## 1 Introduction
These entrapped molecules' H-H stretch modes are always reduced in frequency compared with their in vacuo counterparts, i.e., redshifted vis-a-vis gas-phase results. The most prominent such shift, \(-34\) cm\({}^{-1}\), was witnessed in the sII THF/H\({}_{2}\) clathrate's Raman modes, arising clearly for the singly occupied \(5^{12}\) cavity [34; 40]. An identical redshift was prominent in the case of pure sII H\({}_{2}\) clathrates, rendering it unambiguous that this arises due to H\({}_{2}\)'s presence in the \(5^{12}\) cavities. Pure H\({}_{2}\) hydrate spectra depicted mode shifting by \(-26\), \(-18\), and \(-11\) cm\({}^{-1}\)[34; 40], denoting contributions from \(5^{12}6^{4}\) cavities with H\({}_{2}\) occupations between two and four. Highly controlled, accurate, and sophisticated experiments, entailing several heating and quenching cycles with careful measurements of H\({}_{2}\) amounts released in each, resulted ultimately in the attribution of these respective redshifts to double, triple, and quadruple \(5^{12}6^{4}\) cage occupations [40]. The H-H stretch shift can be rationalized as an effective coupling between two types of interactions [40]. Of these, one, i.e., the attractive interaction between H\({}_{2}\) and the cavity, softens the H-H stretch potential and reduces its value vis-a-vis the result in vacuo. With increasing \(5^{12}6^{4}\) cavity occupation, there is, naturally, closer H\({}_{2}\)-H\({}_{2}\) packing, and these essentially repulsive H\({}_{2}\)-H\({}_{2}\) interactions result in a rising H-H stretch mode and a decline in the extent of redshift. The vibrational H-H mode is redshifted even for the quadruple occupation case in the \(5^{12}6^{4}\) cavity, indicating that the predominantly attractive guest-host interactions dominate over repulsive H\({}_{2}\)-H\({}_{2}\) ones. Accurate calculation of the intramolecular vibrational frequencies of the H\({}_{2}\) molecule(s) inside the hydrate cages has proven to be a most challenging problem. In the last decade or so, various methodologies entailing a range of approximations have been adopted to tackle this challenging conundrum. In one approach [41; 42], the cage-entrapped hydrogen molecules were treated as frozen in the minimal-energy geometry corresponding to the minimum energy of the system and at 0 K. Consequently, nuclear quantal effects were neglected--indeed, most pointedly in the case of \"smearing out\" the guests' large-amplitude vibrations. Moreover, given that only isolated cavities were modelled, the surrounding condensed-matter milieu's influence was not taken into account. To tackle this omission of the background lattice, a judicious mixture of both classical and path integral molecular dynamics (MD and PIMD) simulations employing Density-Functional Theory (DFT) at hybrid-functional level (B3 LYP) and second-order Moller-Plesset perturbation (MP2) prescriptions was used [43]. The H-Hstretch modes found under 1D conditions for the H\({}_{2}\) coordinates sampled from many MD simulation snapshots encapsulated a wide spread of frequencies extending to the free molecule at 4155 cm\({}^{-1}\). Indeed, the distributions' peaks were reasonably consistent with experimental values after a scaling factor was employed in the simulations [43]. Drawing from experiments in [44], classically propagated MD calculations employing DFT were carried out for a full sII clathrate unit cell, and the H-H stretch spectra were computed via Fourier transformation of the H-H bond length autocorrelation [45]. A shortcoming of this approach is the classical treatment of the underlying dynamics, leading to an overestimation of the H-H stretch modes by \(\sim\)250 cm\({}^{-1}\) vis-a-vis Raman measurements at 260 K.  
Various flavors of PIMD, e.g., centroid and ring-polymer approaches, are of use in modelling the dynamical vibrational properties of fluids and materials, capturing, in some cases, semi-quantitative agreement with experimental results [46; 47; 48].