deletions | additions       diff --git a/section_Computational_and_Theoretical_Details__.tex b/section_Computational_and_Theoretical_Details__.tex  index 301752b..07f8722 100644  --- a/section_Computational_and_Theoretical_Details__.tex  +++ b/section_Computational_and_Theoretical_Details__.tex 
...
The energy transfered to the electrons of the host atom ($\mathrm{Cu}$) from a constant velocity moving proton is carefully monitored. The energy loss of the proton is negligible hence total energy of the system is not conserved. This is because at the time scales of the simulations, the large mass of the proton guarantees a negligible decline in its velocity. As the proton moves, the time-dependent Kohn-Sham (TDKS) equation\cite{Runge_1984} describes 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 Cu nuclei. The TDKS equation can be written as (Hartree atomic units are used here):  \begin{equation}  \text{i}\frac{\partial}{\partial t}\psi_{i}(\textbf \mathrm i\frac\partial{\partial t}\psi_i(\textbf  r, t) = \left\{-\hbar^2\frac{\nabla^2}{2m} \left\{-\frac{\hbar^2\nabla^2}{2m}  + V_\text{KS}[n](\mathbf V_\text{KS}[n(t)](\mathbf  R(t), \mathbf r, t)\right\}\psi_i(\textbf r, t) \label{eq:tdks1} \end{equation}  The Kohn-Sham (KS) effective potential $V_{KS}[n](\textbf R, \textbf r, t)$ is given as  \begin{equation}  V_\text{KS}[n](\textbf V_\text{KS}[n](\{\textbf R_i(t)\}_i, \textbf  r, t) = \textit{V}_{ext}(\textbf \textit{V}_{ext}(\{\textbf R_i(t)\}_i, \textbf  r, t) + \textit{V}_{H}(\textbf \textit{V}_\text{H}[n](\textbf  r, t) + \textit{V}_{xc}(\textbf \textit{V}_{XC}[n](\textbf  r, t) \label{eq:tdks3}  \end{equation}