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During the course of the simulation, we monitored the energy transferred to the electrons of the target due to a constant velocity moving proton.   %At the time scales of the simulations, the large mass of the proton guarantees a change in its velocity that is relatively small.  For simplicity, and since the eletronic stopping is a velocity-resolved quantity  the proton is constrained to move at constant velocity, hence the 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 the evolution of the 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 $\mathrm{Cu}$ nuclei.   These states are evolved in time with a self-consistent Hamiltonian that is a functional of the density: