deletions | additions       diff --git a/forces2.tex b/forces2.tex  index 9e3b252..289c6e1 100644  --- a/forces2.tex  +++ b/forces2.tex 
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This alternative formulation of the forces can be extended to obtain the second-order derivatives of the energy with respect to the atomic displacements~\cite{Andrade2010thesis}, which are required to calculate vibrational properties as discussed in section~\ref{sec:sternheimer}. In general, the perturbation operator associated with a an  ionic displacement can be written as \begin{equation}  \label{eq:ionicpertmod} 
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%  \begin{multline}  \left< \varphi_n \left| \frac{\partial \hat{v}_{\alpha}}{\partial R_{i\alpha}} \right| \frac{\partial \varphi_n}{\partial R_{j \beta}} \right> = -\left[  \left< \varphi_n \left| \hat{v}_{\alpha} \right| \frac{\partial^2 \varphi_n}{\partial R_{j \beta} \partial r_i} \right> \right.\\ + \left. \left< \frac{\partial \varphi_n}{\partial r_i}\left| \hat{v}_{\alpha} \right| \frac{\partial \varphi_n}{\partial R_{j \beta}} \right>\right] + {\rm cc.}\ c.c.}\  , \end{multline}  and  \begin{multline}   \left< \varphi_n \left| \frac{\partial^2 \hat{v}_{\alpha}}{\partial R_{i\alpha} \partial R_{j\alpha}} \right| \varphi_n \right> =  \left[ \left< \frac{\partial^2 \varphi_n}{\partial r_i \partial r_j} \left| \hat{v}_{\alpha} \right| \varphi_n \right>\right.\\ +   \left. \left< \frac{\partial \varphi_n}{\partial r_i} \left| \hat{v}_{\alpha}\right| \frac{\partial \varphi_n}{\partial r_j} \right>\right] + {\rm cc.}\ c.c.}\  . \end{multline}