deletions | additions       diff --git a/section_Computational_and_Theoretical_Details__.tex b/section_Computational_and_Theoretical_Details__.tex  index 1a54ab7..1d11f86 100644  --- a/section_Computational_and_Theoretical_Details__.tex  +++ b/section_Computational_and_Theoretical_Details__.tex 
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where the external potential is $V_\text{ext}(\{\textbf R_i(t)\}_i, \textbf r, t)$ due to ionic core potential (with ions at positions $\mathbf R_i(t)$), $V_{H}(\textbf r, t)$ is the Hartree potential comprising the classical electrostatic interactions between electrons and $\textit{V}_{xc}(\textbf r, t)$ 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 $n(t)$.  The exchange-correlation potential used in this study is due to Perdew-Burke-Ernzerhof (PBE) ~\cite{Perdew_1992,Perdew_1996}, using a norm-conserving Troullier-Martins  pseudopotential, with $17$ explicit electrons per Cu atom in the valence band, the coulomb potential is generated. The simulations of the collisions consist of a well-defined trajectories of the projectile (proton) in the metallic bulk.   The calculations were done using the code \textsc{Qbox} \cite{Gygi_2008} with time-dependent modifications \cite{Schleife_2012}.   The Kohn-Sham (KS) orbitals are expanded in a supercell plane-wave basis.   These KS orbitals are evolved in time with a self-consistent Hamiltonian that is a functional of the density. Thealgorithm for evolution of the orbitals is done using the  fourth-order Runge-Kutta scheme (RK4) \cite{Schleife_2012}. \cite{Schleife_2012} is used to propagate these orbitals.  The advantages of using plane-wave approach is that, it conquers basis-size effects which was a drawback for earlier approaches \cite{Pruneda_2007} and finite-size error in the simulations are overcome overcomed  by considering large simulation cells \cite{Schleife_2015}.The Perdew-Burke-Ernzerhof (PBE) ~\cite{Perdew_1992,Perdew_1996} exchange-correlation is used, and the core electrons are represented using norm-conserving Troullier-Martins pseudopotentials \cite{Troullier_1991}.  Periodic boundary conditions are implemented throughout this study. The best supercell size was selected so as to reduce the specious effects of the duplication while maintaining controllable computational demands.   The calculations used $(3\times3\times3)$ supercells containing 108 host Copper $\mathrm{Cu}$  atoms plus a proton, and $\mathrm{H^+}$.%  also represented by a Troullier-Martins pseudopotential ($17$ valence electrons per copper atom are explicitly considered). A To integrate the Brillouin zone a  single \textit{k} point ($\Gamma$) was used for integrations in the Brillouin zone. used.  The density is sampled with a  $130~\mathrm{Ry}$ energy cutoff. The projectile $\mathrm{H^+}$  was initially placed in the crystal and the time-independent DFT calculation was implemented completed  to obtain the solutions for the initial condition of the electronic system for the converged  ground state results that are required  for subsequent evolution. We then perform TDDFT calculations on the electronic system. The Following the method introduced by Pruneda {\emph et al.} \cite{Pruneda_2005} the  projectile is moved then allowed to move  with a constant velocity subjected to a straight uniform movement along a [100] channeling trajectory (also called hyper-channeling trajectory) following the method introduced by Pruneda {\emph et al.} \cite{Pruneda_2005}. trajectory).  This is done to minimize minimizes  the collision of the projectile with the host atoms. Also In the off-channeling case  the projectile is subjected to takes  a random trajectory through the host material (also called off-channeling) in order to assess sensitivity to the ideal hyper-channeling conditions.   In this case, there is yielding  a stronger interaction between the  projectile andhost atoms because  the host atom due large  charge density in the proximity of close to  the host atoms is larger. target.  The time-dependent KS Following the scheme \cite{Schleife_2012} the TDKS  equation (see \ref{eq:tdks1})  was then solved numerically by explicit time-integration scheme as described in \cite{Schleife_2012}. numerically.  A time step of $0.121~\mathrm{attoseconds}$ was used, which is below the stability limit for the numerical explicit time-integration scheme for these type of basis set. The resultant  wave functions were then propagated for several femtoseconds. The total electronic energy ($E$) of theelectronic  system changes as a function of the projectile position ($x$) since the projectile (forced to maintain its velocity) deposits energy into the electronic system as it moves through the host atoms. The increase of $E$ as a function of projectile displacement $x$ enables us to extract the electronic stopping power.  \begin{equation}