indicates that the phonon $i$ at wave vector $\mathbf{k}$ contributes to negative thermal expansion and, if a sufficient number of modes with a large enough negative value of $\gamma_{\mathbf{k},i}$ exist, the overall outcome for the system is a negative value of $\alpha_V$, that is, NTE.

Application of lattice dynamics and Grüneisen theory
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Since Y$_2$W$_3$O$_{12}$ has an orthorhombic lattice, changes in cell size ($\Delta V$) were achieved through variable-cell geometry optimisation at an applied hydrostatic pressure. Testing showed that an applied pressure of $-0.06$ GPa resulted in the desired cell volume change of approximately 0.1%. The structure was optimised at both this pressure and zero pressure, with phonon frequencies also calculated at both pressures for a set of random wave vectors and for wave vectors along the key symmetry directions in reciprocal space.

In order to calculate the change in frequency for a particular mode with change in volume, $\Delta \omega_{\mathbf{k},i}$, we used an eigenvector matching algorithm to ensure that we calculated $\omega_{\mathbf{k},i}$ for equivalent modes (\[draw\]). From these results we generated the set of mode Grüneisen parameters numerically as $\gamma_{\mathbf{k},i} =  (\Delta \omega_{\mathbf{k},i} / \omega_{\mathbf{k},i}) / (\Delta V/V)$.

Flexibility models {#y2w3o12flexmodelsection}
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Our approach was to identify the origin 