Phonon frequencies

The central quantity in the calculation of the phonon frequencies is the force-constant matrix $\Phi_{i s \alpha , j t \beta}$, since the frequencies at wavevector ${\bf k}$ are the eigenvalues of the dynamical matrix $D_{s \alpha , t \beta}$, defined as:

$$D_{s \alpha , t \beta} ( {\bf k} ) = \frac{1}{\sqrt{M_s M_t}}... ...R}_j^0 + {\bf\tau}_t - {\bf R}_i^0 - {\bf\tau}_s) \right] \; .$$ (1)

where ${\bf R}_i^0$ is a vector of the lattice connecting different primitive cells and ${\bf\tau}_s$ is the position of the atom s in the primitive cell. If we have the complete force-constant matrix, then $D_{s \alpha , t \beta}$ and hence the frequencies $\omega_{{\rm k} s}$ can be obtained at any ${\bf k}$. In principle, the elements of $\Phi_{i s \alpha , j t \beta}$ are non-zero for arbitrarily large separations $\mid {\bf R}_j^0 + {\bf\tau}_t - {\bf R}_i^0 - {\bf \tau}_s \mid$, but in practice they decay rapidly with separation, so that a key issue in achieving our target precision is the cut-off distance beyond which the elements can be neglected.