deletions | additions       diff --git a/Each_simulation_of_the_ion__.tex b/Each_simulation_of_the_ion__.tex  index 615e4bc..732935c 100644  --- a/Each_simulation_of_the_ion__.tex  +++ b/Each_simulation_of_the_ion__.tex 
...
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 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 within the earlier observation \cite{Schleife_2015}.  %Finite-size errors 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.   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.   The basis set is sampled with a $130~\mathrm{Ry}$ energy cutoff. We tested for k-point convergence in a $3\times3\times3$ Monkhorst-Pack $(3\times3\times3)$ Morkost-Pack  grid\cite{Monkhorst_1976}  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 for subsequent evolution.   We then perform TDDFT calculations on the electronic system with the moving proton. 
...
In the off-channeling case the projectile takes random trajectories through the host crystal yielding a stronger interaction between the projectile and the core of the host atom due to the larger proximity.   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}.%