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Pierre-Francois Loos 2021-07-23 18:25:16 +02:00
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@ -384,7 +384,7 @@ More details can be found in Ref.~\onlinecite{Nocedal_1999}.
\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 five-point weighted linear fit using the four largest variational wave functions for each set is depicted as a dashed black line.
The five-point weighted linear fit using the five largest variational wave functions for each set is depicted as a dashed black line.
The weights are taken as the inverse square of the perturbative corrections.
The CCSDTQ correlation energy is also represented as a thick black line.
\label{fig:vsEPT2}}
@ -581,9 +581,8 @@ As compared to natural orbitals (solid red lines), one can see that, for a given
Adding the perturbative correction $\EPT$ yields similar curves for both sets of orbitals (dashed lines).
This indicates that, for a given number of determinants, $\EPT$ (which, we recall, provides a qualitative idea to the distance to the FCI limit) is much smaller for optimized orbitals than for natural orbitals.
This is further evidenced in Fig.~\ref{fig:vsEPT2} where we show the behavior of $\Delta \Evar$ as a function of $\EPT$ for both sets of orbitals.
The five-point weighted linear fit using the four largest variational wave functions are also represented (dashed black lines).
The CCSDTQ correlation energy (solid black line) is also reported for comparison purposes in Figs.~\ref{fig:vsNdet} and \ref{fig:vsEPT2}
From Fig.~\ref{fig:vsEPT2}, it is clear that the behavior of $\Delta \Evar$ is much more linear and produces smaller $\EPT$ values, hence facilitating the extrapolation procedure to the FCI limit (see below).
From Fig.~\ref{fig:vsEPT2}, it is clear that, using optimized orbitals, the behavior of $\Delta \Evar$ is much more linear and produces smaller $\EPT$ values, hence facilitating the extrapolation procedure to the FCI limit (see below).
The five-point weighted linear fit using the five largest variational wave functions are also represented (dashed black lines), while the CCSDTQ correlation energy (solid black line) is reported for comparison purposes in Figs.~\ref{fig:vsNdet} and \ref{fig:vsEPT2}.
%%% FIG 4 %%%
\begin{figure}
@ -595,19 +594,19 @@ From Fig.~\ref{fig:vsEPT2}, it is clear that the behavior of $\Delta \Evar$ is m
\end{figure}
%%% %%% %%%
Figure~\ref{fig:BenzenevsNdet} compares the convergence of $\Delta \Evar$ for the natural, localized, and optimized sets of orbitals in the particular case of benzene.
As mentioned in Sec.~\ref{sec:compdet}, although both sets break the spatial symmetry to take advantage of the local nature of electron correlation, optimized orbitals further improve on the use of localized orbitals.
Figure \ref{fig:BenzenevsNdet} compares the convergence of $\Delta \Evar$ for the natural, localized, and optimized sets of orbitals in the particular case of benzene.
As mentioned in Sec.~\ref{sec:compdet}, although both the localized and optimized orbitals break the spatial symmetry to take advantage of the local nature of electron correlation, the latter set further improve on the use of former set.
More quantitatively, optimized orbitals produce the same variational energy as localized orbitals with, roughly, a ten-fold reduction in the number of determinants.
A similar improvement is observed going from natural to localized orbitals.
\titou{Comment on PT2 for 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 four largest variational wave functions (see Fig.~\ref{fig:vsEPT2}).
The fitting weights have been taken as the inverse square of the perturbative correction at each point.
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.
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 8.
Although we cannot provide a mathematically rigorous error bar, the data of 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.
Logically, the FCI estimates for the five-membered rings seem slightly more accurate than for the larger six-membered rings.
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.
Logically, the FCI estimates for the five-membered rings seem slightly more accurate than for the (larger) six-membered rings.
Note that it is pleasing to see that, although different geometries are considered, our present estimate of the frozen-core correlation energy of the benzene molecule in the cc-pVDZ basis is very close to the one reported in Refs.~\onlinecite{Eriksen_2020,Loos_2020e}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -679,20 +678,19 @@ Key statistical quantities [mean absolute error (MAE), mean signed error (MSE),
First, we investigate the ``complete'' and well-established series of methods CCSD, CCSDT, and CCSDTQ.
Unfortunately, CC with singles, doubles, triples, quadruples and pentuples (CCSDTQP) calculations are out of reach here. \cite{Hirata_2000,Kallay_2001}
As expected, going from CCSD to CCSDTQ, one improves systematically and quickly the correlation energies with MAEs of $39.4$, $4.5$, \SI{1.8}{\milli\hartree} for CCSD, CCSDT, and CCSDTQ, respectively.
As usually observed, CCSD(T) (MAE of \SI{4.5}{\milli\hartree}) provides similar correlation energies than the more expensive CCSDT method by computing perturbatively the triple excitations.
As expected for the present set of weakly correlated systems, going from CCSD to CCSDTQ, one improves systematically and quickly the correlation energies with MAEs of $39.4$, $4.5$, \SI{1.8}{\milli\hartree} for CCSD, CCSDT, and CCSDTQ, respectively.
As usually observed, CCSD(T) (MAE of \SI{4.5}{\milli\hartree}) provides similar correlation energies than the more expensive CCSDT method by computing perturbatively (instead of iteratively) the triple excitations.
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 sole 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.
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}.
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.
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}.
Second, we investigate the approximate CC series of methods CC2, CC3, and CC4.
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.
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, which shows that CC3 is particularly effective for ground-state energetics and outperforms both CCSD(T) and CCSDT.
As a final remark, we would like to mention that even if the present perturbative and coupled-cluster methods are known to non-variational, they very rarely produce a lower energy than the FCI estimate as illustrated by the systematic equality between MAEs and MSEs.
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.
%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusion}