deletions | additions       diff --git a/Sternheimer.tex b/Sternheimer.tex  index 0ec0179..d8b15b0 100644  --- a/Sternheimer.tex  +++ b/Sternheimer.tex 
...
\end{align}  The $\vec{k} \cdot \vec{p}$ wavefunctions can be used to compute response to electric fields in periodic systems.  In finite systems, a homogeneous electric field can be represented simply via the position operator $\vec{x}$. $\vec{r}$.  However, this operator is not well defined in a periodic system and cannot be used. According to the quantum theory of polarization,  the solution is to replace $\vec{r} \varphi$ with $-i d/dk \varphi$. % cite Baroni, also Gonze papers  The response to this perturbation can be used to compute the polarizability using  from eq.~\cite{eq:sternheimer_polarizability} eq.~\ref{eq:sternheimer_polarizability}  \begin{align}  \alpha_{ij} \left( \omega \right) = i \sum_k f_k \left[ \left. \left< \frac{\partial \varphi_k}{\partial k_i} \right| \frac{\partial \varphi_k}{\partial \mathcal{E}_{j, \omega}} \right> + \left. \left< \frac{\partial \varphi_k}{\partial \mathcal{E}_{j, -\omega}} \right| \frac{\partial \varphi_k}{\partial k_j} \right> \right]  \end{align}