modulus, \(B\), does not change significantly due to the presence of defects. The second assumption is that the charge is localized at the defect site within the simulated supercell size. Our computed value of the bulk modulus of UO\({}_{2}\) with a U vacancy (in a 144 atom supercell) is 203.3 GPa compared to the bulk value of 220.5 GPa. This difference gives an error of 0.01-0.03 A\({}^{3}\) in the estimation of \(\Delta\Omega_{corr}^{image}\).