## 4 Conclusions  
The main aim of this article was to explore the inter-cage hopping of hydrogen molecules in the bulk structure of type II hydrate. Using classic molecular dynamics simulations, the effects of nominal (initial) large-cage composition and temperature on the number of inter-cage transitions and the dependence on cage occupancy were studied. In parallel, by the substitution of H\({}_{2}\) molecules with D\({}_{2}\) molecules, the behaviour of the heavier isotope was also investigated. The activation energy for the guest-molecule (inter-cage) diffusion was found to be the highest for the 1-occupancy model, whilst this number for the 2- and 3-occupancy was almost at the same level, but still lower than the 1-occupancy model. For the lattice-strained 4-occupancy system, this number was about two times lower than the 1-occupancy system, and almost 40% lower than the systems with 2- and 3-occupancy, owing to the larger amplitude of (thermally-activated) guest-guest collisions inducing H\({}_{2}\) and D\({}_{2}\) to'spill out' of large cages and ingress partly into small ones.  
The average occupancy for both the large and small cages for all of the systems showed the overall preference of the system to minimise the occupancy of small cages and overload the larger cages. It was found that, for the system starting with four guest molecules, the system allowed some of the small cages to have two guest molecules in order to decrease the average occupancy of the large cages to lower than that of the initial structure. Furthermore, by developing the Markov model, we confirmed the preference of hydrogen molecules to travel through the hexagonal ring, via large-to-large (LL) cage hopping, rather than the less-probable large-to-small (LS) or small-to-small (SS) transitions. The D\({}_{2}\) molecules showed a statistically significant lower diffusion rate, while the diffusion activation energies for both cases were almost the same. Further studies, such as the one reported by [PERSON] et al. [50] on Gibbs-ensemble MC calculations to ascertain the right cage occupancies, would be ripe for future MSD and hopping/Markov analysis, along the lines of what we have performed here.  
The following are available online at [[https://www.mdpi.com/2076-341/7/11/1/282/s1](https://www.mdpi.com/2076-341/7/11/1/282/s1)]([https://www.mdpi.com/2076-341/7/11/1/282/s1](https://www.mdpi.com/2076-341/7/11/1/282/s1)), Figure S1: Mean Square Displacement for Hydrogen in clathrate hydrate, Figure S2: Mean Square Displacement for Deuterium in clathrate hydrate, Figure S3: Averaged cage radius with respect to temperature, Figure S4: Simulation box pressure vs temperature, Table S1: Markov chain model for Hydrogen, Table S2: Markov chain model for Deuterium, Table S3: Diffusion constants for D\({}_{2}\) with respect to temperature, Table S4: Diffusion constants for H\({}_{2}\) with respect to temperature.  
Conceptualization, M.R.G. and Y.K.; methodology, M.R.G. and Y.K.; software, Y.K.; validation, M.R.G. and Y.K.; formal analysis, Y.K.; investigation, M.R.G., Y.K., A.D. and N.J.E.; writing--original draft preparation, M.R.G. and Y.K.; writing--review and editing, M.R.G., Y.K., A.D. and N.J.E.; supervision, M.R.G. and N.J.E.; project administration, N.J.E.; funding acquisition, M.R.G. and N.J.E. All authors have read and agreed to the published version of the manuscript.  
This research was funded by Science Foundation Ireland (SFI 17/NSFC/5229) and Irish Research Council (GOIPD/2016/365).  
Not applicable.  
The data presented in this study are available on request from the corresponding author.  
N.J.E. and Y.K. thank Science Foundation Ireland (SFI 17/NSFC/5229). M.R.G. thanks the Irish Research Council for a Government of Ireland postdoctoral fellowship (GOIPD/2016/365), whilst N.J.E. also thanks Science Foundation Ireland for funding under grant SFI 15/ERC-I3142. All authors thank [PERSON] and [PERSON] for computational and technical assistance, as well as interesting and stimulating discussions.  
The authors declare no conflict of interest.