deletions | additions       diff --git a/Along_the_simulation_we_monitor__.tex b/Along_the_simulation_we_monitor__.tex  index 694aa24..cc77fd9 100644  --- a/Along_the_simulation_we_monitor__.tex  +++ b/Along_the_simulation_we_monitor__.tex 
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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 $\mathbf R_J(t)$), $V_\text{H}(\mathbf{r}, t)$ is the Hartree potential comprising the classical electrostatic interactions between electrons and $V_\text{XC}(\mathbf{r}, t)$ 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 $n(\mathbf{r}, t) = \sim_i |\psi(\mathbf{r}, \sum_i |\psi_i(\mathbf{r},  t)|^2$. The XC potential used in this study is due to Perdew-Burke-Ernzerhof (PBE) ~\cite{Perdew_1992,Perdew_1996}, 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).