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47
Manuscript/Ex-srDFT.tex
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@ -705,7 +705,7 @@ However, these results also clearly evidence that special care has to be taken f
\subsection{Doubly-Excited States of the Carbon Dimer} \subsection{Doubly-Excited States of the Carbon Dimer}
\label{sec:C2} \label{sec:C2}
%======================= %=======================
In order to have a miscellaneous test set of excitations, in a fourth time, we propose to study some doubly-excited states of the carbon dimer \ce{C2}, a prototype system for strongly correlated and multireference systems. \cite{AbrShe-JCP-04, AbrShe-CPL-05, Var-JCP-08, PurZhaKra-JCP-09, AngCimPas-MP-12, BooCleThoAla-JCP-11, Sha-JCP-15, SokCha-JCP-16, HolUmrSha-JCP-17, VarRoc-PTRSMPES-18} In order to have a miscellaneous test set of excitations, in a third time, we propose to study some doubly-excited states of the carbon dimer \ce{C2}, a prototype system for strongly correlated and multireference systems. \cite{AbrShe-JCP-04, AbrShe-CPL-05, Var-JCP-08, PurZhaKra-JCP-09, AngCimPas-MP-12, BooCleThoAla-JCP-11, Sha-JCP-15, SokCha-JCP-16, HolUmrSha-JCP-17, VarRoc-PTRSMPES-18}
These two valence excitations --- $1\,^{1}\Sigma_g^+ \ra 1\,^{1}\Delta_g$ and $1\,^{1}\Sigma_g^+ \ra 2\,^{1}\Sigma_g^+$ --- are both of $(\pi,\pi) \ra (\si,\si)$ character. These two valence excitations --- $1\,^{1}\Sigma_g^+ \ra 1\,^{1}\Delta_g$ and $1\,^{1}\Sigma_g^+ \ra 2\,^{1}\Sigma_g^+$ --- are both of $(\pi,\pi) \ra (\si,\si)$ character.
They have been recently studied with state-of-the-art methods, and have been shown to be ``pure'' doubly-excited states as they do not involve single excitations. \cite{LooBogSceCafJac-JCTC-19} They have been recently studied with state-of-the-art methods, and have been shown to be ``pure'' doubly-excited states as they do not involve single excitations. \cite{LooBogSceCafJac-JCTC-19}
The vertical excitation energies associated with these transitions are reported in Table \ref{tab:Mol} and represented in Fig.~\ref{fig:C2}. The vertical excitation energies associated with these transitions are reported in Table \ref{tab:Mol} and represented in Fig.~\ref{fig:C2}.
@ -723,48 +723,29 @@ In other words, the UEG on-top density used in the LDA and PBE functionals (see
\end{figure} \end{figure}
%%% %%% %%% %%% %%% %%%
It is interesting to study the behavior of \manu{the key quantities involved in the basis set correction} for different states as the basis set incompleteness error is obviously state specific. It is interesting to study the behavior of the key quantities involved in the basis set correction for different states as the basis set incompleteness error is obviously state specific.
%\manu{To do so, we report the value of the range separation parameter in real space $\rsmu{}{\Bas}(\br{})$, the value of the energetic correction $\be{\text{c,md}}{\sr,\PBE}\qty(\n{}{}(\br{}),s(\br{}),\zeta(\br{}),\rsmu{}{\Bas}(\br{}) $ and the on-top pair density $\n{2}{\Bas}(\br{},\br{})$ computed with different basis sets for the ground state and second excited state of the carbon dimer which are both of $\Sigma_g^+$ symmetry, in Figures . } %\manu{To do so, we report the value of the range separation parameter in real space $\rsmu{}{\Bas}(\br{})$, the value of the energetic correction $\be{\text{c,md}}{\sr,\PBE}\qty(\n{}{}(\br{}),s(\br{}),\zeta(\br{}),\rsmu{}{\Bas}(\br{}) $ and the on-top pair density $\n{2}{\Bas}(\br{},\br{})$ computed with different basis sets for the ground state and second excited state of the carbon dimer which are both of $\Sigma_g^+$ symmetry, in Figures . }
We report $\rsmu{}{\Bas}(z)$, along the nuclear axis ($z$) for these two electronic states computed with the AVDZ, AVTZ and AVQZ basis sets. We report $\rsmu{}{\Bas}(z)$, along the nuclear axis ($z$) for the two $^1 \Sigma_g^+$ electronic states of \ce{C2} computed with the AVDZ, AVTZ and AVQZ basis sets.
\manu{These figures illustrate several important things: \manu{These figures illustrate several important things:
i) the maximal values of $\rsmu{}{\Bas}(\br{})$ are systematically close to the nuclei, a signature of the atom-centered basis set, i) the maximal values of $\rsmu{}{\Bas}(\br{})$ are systematically close to the nuclei, a signature of the atom-centered basis set,
ii) the overall values of $\rsmu{}{\Bas}(\br{})$ increase with the basis set, which reflects the improvement of the description of the correlation effects when enlarging the basis set, ii) the overall values of $\rsmu{}{\Bas}(\br{})$ increase with the basis set, which reflects the improvement of the description of the correlation effects when enlarging the basis set,
iii) the value of $\rsmu{}{\Bas}(\br{})$ are slightly larger near the oxygen atom, which traduces the fact that the inter-electronic distance is higher than close to the carbon atom due to a higher nuclear charge. } iii) the value of $\rsmu{}{\Bas}(\br{})$ are slightly larger near the oxygen atom, which traduces the fact that the inter-electronic distance is higher than close to the carbon atom due to a higher nuclear charge. }
%%% FIG 4 %%% %%% FIG 4 %%%
\begin{figure} \begin{figure*}
\includegraphics[width=\linewidth]{C2_mu} \includegraphics[height=0.45\linewidth]{C2_mu}
\caption{ C$_2$: $\rsmu{}{\Bas}(z)$ along the molecular axis ($z$) for the ground state (black curve) and second singlet excited state (red curve) which are both of $\Sigma_g^+$ symmetry for various basis sets $\Bas$. The two carbon nuclei are located at $z=-1.180$ and $z=1.180$ bohr, respectively, and are represented by the thin black lines.} \hspace{0.5cm}
\includegraphics[height=0.45\linewidth]{C2_PBEot}
\includegraphics[height=0.45\linewidth]{C2_PBE.pdf}
\hspace{0.5cm}
\includegraphics[height=0.45\linewidth]{C2_n2.pdf}
\caption{$\rsmu{}{\Bas}$ (top left), $\n{}{\Bas} \be{\text{c,md}}{\sr,\PBEot}$ (top right), $\n{}{\Bas} \be{\text{c,md}}{\sr,\PBE}$ (bottom left) and $\n{2}{\Bas}$ (bottom right) along the molecular axis ($z$) for the ground state (black curve) and second doubly-excited state (red curve) of \ce{C2} for various basis sets $\Bas$.
The two electronic states are both of $\Sigma_g^+$ symmetry.
The carbon nuclei are located at $z= \pm 1.180$ bohr and represented by the thin black lines.}
\label{fig:C2_mu} \label{fig:C2_mu}
\end{figure} \end{figure*}
%%% %%% %%% %%% %%% %%%
%%% FIG 4 %%%
\begin{figure}
\includegraphics[width=\linewidth]{C2_PBEot.pdf}
\caption{ C$_2$: $\pbeotint$ along the molecular axis ($z$) for the ground state (black curve) and second singlet excited state (red curve) which are both of $\Sigma_g^+$ symmetry for various basis sets $\Bas$. The two carbon nuclei are located at $z=-1.180$ and $z=1.180$ bohr, respectively, and are represented by the thin black lines.}
\label{fig:C2_PBEot}
\end{figure}
%%% %%% %%%
%%% FIG 4 %%%
\begin{figure}
\includegraphics[width=\linewidth]{C2_PBE.pdf}
\caption{ C$_2$: $\pbeint$ along the molecular axis ($z$) for the ground state (black curve) and second singlet excited state (red curve) which are both of $\Sigma_g^+$ symmetry for various basis sets $\Bas$. The two carbon nuclei are located at $z=-1.180$ and $z=1.180$ bohr, respectively, and are represented by the thin black lines.}
\label{fig:C2_PBE}
\end{figure}
%%% %%% %%%
%%% FIG 4 %%%
\begin{figure}
\includegraphics[width=\linewidth]{C2_n2.pdf}
\caption{ C$_2$: $\n{2}{\Bas}(\br{})$ along the molecular axis ($z$) for the ground state (black curve) and second singlet excited state (red curve) which are both of $\Sigma_g^+$ symmetry for various basis sets $\Bas$. The two carbon nuclei are located at $z=-1.180$ and $z=1.180$ bohr, respectively, and are represented by the thin black lines.}
\label{fig:C2_n2}
\end{figure}
%%% %%% %%%
%======================= %=======================
\subsection{Ethylene} \subsection{Ethylene}
\label{sec:C2H4} \label{sec:C2H4}

32
Manuscript/SI/Ex-srDFT-SI.tex
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@ -200,13 +200,13 @@ C 0.000000 0.000000 -0.624021
\end{verbatim} \end{verbatim}
%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Carbon monoxyde} %\subsection{Carbon monoxyde}
%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{verbatim} %\begin{verbatim}
C 0.000000 0.000000 -1.249421 %C 0.000000 0.000000 -1.249421
0 0.000000 0.000000 0.892667 %0 0.000000 0.000000 0.892667
\end{verbatim} %\end{verbatim}
%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Ethylene} \subsection{Ethylene}
@ -449,17 +449,17 @@ Here, we report the absolute energetic corrections for each state of each molecu
& -0.049\,208 & -0.021\,292 & -0.01\,0257 & -0.049\,208 & -0.021\,292 & -0.01\,0257
\\ \\
\\ \\
Carbon monoxyde & $1\,^{1}\Sigma^+$ % Carbon monoxyde & $1\,^{1}\Sigma^+$
& -0.074\,328 & -0.031\,117 & -0.015\,510 % & -0.074\,328 & -0.031\,117 & -0.015\,510
& -0.084\,655 & -0.035\,318 & -0.017\,142 % & -0.084\,655 & -0.035\,318 & -0.017\,142
& -0.076\,668 & -0.077\,437 & -0.018\,768 % & -0.076\,668 & -0.077\,437 & -0.018\,768
\\ % \\
& $1\,^{1}\Pi$ % & $1\,^{1}\Pi$
& -0.075\,790 & -0.031\,456 & -0.016\,083 % & -0.075\,790 & -0.031\,456 & -0.016\,083
& -0.085\,494 & -0.035\,255 & -0.017\,182 % & -0.085\,494 & -0.035\,255 & -0.017\,182
& -0.036\,301 & -0.036\,359 & -0.018\,855 % & -0.036\,301 & -0.036\,359 & -0.018\,855
\\ % \\
\\ % \\
Water & $1\,^{1}A_1$ Water & $1\,^{1}A_1$
& -0.058\,765 & -0.024\,014 & -0.011\,990 & -0.058\,765 & -0.024\,014 & -0.011\,990
& -0.066\,603 & -0.027\,236 & -0.013\,127 & -0.066\,603 & -0.027\,236 & -0.013\,127