WO$_4$ tetrahedra. Our flexibility analysis reported here has found that the NTE phonons in this system involve motion of rigid Y–O and W–O rods as per the tension effect enabled by relatively stiff cation-oxygen bonds.

However, the tension effect appears to act here as part of a more complex process. In the 0–5 THz range (which contributes strongly to NTE) the WO$_4$ tetrahedra remain effectively rigid with minimal O–W–O bond angle bending while the YO$_6$ octahedra undergo significant O–Y–O bond angle deformation. In the 9–10 THz range (which contributes less to NTE) both O–W–O and O–Y–O bond angles undergo non-trivial deformation although the O–Y–O angles actually deform less than the O–W–O. All PTE regions involve non-trivial levels of Y–O and W–O bond stretching.

As noted earlier, the RUM model is only applicable to systems with stiff coordination polyhedra and is therefore not applicable to Y$_2$W$_3$O$_{12}$. The results of this study show that, in Y$_2$W$_3$O$_{12}$, such very low frequency deformations are made possible by the relatively small energy cost of O–Y–O bond angle bending in YO$_6$ units. The combination of the flexibility of the O–Y–O bond angle and the existence of effectively-rigid WO$_4$ tetrahedra and Y–O rods is such that macroscopic 