deletions | additions       diff --git a/Sternheimer2.tex b/Sternheimer2.tex  index 8fe111d..f5ba42a 100644  --- a/Sternheimer2.tex  +++ b/Sternheimer2.tex 
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We can solve for linear response to various different perturbations. The most straight-forward case is the response of a finite system to an electric field \(\mathcal{E}_{i,\omega}\) with frequency \(\omega\) in the direction \(i\) , where the perturbation operator is \(\delta \hat v = \hat r_i\)~\cite{Andrade_2007}. In this case the polarizability can be calculated as  %  \begin{align}  \alpha_{ij} \left( \omega \right) = - \sum_n f_n \left[ \langle \varphi_n \vert r_i \vert \frac{\partial \varphi_n}{\partial \mathcal{E}_{j, \omega}} \rangle + \langle \frac{\partial \varphi_n}{\partial \varphi_{n}}{\partial  \mathcal{E}_{j, -\omega}} \vert r_i \vert \varphi_n \rangle \right] \label{eq:sternheimer_polarizability}\ . \end{align}  %  The calculations of the polarizability yield optical response properties~\cite{Andrade_2007,Vila_2010} and, for imaginary frequencies, van der Waals coefficients~\cite{Botti_2008}. 
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%  \begin{multline}  m^{-1}_{ijn\vec{k}} = \frac{\partial^2 \epsilon_{n\vec{k}}}{\partial k_i \partial k_j}  = \delta_{ij} + \left< \varphi_{nk} \varphi_{n\vec{k}}  \left| \frac{\partial \hat{H}}{\partial \vec{k}} \right| \frac{\partial \varphi_{n\vec{k}}{\partial \varphi_{n\vec{k}}}{\partial  \vec{k}} \right> + \mathrm{cc.} %\\  %+ \\  +  \left< \varphi_{n\vec{k}} \left| \left[ \hat{\vec{r}}, \left[ \hat{\vec{r}}, \hat{v}_{\alpha} \right] \right] \right| \varphi_{n\vec{k}} \right>\ . \end{multline}  The $\vec{k} \cdot \vec{p}$ wavefunctions can be used to compute response to electric fields in periodic systems.