through to black ($m_{\mathbf{k},i}=1$). Bins that make up the density of states are shaded according to the average $m_{\mathbf{k},i}$ for each bin using the same colour scale.[]{data-label="Y2W3O12Flexibility"}](figure_3.pdf){width="\textwidth"}

shows six plots of the 156 Y$_2$W$_3$O$_{12}$ *ab initio* dispersion curves for frequencies between 0–12 THz (as in ) together with their corresponding densities of states. In each case the data are shaded in greyscale according to the degree to which one of the six flexibility models can reproduce the eigenvectors of each phonon in the *ab initio* calculations. Black shading indicates a perfect match between the *ab inito* phonons and the flexibility model in question; white shading indicates no relation between the two. This approach offers a convenient, visual sense of the atomic motions associated with different regions of the complex Y$_2$W$_3$O$_{12}$ phonon spectrum. Further details on how these ‘match’ values were calculated are given in \[flexibility\_models\].

In what follows we use the information presented in and to correlate the NTE phonons identified in with the vibrations permitted by the six flexibility models.

### Rigid W–O and Y–O rods with no rigid polyhedra.

As discussed in \[maxwell\_counting\], a flexibility model of Y$_2$W$_3$O$_{12}$ consisting of rigid W–O and Y–O rods has 96 excess 