deletions | additions       diff --git a/Sternheimer2.tex b/Sternheimer2.tex  index c551c72..f43ee17 100644  --- a/Sternheimer2.tex  +++ b/Sternheimer2.tex 
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The Sternheimer equation can be used in conjunction with $\vec{k} \cdot \vec{p}$ perturbation theory~\cite{Cardona_1966} to obtain  band velocities and effective masses, as well as to apply electric fields via the quantum theory of polarization. In this case the perturbation is a displacement in the \(k\)-point. Using the effective Hamiltonian for the \(k\)-point  \begin{equation}  H_{\vec{k}} = e^{-i \vec{k} \cdot \vec{r}} H e^{-i e^{i  \vec{k} \cdot \vec{r}} \end{equation}  the perturbation is represented by the operator  % 
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Unfortunately the $\vec{k} \cdot \vec{p}$ perturbation is not usable to calculate the polarization \cite{Resta_2007}, and a sum over strings of k-points on a finer grid is required. We have implemented the special case of a $\Gamma$-point calculation for a large supercell, where the single-point Berry phase can be used \cite{Yaschenko1998}. For cell sizes $L_i$ in each direction, the dipole moment is derived from the determinant of a matrix whose basis is the occupied KS orbitals:  \begin{align}  \mu_i = - \frac{e L_i}/{2 L_i}{2  \pi} \mathcal{I} {\rm ln}\ {\rm det}\ \left< \varphi_n \left| exp(- \exp(-  2 \pi i x_i/L_i) \right| \varphi_m \right> \end{align}  % magnetic is non-self-consistent if wfns are real.