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corrections Secs. IV and V

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kossoski 2021-07-28 13:48:05 +02:00
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Manuscript/Ec.tex
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@ -593,10 +593,10 @@ The five-point weighted linear fit using the five largest variational wave funct
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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.
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, {\color{red} [I don't understand what is meant here:]} 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.
Accordingly to these observations, all our FCI correlation energy estimates have been produced with the set of optimized orbitals.
According 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 corrections.
@ -644,7 +644,7 @@ Note that it is pleasing to see that, although different geometries are consider
\label{tab:stats}}
\begin{ruledtabular}
\begin{tabular}{lcdddd}
Method & Scaling & \tabc{MAE} & \tabc{MSE} & \tabc{Min} & \tabc{Max} \\
Method & Scaling & \tabc{MAE} & \tabc{MSE} & \tabc{Max} & \tabc{Min} \\
\hline
MP2 & $\order{N^5}$ & 68.4 & 68.4 & 80.6 & 57.8 \\
MP3 & $\order{N^6}$ & 46.5 & 46.5 & 58.4 & 37.9 \\
@ -675,12 +675,12 @@ The FCI correlation energy estimate is represented as a black line for reference
Key statistical quantities [mean absolute error (MAE), mean signed error (MSE), and minimum (Min) and maximum (Max) absolute errors with respect to the FCI reference values] are also reported in Table \ref{tab:stats} for each method as well as their formal computational scaling.
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}
Unfortunately, CC with singles, doubles, triples, quadruples, and pentuples (CCSDTQP) calculations are out of reach here. \cite{Hirata_2000,Kallay_2001}
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, while CCSD(T) and CR-CC(2,3) performs equally well.
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, while CCSD(T) and CR-CC(2,3) perform equally well.
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}
Second, let us look into the series of MP approximations which is known, as mentioned in Sec.~\ref{sec:intro}, to potentially exhibit ``surprising'' behaviors 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 weakly-correlated systems.