An example of this analysis is discussed in SI Figure . One can assume that the small patches of configurational space surrounding the singularities in each molecular configuration are negligible compared to the total volume of configurational space. Thus, the errors associated with samples flowing near these regions may, on average, yield a negligible contribution during reweighting. Furthermore, since s does not preserve the antipodal flip-symmetry at the edge of the hemisphere, we note that in the systems we examined so far it was possible to rotate the global reference frame of the supercell data such that, for each molecule in the dataset, the marginal distribution in θ 0 is shifted away from both 0 (one of the singularities) and π 2 (edge of the hemisphere). More generally, one can also note that in a stable crystal system where none of the molecules samples significantly larger rotational angles, and thus their quaternion distribution on S 3 is quite narrow, one can define an arbitrary per-molecule rotation (i.e., a translation of the MD dataset on S 3 ) such that each molecule maps to a formally well behaved joint distribution in the hyperspherical coordinates (s). In this regime, the three marginal variables of s can be transformed and treated independently.