## 3 Results and Discussion
More specifically, in reference [11], we calculated free energy profiles of the molecules over the volume of the 51264 cages' interiors at 50, 100 and 200 K. We showed that the associated barrier reduced almost linearly for 1-3 molecules per large cage but grew larger than expected for quadruple occupancy, with this departure from linearity becoming more pronounced at lower temperatures [11]. The barriers tended to rise with rising temperature, with nuclear quantum effects (NQEs) raising further due to quantal delocalisation [11]. In terms of intra-cage dynamical motion (to study classical site-jump phenomena), we first studied the vibrational (proton) velocity-spectra of the guest and host species (i.e., H\({}_{2}\) in H\({}_{2}\)O and D\({}_{2}\) in D\({}_{2}\)O) at 80, 100 and 125 K for 200 ps. These are provided in the Supplementary Information (Figures S1 and S2), and they obey the expected isotope vibrational effect quite well, i.e., doubling the mass in going from H\({}_{2}\) to D\({}_{2}\) (and H\({}_{2}\)O to D\({}_{2}\)O) reduces the frequency by \(\sqrt{2}\). For double occupation of the 51,264 cages, the double exponential decay fit for the site-residence distribution h(t), from Equation (1), gives an expected relaxation time \(\tau\)2 at 100 K of around 10 and 13 ps for H2 and D2, respectively, with r2 values of over 0.95. The shorter \(\tau\)1 times, reflective of localised rattling, were about 2.0 and 3.1 ps for H\({}_{2}\) and D\({}_{2}\), respectively. The h(t) data is shown in Figure 2 at 100 K for both D\({}_{2}\) and H\({}_{2}\) guests, with the H\({}_{2}\)-fit as an example. QENS signals observed in the spectra of reference [23] indicate that increasing temperature in the H\({}_{2}\)O-H\({}_{2}\) sII clathrate corresponds to a change in H\({}_{2}\) dynamics from a strictly quantum to a more mixed quantum-classical regime around 10-15 K. At all investigated temperatures up to around 50 K, the quasielastic broadening width was \(Q\)-independent [23], suggesting, as expected in cage/confined dynamics [26; 28; 29; 30], a localised rattling motion (reflective of the shorter-time transient relaxation process \(\tau_{1}\) in Equation (1)), as opposed to more jump-/hop-diffusion exploration of other tetrahedral sites. For temperatures between 25 and 45 K, it was found (using QENS) that the width of this hybrid quantum-classical process ranged between approximately 0.55 and 0.65 meV, which corresponded to between 8.2 and 6.4 ps [23]. At around 50 K, there was an onset of fully classical site-jump diffusional behaviour, i.e., with \(Q\)-dependence of the quasielastic broadening width emerging clearly [23], and so extrapolation of the QENS width to the fully classical hop-diffusion regime at 100 K (in the present work) was expected to yield localised rattling-timescales of no more than a few picoseconds. This is not inconsistent with our present \(\tau\)1 estimates (i.e., 2.0 and 3.1 ps for H2 and D2, resp.). In addition, taking into account shorter time residences of the tetrahedral sites (<5 ps), the present work's (time-weighted) means of the dwell time distributions at 100 K were 1.38 and 1.21 ps for H\({}_{2}\) and D\({}_{2}\), respectively. This is broadly in line with site-specific timescales in a classical jump-diffusion regime at 100 K from neutron scattering experiments [23; 26]. In terms of temperature dependence of \(\tau_{1}\) and \(\tau_{2}\) bi-exponential times, Arrhenius plots are provided in Figure S3 for 80, 100 and 125 K, revealing activation energies of 0.45, 0.50, 0.59 and 0.66 kJ/mol for \(\tau_{1}\) (H\({}_{2}\), D\({}_{2}\)) and \(\tau_{2}\) (H\({}_{2}\), D\({}_{2}\)), respectively. As expected, this is a deal smaller than for inter-cage phenomena [11; 19; 27]. The relative linearity of the Arrhenius fits shows that they are essentially undergoing a quasi-classical-type jump diffusion as the basic mechanism. The underlying \(\tau_{1}\) (H\({}_{2}\), D\({}_{2}\)) and \(\tau_{2}\) (H\({}_{2}\), D\({}_{2}\)) values are summarised below in Table 1, with \(r^{2}\) values well over 0.95.