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@ -160,6 +160,7 @@ The performance of the ground-state gold standard CCSD(T) is also investigated.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Computational details} \section{Computational details}
\label{sec:compdet}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The geometries of the twelve systems considered in the present study have been all obtained at the CC3/aug-cc-pVTZ level of theory and have been extracted from a previous study. \cite{Loos_2020a} The geometries of the twelve systems considered in the present study have been all obtained at the CC3/aug-cc-pVTZ level of theory and have been extracted from a previous study. \cite{Loos_2020a}
Note that, for the sake of consistency, the geometry of benzene considered here is different from one of Ref.~\onlinecite{Loos_2020e} which has been computed at a lower level of theory [MP2/6-31G(d)]. \cite{Schreiber_2008} Note that, for the sake of consistency, the geometry of benzene considered here is different from one of Ref.~\onlinecite{Loos_2020e} which has been computed at a lower level of theory [MP2/6-31G(d)]. \cite{Schreiber_2008}
@ -185,6 +186,7 @@ We have found that $\expval*{\Hat{S}^2}$ is, nonetheless, very close to zero for
%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%
\section{CIPSI with optimized orbitals} \section{CIPSI with optimized orbitals}
\label{sec:OO-CIPSI}
%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%
Here, we provide key details about the CIPSI method \cite{Huron_1973,Garniron_2019} as well as the orbital optimization procedure which has been shown to be highly effective in the context of SHCI by Umrigar and coworkers. \cite{Eriksen_2020,Yao_2020,Yao_2021} Here, we provide key details about the CIPSI method \cite{Huron_1973,Garniron_2019} as well as the orbital optimization procedure which has been shown to be highly effective in the context of SHCI by Umrigar and coworkers. \cite{Eriksen_2020,Yao_2020,Yao_2021}
@ -320,6 +322,49 @@ More details can be found in Ref.~\onlinecite{Nocedal_1999}.
\label{sec:res} \label{sec:res}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure*}
\includegraphics[width=0.24\textwidth]{Cyclopentadiene_EvsNdet}
\includegraphics[width=0.24\textwidth]{Furan_EvsNdet}
\includegraphics[width=0.24\textwidth]{Imidazole_EvsNdet}
\includegraphics[width=0.24\textwidth]{Pyrrole_EvsNdet}
\\
\includegraphics[width=0.24\textwidth]{Thiophene_EvsNdet}
\includegraphics[width=0.24\textwidth]{Benzene_EvsNdet}
\includegraphics[width=0.24\textwidth]{Pyrazine_EvsNdet}
\includegraphics[width=0.24\textwidth]{Pyridazine_EvsNdet}
\\
\includegraphics[width=0.24\textwidth]{Pyridine_EvsNdet}
\includegraphics[width=0.24\textwidth]{Pyrimidine_EvsNdet}
\includegraphics[width=0.24\textwidth]{Tetrazine_EvsNdet}
\includegraphics[width=0.24\textwidth]{Triazine_EvsNdet}
\caption{$\Delta \Evar$ (solid) and $\Delta \Evar + \EPT$ (dashed) as functions of the number of determinants $\Ndet$ in the variational space for the twelve cyclic molecules represented in Fig.~\ref{fig:mol}.
Two sets of orbitals are considered: natural orbitals (NOs, in red) and optimized orbitals (OOs, in blue).
The CCSDTQ correlation energy is represented as a thick black line.
\label{fig:vsNdet}}
\end{figure*}
\begin{figure*}
\includegraphics[width=0.24\textwidth]{Cyclopentadiene_EvsPT2}
\includegraphics[width=0.24\textwidth]{Furan_EvsPT2}
\includegraphics[width=0.24\textwidth]{Imidazole_EvsPT2}
\includegraphics[width=0.24\textwidth]{Pyrrole_EvsPT2}
\\
\includegraphics[width=0.24\textwidth]{Thiophene_EvsPT2}
\includegraphics[width=0.24\textwidth]{Benzene_EvsPT2}
\includegraphics[width=0.24\textwidth]{Pyrazine_EvsPT2}
\includegraphics[width=0.24\textwidth]{Pyridazine_EvsPT2}
\\
\includegraphics[width=0.24\textwidth]{Pyridine_EvsPT2}
\includegraphics[width=0.24\textwidth]{Pyrimidine_EvsPT2}
\includegraphics[width=0.24\textwidth]{Tetrazine_EvsPT2}
\includegraphics[width=0.24\textwidth]{Triazine_EvsPT2}
\caption{$\Delta \Evar$ as a function of $\EPT$ for the twelve cyclic molecules represented in Fig.~\ref{fig:mol}.
Two sets of orbitals are considered: natural orbitals (NOs, in red) and optimized orbitals (OOs, in blue).
The four-point linear fit using the four largest variational wave functions for each set is depicted as a dashed black line.
The CCSDTQ correlation energy is also represented as a thick black line.
\label{fig:vsEPT2}}
\end{figure*}
\begin{table*} \begin{table*}
\caption{Total energy $E$ (in \SI{}{\hartree}) and correlation energy $\Delta E$ (in \SI{}{\milli\hartree}) for the frozen-core ground state of five-membered rings in the cc-pVDZ basis set. \caption{Total energy $E$ (in \SI{}{\hartree}) and correlation energy $\Delta E$ (in \SI{}{\milli\hartree}) for the frozen-core ground state of five-membered rings in the cc-pVDZ basis set.
\label{tab:Tab5-VDZ}} \label{tab:Tab5-VDZ}}
@ -329,7 +374,7 @@ More details can be found in Ref.~\onlinecite{Nocedal_1999}.
\cline{2-3} \cline{4-5} \cline{6-7} \cline{8-9} \cline{10-11} \cline{2-3} \cline{4-5} \cline{6-7} \cline{8-9} \cline{10-11}
Method & $E$& $\Delta E$ & $E$ & $\Delta E$ & $E$ & $\Delta E$ & $E$ & $\Delta E$ & $E$ & $\Delta E$ \\ Method & $E$& $\Delta E$ & $E$ & $\Delta E$ & $E$ & $\Delta E$ & $E$ & $\Delta E$ & $E$ & $\Delta E$ \\
\hline \hline
HF & $-192.8083$ & & $-228.6433$ & & $-224.8354$ & & $-208.8286$ & & -551.3210 & \\ HF & $-192.8083$ & & $-228.6433$ & & $-224.8354$ & & $-208.8286$ & &$-551.3210$ & \\
\hline \hline
MP2 & $-193.4717$ & $-663.4$ & $-229.3508$ & $-707.5$ & $-225.5558$ & $-720.4$ & $-209.5243$ & $-695.7$ & $-551.9825$ & $-661.5$ \\ MP2 & $-193.4717$ & $-663.4$ & $-229.3508$ & $-707.5$ & $-225.5558$ & $-720.4$ & $-209.5243$ & $-695.7$ & $-551.9825$ & $-661.5$ \\
MP3 & $-193.5094$ & $-701.0$ & $-229.3711$ & $-727.8$ & $-225.5732$ & $-737.8$ & $-209.5492$ & $-720.6$ & $-552.0104$ & $-689.4$ \\ MP3 & $-193.5094$ & $-701.0$ & $-229.3711$ & $-727.8$ & $-225.5732$ & $-737.8$ & $-209.5492$ & $-720.6$ & $-552.0104$ & $-689.4$ \\
@ -386,48 +431,11 @@ More details can be found in Ref.~\onlinecite{Nocedal_1999}.
\end{table*} \end{table*}
\end{squeezetable} \end{squeezetable}
\begin{figure*} We first study the convergence of the variational energy as a function of the number of determinants.
\includegraphics[width=0.24\textwidth]{Cyclopentadiene_EvsNdet} For the natural and optimized orbital sets we report, in Fig.~\ref{fig:vsNdet}, the evolution of the variational correlation energy $\Delta \Evar$ with respect to the number of determinants for the set of twelve cyclic molecules represented in Fig.~\ref{fig:mol}.
\includegraphics[width=0.24\textwidth]{Furan_EvsNdet} As one can see, the use of optimized orbitals greatly facilitate the convergence towards the FCI limit.
\includegraphics[width=0.24\textwidth]{Imidazole_EvsNdet} This is further evidenced in Fig.~\ref{fig:vsEPT2} where we show the behavior of $\Delta \Evar$ as a function of $\EPT$ as well as its 4-point linear fit using the four largest variational wave functions.
\includegraphics[width=0.24\textwidth]{Pyrrole_EvsNdet} In both cases, the CCSDTQ correlation energy is also represented for comparison purposes.
\\
\includegraphics[width=0.24\textwidth]{Thiophene_EvsNdet}
\includegraphics[width=0.24\textwidth]{Benzene_EvsNdet}
\includegraphics[width=0.24\textwidth]{Pyrazine_EvsNdet}
\includegraphics[width=0.24\textwidth]{Pyridazine_EvsNdet}
\\
\includegraphics[width=0.24\textwidth]{Pyridine_EvsNdet}
\includegraphics[width=0.24\textwidth]{Pyrimidine_EvsNdet}
\includegraphics[width=0.24\textwidth]{Tetrazine_EvsNdet}
\includegraphics[width=0.24\textwidth]{Triazine_EvsNdet}
\caption{$\Delta \Evar$ (solid) and $\Delta \Evar + \EPT$ (dashed) as functions of the number of determinants $\Ndet$ in the variational space for the twelve cyclic molecules represented in Fig.~\ref{fig:mol}.
Two sets of orbitals are considered: natural orbitals (NOs, in red) and optimized orbitals (OOs, in blue).
The CCSDTQ correlation energy is represented as a thick black line.
\label{fig:vsNdet}}
\end{figure*}
\begin{figure*}
\includegraphics[width=0.24\textwidth]{Cyclopentadiene_EvsPT2}
\includegraphics[width=0.24\textwidth]{Furan_EvsPT2}
\includegraphics[width=0.24\textwidth]{Imidazole_EvsPT2}
\includegraphics[width=0.24\textwidth]{Pyrrole_EvsPT2}
\\
\includegraphics[width=0.24\textwidth]{Thiophene_EvsPT2}
\includegraphics[width=0.24\textwidth]{Benzene_EvsPT2}
\includegraphics[width=0.24\textwidth]{Pyrazine_EvsPT2}
\includegraphics[width=0.24\textwidth]{Pyridazine_EvsPT2}
\\
\includegraphics[width=0.24\textwidth]{Pyridine_EvsPT2}
\includegraphics[width=0.24\textwidth]{Pyrimidine_EvsPT2}
\includegraphics[width=0.24\textwidth]{Tetrazine_EvsPT2}
\includegraphics[width=0.24\textwidth]{Triazine_EvsPT2}
\caption{$\Delta \Evar$ as a function of $\EPT$ for the twelve cyclic molecules represented in Fig.~\ref{fig:mol}.
Two sets of orbitals are considered: natural orbitals (NOs, in red) and optimized orbitals (OOs, in blue).
The four-point linear fit using the four largest variational wave functions for each set is depicted as a dashed black line.
The CCSDTQ correlation energy is also represented as a thick black line.
\label{fig:vsNdet}}
\end{figure*}
%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusion} \section{Conclusion}