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Unfortunately the $\vec{k} \cdot \vec{p}$ perturbation is not usable to calculate the polarization \cite{Resta_2007}, and a sum over strings of k-points on a finer grid is required. We have implemented the special case of a $\Gamma$-point calculation for a large supercell, where the single-point Berry phase can be used \cite{Yaschenko1998}. For cell sizes $L_i$ in each direction, the dipole moment is derived from the determinant of a matrix whose basis is the occupied KS orbitals:  \begin{align}  \mu_i = - \frac{e L_i}{2 \pi} \mathcal{I} \mathcal{I}\  {\rm ln}\ {\rm det}\ \left< \varphi_n \left| \exp(- e^{-  2 \pi i x_i/L_i) x_i/L_i}  \right| \varphi_m \right> \end{align}  % magnetic is non-self-consistent if wfns are real.