deletions | additions       diff --git a/forces2.tex b/forces2.tex  index aa160e5..ec0650a 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 second-order  derivatives of the energy with respect to the atomic displacements~\cite{Andrade2010thesis}, that which  are required to calculate vibrational properties as discussed in section~\ref{sec:sternheimer}. In general, the perturbation operator associated with a ionic displacement can be written as \begin{equation}  \label{eq:ionicpertmod}