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minor modifications in SI

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Pierre-Francois Loos 2019-09-11 09:38:41 +02:00
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Manuscript/SI/eDFT-SI.tex
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@ -548,7 +548,7 @@ The numerical values of the correlation energy for various $R$ are reported in T
\begin{figure} \begin{figure}
\includegraphics[width=\linewidth]{Ec} \includegraphics[width=\linewidth]{Ec}
\caption{ \caption{
$\e{c}{(I)}$ [see Eq.~\eqref{eq:ec}] as a function of $R = 1/(\pi n)$ for the ground state ($I=0$), the first singly-excited state ($I=1$), and the first doubly-excited state ($I=2$) of the (spin-polarized) two-electron ringium system. Reduced (i.e., per electron) correlation energy $\e{c}{(I)}$ [see Eq.~\eqref{eq:ec}] as a function of $R = 1/(\pi n)$ for the ground state ($I=0$), the first singly-excited state ($I=1$), and the first doubly-excited state ($I=2$) of the (spin-polarized) two-electron ringium system.
The data gathered in Table \ref{tab:Ref} are also reported. The data gathered in Table \ref{tab:Ref} are also reported.
} }
\label{fig:Ec} \label{fig:Ec}
@ -581,10 +581,10 @@ The numerical values of the correlation energy for various $R$ are reported in T
Based on these highly-accurate calculations, one can write down, for each state, an accurate analytical expression of the reduced correlation energy \cite{Loos_2013a, Loos_2014a} via the following Pad\'e approximant Based on these highly-accurate calculations, one can write down, for each state, an accurate analytical expression of the reduced correlation energy \cite{Loos_2013a, Loos_2014a} via the following Pad\'e approximant
\begin{equation} \begin{equation}
\label{eq:ec} \label{eq:ec}
\e{c}{(I)}(n) = \frac{a^{(I)}\,n}{n + b^{(I)} \sqrt{n} + c^{(I)}}, \e{c}{(I)}(n) = \frac{c_1^{(I)}\,n}{n + c_2^{(I)} \sqrt{n} + c_3^{(I)}},
\end{equation} \end{equation}
where $b^{(I)}$ and $c^{(I)}$ are state-specific fitting parameters, which are provided in Table I of the manuscript. where $c_2^{(I)}$ and $c_3^{(I)}$ are state-specific fitting parameters, which are provided in Table I of the manuscript.
The value of $a^{(I)}$ is obtained via the exact high-density expansion of the correlation energy. \cite{Loos_2013a, Loos_2014a} The value of $c_1^{(I)}$ is obtained via the exact high-density expansion of the correlation energy. \cite{Loos_2013a, Loos_2014a}
Equation \eqref{eq:ec} is depicted in Fig.~\ref{fig:Ec} for each state alongside the data gathered in Table \ref{tab:Ref}. Equation \eqref{eq:ec} is depicted in Fig.~\ref{fig:Ec} for each state alongside the data gathered in Table \ref{tab:Ref}.