Dynamical matrix

Dnamical matrix at a q-point is constructed by supercell force constants and is written as,

$$D_{\alpha\beta}(j j',\mathbf{k}) = \frac{1}{(m_j m_{j'})^{1/2}} \... ...\ ll' \\ \end{pmatrix} \exp(i\mathbf{k}\cdot[\mathbf{r}(j'l')-\mathbf{r}(j0)]).$$ (3.9)

An element of a dynamical matrix is calculated by the sum of force constants multiplied by phase factor which is a vector product of the q-point and a vector between focused atoms. $l'$ runs through all unitcells contained in a supercell. In output of fropho, a column ($j$) of a dynamical matrix corresponds to a displaced atom number and a row ($j'$) of that corresponds to an atom number where force is affected. If an atom of a force constant is on a rim of the supercell, the force constant is divided by the multiplicty which the atom is shared by the neighboring supercells.