deletions | additions       diff --git a/The_simulations_of_the_collisions__.tex b/The_simulations_of_the_collisions__.tex  index 69e6e95..fd3f707 100644  --- a/The_simulations_of_the_collisions__.tex  +++ b/The_simulations_of_the_collisions__.tex 
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Following the scheme of Ref.~\cite{Schleife_2012} the TDKS equation (see Eq. \ref{eq:tdks1}) was solved numerically   with a time step of, at most, $0.121~\mathrm{attoseconds}$ was used (which is within the stability limit for the numerical explicit time-integration scheme for these type of basis set). High velocity points are simulated with smaller time steps.  The wavefunctions were then propagated for up to several tens of  femtoseconds. The total electronic energy ($E$) of the system changes as a function of the projectile position ($x$) since the projectile   (forced to maintain its velocity) deposits energy into the electronic system as it moves through the host atoms.   The increase of $E$ as a function of projectile displacement $x$ enables us to extract the electronic stopping power as an average quantity.  \begin{equation}  S_\text{e} = \bar{\mathrm{d}E(x)/\mathrm{d}x} \overline{\mathrm{d}E(x)/\mathrm{d}x}  \label{eq:stopping}   \end{equation}