deletions | additions       diff --git a/section_Computational_and_Theoretical_Details__.tex b/section_Computational_and_Theoretical_Details__.tex  index a6d4378..1a54ab7 100644  --- a/section_Computational_and_Theoretical_Details__.tex  +++ b/section_Computational_and_Theoretical_Details__.tex 
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\end{equation}  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 comprising the classical electrostatic interactions between electrons and $\textit{V}_{xc}(\textbf r, t)$ denotes the exchange-correlation (XC) potential. The spatial and time coordinates are represented by $\mathbf r$ and $t$ respectively.   The instantaneous density at At  time $t$ the instantaneous density  is given by $n(t)$. The exchange-correlation potential used in this study is due to Perdew-Burke-Ernzerhof (PBE) ~\cite{Perdew_1992,Perdew_1996}, using a norm-conserving pseudopotential, with $17$ explicit electrons per Cu atom in the valence band, the coulomb potential is generated.  The simulations of the collisions consist of a well-defined trajectories of the projectile (proton) in the metallic bulk.