diff --git a/Manuscript/Ec.tex b/Manuscript/Ec.tex index 4bac422..7d337dc 100644 --- a/Manuscript/Ec.tex +++ b/Manuscript/Ec.tex @@ -160,6 +160,7 @@ The performance of the ground-state gold standard CCSD(T) is also investigated. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \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} 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} +\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} @@ -320,6 +322,49 @@ More details can be found in Ref.~\onlinecite{Nocedal_1999}. \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*} \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}} @@ -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} Method & $E$& $\Delta E$ & $E$ & $\Delta E$ & $E$ & $\Delta E$ & $E$ & $\Delta E$ & $E$ & $\Delta E$ \\ \hline - HF & $-192.8083$ & & $-228.6433$ & & $-224.8354$ & & $-208.8286$ & & -551.3210 & \\ + HF & $-192.8083$ & & $-228.6433$ & & $-224.8354$ & & $-208.8286$ & &$-551.3210$ & \\ \hline MP2 & $-193.4717$ & $-663.4$ & $-229.3508$ & $-707.5$ & $-225.5558$ & $-720.4$ & $-209.5243$ & $-695.7$ & $-551.9825$ & $-661.5$ \\ MP3 & $-193.5094$ & $-701.0$ & $-229.3711$ & $-727.8$ & $-225.5732$ & $-737.8$ & $-209.5492$ & $-720.6$ & $-552.0104$ & $-689.4$ \\ @@ -386,48 +431,11 @@ More details can be found in Ref.~\onlinecite{Nocedal_1999}. \end{table*} \end{squeezetable} -\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:vsNdet}} -\end{figure*} +We first study the convergence of the variational energy as a function of the number of determinants. +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}. +As one can see, the use of optimized orbitals greatly facilitate the convergence towards the FCI limit. +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. +In both cases, the CCSDTQ correlation energy is also represented for comparison purposes. %%%%%%%%%%%%%%%%%%%%%%%%% \section{Conclusion}