deletions | additions       diff --git a/Sternheimer2.tex b/Sternheimer2.tex  index f48ca1b..b15f239 100644  --- a/Sternheimer2.tex  +++ b/Sternheimer2.tex 
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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 normal-mode  eigenvector $x$ \begin{align}  I_i = \sum_{j \alpha} Z^{*}_{ij \alpha} x_{j \alpha}\ .  \end{align}