of negative $\gamma_{\mathbf{k},i}$ values in terms of the flexibility of the structure. We therefore constructed a number of flexibility models in which we enforced a high degree of rigidity for certain bonds or structural units, whilst maintaining force constants of zero value for other bond stretching or bending motions. This meant that any phonon calculated for a given flexibility model would have zero or near-zero frequency if that phonon did not violate the constraints of that model, and high frequency otherwise.

The objective was to compare eigenvectors of phonons from the *ab initio* lattice dynamics calculations with eigenvectors of those zero or near-zero frequency modes calculated for the constructed flexibility models. This allowed us to pinpoint different types of atomic-scale motion in the highly complex *ab initio* dispersion curves (including those regions which drive NTE) with relative ease. The flexibility models considered are best described as combinations of separate components that represent different structural units. Hence, the YO$_6$ units were represented by one of

1.  Rigid Y–O rods

2.  Rigid YO$_6$ octahedra

while the WO$_4$ units were represented by one of

1.  Rigid W–O rods

2.  Rigid WO$_4$ tetrahedra

The construction and application of these models is discussed in \[flexibility\_models\].

