deletions | additions       diff --git a/The_energy_transfered_to_the__1.tex b/The_energy_transfered_to_the__1.tex  index 8de4409..7471561 100644  --- a/The_energy_transfered_to_the__1.tex  +++ b/The_energy_transfered_to_the__1.tex 
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At the time scales of the simulations, the large mass of the proton guarantees a change in its velocity is relatively small.  For simplicity, the proton is 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 $\mathrm{Cu}$  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)