## 3 Results and Discussion  
Figure 1 depicts the Schrodinger and classical MD sampled H-H spectra for the four large-cage occupations. A systematic redshift of \(\sim\)350 cm\({}^{-1}\) is clearly evident for the quantum case vis-a-vis its classical counterpart for all occupations, regardless of cage occupation or type (cf. Figure 2). This puts the Schrodinger-solved H-H stretch results in slightly superior agreement with Raman-measured data at 260 K of \(\sim\)4050-4150 cm\({}^{-1}\)[34, 45, 50], although these quantal results systematically underpredict H-H stretch modes, whilst classical MD overpredicts. This \"reversed error sign\" situation is not altogether ideal if one seeks true quantitative accuracy, but several mitigating factors must be borne in mind in this respect. Although ref. 4 presents a relatively accurate pairwise potential model, it is, of course, imperfect and cannot capture all of the subtleties of H\({}_{2}\)-H\({}_{2}\) and H\({}_{2}\)-water interactions. Although care has been taken to define the H-H stretch coordinate, there may be some residual background translational and rotational H\({}_{2}\) motion; for the case of quadruple occupation, with the H\({}_{2}\) molecules more localized spatially--essentially \"hemmed in\" to their tetrahedral configuration [2, 3, 35, 36]--this point is less of a difficultly. Naturally, depending on occupation, one to four independent one-dimensional Schrodinger equations are solved per large cavity, and it is assumed that each H-H vibration is independent of others. This assumption may be somewhat questionable and serves to further explain the lack of full quantitative agreement with experimental results [34, 40, 45]. Of course, the distribution of experimental cage occupations is not well characterized or known [34, 40, 45], which hampers further attempts for quantitative comparison between measured Raman spectra and H-H vibrational spectra; in the present work, and the DFT-based work of ref. [45], there were found to be distinct and distinguishable differences between large-cage frequency modes and occupations (as will be discussed below). Interestingly, there was no evidence of peak \"bifurcation\" or splitting in each cage type, except for the case of quadruple occupation in the large cage for classical MD, which was suppressed in the Schrodinger case. Of course, to study mode-splitting effects with greater rigor, state-of-the art DRSLL is very helpful, albeit that PIMD or a Schrodinger equation approach therewith would be very computationally demanding (as discussed further below). Figure 3 shows BO-MD-sampled H-H spectra for large-cage occupancies of 1-4, with single small-cavity occupation; here, the dynamics are propagated classically. The total spectrum is shown, in addition to individual large- and small-cage spectral decompositions. For all occupations, as with the empirical model results of Figures 1 and 2, the low-frequency part of the vibrational band originates from H\({}_{2}\) entrapped in small cages, whilst the large-cage hydrogen molecules contribute to the higher-frequency peaks. This clearly demonstrates the important effect of the molecular environment in influencing the spectral pattern. As discussed in ref. 45, the ratio of the peak areas corresponds to number of the H\({}_{2}\) molecules located in the large/small cages. Study of the broad panoply of individual-molecule spectra is revealing (cf. the lower insets for each occupation case in Figure 3). For some H\({}_{2}\) molecules, single-occupation spectra in the large cages (top left) show some evidence of peak splitting, as do those in small cavities for all large-cage occupancies (with single occupation therein irrespective of large-cage occupancy). However, it must be noted that no such fundamental H-H stretch  
splitting was observed in quantum 6D calculations of H\({}_{2}\) in both small and large cages [51]; we remark on possible reasons for potentially spurious peak splitting below.