deletions | additions       diff --git a/Each_simulation_of_the_ion__.tex b/Each_simulation_of_the_ion__.tex  index bdd006c..022a65d 100644  --- a/Each_simulation_of_the_ion__.tex  +++ b/Each_simulation_of_the_ion__.tex 
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The advantages of using the plane-wave approach is that it systematically deals with basis-size effects, which was a drawback for earlier approaches \cite{Pruneda_2007, Correa_2012}.  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}.  Periodic boundary conditions along with Ewald summation~\cite{Amisaki_2000,Roy_2007} are used throughout this study to ensure finite range due to screening length of $\mathrm{Cu}$ is close to the interatomic spacing. study.  The supercell size was chosen so as to reduce the specious size effects while maintaining controllable computational demands.   Since the larger size effects are negligible this calculation 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. used, except for test cases.  The screening length of $\mathrm{Cu}$ is close to the interatomic spacing, which reduces the range of Coulomb interactions and makes it controllable in a periodic representation.  The basis set is sampled with a $130~\mathrm{Ry}$ energy cutoff.   We also tested for k-point convergence in a $(3\times3\times3)$ Morkost-Pack grid (for the 108-supercell), that would be equivalent to a 2916   ($108\times 27$ simulation cell of an hypothetical periodic system, including replicas of the proton), for a selected velocities with negligible differences of 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. evolution with the moving projectile.  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~\cite{Pruneda_2007,Correa_2012,Schleife_2015}.   In the off-channeling case the projectile takes random trajectories through the host crystal yielding a occasionally  stronger interaction between the projectile and the core tightly bound electrons  of the host atom due to the larger proximity. atom.  The need to take into account off-channeling trajectories was described in Ref.~\cite{Schleife_2015}.  %Finite-size errors in the simulations are overcome by considering large simulation cells \cite{Schleife_2015}.%