During the course of the simulation, we monitored the energy transferred to the electrons of the target due to a constant velocity moving proton. For simplicity, and since the eletronic stopping is a velocity-resolved quantity the proton is constrained to move at constant velocity, hence the total energy of the system is not conserved. 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: \[\mathrm i\hbar\tfrac\partial{\partial t}\psi_i(\mathbf{r}, t) = \left\{-\tfrac{\hbar^2\nabla^2}{2m} + V_\text{KS}[n(t), \{\mathbf{R}_J(t)\}_J](\mathbf{r}, t)\right\}\psi_i(\mathbf{r}, t) \label{eq:tdks1}\]
The KS effective potential \(V_\text{KS}[n(t), \{\mathbf {R}_J(t)\}_J](\mathbf{r}, t)\) is given by \[\begin{aligned} V_\text{KS}[n, \{\mathbf{R}_J(t)\}_J] = V_\text{ext}[\{\mathbf{R}_J(t)\}_J] + V_\text{H}[n] + V_\text{XC}[n] \label{eq:tdks3} \end{aligned}\]
where the external potential is \(V_\text{ext}[\{\mathbf{R}_J(t)\}_J](\mathbf{r}, t)\) due to ionic core potential (with ions at positions \(\mathbf R_J(t)\)), \(V_\text{H}[n](\mathbf{r}, t)\) is the Hartree potential comprising the classical electrostatic interactions between electrons and \(V_\text{XC}[n](\mathbf{r}, t)\) denotes the exchange-correlation (XC) potential. The spatial and time coordinates are represented by \(\mathbf{r}\) and \(t\) respectively. At time \(t\) the instantaneous density is given by the sum of individual electron probabilities \(n(\mathbf{r}, t) = \sum_i |\psi_i(\mathbf{r}, t)|^2\). The XC potential used in this study is due to Perdew-Burke-Ernzerhof (PBE) \cite{Perdew_1992,Perdew_1996}, and we used norm-conserving Troullier-Martins pseudopotential to represent \(V_\text{ext}\), with \(17\) explicit electrons per \(\mathrm{Cu}\) atom (not necessarily all 17 electrons participate in the process as we will discuss later).