likely to require higher energies due to the electrostatic repulsion between oxygens at the polyhedral vertices or to the high sensitivity of d-orbital bonding around a metal coordination centre with respect to bonding geometry. Therefore, any tension effect that avoids or minimises deformation of those polyhedra is more likely to be acting at a low enough frequency to drive macroscopic NTE.

The conditions outlined above describe a *Rigid Unit Mode* (RUM) [@Pryde_1996; @Giddy_RUMs; @Hammonds_RUMs; @Dove_RUMs]: a low frequency phonon with rotations and translations of effectively rigid coordination polyhedra. The RUM model can be used to explain why structures with stiff coordination polyhedra have tension effect vibrations with sufficiently large amplitude to drive NTE [@NTE_theory_Welche]. In cases such as ScF$_3$ [@ScF3_phonons], Zn(CN)$_2$ [@Hong_Zn(CN)2] and Cu$_2$O [@Leila_Cu2O], lattice dynamics calculations have shown that the phonons responsible for NTE are RUMs involving motion of ScF$_6$ octahedra, Zn(C/N)$_4$ tetrahedra and CuO$_2$ rods respectively moving as rigid units. The case of ZrW$_2$O$_8$ is one that we will consider in a separate publication but, as a prelude to this, we consider here the case of Y$_2$W$_3$O$_{12}$—a material which, intriguingly, *cannot* support RUMs [@Hammonds_octahedra] but *does* exhibit NTE [@Forster_Sleight_1999_IntJInorgMat_Y2W3O12NTEExpt; @Sumithra_2005].

![The Y$_2$W$_3$O$_{12}$ structure [@Forster_Sleight_1999_IntJInorgMat_Y2W3O12NTEExpt]. (a) Shows the 