deletions | additions       diff --git a/The_simulations_of_the_collisions__.tex b/The_simulations_of_the_collisions__.tex  index fd3f707..d97f5bd 100644  --- a/The_simulations_of_the_collisions__.tex  +++ b/The_simulations_of_the_collisions__.tex 
...
The simulations of the collisions collision  consist of a well-defined trajectories of the projectile (proton) in the FCC metallic bulk with experimental density. The calculations were done using the code \textsc{Qbox} \cite{Gygi_2008} with time-dependent modifications \cite{Schleife_2012}.   The Kohn-Sham (KS) orbitals are expanded in a supercell plane-wave basis.   These KS orbitals are evolved in time with a self-consistent Hamiltonian that is a functional of the density. The fourth-order Runge-Kutta scheme (RK4) \cite{Schleife_2012} is used to propagate these orbitals in time. The advantages of using plane-wave approach is that, it systematically deals with basis-size effects which was a drawback for earlier approaches \cite{Pruneda_2007, Correa_2012} and finite-size error in the simulations are overcomed overcome  by considering large simulation cells \cite{Schleife_2015}. Periodic boundary conditions are used throughout this study.   Thebest  supercell size was selected chosen  so as to reduce the specious size effects while maintaining controllable computational demands. The calculations used $(3\times3\times3)$ conventional cells containing 108 host $\mathrm{Cu}$ atoms and $\mathrm{H^+}$.  % also represented by a Troullier-Martins pseudopotential ($17$ valence electrons per copper atom are explicitly considered).   To integrate the Brillouin zone a single $k$-point ($\Gamma$) was used.   The basis set is sampled with a $130~\mathrm{Ry}$ energy cutoff.  The projectile $\mathrm{H^+}$ was initially placed in the crystal and a time-independent DFT calculation was completed to obtain the converged ground state results that are required for subsequent evolution. We then perform TDDFT calculations on the electronic system with the moving proton.  Following the method introduced by Pruneda {\emph et \emph{et  al.} \cite{Pruneda_2005} the projectile is then allowed to move with a constant velocity subjected to a straight uniform movement along a [100] $\langle 100$\rangle$  channeling trajectory (also called hyper-channeling trajectory). This trajectory, which  minimizes the collision of the projectile with the host atoms. atoms).  In the off-channeling case the projectile takes a random trajectory through the host material yielding a stronger interaction between the projectile and the host atom due large charge density close to the target.   The need to take into account off-channeling trajectories is described in \cite{Schleife_2015}.  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.