Each simulation of the ion-solid collision consists of a well-defined trajectory of the projectile in the FCC metallic bulk sample with experimental density. The calculations were done using the code Qbox \cite{Gygi_2008} with time-dependent modifications \cite{Schleife_2012}\cite{Draeger_2016}. The KS orbitals are expanded in a supercell plane-wave basis. The advantages of using the plane-wave approach is that it systematically deals with basis-size effects, which was a drawback for earlier approaches \cite{Pruneda_2007, Correa_2012}.

Periodic boundary conditions along with Ewald summation \cite{Amisaki_2000,Roy_2007} are used throughout this study. The supercell size was chosen so as to reduce the specious size effects while maintaining controllable computational demands. Since the larger size effects are negligible this calculation used \((3\times3\times3)\) conventional cells containing \(108\) host \(\mathrm{Cu}\) atoms and \(\mathrm{H^+}\). To integrate the Brillouin zone a single \(k\)-point (\(\Gamma\)) was used, except for test cases. The screening length of \(\mathrm{Cu}\) is close to the interatomic spacing, which reduces the range of Coulomb interactions and makes it controllable in a periodic representation.

Finite size effects are studied between 108 and 256 atoms in a supercell of \((3\times3\times3)\) and \((4\times4\times4)\) respectively in this simulation, the errors remain within 3% in conformity with the earlier observation \cite{Schleife_2015}.

The plane-wave basis set is sampled accurately with a \(130~\mathrm{Ry}\) energy cutoff. We also tested for k-point convergence in a \((3\times3\times3)\)-Monkhorst-Pack grid (for the cubic 108-supercell), which would be equivalent to a 2916 (\(108\times 27\) simulation cell of an hypothetical periodic system, including replicas of the proton), for selected velocities with negligible differences within 0.08%.

The projectile \(\mathrm{H^+}\) is initially placed in the crystal and a time-independent DFT calculation was completed to obtain the converged ground state results that are required as the initial condition for subsequent evolution with the moving projectile. We then perform TDDFT calculations on the electronic system with the moving proton in the channeling and off-channeling geometries. Following the method introduced by Pruneda et al. \cite{Pruneda_2005} the projectile is put in motion with a constant velocity in a straight along a \(\langle 100\rangle\) channeling trajectory (also called hyper-channeling) which minimizes the collision of the projectile with the host atoms \cite{Pruneda_2007,Correa_2012,Schleife_2015}.

In the off-channeling case the projectile takes random trajectory directions through the host crystal yielding occasionally stronger interaction between the projectile and the tightly bound electrons of the host atom. The use of off-channeling trajectories was introduced in Ref. \cite{Schleife_2015}.