deletions | additions       diff --git a/Forces and geometry optimization.tex b/Forces and geometry optimization.tex  index 71b0b60..6eff477 100644  --- a/Forces and geometry optimization.tex  +++ b/Forces and geometry optimization.tex 
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% \vec{R}_\ia)\frac{\partial \phi_{\io}(\vec{r})}{\partial \vec{r}}  \right]\ .  \end{equation}  %  As we show in section \ref{sec:numerical:forces}, this last form has  advantages for the numerical implementation and for the precision of  the result.  This previous transformation has two advantages for numerical implementations:  First, it is not necessary to implement the derivative of the ionic