deletions | additions       diff --git a/Sternheimer2.tex b/Sternheimer2.tex  index 7632f53..8e63b0b 100644  --- a/Sternheimer2.tex  +++ b/Sternheimer2.tex 
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%  \begin{multline}  D_{i \alpha, j \beta} = \frac{\partial^2 E}{\partial R_{i \alpha} \partial R_{j \beta}} = D_{i \alpha, j \beta}^{\rm ion-ion} \\  + \sum_n^{\rm occ} \left[ \left< \varphi_k \varphi_n  \left| \frac{\partial V_{\alpha}}{\partial r_i} \right| \frac{\partial \varphi_n}{\partial R_{j \beta}} \right> + {\rm cc.} + \delta_{\alpha \beta} \left< \varphi_n \left| \frac{\partial^2 V_{\alpha}}{\partial r_i \partial r_j} \right| \varphi_n \right> \right] \label{eq:dynmatrix}  \end{multline}  % 
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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}} \right| \frac{\partial \varphi_n}{\partial \mathcal{E}_j} \right> + {\rm cc.} \right]  \end{align}  This expression is easily generalized to non-zero frequency too.  We can also make the above 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:  \begin{align}