## 2 Theoretical Framework
### Volume of Individual Species Corprising the Defect  
As described in Equation (5), computation of the defect formation volume needs to take into account the reference volume, \(\Omega_{0}\), of the individual atomic species constituting the defect reaction. For a single-component system like silicon, \(\Omega_{0}\) is equal to the cohesive volume of an atom in bulk silicon. However, for a multicomponent system, \(\Omega_{0}\) is given by the partial molar volume, which measures the variation of volume as chemical components are added or removed from the reservoir during the defect reaction. The choice of reservoir for partial molar volume is exactly analogous to the atomic chemical potential in the definition of defect formation energy. In a multicomponent system, defect reactions can be of two types:  
1. _Stoichiometric reactions_ comprising of defects like the Schottky trio. In this case, \(\Omega_{0}\) is the volume per formula unit of the multicomponent system or, analogous to the monoatomic case, the cohesive volume of a formula unit. For example, in UO\({}_{2}\) this is computed to be \(\Omega_{\text{UO}_{2}}=\) 40.3 A\({}^{3}\) (40.9 A\({}^{3}\) in experiment [55]) and in GaAs it is \(\Omega_{\text{GaAs}}=\) 44.5 A\({}^{3}\);
2. _Non-stoichiometric reactions_ comprised of defects such as vacancies and interstitials. Here \(\Omega_{0}\) is the volume per unit of the reference. For example, in GaAs where solid bulk Ga and As can be considered as the reference (or atomic reservoir) for respective defects, the computed volumes per atom are \(\Omega_{\text{Ga}}=\) 20.3 A\({}^{3}\) (19.3 A\({}^{3}\) in experiment [56]) and \(\Omega_{\text{As}}=\) 22.4 A\({}^{3}\) (21.3 A\({}^{3}\) in experiment [57]), respectively. That is, a reference is defined for each species that depends on the experimental conditions of interest. In this example, we chose bulk solids for each element. However, other reference states are possible or desired depending on the materials system and what experimental conditions are being matched. For example, in UO\({}_{2}\), bulk U metal in the orthorhombic structure (\(\alpha\)-phase) is considered as a reference for U vacancy and interstitial, with computed volume of 19.9 A\({}^{3}\) (20.6 A\({}^{3}\) in experiment [58]). For O defects, we choose orthorhombic \(\alpha\)-U\({}_{3}\)O\({}_{8}\) as reference, with a computed volume \(\Omega_{\text{U}_{3}\text{O}_{8}}=\) 56.9 A\({}^{3}\) per U atom; this is close to the experimental value of 55.5 A\({}^{3}\) per U atom [59]. From this, we derive the reference volume for O atom to be 22.6 A\({}^{3}\) from \(\Omega_{\text{UO}_{2}}\) and \(\Omega_{\text{U}_{3}\text{O}_{8}}\) as discussed in detail in Section 3.3. In all of these cases, the reference explicitly depends on the experimental conditions and is not something directly computable from electronic structure methods. That is, once a reference is chosen, the relevant thermodynamics properties can be calculated, but the reference itself is not something that can be determined solely from the calculations; it must be asserted or assumed. We use Equations (5) and (6) to calculate the defect formation and relaxation volumes. In compounds this calculation is complicated by the fact that the stoichiometry of the system changes when individual point defects such as vacancies or interstitials are formed. For charged defects, the issue is further exacerbated as the electrons or holes added to the defect system also contribute to both the defect formation energy and volume. While the contribution of net charge to the defect formation energy has been well studied in the past and several correction terms have been developed, the contribution of the net charge to the defect formation volume has only recently been recognized [14; 15]. Indeed, in those seminal works, the authors have shown that defect formation volumes are large, and propose various approaches for considering artificial effects of net charge on formation volume.