deletions | additions       diff --git a/section_Computational_and_Theoretical_Details__1.tex b/section_Computational_and_Theoretical_Details__1.tex  index d1365b4..a5e2b77 100644  --- a/section_Computational_and_Theoretical_Details__1.tex  +++ b/section_Computational_and_Theoretical_Details__1.tex 
...
%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 the evolution of the electronic density and energy of the system 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:  \begin{equation}  \mathrm i\hbar\tfrac\partial{\partial t}\psi_i(\mathbf{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(\mathbf{r}, t)