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Pierre-Francois Loos 2021-07-25 22:29:36 +02:00
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Manuscript/Ec.tex
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@ -594,7 +594,7 @@ The five-point weighted linear fit using the five largest variational wave funct
\begin{figure}
\includegraphics[width=\linewidth]{Benzene_EvsNdetLO}
\caption{$\Delta \Evar$ (solid) and $\Delta \Evar + \EPT$ (dashed) as functions of the number of determinants $\Ndet$ in the variational space for the benzene molecule.
Three sets of orbitals are considered: natural orbitals (NOs, in red), localized orbitals (LOs, in red), and optimized orbitals (OOs, in blue).
Three sets of orbitals are considered: natural orbitals (NOs, in red), localized orbitals (LOs, in green), and optimized orbitals (OOs, in blue).
The CCSDTQ correlation energy is represented as a thick black line.
\label{fig:BenzenevsNdet}}
\end{figure}
@ -608,7 +608,7 @@ A similar improvement is observed going from natural to localized orbitals.
Accordingly to these observations, all our FCI correlation energy estimates have been produced with the set of optimized orbitals.
To do so, we have then extrapolated the orbital-optimized variational CIPSI correlation energies to $\EPT = 0$ via a weighted five-point linear fit using the five largest variational wave functions (see Fig.~\ref{fig:vsEPT2}).
The fitting weights have been taken as the inverse square of the perturbative correction.
The fitting weights have been taken as the inverse square of the perturbative corrections.
Our final FCI correlation energy estimates are reported in Tables \ref{tab:Tab5-VDZ} and \ref{tab:Tab6-VDZ} for the five- and six-membered rings, respectively, alongside their corresponding fitting error.
The stability of these estimates are illustrated by the results gathered in Table \ref{tab:fit} where we report, for each system, the extrapolated correlation energies $\Delta \Eextrap$ and their associated fitting errors obtained via weighted linear fits varying the number of fitting points from $3$ to $7$.
Although we cannot provide a mathematically rigorous error bar, the data provided by Table \ref{tab:fit} show that the extrapolation procedure is robust and that our FCI estimates are very likely accurate to a few tenths of a millihartree.
@ -638,8 +638,8 @@ Note that it is pleasing to see that, although different geometries are consider
\includegraphics[width=0.32\textwidth]{Tetrazine_MPCC}
\includegraphics[width=0.32\textwidth]{Triazine_MPCC}
\caption{Convergence of the correlation energy (in \SI{}{\milli\hartree}) as a function of the computational cost for the twelve cyclic molecules represented in Fig.~\ref{fig:mol}.
Three series of methods are considered: i) MP2, MP3, MP4, and MP5 (blue), ii) CC2, CC3, and CC4 (green), and iii) CCSD, CCSDT, CCSDTQ (red)
The CIPSI estimate of the correlation energy is represented as a black line.
Three series of methods are considered: i) MP2, MP3, MP4, and MP5 (blue), ii) CC2, CC3, and CC4 (green), and iii) CCSD, CCSDT, CCSDTQ (red).
The CIPSI estimate of the correlation energy is represented as a black line for reference.
\label{fig:MPCC}}
\end{figure*}
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@ -690,13 +690,13 @@ As usually observed, CCSD(T) (MAE of \SI{4.5}{\milli\hartree}) provides similar
Second, let us look into the series of MP approximations which is known, as mentioned in Sec.~\ref{sec:intro}, to potentially exhibit ``surprising'' behavior depending on the type of correlation at play.\cite{Laidig_1985,Knowles_1985,Handy_1985,Gill_1986,Laidig_1987,Nobes_1987,Gill_1988,Gill_1988a,Lepetit_1988,Malrieu_2003}
(See Ref.~\onlinecite{Marie_2021} for a detailed discussion).
For each system, the MP series decreases monotonically up to MP4 but raises quite significantly when one takes into account the fifth-order correction.
We note that the MP4 correlation energy is always quite accurate (MAE of \SI{2.1}{\milli\hartree}) and is only a few millihartree higher than the FCI value (except in the case of tetrazine where the MP4 number is very slightly below the reference value): MP5 (MAE of \SI{9.4}{\milli\hartree}) is thus systematically worse than MP4 for these systems.
We note that the MP4 correlation energy is always quite accurate (MAE of \SI{2.1}{\milli\hartree}) and is only a few millihartree higher than the FCI value (except in the case of tetrazine where the MP4 number is very slightly below the reference value): MP5 (MAE of \SI{9.4}{\milli\hartree}) is thus systematically worse than MP4 for these weakly-correlated systems.
Importantly here, one notices that MP4 [which scales as $\order*{N^7}$] is systematically on par with the more expensive $\order*{N^{10}}$ CCSDTQ method which exhibits a slightly smaller MAE of \SI{1.8}{\milli\hartree}.
Third, we investigate the approximate CC series of methods CC2, CC3, and CC4.
As observed in our recent study on excitation energies, \cite{Loos_2021} CC4, which returns a MAE of \SI{1.5}{\milli\hartree}, is an outstanding approximation to its CCSDTQ parent and is, in the present case, even slightly more accurate.
As observed in our recent study on excitation energies, \cite{Loos_2021} CC4, which returns a MAE of \SI{1.5}{\milli\hartree}, is an outstanding approximation to its CCSDTQ parent and is, in the present case, even slightly more accurate in terms of mean errors as well as maximum and minimum absolute errors.
Moreover, we observe that CC3 (MAE of \SI{2.7}{\milli\hartree}) and CC4 provide correlation energies that only deviate by one or two millihartree, showing that the iterative CC3 method is particularly effective for ground-state energetics and outperforms both the perturbative CCSD(T) and iterative CCSDT models.
As a final remark, we would like to mention that even if the two families of CC methods studied here are known to be non-variational, for the present set of weakly-correlated molecular systems, they never produce a lower energy than the FCI estimate as illustrated by the systematic equality between MAEs and MSEs.
As a final remark, we would like to mention that even if the two families of CC methods studied here are known to be non-variational (see Sec.~\ref{sec:intro}), for the present set of weakly-correlated molecular systems, they never produce a lower energy than the FCI estimate as illustrated by the systematic equality between MAEs and MSEs.
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\section{Conclusion}