H ₂ , HD, and D ₂ in the small cage of structure II clathrate hydrate: Vibrational frequency shifts from fully coupled quantum six-dimensional calculations of the vibration-translation-rotation eigenstates

We report the first fully coupled quantum six-dimensional (6D) bound-state calculations of the vibration-translation-rotation eigenstates of a flexible H ₂ , HD, and D ₂ molecule confined inside the small cage of the structure II clathrate hydrate embedded in larger hydrate domains with up to 76 H ₂ O molecules, treated as rigid. Our calculations use a pairwise-additive 6D intermolecular potential energy surface for H ₂ in the hydrate domain, based on an ab initio 6D H ₂ -H ₂ O pair potential for flexible H ₂ and rigid H ₂ O. They extend to the first excited (v = 1) vibrational state of H ₂ , along with two isotopologues, providing a direct computation of vibrational frequency shifts. We show that obtaining a converged v = 1 vibrational state of the caged molecule does not require converging the very large number of intermolecular translation-rotation states belonging to the v = 0 manifold up to the energy of the intramolecular stretch fundamental (≈4100 cm ⁻¹ for H ₂ ). Only a relatively modest-size basis for the intermolecular degrees of freedom is needed to accurately describe the vibrational averaging over the delocalized wave function of the quantum ground state of the system. For the caged H ₂ , our computed fundamental translational excitations, rotational j = 0 → 1 transitions, and frequency shifts of the stretch fundamental are in excellent agreement with recent quantum 5D (rigid H ₂ ) results [A. Powers et al., J. Chem. Phys. 148, 144304 (2018)]. Our computed frequency shift of −43 cm ⁻¹ for H ₂ is only 14% away from the experimental value at 20 K.