## 2 Theoretical Framework
### Defect Formation and Relaxation Volumes  
The formation volume of a defect \(\Delta\Omega^{f}\) is given by the change in Gibbs free energy, \(\Delta G^{f}\) (\(p\), \(T\)), for the defect formation reaction at constant pressure and temperature via [5]:  
\[\Delta G^{f}(p,T)=\Delta E^{f}(p,T)+p\Delta\Omega^{f}-T\Delta S(p,T) \tag{2}\]  
where \(\Delta E^{f}\) denotes the change in the internal energy, \(\Delta\Omega^{f}\) the formation volume, and \(\Delta S\) the change in entropy due to the formation of the defect at constant pressure and temperature. The defect formation volume is the difference in volume across the defect formation reaction and is related to the Gibbs free energy of defect formation by:  
\[\Delta\Omega^{f}= \left(\frac{\partial\Delta G^{f}(p,T)}{\partial\text{P}}\right)_ {T,n_{i}} \tag{3}\]  
At \(T=0\) K, the change in Gibbs free energy, \(\Delta G^{f}\) (\(p\), 0) reduces to the change in enthalpy, \(\Delta H^{f}\). However, at higher temperatures, the entropy contribution can also become relevant [52]. The defect formation enthalpy, \(\Delta H^{f}\), from DFT calculations for a defect \(X\), in charge state \(q\) is given as [52; 53]:  
\[\Delta H^{f}(X,q)=\left[E_{tot}^{DFT}(X,q)+E_{corr}^{Image}+q\Delta V \right]-E_{tot}^{DFT}(bulk)-\sum_{i}n_{i}\mu_{i}+q(\varepsilon_{V}+\varepsilon _{F})\] \[+p\Delta\Omega^{f} \tag{4}\]  
where, \(E_{tot}^{DFT}(X,q)\) is the total energy of the DFT supercell with the charged defect, \(E_{tot}^{DFT}(bulk)\) is the total energy of the defect-free bulk supercell, and \(n_{i}\) and \(\mu_{i}\) are the number and chemical potential of each element contributing to the defect, \(\varepsilon_{v}\) is the energy of the valence band maximum (VBM) calculated in the defect free bulk supercell [49], and \(\varepsilon_{F}\) is the Fermi energy with respect to the VBM. The term \(q(\varepsilon_{v}+\varepsilon_{F})\) is the chemical potential of electrons, i.e., it accounts for exchange of electrons with the reservoir. The \(\Delta\Omega^{f}\) contribution to the enthalpy in Equation (4), is usually very small at ambient conditions (\(\sim\)10\({}^{-5}\) eV per defect at \(p=1\) atm) and often neglected. The terms \(E_{corr}^{Image}\) and \(\Delta V\) are the charged image interaction and potential alignment corrections, respectively. These corrections are needed to correct for finite-size artifacts within the DFT supercell approach to calculate the defect formation energies. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**System** & \(a_{0}\) & \(C_{11}\) & \(C_{12}\) & \(C_{44}\) & \(B\) & \(\varepsilon\) \\ \hline \(\mathbf{U}\mathbf{O}_{2}\) & 5.451 & 377.8 & 133.2 & 78.2 & 220.5 & 21.53 \\ DFT [44] & 5.448 & 380.9 & 140.4 & 63.2 & 220.6 & 20.9 \\ Expt. [45] & 5.473 & 389.3 & 118.7 & 59.7 & 209.0 & 23.6 \\ \(\mathbf{S}\)I & 5.468 & 153.1 & 56.8 & 74.6 & 88.9 & 13.1 \\ DFT [40; 46] & 5.47 & 154.6 & 58.1 & 74.4 & 90.2 & 12.98 \\ Expt. [47; 48] & 5.431 & 167.7 & 65.0 & 80.4 & 99.2 & 11.17 \\ _GaAs_ & 5.628 & 117.4 & 54.1 & 56.0 & 75.2 & 18.7 \\ DFT [38; 49] & 5.627 & 124.2 & 51.4 & 63.4 & 75.6 & 17.9 \\ Expt. [50; 51] & 5.654 & 121.7 & 54.6 & 61.6 & 76.9 & 13.1 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Calculated lattice parameters, \(a_{0}\) in Å, elastic constants, \(C_{ij}\), and bulk modulus, \(B\), in GPa and the static (electronic + ionic contribution) dielectric constant, \(\varepsilon\), from density-functional theory (DFT) calculations in UO\({}_{2}\) (local spin density approximation; LSDA + U), Si (GGA) and GaAs (local density approximation, LDA). Our results are indicated in italics while literature values are provided for comparison.