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The energy transfered to the electrons of the host atom ($\mathrm{Cu}$) from a constant velocity moving proton is carefully monitored. The energy loss of the proton is negligible hence total energy of the system is not conserved. This is because at the time scales of the simulations, the large mass of the proton guarantees a negligible decline in its 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 (Hartree atomic units are used here):  \begin{equation}  \text{i}\frac{\partial}{\partial t}\Psi_{i}(\textbf r, t) = \left\{-\hbar^2\frac{1}{2}\nabla^{2 m} + V_{KS}[n](\textbf V_\text{KS}[n](\mathbf R(t), \mathbf  r, t)\right\}\Psi_i(\textbf r, t) \label{eq:tdks1}   \end{equation}  The Kohn-Sham (KS) effective potential $V_{KS}[n](\textbf R, \textbf  r, t)$ is given as \begin{equation}  V_{KS}[n](\textbf V_\text{KS}[n](\textbf  r, t) = \textit{V}_{ext}(\textbf r, t) + \textit{V}_{H}(\textbf r, t) + \textit{V}_{xc}(\textbf r, t) \label{eq:tdks3}  \end{equation}  where the external potential is $\textit{V}_{ext}(\textbf $V_\text{ext}(\{\textbf R_i(t)\}_i, \textbf  r, t)$ due to ionic core potential, $\textit{V}_{H}(\textbf 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 $\textbf $\mathbf  r$ and $t$ respectively. The instantaneous density at time $t$ is denoted by $n(=n(t))$.  Exchange-correlation potential is approximated by the LDA (cite PerdewZunger here), the singular (Coulomb) external potential is approximated by a norm-conserving pseudopotential, with $XX$ 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}.