deletions | additions       diff --git a/Sternheimer.tex b/Sternheimer.tex  index 2dd9d50..e95da37 100644  --- a/Sternheimer.tex  +++ b/Sternheimer.tex 
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We can solve for linear response to various different perturbations. To compute vibrational frequencies,  we calculate response to the ionic displacement perturbation  $\partial H/\partial R_{i \alpha} = \partial V_{\alpha}/\partial x_i$, for each direction $i$ and atom $\alpha$.  % FIXME: make clear this is a non-local operator in general  currently only implemented in finite case  The elements of the dynamical matrix are  \begin{align}  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_k \left[ f_k \left< \varphi_k \left| \frac{\partial V_{\alpha}}{\partial x_i} \right| \frac{\partial \varphi_k}{\partial R_{j \beta}} \right> + {\rm cc.}  + \delta_{\alpha \beta} \left< \varphi_k \left| \frac{\partial^2 V_{\alpha}}{\partial x_i \partial x_j} \right| \varphi_k \right> \right]  \end{align}  The ion-ion term is  \begin{align}  D_{i \alpha, j \beta}^{\rm ion-ion} =  \begin{cases} 
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-Z_\alpha Z_\beta \left[ \frac{\delta_{ij}}{\left|R_\alpha - R_\beta \right|^3} - 3 \frac{\left( R_{i \alpha} - R_{i \beta} \right)}{\left|R_\alpha - R_\gamma \right|^4} \right] & \textrm{if}\ \alpha \ne \beta  \end{cases}  \end{align}  where $Z_\alpha$ is the ionic charge of atom $\alpha$.  The matrix elements of the perturbation are best calculated by differentiating the wavefunctions, as discussed for the forces in section XXX. % FIXME  \begin{align}  \left< \varphi_k \left| \frac{\partial V_{\alpha}}{\partial x_i} \right| \frac{\partial \varphi_k}{\partial R_{j \beta}} \right> =