We report the first fully coupled quantum six-dimensional (6D) bound-state calculations of the vibration-translation-rotation (VTR) eigenstates of a flexible H 2 , HD, and D 2 molecule confined inside the small cage of the sII 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 (PES) 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 TR 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.,