deletions | additions       diff --git a/The_simulations_of_the_collisions__.tex b/The_simulations_of_the_collisions__.tex  index 5c037bd..71cfe9c 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 resultant wave functions wavefunctions  were then propagated for up to  several 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.  \begin{equation}