deletions | additions       diff --git a/Each_simulation_of_the_ion__.tex b/Each_simulation_of_the_ion__.tex  index f1bdcb4..e5e1fd5 100644  --- a/Each_simulation_of_the_ion__.tex  +++ b/Each_simulation_of_the_ion__.tex 
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Finite size effects are studied between 108 and 256 atoms in a supercell of $(3\times3\times3)$ and $(4\times4\times4)$ respectively in this simulation, the errors remain within 3\% in conformity with the earlier observation \cite{Schleife_2015}.  The plane-wave basis set is sampled accurately  with a $130~\mathrm{Ry}$ energy cutoff. We also tested for k-point convergence in a $(3\times3\times3)$ Morkost-Pack $(3\times3\times3)$-Morkost-Pack  grid (for the cubic 108-supercell), which would be equivalent to a 2916 ($108\times 27$ simulation cell of an hypothetical periodic system, including replicas of the proton), for selected velocities with negligible differences within 0.08\%.  The projectile $\mathrm{H^+}$ is initially placed in the crystal and a time-\emph{independent} DFT calculation was completed to obtain the converged ground state results that are required as the initial condition for subsequent evolution with the moving projectile.  We then perform TDDFT calculations on the electronic system with the moving proton. proton in the channeling and off-channeling geometries.  Following the method introduced by Pruneda \emph{et al.} \cite{Pruneda_2005} the projectile is then allowed to move put in motion  with a constant velocity subjected to in  a straightuniform movement  along a $\langle 100\rangle$ channeling trajectory (also called hyper-channeling) which minimizes the collision of the projectile with the host atoms~\cite{Pruneda_2007,Correa_2012,Schleife_2015}. In the off-channeling case the projectile takes random trajectories trajectory directions  through the host crystal yielding occasionally stronger interaction between the projectile and the tightly bound electrons of the host atom. The need to take into account use of  off-channeling trajectories was described introduced  in Ref.~\cite{Schleife_2015}. %Finite-size errors in the simulations are overcome by considering large simulation cells \cite{Schleife_2015}.%