## 1 Introduction
Strictly, however, the use of these methods is less appropriate for estimating time-dependent properties and correlation functions, as argued with acuity by [PERSON] et al. in the formulation of Matsubara dynamics [49] and its formal relationship with forms of PIMD [50]; Matsubara dynamics may, in time, be shown to offer one pathway towards more rigorous and pragmatic approximations for quantal dynamics [50]. In principle, the most appropriate and rigorous quantal treatment of the H-H intramolecular stretch vibrational dynamics involves high-dimensional quantum calculations, with full coupling, of the vibration-rotation-translation (VRT) eigenstates of one or more H\({}_{2}\) molecules confined inside a clathrate hydrate cage. Such calculations face two serious obstacles. One is tackling the high dimensionality of the quantum bound state that requires a solution even when a clathrate hydrate is treated as rigid; it is 6\(n\), with \(n\) as the number of H\({}_{2}\) molecules in the hydrate cage. A second obstacle hinges on the belief, widely held until recently, that molecular systems featuring both high-frequency intramolecular vibrational mode(s) and low-frequency intermolecular ones would present a hugely difficult and computationally prohibitive task; this centers on H\({}_{2}\) in hydrate cavities and hydrogen-bonded/van der Waals (vdW) complexes, with rigorous computation of fundamental (and overtone) excitation(s) of their intramolecular vibrational mode(s). The principal source of this challenge was thought to be the large number of highly excited intermolecular vibrational eigenstates in the manifold of intramolecular ground states located below the energy of the intramolecular vibrational excitation(s), together with assumptions that convergence is required to afford their accurate calculation. However, recently [51] it was shown, contrary to this expectation, that accurate calculation of the small-cage-bound H-H stretch fundamental in sII hydrate at \(\sim\)4100 cm\({}^{-1}\) needs only a modest number of converged TR states in the v = 0 manifold up to no more than 400-450 cm\({}^{-1}\) above the ground state, and certainly none within several thousand wave-numbers of the intramolecular fundamental. This unexpected conclusion was rationalized by very weak coupling of excited intermolecular v = 0 TR states in the locale of the H\({}_{2}\) stretch fundamental, despite their large number. Therefore, excluding these in calculations affects the accuracy of calculating intramolecular vibrational excitation in a negligible manner. Naturally, this resulted in a substantial drop in the basis size needed and converted a computationally Herculean project into a readily tractable undertaking [51]. For the biggest (rigid) clathrate domain modelled, featuring 76 water molecules (i.e., constituting the first three hydration shells around H\({}_{2}\)), the mode change of \(-\)43 cm\({}^{-1}\) for H\({}_{2}\) is just 14% away from the experimental value at 20 K.  
The above insight is at the heart of the rigorous and efficient methodology allowing full-dimensional quantum computations, with full coupling, of the intra- and intermolecular rovibrational states of dimers of molecules featuring weak intermolecular bonding [52]. This methodology was utilized to calculate the H-H stretch fundamentals of (p-H\({}_{2}\))\({}_{2}\) inside the 51264 cavity [53]. The 8D quantum methodology is predicated on rotationless (j = 0) hydrogen moieties and a frozen host lattice, but coupled fully in every other sense. The largest hydrate domain considered had 98 H\({}_{2}\)O molecules, all those featuring O atoms within 9 A of the center of the cage. In this case, both H-H stretch fundamental modes were observed to be site-split by \(\sim\)0.5 cm\({}^{-1}\) and shifted by approximately 24 cm\({}^{-1}\), i.e., in good accord with the experimental result of 26 cm\({}^{-1}\).