parametrised interatomic potential calculations [@Sumithra_etal_2007_PRB_Y2W3O12NTEPhononsMechanism] have suggested possible NTE phonons between 1 and 2.5 THz that may correspond to the tension effect. However, this model was only fitted to the structure (sensitive to the first-order differentials of the interatomic potential) and not to any property that will give information about the important second-order differentials (essential for vibrational frequencies). Moreover, the model includes interactions that depend on the Y–O–W bonds but nothing dependent on the more useful O–Y–O or O–W–O bonds.

In this paper we investigate exactly where in the Y$_2$W$_3$O$_{12}$ phonon spectrum the structure flexibility manifests itself and, since the RUM model is not applicable, how NTE phonons exist in enough of the spectrum (and at low enough frequencies) so as to give rise to an overall negative coefficient of thermal expansion. To do this, we use an approach similar to that used in our studies of Zn(CN)$_2$ [@Hong_Zn(CN)2], Cu$_2$O [@Leila_Cu2O] and MOF-5 [@Leila_MOF5], that is we perform *ab initio* calculations of the material’s phonon spectrum that are then compared to models of structural flexibility. Lattice dynamics calculations have recently been reported for the related material Y$_2$Mo$_2$O$_{12}$ [@Wang_Y2Mo3O12; @Romao_Y2Mo3O12] which, like Y$_2$W$_3$O$_{12}$, also shows NTE along all three axes [@Marinkovic_Y2Mo3O12]. 