deletions | additions       diff --git a/Sternheimer2.tex b/Sternheimer2.tex  index 8e63b0b..f48ca1b 100644  --- a/Sternheimer2.tex  +++ b/Sternheimer2.tex 
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The Born effective charges can be computed from the response of the dipole moment to ionic displacement:  \begin{align} \label{eq:Born}  Z^{*}_{i j \alpha} = -\frac{\partial^2 E}{\partial \mathcal{E}_i \partial R_{j \alpha}}  = \frac{\partial \mu_i}{\partial R_{j \alpha}} = Z_{\alpha} \delta_{ij} + -  \sum_n^{\rm occ} \left< \varphi_n \left| r_i \right| \frac{\partial \varphi_n}{\partial R_{j \alpha}} \right> \end{align}  The intensities for each mode for absorption of radiation polarized in direction $i$, which can be used to predict infrared spectra, are calculated by multiplying by the normal mode eigenvector $x$  \begin{align} 
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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} \\ \nonumber  = Z_\alpha \delta_ij} + -  \sum_n^{\rm occ} \left[ \left< \varphi_n \left| \frac{\partial V_{\alpha}}{\partial R_{i \alpha}} r_i}  \right| \frac{\partial \varphi_n}{\partial \mathcal{E}_j} \right> + {\rm cc.} \right] \end{align}  This expression can be evaluated with the same approach as for the dynamical matrix elements, and  is easily generalized to non-zero frequency too. We can also make the previous expression Eq. \ref{eq:Born} for Born charges from the vibrational perturbation usable in a periodic system with the replacement $\vec{r} \varphi \rightarrow -i \partial \varphi/\partial k$.  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: