## 1 Introduction
Thus, the dipole tensor provides a link between atomistic calculations of defects in an ideal bulk environment and in the more realistic microstructure of the material during application, allowing one to up-scale results to more applied studies, such as irradiation-induced creep which relies on the treatment of defects interacting with the existing microstructure. To establish this link, one must be able to calculate quantitatively accurate values for the dipole tensor or, related to this, the relaxation volume of defects. Building on our previous work on point defect behavior in the presence of external strain [11], here we find the relaxation volumes of individual charged point defects in uranium dioxide (UO\({}_{2}\)) to be much larger than those of the corresponding neutral defects. Furthermore, when we calculate the corresponding elastic dipole tensor, the predicted changes in energy of the defect under elastic strain fields are extremely large; indeed, they are so large that they seem to be unphysical. As the dipole tensor is directly related to defect relaxation volumes, this result motivates us to examine this issue in greater depth. The issue of large DFT relaxation volumes of charged defects as well as discrepancies between theory and experiments on the topic has been recognized in the past [4; 9; 14; 15; 25]. However, we argue that the issue is not completely resolved. We suggest that by considering the change in volume of the electron reservoir in the formation reaction of the charged defects, we can renormalize the formation volumes of charged defects, and this approach represents one possible route for resolving the issue, in addition to using a net neutral defect reaction model to determine relaxation volumes. In this work we focus on the issue of the relaxation volume of charged defects, focusing on the actinide material UO\({}_{2}\), both because of its complex electronic structure and because its fluorite crystal structure is common to many technologically important materials. Defect calculations [26; 27; 28; 29] in UO\({}_{2}\) have established that defects such as Frenkel pairs and Schottky trios and fission product complexes are likely to consist of individual components that are charged. The individual volume change depends not only on the size mismatch of the defect species, but also on the more complex charge transfer mechanisms occurring in the presence of the defect. In addition to UO\({}_{2}\), we present results and comparisons for relaxation volumes of charged defects in Si and GaAs, for the purpose of validating against existing data in these systems and uncovering behavior that applies to charged systems more generally. The paper is arranged as follows. Section 2.1 presents the density-functional theory (DFT) methodology used in calculating defect properties. In Section 2.2, we discuss the definitions of defect formation and relaxation volumes. Various schemes to correct the total energy of charged supercells simulated using DFT and their relation to defect relaxation volumes are investigated in Section 3.1. Our results on the details of the lattice distortion around individual charged and neutral defects are presented in Section 3.2. In Section 3.3, we discuss the defect reaction approach for calculating the defect relaxation and formation volumes for complete defect formation reactions. In Section 4, we discuss the significance of our results and present our conclusions in Section 5.