deletions | additions       diff --git a/The_energy_transfered_to_the__1.tex b/The_energy_transfered_to_the__1.tex  index b97f0c3..e10151e 100644  --- a/The_energy_transfered_to_the__1.tex  +++ b/The_energy_transfered_to_the__1.tex 
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\end{aligned}  \end{equation}  where the external potential is $V_\text{ext}(\{\textbf R_J(t)\}_J, \textbf r, $V_\text{ext}(\{\mathbf{R}_J(t)\}_J, \mathbf{r},  t)$ due to ionic core potential (with ions at positions $\mathbf R_J(t)$), $V_\text{H}(\textbf r, t)$ is the Hartree potential comprising the classical electrostatic interactions between electrons and $\textit{V}_\text{XC}(\textbf r, t)$ 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 $n(t)$. $n(\mathbf{r}, t) = \sim_i \psi($.  The XC 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 $\mathrm{Cu}$ atom (not necessarily all 17 electrons participate in the process as we will discuss later).