The recent measurements by Cantero et al. \cite{Cantero_2009} and by Markin et al. \cite{Markin_2009} of slow (\(v \leq 0.6~\mathrm{a.u.}\)) \(\mathrm{H^+}\) in \(\mathrm{Cu}\) give a glimpse of the interesting extreme low velocity limit. Although disagreeing with each other in absolute scale by \(\sim 40\%\) (Fig. \ref{fig:stopping_power}), both reveal the stopping due to conduction band electronic excitations at lower velocity, evidenced as a change in slope near \(v=0.15\) or \(0.10~\mathrm{a.u.}\) respectively. The change of slope was deduced qualitatively to be caused by the participation of \(\mathrm{d}\)-electrons \cite{Goebl_2013}. In this paper we will address the problem of theoretical calculation of \(S_\text{e}\) of protons in crystalline \(\mathrm{Cu}\) for a wide range of available experimental velocities (\(0.02~\mathrm{a.u.} \leq v \leq 10~\mathrm{a.u.}\)). We perform our calculations by directly simulating the process of a proton traversing a crystal of \(\mathrm{Cu}\) atoms, producing individual and collective electronic excitations within the TDDFT framework \cite{Correa_2012,Schleife_2012,Schleife_2014} including Ehrenfest molecular dynamics (EMD) \cite{Gross_1996,Calvayrac_2000,Mason_2007,Alonso_2008,Andrade_2009}. This method is used to calculate most microscopic quantities along the process (forces, electronic density, charges, etc); in particular, we concentrate here in the calculation of \(S_\text{e}\). A quantitative explanation and interpretation of our results are furnished along with a detailed experimental comparison.