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Edwin E. Quashie edited Along_the_simulation_we_monitor__.tex
over 8 years ago
Commit id: ae650ae33deb47b2e3793c19d958255d5ed750ea
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...
\label{eq:tdks1}
\end{equation}
The Kohn-Sham (KS) effective potential
$V_\text{KS}[n(t), \{\mathbf R_J(t)\}_J](\mathbf r, t)$ is given as
\begin{equation}
\begin{aligned}
V_\text{KS}[n, \{\textbf R_J(t)\}_J] V_\text{KS}[n(t), \{\mathbf R_J(t)\}_J](\mathbf r, t) = \textit{V}_\text{ext}[\mathbf{R}_J(t)\}_J] + \textit{V}_\text{H}[n] + V_\text{XC}[n]
\label{eq:tdks3}
\end{aligned}
\end{equation}
...
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 $V_\text{XC}[n](\mathbf{r}, t)$ 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 $n(\mathbf{r}, t) = \sum_i |\psi_i(\mathbf{r}, t)|^2$.
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).
%V_\text{KS}[n, \{\textbf R_J(t)\}_J] = \textit{V}_\text{ext}[\mathbf{R}_J(t)\}_J] + \textit{V}_\text{H}[n] + V_\text{XC}[n]