deletions | additions       diff --git a/During_the_course_of_the__.tex b/During_the_course_of_the__.tex  index 2388330..2218be7 100644  --- a/During_the_course_of_the__.tex  +++ b/During_the_course_of_the__.tex 
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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.   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}\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(\textbf r, \{\mathbf{R}_J(t)\}_J](\mathbf{r}, t)\right\}\psi_i(\mathbf{r},  t) \label{eq:tdks1}   \end{equation}