S and T). Acoustic modes also contribute to NTE, as is the case in some other systems recently studied [@Leila_Cu2O; @Hong_Zn(CN)2; @Leila_MOF5]. We note that recent *ab initio* calculations of the mode Grüneisen parameters at the $\Gamma$ point in Y$_2$Mo$_3$O$_{12}$ using the frozen-phonon method also show a large number of modes with negative values, but see smaller values except for the lowest-frequency mode [@Wang_Y2Mo3O12].

A second group of strongly-NTE phonons exists around 1.8 THz but, more generally, it can be seen that all the phonons up to around 5 THz have negative $\gamma_{\mathbf{k},i}$ (albeit approaching zero values as the frequency increases to 5 THz). There is also a higher frequency band of weak NTE modes around 9–10 THz $\gamma_{\mathbf{k},i}\sim0$. Weak, low frequency, PTE modes exist around 5–8 THz and for frequencies larger than 10 THz.

Flexibility model mapping
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![Flexibility analysis of Y$_2$W$_3$O$_{12}$ phonon dispersion curves and densities of states. All flexibility models shown here describe YO$_6$ units in terms of corner-sharing rigid YO$_6$ octahedra. The notation of Bradley and Cracknell denotes special Brillouin zone points [@Bradley_Cracknell_1972_CrystalBZSymmetryBook]. All data are shaded according to the value of $m_{\mathbf{k},i}$ at each mode for each wave vector. The shading ranges from white ($m_{\mathbf{k},i}=0$) 