deletions | additions       diff --git a/The_energy_transfered_to_the__1.tex b/The_energy_transfered_to_the__1.tex  index 038c3a1..52a1d74 100644  --- a/The_energy_transfered_to_the__1.tex  +++ b/The_energy_transfered_to_the__1.tex 
...
The energy transfered to the electrons of the targetatom  due to a constant velocity moving proton is monitored. At the time scales of the simulations, the large mass of the proton guarantees athe  change in its velocity is relatively small. For simplicity simplicity,  the proton is forced constrained  to move at constant velocity, hence total energy of the system is not conserved. The excess in total energy is instead used as a measure of the stopping power as a function of the proton velocity.  As the proton moves, the time-dependent Kohn-Sham (TDKS) equation \cite{Runge_1984} describes electronic density and energy of the system due to the dynamics of effective single particle states under the external potential generated by the proton and the crystal of Cu nuclei. The TDKS equation can be written as: 
...
\label{eq:tdks1}   \end{equation}  The Kohn-Sham (KS) effective potential $V_{KS}[n](\textbf $V_\text{KS}[n](\textbf  R, \textbf r, t)$ is given as \begin{equation}  V_\text{KS}[n](\{\textbf R_i(t)\}_i, \textbf r, t) = \textit{V}_\text{ext}(\{\textbf R_i(t)\}_i, \textbf r, t) + \textit{V}_\text{H}[n](\textbf r, t) + \textit{V}_\text{XC}[n](\textbf r, t)  \label{eq:tdks3}  \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 $V_\text{H}(\textbf  r, t)$ 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. 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 Troullier-Martins pseudopotential, with $17$ explicit electrons per Cu atom.