deletions | additions       diff --git a/section_Computational_and_Theoretical_Details__.tex b/section_Computational_and_Theoretical_Details__.tex  index 5c65681..7f617b2 100644  --- a/section_Computational_and_Theoretical_Details__.tex  +++ b/section_Computational_and_Theoretical_Details__.tex 
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where the external potential is $V_\text{ext}(\{\textbf R_i(t)\}_i, \textbf r, t)$ due to ionic core potential (with ions at positions $\mathbf R_i(t)$), $V_{H}(\textbf r, t)$ is the Hartree potential which describes the classical electrostatic interactions between electrons and $\textit{V}_{xc}(\textbf r, t)$ is the exchange-correlation (XC) potential. The spatial and time coordinates are represented by $\mathbf r$ and $t$ respectively.   The instantaneous density at time $t$ is denoted by $n(t)$.  Exchange-correlation potential is approximated by the LDA (cite PerdewZunger here), The Perdew-Burke-Ernzerhof (PBE)~\cite{Perdew_1992,Perdew_1996},  the singular (Coulomb) external potential is approximated by a norm-conserving pseudopotential, with $17$ explicit electrons per Cu atom in the valence band. The simulations of the collisions consist of a well-defined trajectories of the projectile (proton) in the metallic bulk.   The calculations were done using the code \textsc{Qbox} \cite{Gygi_2008} with time-dependent modifications \cite{Schleife_2012}.   The Kohn-Sham (KS) orbitals are expanded in a supercell plane-wave basis.   These KS orbitals are evolved in time with a self-consistent Hamiltonian that is a functional of the density.   The algorithm for evolution of the orbitals is done using the fourth-order Runge-Kutta scheme (RK4) \cite{Schleife_2012}. The advantages of using plane-wave approach is that, it conquers basis-size effects which was a drawback for earlier approaches \cite{Pruneda_2007} and finite-size error in the simulations are overcome by considering large simulation cells \cite{Schleife_2015}.   The Perdew-Zunger's Perdew-Burke-Ernzerhof (PBE)~\cite{Perdew_1992,Perdew_1996}  exchange-correlation\cite{Perdew_1992}  is used, and the core electrons are represented using norm-conserving Troullier-Martins pseudopotentials \cite{Troullier_1991}. Periodic boundary conditions were used throughout. The best supercell size was selected so as to reduce the specious effects of the duplication while maintaining controllable computational demands.   The calculations used $(3\times3\times3)$ supercells containing 108 host Copper atoms plus a proton, also represented by a Troullier-Martins pseudopotential ($17$ valence electrons per copper atom are explicitly considered).