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Pierre-Francois Loos 2020-04-08 10:41:49 +02:00
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FarDFT.nb
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Manuscript/FarDFT.tex
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@ -364,7 +364,7 @@ makes the ensemble much more linear (see Fig.~\ref{fig:Ew_H2}), and the excitati
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 $1/2 < \ew{} \le 1$.
We ensure that the weight-dependent functional does not affect the two ghost-interaction-free limit at $\ew{} = 0$ and $1$. We ensure that the weight-dependent functional does not affect the two ghost-interaction-free limit at $\ew{} = 0$ and $1$.
It is interesting to note that, around $\ew{} = 0$, the behavior of Eq.~\eqref{eq:Cxw} is linear
\begin{figure} \begin{figure}
\includegraphics[width=0.8\linewidth]{Cx_H2} \includegraphics[width=0.8\linewidth]{Cx_H2}
\caption{ \caption{
@ -373,7 +373,7 @@ We ensure that the weight-dependent functional does not affect the two ghost-int
} }
\end{figure} \end{figure}
In a third time, we add up correlation via the VWN5 local correlation functional. \cite{Vosko_1980} Third, we add up correlation via the VWN5 local correlation functional. \cite{Vosko_1980}
For the sake of clarity, the explicit expression of the VWN5 functional is not reported here but it can be found in Ref.~\onlinecite{Vosko_1980}. For the sake of clarity, the explicit expression of the VWN5 functional is not reported here but it can be found in Ref.~\onlinecite{Vosko_1980}.
The combination of the Slater and VWN5 functionals (SVWN5) yield a highly non-linear ensemble energy (as shown in Fig.~\ref{fig:Ew_H2}), while the combination of the MSFL and VWN5 functionals exhibit a small curvature and improved excitation energies, especially at small weights. The combination of the Slater and VWN5 functionals (SVWN5) yield a highly non-linear ensemble energy (as shown in Fig.~\ref{fig:Ew_H2}), while the combination of the MSFL and VWN5 functionals exhibit a small curvature and improved excitation energies, especially at small weights.
@ -608,38 +608,38 @@ $\pdv{\E{\xc}{(1-\xi,\xi)}[\n{}{}]}{\xi}$ dans l'intégrale, non ? Can it be :
%%% TABLE I %%% %%% TABLE I %%%
\begin{table*} \begin{table*}
\caption{ \caption{
Excitation energies (in eV) of \ce{H2} with $\RHH = 1.4$ bohr for various methods and basis sets. Excitation energies (in eV) associated with the lowest double excitation of \ce{H2} with $\RHH = 1.4$ bohr for various methods and basis sets.
\label{tab:Energies} \label{tab:Energies}
} }
\begin{ruledtabular} \begin{ruledtabular}
\begin{tabular}{llccccc} \begin{tabular}{llccccc}
\mc{2}{c}{xc functional} \\ \mc{2}{c}{xc functional} & & \mc{2}{c}{GOK} \\
\cline{1-2} \cline{1-2} \cline{4-5}
exchange & correlation & Basis & GOK($\ew{} = 0$) & GOK($\ew{} = 1/2$) & LIM & MOM \\ exchange & correlation & Basis & $\ew{} = 0$ & $\ew{} = 1/2$ & LIM & MOM \\
\hline \hline
HF & & aug-cc-pVDZ & 38.52 & 30.86 & 34.55 & 28.65 \\ HF & & aug-cc-pVDZ & 38.52 & 30.86 & 34.55 & 28.65 \\
& & aug-cc-pVTZ & 38.58 & 35.82 & 35.68 & 28.65 \\ & & aug-cc-pVTZ & 38.58 & 35.82 & 35.68 & 28.65 \\
& & aug-cc-pVQZ & 39.12 & 35.94 & 35.64 & 28.65 \\ & & aug-cc-pVQZ & 39.12 & 35.94 & 35.64 & 28.65 \\
\\ \\
S & & aug-cc-pVDZ & 38.40 & 27.35 & 23.54 & 26.60 \\ S & & aug-cc-pVDZ & 38.40 & 27.35 & 23.54 & 26.60 \\
& & aug-cc-pVTZ & 39.21 & 27.42 & 23.62 & 26.67 \\ & & aug-cc-pVTZ & 39.21 & 27.42 & 23.62 & 26.67 \\
& & aug-cc-pVQZ & 39.78 & 27.42 & 23.62 & 26.67 \\ & & aug-cc-pVQZ & 39.78 & 27.42 & 23.62 & 26.67 \\
\\ \\
S & VWN5 & aug-cc-pVDZ & 38.10 & 27.76 & 24.40 & 27.10 \\ S & VWN5 & aug-cc-pVDZ & 38.10 & 27.76 & 24.40 & 27.10 \\
& & aug-cc-pVTZ & 38.54 & 27.81 & 24.46 & 27.17 \\ & & aug-cc-pVTZ & 38.54 & 27.81 & 24.46 & 27.17 \\
& & aug-cc-pVQZ & 38.81 & 27.81 & 24.46 & 27.17 \\ & & aug-cc-pVQZ & 38.81 & 27.81 & 24.46 & 27.17 \\
\\ \\
MSFL & & aug-cc-pVDZ & 26.83 & 26.51 & 26.53 & 26.60 \\ MSFL & & aug-cc-pVDZ & 26.83 & 26.51 & 26.53 & 26.60 \\
& & aug-cc-pVTZ & 26.88 & 26.59 & 26.61 & 26.67 \\ & & aug-cc-pVTZ & 26.88 & 26.59 & 26.61 & 26.67 \\
& & aug-cc-pVQZ & 26.82 & 26.60 & 26.62 & 26.67 \\ & & aug-cc-pVQZ & 26.82 & 26.60 & 26.62 & 26.67 \\
\\ \\
MSFL & VWN5 & aug-cc-pVDZ & 28.54 & 26.94 & 27.48 & 27.10 \\ MSFL & VWN5 & aug-cc-pVDZ & 28.54 & 26.94 & 27.48 & 27.10 \\
& & aug-cc-pVTZ & 28.66 & 27.00 & 27.56 & 27.17 \\ & & aug-cc-pVTZ & 28.66 & 27.00 & 27.56 & 27.17 \\
& & aug-cc-pVQZ & 28.64 & 27.00 & 27.56 & 27.17 \\ & & aug-cc-pVQZ & 28.64 & 27.00 & 27.56 & 27.17 \\
\\ \\
MSFL & MSFL & aug-cc-pVDZ & 28.78 & 27.10 & 27.56 & 27.27 \\ MSFL & MSFL & aug-cc-pVDZ & 28.78 & 27.10 & 27.56 & 27.27 \\
& & aug-cc-pVTZ & 28.90 & 27.16 & 27.64 & 27.34 \\ & & aug-cc-pVTZ & 28.90 & 27.16 & 27.64 & 27.34 \\
& & aug-cc-pVQZ & 28.89 & 27.16 & 27.65 & 27.34 \\ & & aug-cc-pVQZ & 28.89 & 27.16 & 27.65 & 27.34 \\
\\ \\
B88 & LYP & aug-mcc-pV8Z & & & & 28.42\fnm[1] \\ B88 & LYP & aug-mcc-pV8Z & & & & 28.42\fnm[1] \\
B3 & LYP & aug-mcc-pV8Z & & & & 27.77\fnm[1] \\ B3 & LYP & aug-mcc-pV8Z & & & & 27.77\fnm[1] \\

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