deletions | additions       diff --git a/Each_simulation_of_the_ion__.tex b/Each_simulation_of_the_ion__.tex  index b1a64be..1757746 100644  --- a/Each_simulation_of_the_ion__.tex  +++ b/Each_simulation_of_the_ion__.tex 
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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 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.   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, 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.  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 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), fora  selected velocities with negligible differences of 0.08\% 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.