deletions | additions       diff --git a/Each_simulation_of_the_ion__.tex b/Each_simulation_of_the_ion__.tex  index 732935c..54954c0 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 within the earlier observation \cite{Schleife_2015}.  Periodic boundary conditions along with Ewald summation~\cite{Amisaki_2000,Roy_2007}  are used throughout this study. study to ensure finite range due to screening length of $\mathrm{Cu}$ is close to the interatomic spacing.  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).