bonds is typically much higher than transverse vibrations of those same bonds. For example, we could label two cations **M** and **M$^\prime$** that form a **M**–O–**M$^\prime$** linkage via a bridging oxygen atom. If the **M**–O and **M$^\prime$**–O bonds are very stiff, transverse displacements of the O atom will pull the **M** and **M$^\prime$** cations towards each other. Since the amplitude of a transverse displacement will increase on heating, this leads to negative thermal expansion. This mechanism is called the *tension effect* [@Barrera_2005_review].

The drive towards positive thermal expansion exists in all materials, including those with overall NTE. Therefore, for a material to show NTE, the amplitude of the oxygen displacement must be sufficiently large to outweigh those mechanisms that drive positive thermal expansion. Given that the amplitude of a phonon is inversely proportional to the square of its frequency, an NTE-driving phonon must also have low frequency to achieve this condition.

All of the NTE-exhibiting material families mentioned above share one significant feature: their crystal structures can all be described in terms of an infinite three-dimensional network of corner-sharing structural polyhedra (such as SiO$_4$, WO$_4$, OCu$_4$ and ZnN$_4$ tetrahedra, and ScF$_6$, ZrO$_6$, and YO$_6$ octahedra). Deformation of coordination polyhedra is 