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Pierre-Francois Loos 2019-06-10 10:28:45 +02:00
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Manuscript/Ex-srDFT.tex
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@ -573,9 +573,9 @@ However, these results also clearly evidence that special care has to be taken f
\\ \\
\\ \\
Carbon monoxide & $1\,^{1}\Sigma^+ \ra 1\,^{1}\Pi$ & Val. & 8.48\fnm[1] & 0.09 & 0.01 & 0.02 Carbon monoxide & $1\,^{1}\Sigma^+ \ra 1\,^{1}\Pi$ & Val. & 8.48\fnm[1] & 0.09 & 0.01 & 0.02
& 0.05 & 0.00 & & 0.05 & 0.00 & 0.00
& 0.07 & 0.01 & & 0.07 & 0.01 & 0.02
& 0.07 & 0.00 & & 0.07 & 0.00 & 0.02
\\ \\
\\ \\
Ethylene & $1\,^{1}A_{1g} \ra 1\,^{1}B_{3u}$ & Ryd. & 7.43\fnm[3] & -0.12 & -0.04 & Ethylene & $1\,^{1}A_{1g} \ra 1\,^{1}B_{3u}$ & Ryd. & 7.43\fnm[3] & -0.12 & -0.04 &
@ -594,17 +594,17 @@ However, these results also clearly evidence that special care has to be taken f
& 0.00 & 0.01 & & 0.00 & 0.01 &
\\ \\
& $1\,^{1}A_{1g} \ra 1\,^{3}B_{1u}$ & Val. & 4.54\fnm[3] & 0.01 & 0.00 & & $1\,^{1}A_{1g} \ra 1\,^{3}B_{1u}$ & Val. & 4.54\fnm[3] & 0.01 & 0.00 &
& 0.07 & 0.03 & & 0.05 & 0.03 &
& 0.10 & 0.04 &
& 0.08 & 0.04 & & 0.08 & 0.04 &
& 0.07 & 0.04 &
\\ \\
& $1\,^{1}A_{1g} \ra 1\,^{3}B_{3u}$ & Val. & 7.28\fnm[4] & -0.12 & -0.04 & & $1\,^{1}A_{1g} \ra 1\,^{3}B_{3u}$ & Val. & 7.28\fnm[4] & -0.12 & -0.04 &
& -0.03 & 0.00 & & -0.04 & 0.00 &
& 0.00 & 0.00 & & 0.00 & 0.00 &
& 0.00 & 0.02 & & 0.00 & 0.02 &
\\ \\
& $1\,^{1}A_{1g} \ra 1\,^{3}B_{1g}$ & Val. & 8.00\fnm[4] & -0.07 & -0.01 & & $1\,^{1}A_{1g} \ra 1\,^{3}B_{1g}$ & Val. & 8.00\fnm[4] & -0.07 & -0.01 &
& 0.01 & 0.03 & & 0.00 & 0.03 &
& 0.04 & 0.03 & & 0.04 & 0.03 &
& 0.05 & 0.04 & & 0.05 & 0.04 &
\\ \\
@ -680,11 +680,11 @@ To do so, we consider the ground state (${}^{1}\Sigma^+$) of carbon monoxide as
The values of the vertical excitation energies obtained for various methods and basis sets are reported in Table \ref{tab:Mol}. The values of the vertical excitation energies obtained for various methods and basis sets are reported in Table \ref{tab:Mol}.
Figure \ref{fig:CO} represents $\rsmu{}{}(\br{})$ for these two electronic states computed with the AVDZ, AVTZ and AVQZ basis sets. Figure \ref{fig:CO} represents $\rsmu{}{}(\br{})$ for these two electronic states computed with the AVDZ, AVTZ and AVQZ basis sets.
%%% FIG 3 %%% %%% FIG 4 %%%
\begin{figure} \begin{figure}
\includegraphics[width=\linewidth]{CO} \includegraphics[width=\linewidth]{CO}
\caption{$\rsmu{}{\Bas}(z)$ along the molecular axis ($z$) for the ground state ${}^{1}\Sigma^+$ and first singlet excited state ${}^{1}\Pi$ of \ce{CO} for various basis sets $\Bas$. \caption{$\rsmu{}{\Bas}(z)$ along the molecular axis ($z$) for the ground state ${}^{1}\Sigma^+$ (black curve) and first singlet excited state ${}^{1}\Pi$ (red curve) of \ce{CO} for various basis sets $\Bas$.
The carbon and oxygen nuclei are located at $z=-1.249$ and $z=0.893$ bohr, respectively.} The carbon and oxygen nuclei are located at $z=-1.249$ and $z=0.893$ bohr, respectively, and are represented by the thin black lines.}
\label{fig:CO} \label{fig:CO}
\end{figure} \end{figure}
%%% %%% %%% %%% %%% %%%
@ -701,7 +701,7 @@ An interesting point here is that one really needs the PBEot to get chemically-a
We believe that the present result is a direct consequence of the multireference character of the \ce{C2} molecule. We believe that the present result is a direct consequence of the multireference character of the \ce{C2} molecule.
In other words, the UEG on-top density used in the LDA and PBE functionals (see Sec.~\ref{sec:func}) is a particularly bad approximation of the true on-top density. In other words, the UEG on-top density used in the LDA and PBE functionals (see Sec.~\ref{sec:func}) is a particularly bad approximation of the true on-top density.
%%% FIG 4 %%% %%% FIG 5 %%%
\begin{figure} \begin{figure}
\includegraphics[width=\linewidth]{C2} \includegraphics[width=\linewidth]{C2}
\caption{Error in vertical excitation energies $\Eabs$ (in eV) for two doubly-excited states of the carbon dimer for various basis sets and methods. \caption{Error in vertical excitation energies $\Eabs$ (in eV) for two doubly-excited states of the carbon dimer for various basis sets and methods.
@ -721,9 +721,10 @@ We refer the interested reader to the work of Feller et al.\cite{FelPetDav-JCP-1
In the present context, ethylene is a particularly interesting system as it contains a mixture of valence and Rydberg excited states. In the present context, ethylene is a particularly interesting system as it contains a mixture of valence and Rydberg excited states.
Our basis set corrected vertical excitation energies are gathered in Table \ref{tab:Mol} and depicted in Fig.~\ref{fig:C2H4}. Our basis set corrected vertical excitation energies are gathered in Table \ref{tab:Mol} and depicted in Fig.~\ref{fig:C2H4}.
Except for one particular excitation (the lowest singlet-triplet excitation $1\,^{1}A_{1g} \ra 1\,^{3}B_{1u}$), the exFCI+PBEot/AVDZ excitation energies are chemically accurate and the errors drop further when one goes to the triple-$\zeta$ basis. Except for one particular excitation (the lowest singlet-triplet excitation $1\,^{1}A_{1g} \ra 1\,^{3}B_{1u}$), the exFCI+PBEot/AVDZ excitation energies are chemically accurate and the errors drop further when one goes to the triple-$\zeta$ basis.
%(Note that one cannot afford exFCI/AVQZ calculations for ethylene.)
Consistently with the previous examples, the LDA and PBE functionals are slightly less accurate, although they still correct the excitation energies in the right direction. Consistently with the previous examples, the LDA and PBE functionals are slightly less accurate, although they still correct the excitation energies in the right direction.
%%% FIG 5 %%% %%% FIG 6 %%%
\begin{figure} \begin{figure}
\includegraphics[width=\linewidth]{C2H4} \includegraphics[width=\linewidth]{C2H4}
\caption{Error in vertical excitation energies $\Eabs$ (in eV) of ethylene for various basis sets and methods. \caption{Error in vertical excitation energies $\Eabs$ (in eV) of ethylene for various basis sets and methods.