deletions | additions       diff --git a/Along_the_simulation_we_monitor__.tex b/Along_the_simulation_we_monitor__.tex  index dde6187..694aa24 100644  --- a/Along_the_simulation_we_monitor__.tex  +++ b/Along_the_simulation_we_monitor__.tex 
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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 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.   The TDKS equation can be written as: These states are evolved in time with a self-consistent Hamiltonian that is a functional of the density:  \begin{equation}  \mathrm i\hbar\tfrac\partial{\partial t}\psi_i(\textbf r, t) = \left\{-\tfrac{\hbar^2\nabla^2}{2m} + V_\text{KS}[n(t), \{\mathbf R_J(t)\}_J](\mathbf r, t)\right\}\psi_i(\textbf r, t)