deletions | additions       diff --git a/Figure_ref_fig_fit_graph_explains__.tex b/Figure_ref_fig_fit_graph_explains__.tex  index 08095ae..a5b51c6 100644  --- a/Figure_ref_fig_fit_graph_explains__.tex  +++ b/Figure_ref_fig_fit_graph_explains__.tex 
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Figure \ref{fig:fit_graph} explains how the slopes of Figure \ref{fig:energy_distance} are evaluated. Here the dot-dash (magenta) curve represents the adiabatic potential energy surface (APES) curve at $v = 0 ~\mathrm{a.u.}$; due to the Born-Oppenheimer approximation. The oscillations in the curves reflect the periodicity of the $\mathrm{Cu}$ lattice. At $v = 0 ~\mathrm{a.u.}$, there is no transfer of energy, just oscillations of the total energy with a zero slope. To obtain the slope for $v = 0.06 ~\mathrm{a.u.}$ which represents the $S_\text{e}$ at this velocity, the adiabatic results ($v = 0 ~\mathrm{a.u.}$) are subtracted from those of $v = 0.06$ to obtain only the non-adiabatic contributions; it is shown as a solid line (red) with little oscillations. This subtraction is done to remove the oscillations that result from the periodic lattice. A linear fit , $y = a + bx$, dash line (blue) bx$ (blue line)  yields a slope of $0.00698913~E_\text{h}/a_0$ with an error of $\pm 6.936 \times 10^{-5}~E_\text{h}/a_0$. We then proceed to do a linear fit in addition to an oscillatory function $y = a + bx + A\cos(k x + \phi)$, dot line (black) \phi)$ (black line)  to capture any remnant oscillation. This oscillation fit generates a slope of $0.00743456~E_\text{h}/a_0$ with an error of $\pm 8.52 \times 10^{-7}~E_\text{h}/a_0$. For higher velocities, a linear fit alone is enough to obtain reasonable error values.%