deletions | additions       diff --git a/Forces and geometry optimization.tex b/Forces and geometry optimization.tex  index b6f4cb0..2020394 100644  --- a/Forces and geometry optimization.tex  +++ b/Forces and geometry optimization.tex 
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gives forces with a considerable oscillation due to the grid, using  the derivative of the orbitals gives a force that is consirably smoother.  This alternative formulation of the forces can be extended to higher orders~\cite{Andrade2010thesis}, to obtain the second order derivatives of the energy with respect to the atomic displacements that are required to calculate vibrational properties as discussed in section~\ref{sec:sternheimer}. In this case the perturbation operator becomes  \begin{equation}  \label{eq:ionicpertmod}  \frac{\partial V_\alpha(\vec{r}-\vec{R}_\alpha)}{\partial R_i\alpha} =   V_{\alpha}(\vec{r}-\vec{R}_\alpha)\frac{\partial}{\partial r_i}  -\frac{\partial}{\partial  r_i}V_{\alpha}(\vec{r}-\vec{R}_\alpha) r_i}V_{\alpha}(\vec{r}-\vec{R}_\alpha)\ .  \end{equation}