deletions | additions       diff --git a/Sternheimer.tex b/Sternheimer.tex  index 284ec37..d4779f5 100644  --- a/Sternheimer.tex  +++ b/Sternheimer.tex 
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\begin{align}  I_i = \sum_{j \alpha} Z^{*}_{ij \alpha} y_{j \alpha}  \end{align}  These intensities can be used to plot the predict  infrared spectrum. spectra.  The Born charges must obey the acoustic sum rule, from translational invariance:  \begin{align} 
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\end{align}  The discrepancy arises from the same causes as the non-zero translational and rotational modes.  The Sternheimer equation can be used in conjunction with $\vec{k} \cdot \vec{p}$ perturbation theory theory~\cite{Cardona_1966}  to obtain band velocities and effective masses, as well as to apply electric fields via the quantum theory of polarization.  We consider the effective Hamiltonian for the k-point. \(\vec{k}\)-point.  % FIXME: use of k for bands interferes with k-points here!!  We compute band group velocities in a periodic system from 
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we use dH/d(ik) so real-> real.  adding in occupied contributions  Inverse effective mass tensors can be calculated by solving the Sternheimer equation with the perturbation. perturbation  \begin{align}  m^{-1}_{ijnk} = \frac{1}{\hbar^2} \frac{\partial^2 \epsilon_{nk}}{\partial k_i \partial k_j}  = \delta_{ij} + \left< \varphi_{nk} \left| dH/dk \right| d \varphi_{nk}/dk \right> + {\rm cc.}  + \left< \varphi_{nk} \left| \left[ \vec{r}, \left[ \vec{r}, V_{\alpha} \right] \right] \right| \varphi_{nk} \right> \right>\ .  \end{align}  cite M Cardona and FH Pollak, Phys. Rev. 142, 530-543 (1966).  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 $\vec{x}$.