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\epsilon_{ij} = \delta_{ij} + \frac{4 \pi}{V} \alpha_{ij}  \end{align}  where $V$ is the volume of the unit cell.  We can compute the Born charges from the electric-field response in either finite or periodic systems (as a complementary approach to using the vibrational response):  \begin{align}  Z^{*}_{i j \alpha} = -\frac{\partial^2 E}{\partial \mathcal{E}_i \partial R_{j \alpha}} = \frac{\partial F_{j \alpha}}{\partial \mathcal{E}_i}  =   \end{align}  single-point %We can compute the Born charges from the electric-field response in either finite or periodic systems (as a complementary approach to using the vibrational response):  %\begin{align}  % Z^{*}_{i j \alpha} = -\frac{\partial^2 E}{\partial \mathcal{E}_i \partial R_{j \alpha}} = \frac{\partial F_{j \alpha}}{\partial \mathcal{E}_i}  % =   %\end{align}  %single-point  Berry phase for polarization. 'symmetrization' %'symmetrization'  of $\epsilon$ and Born charge tensors solvers: %solvers:  cg, bicgstab, qmr. when applicable. % ought to discuss dichroism?  % Arto's stuff  % magnetic is non-self-consistent if wfns are real.  cite %cite  my chapter on linear response