## 3 Result and Discussion  
The calculated diffusion coefficients at the different temperatures for each of the simulated systems are presented in Figure 2; for the H\({}_{2}\) and D\({}_{2}\) systems, the values are tabled in Tables S3 and S4, respectively (see SI). The clathrate hydrates are of special interest because they are less dense than regular ice and, under negative pressure, are expected to be stable [29; 40; 41; 42; 43; 44; 45]; the density of the clathrate hydrate is also less in low temperatures, and the density increases with the temperature, as shown in the Table S4 (see SI). The diffusion coefficient is a physical constant dependent on the molecule size and other properties of the diffusing substance, as well as on temperature and pressure. The diffusion coefficient increases with the temperature, as can be expected for thermally-activated Brownian motion. The diffusion coefficient can be connected to the temperature by the Arrhenius expression, in which the amount of existing potential barriers (activation energy) plays the main role. In the double occupancy configuration, the diffusion coefficient does not change in low temperatures, and it has higher D values in high temperatures from 200 K to 240 K compared to the single occupancy. The triple occupancy showed higher D values than the single and double occupancies. Indeed, this is not unexpected, in that there are more frequent guest-guest collisions inside cages, which propel and push them to rearrange themselves dynamically, and leads to a greater propensity for a'momentum kick' to hop to  
Figure 1: Schematic of a hydrogen/Deuterium hydrate in a bulk simulation box. a neighbouring cage (with [PERSON]'s Third Law providing equal and opposite pairwise reaction-kick forces on the other guests). This phenomenon of 'crowding out' becomes even more pronounced for quadruple occupation, and this same increase of the self-diffusivity is also seen in Figure 2.  
The activation energy for each system was derived based on the different D values at different temperatures using the Arrhenius equation (Figure S3b,d), and the values are summarized in the Table 1 for both the hydrogen and deuterium systems. The calculated energy barrier (E\({}_{\text{a}}\)) for the diffusion inside the bulk hydrate structure revealed that the inter-cage hopping in the model with four occupancies is more favorable, from a thermodynamic point of view, for both the H\({}_{2}\) and D\({}_{2}\) systems. Despite the statistically significantly different diffusion coefficient for H\({}_{2}\) and D\({}_{2}\), the activation energies for both systems are almost similar. This similarity shows that the dynamics of the water molecules in the hydrate structure are the controlling parameters during the hopping process, and they do not depend greatly on the traveling gas type. Such an observation is confirmed by recent experimental results obtained on the H\({}_{2}\) long-range diffusion in gas hydrate-containing H-bond defects. In such ionic hydrates, the H\({}_{2}\)O relaxation occurs on a shorter timescale than hydrates without H-bond defects [46; 47], facilitating cage deformation [48]; thus, the H\({}_{2}\) diffusion is measured to be faster than in the same hydrate without H-bond defects [49]. Figure 2: Diffusion coefficient for the different configurations at various temperature. (**a**) H\({}_{2}\), (**b**) Arrhenius fitting for H\({}_{2}\) (**c**) D\({}_{2}\) and (**d**) Arrhenius fitting for D\({}_{2}\). The area selected for the Arrhenius fitting is highlighted in yellow. The dashed lines present the fitted line for the Arrhenius equation. The most probable origin of a lower activation energy for the higher-occupancy systems is the deformation of the cage due to the increased cage occupancy. Figure 3 represents the average cage radius for small and large cages, and the increase in the radius in going from triple to quadruple occupation is clearly evident, with the increasing temperature serving to exaggerate this cage stretching further (see SI Figure S3.