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 explicitly preserve the antipodal flip-symmetry near the edge of the hemisphere, we note that in the systems we examined so far it was always 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). In other words, there often exists a global rotation to an MD crystal dataset that makes the support of the distribution in s simply connected for all molecules. In this scenario, the three marginal variables of s can be transformed and treated independently. Alternatively, if finite density near θ 0 = π 2 cannot be avoided, one can anticipate that the associated features of the densities on θ 1 and ϕ can be learned from the data by brute force. However, a non-trivial embedding may then be necessary when conditioning transformations of other DOFs on any s that is associated with a disconnected distribution. After each unit-quaternions is transformed to s, the combined set of 3n mol rotational DOFs of the crystal were concatenated to the rest of the marginal variables mentioned earlier to give x H shown in Fig. .