## 3 Result and Discussion
The extent of the deformation also becomes more evident at high temperatures, with Figure 3 showing larger cage-radius deviations for higher occupancies and temperatures; there is greater distortion to the cages, which is sometimes anisotropic, as hexagonal rings, in particular, flex and stretch to allow the passage of cage-hopping H\({}_{2}\) and D\({}_{2}\) molecules, which increase in frequency at higher temperatures and cage occupancies owing to the greater amplitude of guest-guest collisions. The cage occupancies for both the large and small cages were calculated for all of the simulated systems over the full simulation time at different temperatures, and are reported in Figure 4. At all of the temperatures for both the H\({}_{2}\) and D\({}_{2}\) containing models, in the case of the 1, 2 and 3-occupancy systems, the average occupancy for small cages was found to be less than one, whilst the large cages contain a higher number of guest molecules with respect to the starting structure. Instead, a system with four guest molecules in large cages exhibits a cage occupancy greater than one for small cages and less than four of the large cages, in order to minimise the level of cage and lattice deformation for hosting such a high number of guests. In this way, the overall energy of the full hydrate structure is lowered over time. It is interesting that the small-cage D\({}_{2}\) and H\({}_{2}\) occupancies are almost identical at around 0.95 for both the nominal (i.e., initial) double- and triple-occupation of the large cages (cf. Figure 4). In this case, we see that there is less 'leakage' of the small-cavity H\({}_{2}\) and D\({}_{2}\) into the large cages than for (initial) single large-cage occupancies, for the essential equalisation of guest chemical potentials across the cage types in the most stable, lowest-energy systems. For the initial quadruple occupancy, with the cage distortion that this entails, there is occasional small-cage double occupation (i.e., _circa_ 1.06, cf. Figure 4), which is energetically less preferred. Still, the large level of guest-guest repulsion in the large cages, especially at 240 K, leads to partial double-filling of small cages (for about 7\(\frac{1}{2}\)% of them at 240 K, for both H\({}_{2}\) and D\({}_{2}\)). \begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Configuration**} & \multicolumn{2}{c}{**Activation Energy (kJ/mol)**} \\ \cline{2-3}  & \(\mathbf{H_{2}}\) & \(\mathbf{D_{2}}\) \\ \hline
1-Occupancy & 20.5 \(\pm\) 0.6 & 20.9 \(\pm\) 1.4 \\
2-Occupancy & 14.7 \(\pm\) 1.7 & 14.2 \(\pm\) 4.3 \\
3-Occupancy & 14.2 \(\pm\) 0.3 & 15.0 \(\pm\) 0.7 \\
4-Occupancy & 9.8 \(\pm\) 0.8 & 9.3 \(\pm\) 0.7 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Activation energy of hydrogen and deuterium for inter-cage diffusion. A Markov chain is a sequence of trials in which the outcome of a successive trial depends only on the immediate predecessor. In a Markov chain, a new state will only be accepted if it is more favourable than the existing state. In the context of a simulation using an ensemble, this usually means that the new trial state is lower in energy. In the present study, one cage-hop travel is enumerated for Markov consideration if a guest molecule moves from one cage to a secondary cage. Each travel was labelled based on the starting and the secondary cage type, i.e., as Large-to-Large (LL), Large-to-Small (LS), Small-to-Large (SL), and Small-to-Small (SS) (cf. Figure 5). The Markov model was calculated for all of the simulated systems at all of the temperatures. It was found that, in our system, LL travel dominates in inter-cage hopping. Each LL cage-hop transition represents a molecule passing through a hexagonal ring shared between each large cage. Keeping in mind the structure of a large cage in type II hydrate (5\({}^{12}6^{4}\)), this means that each large cage has four 'gates' that are preferable for guest hopping, and twelve pentagonal portals with a much smaller priority and probability (Table 2).