stretching of W–O and Y–O bonds). This observation is a good quantification of the tension effect giving rise to NTE in Y$_2$W$_3$O$_{12}$.

The addition of the constraint that keeps the Y…W distance fixed yields similar results. This flexibility model has 60 excess degrees of freedom (reduced from the original 96; see \[maxwell\_counting\]). shows that the same eigenvector mixing that was observed previously is still present, and that the reduced flexibility of this model manifests itself in a reduction in the match with the *ab initio* phonons (seen in the lighter greyscale shading in ) across the whole phonon spectrum. We can conclude that the eigenvectors of all phonons with frequencies in the range 0–12 THz in Y$_2$W$_3$O$_{12}$ involve some changes in Y…W distance.

### Rigid WO$_4$ tetrahedra and Y–O bonds.

The analysis of \[maxwell\_counting\] shows that the flexibility model for Y$_2$W$_3$O$_{12}$ consisting of rigid WO$_4$ tetrahedra and Y–O rods has 48 excess degrees of freedom. This model is still highly flexible, albeit less flexible than the two rigid rod models discussed above.

shows, once again, that a high degree of eigenvector mixing occurs for this system with each flexibility model phonon spread across multiple *ab initio* phonons. It can also 