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Among the measurable quantities associated to the interaction between ions and solids, the stopping power $\mathrm(S)$ \cite{Ferrell_1977} has received much attention; it provides information regarding the energy transfer between the incoming projectile and the solid target in what constitutes a fundamentally non-adiabatic process.  %Various Various  models and theories have been proposed to calculate stopping cross sections due to electrons. Employing the First Born Approximation, Bethe \cite{Bethe_1930_EN} has introduced the first calculations of inelastic and ionization cross section. The Bloch correction \cite{Bloch_1933} provides a convenient link between the Bohr and the Bethe scheme. %Fermi Fermi  and Teller \cite{Fermi_1947} using electron gas models had reported electronic stopping for various targets. %The The  Bethe formula for stopping has been studied in details by Lindhard and Winther \cite{Lindhard_Winther} on the basis of the generalized linear-response theory. %  %In the low energy region the energy loss in metal is due to the excitation of a portion of electrons around the Fermi level to empty states in the conducting band at higher energies, a minimum momentum transfer due to the limited response time of target electrons to the projectile ions. In this paper we concentrate in the near and below the maximum of stopping down to the low velocity regime for $\mathrm{H}$ in $\mathrm{Cu}$.  %  In particular, the condensed matter community has introduced sophisticated numerical computer simulation techniques for this fundamentally non-adiabatic problem as spearheaded by Echenique \emph{et al.} ~\cite{Echenique_1989} aimed to overcome limitations of historical approaches \cite{Bethe_1930,Bloch_1933,Fermi_1947,Lindhard_Winther}.  A unified \emph{ab initio} theoretical approach suitable for different projectiles and energies is in its developing stages \cite{Pruneda_2007,Schleife_2015,Ullah_2015}.   A review on the topic can be found in Ref.~\cite{Race_2010} and references therein.  Using a Kohn-Sham (KS) scheme of time-dependent density functional theory (TDDFT) where the KS wave functions are expanded in a basis set of spherical harmonics, Quijada {\emph et al} \emph{et al.}  \cite{Quijada_2007} have studied the energy loss of protons and anti-protons moving inside metallic $\mathrm{Al}$ (spherical Jellium) clusters and obtained good results for the projectile-target energy transfer over a restricted energy range. Recently Uddin \emph{et al.} \cite{Alfaz_Uddin_2013} and Haque \emph{et al.} \cite{Haque_2015} have calculated stopping cross sections for various media using atomic density functions from Dirac-Hartree-Fock-Slater wave functions in the Lindhard-Schraff theory \cite{Lindhard_Scharff} with fitted parameters and obtained close agreement with the experimental and \textsc{Srim} data.  \textsc{Srim} \cite{Ziegler_2010} provides both a fitted model for electronic stopping as well as a large set of experimental points.