The fourth-order Runge-Kutta scheme (RK4) \cite{Schleife_2012} is used to propagate the electronic orbitals in time, following the scheme of Ref. \cite{Schleife_2012} the TDKS equation (see Eq. \ref{eq:tdks1}) with a time step of, at most, \(0.121~\mathrm{attoseconds}\) (which is within the numerical stability limit time-integration scheme for this basis set). High velocity points were simulated with smaller time steps. The wavefunctions were then propagated for up to 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 a time-averaged quantity for each velocity, \[S_\text{e} = \overline{\mathrm{d}E(x)/\mathrm{d}x} \label{eq:stopping}\] \(S_\text{e}\) has the dimension of a force (e.g. \(E_\text{h}/a_0\)) and it has the interpretation of a drag force acting on the projectile.