phonons were calculated for 490 wave-vectors spaced randomly throughout the Brillouin zone for the production of the vibrational density of states. This level of sampling of wave vectors proved to be sufficient to achieve convergence of phonon properties.

Lattice dynamics calculations and thermal expansion
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The role of individual phonons in thermal expansion is represented by the mode Grüneisen parameter $\gamma_{\mathbf{k},i}$ [@Barrera_2005_review], which quantifies the change in phonon frequency with volume through the relation $$\gamma_{\mathbf{k},i} = -\frac{\partial \ln \omega_{\mathbf{k},i}}{\partial \ln V}$$ where $V$ is the crystal volume and $\omega_{\mathbf{k},i}$ is the angular frequency of the phonon of branch $i$ and wave vector $\mathbf{k}$.

Within the quasi-harmonic approximation (the approximation in which all temperature dependence of phonon frequencies occurs indirectly through changes in force constants arising from changes in volume) the coefficient of volumetric thermal expansion $\alpha_V$ is given as $$\alpha_V = \frac{\partial \ln V}{\partial T} = \frac{1}{BV} \sum_{\mathbf{k},i} c_{\mathbf{k},i} \gamma_{\mathbf{k},i}$$ where $$c_{\mathbf{k},i} = \hbar \omega_{\mathbf{k},i} \frac{\partial n(T,\omega_{\mathbf{k},i})}{\partial T}$$ $c_{\mathbf{k},i}$ is the contribution of the individual phonon mode to the overall heat capacity, $T$ is the temperature, $B$ is the bulk modulus, $\hbar$ is the reduced Planck’s constant, and $n(T,\omega)$ is the Bose–Einstein distribution for phonons. A negative value of $\gamma_{\mathbf{k},i}$ 