deletions | additions       diff --git a/The_energy_transfered_to_the__1.tex b/The_energy_transfered_to_the__1.tex  index 7471561..5c90f95 100644  --- a/The_energy_transfered_to_the__1.tex  +++ b/The_energy_transfered_to_the__1.tex 
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The energy transfered to the electrons of the target due to a constant velocity moving proton is monitored.   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, 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 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. 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)