deletions | additions       diff --git a/Each_simulation_of_the_ion__.tex b/Each_simulation_of_the_ion__.tex  index 4d9a3bf..c141845 100644  --- a/Each_simulation_of_the_ion__.tex  +++ b/Each_simulation_of_the_ion__.tex 
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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.   The fourth-order Runge-Kutta scheme (RK4) \cite{Schleife_2012} is used to propagate these orbitals in time.   The advantages of using the  plane-wave approach is that, that  it systematically deals with basis-size effects which was a drawback for earlier approaches \cite{Pruneda_2007, Correa_2012} and finite-size Correa_2012}.   Finite-size  error in the simulations are overcome by considering large simulation cells \cite{Schleife_2015}. Periodic boundary conditions are used throughout this study.   The supercell size was chosen so as to reduce the specious size effects while maintaining controllable computational demands.  
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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 is  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 al.} \cite{Pruneda_2005} the projectile is then allowed to move with a constant velocity subjected to a straight uniform movement along a $\langle 100\rangle$ channeling trajectory (also called hyper-channeling) which minimizes the collision of the projectile with the host atoms.   In the off-channeling case the projectile takesa  random trajectory trajectories  through the host material crystal  yielding a stronger interaction between the projectile and the core of the  host atom duelarge charge density close  to the target. larger proximity.  The need to take into account off-channeling trajectories is described in \cite{Schleife_2015}.