deletions | additions       diff --git a/Figure_ref_fig_force_on_neighbor_shows__.tex b/Figure_ref_fig_force_on_neighbor_shows__.tex  index 1eb1bd4..cd3c566 100644  --- a/Figure_ref_fig_force_on_neighbor_shows__.tex  +++ b/Figure_ref_fig_force_on_neighbor_shows__.tex 
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Figure \ref{fig:force_on_neighbor} shows the radial force exerted on a neighboring Cu atom closest to $\mathrm{H^+}$ trajectory as a function of parallel distance to the projectile at different projectile velocities along the [100] channeling trajectory. The forces on the nuclei are evaluated using the time-dependent electron density, $n(\mathbf{r}, t)$.  The force is obtained by the application of the Hellmann-Feynman theorem \cite{Hellmann_2015,1941,Feynman_1939} \cite{Hellmann_2015,1941,Feynman_1939}.%  but not applied to the ground state in this case. The adiabatic force is recovered for $v \to 0$ with no oscillations as expected.   The maximum value for the force is obtained at the closest distance between $\mathrm{H^+}$ and neighbor $\mathrm{Cu}$ atom.  As the proton movesfurther  away from the $\mathrm{Cu}$ atom, the force decreases significantly and eventually goes reduces  to zero. As the velocity increases the position of the maximum value of the force shifts due to the complex structure obtained by the shape shape-structure  of the curves and oscillations are created. results oscillations.  These oscillations becomes persistent as the velocity of the proton increases.% %