## 3 Results and Discussion
Returning to the present work, even for the total, aggregate spectra (top panels in Figure 3), there is a broader distribution and heterogeneity in peaks vis-a-vis the empirical model case (cf. Figure 1). In the case of multiple large-cavity occupancy, there is less evidence of peak-splitting in individual molecules, particularly at higher occupations (cf. blue curves in Figure 3 in comparison to the single-occupancy case on the top left). This suggests an environmental effect of spatial heterogeneity within the cavities in determining, or at least influencing, the H-H vibrational frequency properties. With increasing large-cage occupation, the greater spatial localization of H\({}_{2}\) molecules into a quasi-tetrahedral arrangement [2; 3; 35; 36] leads to less ability to move freely within the cavity, sampling it in a reasonably uniform way; this implies that the local electrostatic environment, possessing its own largely lattice-determined (and slightly temporally fluctuating) intrinsic electric field [54; 55] at each molecule's proximal location, may determine underlying H-H stretch modes. Certainly, the broad spread of individual-molecule peaks (blue curves in the lower insets of Figure 3) points to the heterogeneity of intrinsic electric field distributions [54; 55]. Figure 1: Schrödinger H–H spectra for large-cage occupations of \(n=1,\ldots,4\) at 100 K, with classical molecular dynamics (MD) results. Black lines represent total spectra over all H\({}_{2}\) molecules, red lines represent spectra of large-cage H\({}_{2}\), green lines represent spectra of small-cage H\({}_{2}\). “Qu” refers to quantum, while “Cl” denotes classical. However interesting--and even appealing--this hypothesis may be of local electrostatic conditions and the electric field environment in influencing H-H stretch frequencies, it is still somewhat tentative. For instance, we have not yet been able to remove H\({}_{2}\)-H\({}_{2}\) interactions, even in neighboring cages in single-occupancy cases; naturally, this may well be important for multiply occupied large cages  
Figure 3: BO-MD-sampled H–H spectra for large-cage occupancies of 1-4, with single small-cavity occupation. Below the total spectrum is shown the large/small individual-cage spectral decomposition. Figure 2: Average frequency positions for classical and Schrödinger large-cage (red) and small-cage (green) spectra. in determining environmental effects (albeit more in terms of structural localization, rather than electrostatic effects). With this in mind, and recalling the pivotal role of the lattice in determining local electric field intensities, we present in Figure 4 the H-H spectra for a solitary H\({}_{2}\) molecule in a large cage, with the rest of the unit cell cages empty, for both the mobile (normally vibrating) lattice and its fixed counterpart. It is readily evident that mode-splitting does indeed occur, to some extent, as the H\({}_{2}\) molecule samples space in the large cage essentially uniformly [2; 35; 36]. The \(\sim\)4325 cm\({}^{-1}\) peak dominates at the cage centre, but the \(\sim\)4280-4310 cm\({}^{-1}\) peaks are suggestive of the lesser amounts of time spent in the other \"rim\" of the centre, closer to the cage faces, with its more confined environment. However, we remark that in the 8D quantum calculations of ref. [53], there was no splitting of the H-H stretch fundamental observed for a single H\({}_{2}\) molecule in the large cage, and we discuss further potential shortcomings of splitting observations below. In any event, these observations of Figure 4 are more akin to the to-lower-frequency redshift witnessed in moving to a single H\({}_{2}\) molecule in a small cavity, i.e., to \(\sim\)4250 cm\({}^{-1}\), featuring more intimate contact with the surrounding water molecules and cavity faces, and the enhanced intrinsic electric field intensity in this small cage borne of these stronger interactions with the lattice.