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Pierre-Francois Loos 2020-04-08 13:27:18 +02:00
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commit 690be93a94
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Manuscript/FarDFT.tex
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@ -373,16 +373,18 @@ Doing so, we have found that the present weight-dependent exchange functional (d
with with
\begin{equation} \begin{equation}
\label{eq:Cxw} \label{eq:Cxw}
\Cx{\ew{}} = \Cx{} \qty{ 1 - \ew{} (1 - \ew{})\qty[ \alpha + \beta (\ew{} - 1/2) + \gamma (w - 1/2)^2 ]} \frac{\Cx{\ew{}}}{\Cx{}} = 1 - \ew{} (1 - \ew{})\qty[ \alpha + \beta (\ew{} - 1/2) + \gamma (w - 1/2)^2 ]
\end{equation} \end{equation}
and and
\begin{subequations}
\begin{align} \begin{align}
\alpha & = + 0.575\,178, \alpha & = + 0.575\,178,
& \\
\beta & = - 0.021\,108, \beta & = - 0.021\,108,
& \\
\gamma & = - 0.367\,189, \gamma & = - 0.367\,189,
\end{align} \end{align}
\end{subequations}
makes the ensemble almost perfectly linear (see Fig.~\ref{fig:Ew_H2}), and the excitation energy much more stable and closer to the full configuration interaction (FCI) reference of $28.75$ eV \cite{Barca_2018a} (see Fig.~\ref{fig:Om_H2}) makes the ensemble almost perfectly linear (see Fig.~\ref{fig:Ew_H2}), and the excitation energy much more stable and closer to the full configuration interaction (FCI) reference of $28.75$ eV \cite{Barca_2018a} (see Fig.~\ref{fig:Om_H2})
As readily seen from Eq.~\eqref{eq:Cxw}, $\Cx{\ew{}}$ reduces to $\Cx{}$ for $\ew{} = 0$. As readily seen from Eq.~\eqref{eq:Cxw}, $\Cx{\ew{}}$ reduces to $\Cx{}$ for $\ew{} = 0$.
Note that we are not only using data from $\ew{} = 0$ to $\ew{} = 1/2$, but we also consider $1/2 < \ew{} \le 1$. Note that we are not only using data from $\ew{} = 0$ to $\ew{} = 1/2$, but we also consider $1/2 < \ew{} \le 1$.
@ -505,7 +507,7 @@ Combining these, we build a two-state weight-dependent correlation functional:
%%% FIG 1 %%% %%% FIG 1 %%%
\begin{figure} \begin{figure}
\includegraphics[width=\linewidth]{fig/fig1} \includegraphics[width=0.8\linewidth]{fig/fig1}
\caption{ \caption{
Reduced (i.e., per electron) correlation energy $\e{\co}{(I)}$ [see Eq.~\eqref{eq:ec}] as a function of $R = 1/(\pi^2 \n{}{})^{1/3}$ for the ground state ($I=0$), and the first doubly-excited state ($I=1$) of the (spin-unpolarised) two-electron FUEG. Reduced (i.e., per electron) correlation energy $\e{\co}{(I)}$ [see Eq.~\eqref{eq:ec}] as a function of $R = 1/(\pi^2 \n{}{})^{1/3}$ for the ground state ($I=0$), and the first doubly-excited state ($I=1$) of the (spin-unpolarised) two-electron FUEG.
The data gathered in Table \ref{tab:Ref} are also reported. The data gathered in Table \ref{tab:Ref} are also reported.