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Edwin E. Quashie edited Along_the_simulation_we_monitor__.tex
over 8 years ago
Commit id: 9cbcab1008d54024dcffba1854a5d4f59cf86981
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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.
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)
\label{eq:tdks1}