deletions | additions       diff --git a/Forces and geometry optimization.tex b/Forces and geometry optimization.tex  index f5458a8..f512c20 100644  --- a/Forces and geometry optimization.tex  +++ b/Forces and geometry optimization.tex 
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\vec{R}_\alpha)\ .  \end{equation}  %  %Using Using  this expression, the terms of the dynamical matrix, eq.~\ref{eq  %The matrix elements of the perturbation eq.~\ref{eq:dynmatrix}  arebest calculated by differentiating the wavefunctions, as discussed for the forces in section \ref{sec:forces}. % FIXME  %\begin{align}  % \begin{align}  \left< \varphi_n \left| \frac{\partial v_{\alpha}}{\partial %r_{i\alpha}} r_{i\alpha}}  \right| \frac{\partial \varphi_n}{\partial R_{j \beta}} %\right> \right>  =%  \left< \varphi_n \left| V_{\alpha} \right| \frac{\partial^2 \varphi_n}{\partial R_{j \beta} \partial x_i} \right> \\%  \left< \varphi_n \left| \frac{\partial^2 V_{\alpha}}{\partial x_i \partial x_j} \right| \varphi_n \right> =%  \left< \frac{\partial^2 \varphi_n}{\partial x_i \partial x_j} \left| V_{\alpha} \right| \varphi_n \right> + {\rm cc.} +%  \left< \frac{\partial \varphi_n}{\partial x_i} \left| V_{\alpha} %\right| V_{\alpha}\right|  \frac{\partial \varphi_n}{\partial x_j} \right> + {\rm cc.} %\end{align} \end{align}