## 2 Methodology  
The empirical model used in this work was the force-matched model (FMM) of [PERSON], [PERSON] and [PERSON] [27]. The model was parameterised against Born-Oppenheimer molecular dynamics (BOMD) trajectories of type II clathrate hydrates containing guest hydrogen molecules. The model was fit using a force-matching method, in which the forces on each particle during the course of the BOMD trajectory were fit. The results had good agreement with the density functional theory results for the free energy barriers for H\({}_{2}\) molecules' passage across large cages. This was true, in particular, along the path connecting pairs of large cages through their shared hexagonal face, through which the guest molecules hopped from one large cage to another. Both classically propagated and 32-bead path-integral molecular dynamics (PIMD) were carried out under periodic boundary conditions for an sII unit cell with single and double occupation in small and large cages of both D\({}_{2}\) and H\({}_{2}\), respectively. The D\({}_{2}\) and H\({}_{2}\) were in respective D\({}_{2}\)O and H\({}_{2}\)O lattices. The molecular-dynamics methodology and other computational details are otherwise very similar to reference [27]. Canonical-ensemble dynamics (classical and 32-bead PI) were performed in an sII unit cell with a vanishingly small dipole moment [27] for 20 ps with a 0.2-fs time step. All cages in the unit-cell simulation box were occupied during molecular dynamics (MD) (single or double, as appropriate). Initially, we wished to average the results over all cages (either \(5^{12}\) or \(5^{12}6^{4}\)) and over all time steps. Doing so first required identifying all the cages of each type, and then placing those cages in a standard position/orientation. To do this required a method for producing an equivalent orthonormal x-, y-, z- axes set for each of the large (and small) cages, such that if the cages only differed by a rotation in the frame of the axes set, each cage would be brought into perfect alignment. To place each of the \(5^{12}6^{4}\) large cages into a standard orientation, we first used an algorithm to identify the vertices of the four hexagonal faces. We then constructed four relative vectors from the cage-centre to the centre of each of the four hexagonal faces, which should, and does, form a tetrahedral vector set. The vectors were then normalised. It was then possible to form an orthonormal axis set from linear combinations of the tetrahedral vectors, which can be written in matrix form as \(\mathbf{a}=\mathbf{M}\mathbf{t}\), where \(\mathbf{t}=\{\mathbf{t}_{1},\mathbf{t}_{2}\mathbf{t}_{3},\mathbf{t}_{4}\}\) are the four tetrahedral vectors, \(\mathbf{a}=\{\mathbf{a}_{0},\mathbf{a}_{1},\mathbf{a}_{2},\mathbf{a}_{3}\}\) contain the orthonormal axis vectors (in \(\mathbf{a}_{1},\mathbf{a}_{2},\mathbf{a}_{3}\)), and the orthogonal \(\mathbf{M}\) matrix is given by:  
\[\mathbf{M}=\frac{1}{2}\left( \begin{array}{cccc}1&1&1&1\\ 1&1&-1&-1\\ 1&-1&1&-1\\ 1&-1&-1&1\end{array} \right)\]  
To place each of the \(5^{12}\) dodecahedral cages in a standard orientation, we first chose an arbitrary starting vertex on each dodecahedron. Then, starting from the initial vertex, we searched for the six second nearest neighbours. From these we constructed six unit vectors pointing from the starting vertex to the six second nearest neighbours. It can be shown that, for a regular dodecahedron, these six vectors comprised two sets of mutually orthonormal triplets, either of which were used to form an axis set. One set was found from the three left turns that were made at the connecting first nearest neighbour (to the second-nearest neighbour), and the other set was found from the three right turns. In fact, the two choices were not equivalent as they corresponded to two different orientationsof the dodecahedron. For this reason, the equivalent axis set for each dodecahedron (choosing consistently either all-left or all-right turns) should be chosen carefully.