deletions | additions       diff --git a/The_energy_transfered_to_the__.tex b/The_energy_transfered_to_the__.tex  index 07beee8..9400b98 100644  --- a/The_energy_transfered_to_the__.tex  +++ b/The_energy_transfered_to_the__.tex 
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The energy transfered to the electrons of the target atom due to a constant velocity moving proton iscarefully  monitored. The energy loss At the time scales of the simulations, the large mass  of the proton guarantees a the change in its velocity  is negligible relatively small.  For simplicity the proton is forced to move at constant velocity,  hence total energy of the system is not conserved. This The excess in total energy  is because at the time scales instead used as a measure  of the simulations, the large mass stopping power as a function  of the protonguarantees 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: \begin{equation}  \mathrm i\hbar\frac\partial{\partial t}\psi_i(\textbf r, t) = \left\{-\frac{\hbar^2\nabla^2}{2m} + V_\text{KS}[n(t)](\{\mathbf R_i(t)\}_i, \mathbf r, t)\right\}\psi_i(\textbf r, t)