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@ -4,7 +4,7 @@
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\newcommand{\ie}{\textit{i.e.}}
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\newcommand{\eg}{\textit{e.g.}}
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\newcommand{\alert}[1]{\textcolor{black}{#1}}
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\newcommand{\alert}[1]{\textcolor{red}{#1}}
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\usepackage[normalem]{ulem}
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\newcommand{\titou}[1]{\textcolor{red}{#1}}
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\newcommand{\denis}[1]{\textcolor{blue}{#1}}
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@ -55,7 +55,9 @@
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\newcommand{\EHF}{E_\text{HF}}
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\newcommand{\Ec}{E_\text{c}}
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\newcommand{\Evar}{E_\text{var}}
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\newcommand{\Efinal}{E_\text{final}}
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\newcommand{\Eextrap}{E_\text{extrap}}
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\newcommand{\Edist}{E_\text{dist}}
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\newcommand{\EPT}{E_\text{PT2}}
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\newcommand{\ECIPSI}{E_\text{CIPSI}}
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@ -167,6 +169,7 @@ This set of molecular systems corresponds to Hilbert spaces with sizes ranging f
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In addition to CIPSI, the performance and convergence properties of several series of methods are investigated.
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In particular, we study i) the MP perturbation series up to fifth-order (MP2, MP3, MP4, and MP5), ii) the CC2, CC3, and CC4 approximate series, and ii) the ``complete'' CC series up to quadruples (\ie, CCSD, CCSDT, and CCSDTQ).
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The performance of the ground-state gold standard CCSD(T) as well as the completely renormalized (CR) CC model, CR-CC(2,3), \cite{Kowalski_2000a,Kowalski_2000b,Piecuch_2002a,Piecuch_2002b,Piecuch_2005} are also investigated.
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\alert{From a theoretical point of view, one would expect the following ranking: MP2 $<$ CC2 $<$ MP3 $<$ CCSD $<$ MP4 $<$ CCSD(T) $<$ CR-CC(2,3) $<$ CC3 $<$ CCSDT $<$ MP5 $<$ CC4 $<$ CCSDTQ. But, as we shall see below, this ranking is slightly altered for the present systems.}
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The present manuscript is organized as follows.
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In Sec.~\ref{sec:OO-CIPSI}, we provide theoretical details about the CIPSI algorithm and the orbital optimization procedure employed here.
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@ -441,93 +444,94 @@ More details can be found in Ref.~\onlinecite{Nocedal_1999}.
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%%% TABLE III %%%
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\begin{squeezetable}
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\begin{table}
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\begin{table*}
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\caption{
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Extrapolated correlation energies $\Delta \Eextrap$ (in \si{\milli\hartree}) computed in the cc-pVDZ basis for the twelve cyclic molecules represented in Fig.~\ref{fig:mol} and their associated fitting errors (in \si{\milli\hartree}) obtained via weighted linear fits with a varying number of points.
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\alert{Extrapolation distance $\Delta \Edist$ (in \si{\milli\hartree}) defined as the difference between the final computed energy $\Delta \Efinal$ (in \si{\milli\hartree}) and the extrapolated correlation energies $\Delta \Eextrap$ (in \si{\milli\hartree}) computed in the cc-pVDZ basis for the twelve cyclic molecules represented in Fig.~\ref{fig:mol} and their associated fitting errors (in \si{\milli\hartree}) obtained via weighted linear fits with a varying number of points.}
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Two sets of orbitals are considered: natural orbitals and optimized orbitals.
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The weights are taken as the inverse square of the perturbative corrections.
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For a $m$-point fit, the $m$ largest variational wave functions are used.
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\label{tab:fit}}
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\begin{ruledtabular}
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\begin{tabular}{lccccc}
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Molecule & Number of & \mc{2}{c}{Natural orbitals} & \mc{2}{c}{Optimized orbitals} \\
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\cline{3-4}\cline{5-6}
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& fitting points & $\Delta \Eextrap$ & Fitting error & $\Delta \Eextrap$ & Fitting error \\
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\begin{tabular}{lccccccccc}
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Molecule & Number of & \mc{4}{c}{Natural orbitals} & \mc{4}{c}{Optimized orbitals} \\
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\cline{3-6}\cline{7-10}
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& fitting points & $\Delta \Efinal$ & $\Delta \Eextrap$ & $\Delta \Edist$ & Fitting error
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& $\Delta \Efinal$ & $\Delta \Eextrap$ & $\Delta \Edist$ & Fitting error \\
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\hline
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Cyclopentadiene & 3 & $-740.639$ & $0.273$ & $-739.295$ & $0.199$ \\
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& 4 & $-740.243$ & $0.306$ & $-739.309$ & $0.088$ \\
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&\bf5 & $-740.047$ & $0.242$ & $\bf-739.230$& $\bf0.074$ \\
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& 6 & $-739.952$ & $0.187$ & $-739.304$ & $0.072$ \\
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& 7 & $-739.761$ & $0.204$ & $-739.292$ & $0.055$ \\
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Cyclopentadiene & 3 & $-728.941$ & $-740.639$ & $11.699$ & $0.273$ & $-731.987$ & $-739.295$ & $7.308$ & $0.199$ \\
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& 4 & $-728.941$ & $-740.243$ & $11.303$ & $0.306$ & $-731.987$ & $-739.309$ & $7.322$ & $0.088$ \\
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&\bf5 & $-728.941$ & $-740.047$ & $11.106$ & $0.242$ &$\bf-731.987$ &$\bf-739.230$ &$\bf7.243$ &$\bf0.074$ \\
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& 6 & $-728.941$ & $-739.952$ & $11.011$ & $0.187$ & $-731.987$ & $-739.304$ & $7.317$ & $0.072$ \\
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& 7 & $-728.941$ & $-739.761$ & $10.820$ & $0.204$ & $-731.987$ & $-739.292$ & $7.305$ & $0.055$ \\
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\hline
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Furan & 3 & $-766.090$ & $0.729$ & $-767.790$ & $0.064$ \\
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& 4 & $-766.445$ & $0.459$ & $-768.104$ & $0.196$ \\
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&\bf5 & $-766.582$ & $0.318$ & $\bf-768.194$ &$\bf0.135$ \\
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& 6 & $-766.366$ & $0.288$ & $-768.060$ & $0.131$ \\
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& 7 & $-766.507$ & $0.254$ & $-768.086$ & $0.101$ \\
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Furan & 3 & $-758.946$ & $-766.090$ & $7.144$ & $0.729$ & $-761.715$ & $-767.790$ & $6.076$ & $0.064$ \\
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& 4 & $-758.946$ & $-766.445$ & $7.499$ & $0.459$ & $-761.715$ & $-768.104$ & $6.389$ & $0.196$ \\
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&\bf5 & $-758.946$ & $-766.582$ & $7.636$ & $0.318$ &$\bf-761.715$ &$\bf-768.194$ &$\bf6.479$ &$\bf0.135$ \\
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& 6 & $-758.946$ & $-766.366$ & $7.420$ & $0.288$ & $-761.715$ & $-768.060$ & $6.345$ & $0.131$ \\
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& 7 & $-758.946$ & $-766.507$ & $7.561$ & $0.254$ & $-761.715$ & $-768.086$ & $6.372$ & $0.101$ \\
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\hline
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Imidazole & 3 & $-778.148$ & $2.197$ & $-778.295$ & $0.356$ \\
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& 4 & $-777.436$ & $1.107$ & $-778.270$ & $0.150$ \\
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&\bf5 & $-776.300$ & $0.996$ & $\bf-778.178$ &$\bf0.105$ \\
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& 6 & $-776.104$ & $0.712$ & $-778.174$ & $0.072$ \\
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& 7 & $-776.098$ & $0.541$ & $-778.051$ & $0.099$ \\
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Imidazole & 3 & $-767.314$ & $-778.148$ & $10.833$ & $2.197$ & $-771.362$ & $-778.295$ & $6.932$ & $0.356$ \\
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& 4 & $-767.314$ & $-777.436$ & $10.122$ & $1.107$ & $-771.362$ & $-778.270$ & $6.908$ & $0.150$ \\
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&\bf5 & $-767.314$ & $-776.300$ & $8.986$ & $0.996$ &$\bf-771.362$ &$\bf-778.178$ &$\bf6.816$ &$\bf0.105$ \\
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& 6 & $-767.314$ & $-776.104$ & $8.789$ & $0.712$ & $-771.362$ & $-778.174$ & $6.812$ & $0.072$ \\
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& 7 & $-767.314$ & $-776.098$ & $8.784$ & $0.541$ & $-771.362$ & $-778.051$ & $6.689$ & $0.099$ \\
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\hline
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Pyrrole & 3 & $-758.309$ & $0.447$ & $-758.650$ & $0.321$ \\
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& 4 & $-758.749$ & $0.393$ & $-758.389$ & $0.174$ \\
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&\bf5 & $-758.405$ & $0.359$ & $\bf-758.460$ &$\bf0.110$ \\
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& 6 & $-758.136$ & $0.334$ & $-758.352$ & $0.100$ \\
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& 7 & $-757.990$ & $0.283$ & $-758.347$ & $0.075$ \\
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Pyrrole & 3 & $-748.961$ & $-758.309$ & $9.348$ & $0.447$ & $-751.862$ & $-758.650$ & $6.788$ & $0.321$ \\
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& 4 & $-748.961$ & $-758.749$ & $9.788$ & $0.393$ & $-751.862$ & $-758.389$ & $6.527$ & $0.174$ \\
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&\bf5 & $-748.961$ & $-758.405$ & $9.444$ & $0.359$ &$\bf-751.862$ &$\bf-758.460$ &$\bf6.598$ &$\bf0.110$ \\
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& 6 & $-748.961$ & $-758.136$ & $9.175$ & $0.334$ & $-751.862$ & $-758.352$ & $6.490$ & $0.100$ \\
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& 7 & $-748.961$ & $-757.990$ & $9.029$ & $0.283$ & $-751.862$ & $-758.347$ & $6.485$ & $0.075$ \\
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\hline
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Thiophene & 3 & $-728.054$ & $0.134$ & $-728.744$ & $0.691$ \\
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& 4 & $-728.240$ & $0.139$ & $-729.052$ & $0.331$ \\
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&\bf5 & $-728.243$ & $0.087$ & $\bf-728.948$ &$\bf0.203$ \\
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& 6 & $-728.242$ & $0.062$ & $-728.987$ & $0.140$ \\
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& 7 & $-728.420$ & $0.144$ & $-729.067$ & $0.117$ \\
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Thiophene & 3 & $-718.769$ & $-728.054$ & $9.285$ & $0.134$ & $-721.757$ & $-728.744$ & $6.987$ & $0.691$ \\
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& 4 & $-718.769$ & $-728.240$ & $9.471$ & $0.139$ & $-721.757$ & $-729.052$ & $7.295$ & $0.331$ \\
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&\bf5 & $-718.769$ & $-728.243$ & $9.474$ & $0.087$ &$\bf-721.757$ &$\bf-728.948$ &$\bf7.191$ &$\bf0.203$ \\
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& 6 & $-718.769$ & $-728.242$ & $9.472$ & $0.062$ & $-721.757$ & $-728.987$ & $7.230$ & $0.140$ \\
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& 7 & $-718.769$ & $-728.420$ & $9.651$ & $0.144$ & $-721.757$ & $-729.067$ & $7.310$ & $0.117$ \\
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\hline
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Benzene & 3 & $-860.350$ & $0.496$ & $-862.325$ & $0.279$ \\
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& 4 & $-861.949$ & $0.811$ & $-863.024$ & $0.424$ \\
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&\bf5 & $-861.807$ & $0.474$ & $\bf-862.890$ &$\bf0.266$ \\
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& 6 & $-861.110$ & $0.539$ & $-862.360$ & $0.383$ \\
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& 7 & $-861.410$ & $0.444$ & $-862.083$ & $0.339$ \\
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Benzene & 3 & $-841.030$ & $-860.350$ & $19.3197$ & $0.496$ & $-848.540$ & $-862.325$ & $13.7847$ & $0.279$ \\
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& 4 & $-841.030$ & $-861.949$ & $20.9186$ & $0.811$ & $-848.540$ & $-863.024$ & $14.4842$ & $0.424$ \\
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&\bf5 & $-841.030$ & $-861.807$ & $20.7772$ & $0.474$ &$\bf-848.540$ &$\bf-862.890$ &$\bf14.3496$ &$\bf0.266$ \\
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& 6 & $-841.030$ & $-861.110$ & $20.0803$ & $0.539$ & $-848.540$ & $-862.360$ & $13.8202$ & $0.383$ \\
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& 7 & $-841.030$ & $-861.410$ & $20.3794$ & $0.444$ & $-848.540$ & $-862.083$ & $13.5435$ & $0.339$ \\
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\hline
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Pyrazine & 3 & $-904.148$ & $0.035$ & $-904.867$ & $1.420$ \\
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& 4 & $-904.726$ & $0.377$ & $-904.588$ & $0.650$ \\
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&\bf5 & $-904.274$ & $0.383$ & $\bf-904.550$ &$\bf0.385$ \\
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& 6 & $-903.980$ & $0.341$ & $-903.982$ & $0.439$ \\
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& 7 & $-903.621$ & $0.370$ & $-903.746$ & $0.359$ \\
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Pyrazine & 3 & $-887.414$ & $-904.148$ & $16.734$ & $0.035$ & $-891.249$ & $-904.867$ & $13.619$ & $1.420$ \\
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& 4 & $-887.414$ & $-904.726$ & $17.312$ & $0.377$ & $-891.249$ & $-904.588$ & $13.340$ & $0.650$ \\
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&\bf5 & $-887.414$ & $-904.274$ & $16.859$ & $0.383$ &$\bf-891.249$ &$\bf-904.550$ &$\bf13.301$ &$\bf0.385$ \\
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& 6 & $-887.414$ & $-903.980$ & $16.566$ & $0.341$ & $-891.249$ & $-903.982$ & $12.734$ & $0.439$ \\
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& 7 & $-887.414$ & $-903.621$ & $16.206$ & $0.370$ & $-891.249$ & $-903.746$ & $12.497$ & $0.359$ \\
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\hline
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Pyridazine & 3 & $-910.856$ & $3.053$ & $-909.292$ & $0.024$ \\
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& 4 & $-908.222$ & $1.834$ & $-908.808$ & $0.230$ \\
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&\bf5 & $-909.282$ & $1.191$ & $\bf-908.820$ &$\bf0.133$ \\
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& 6 & $-912.566$ & $1.727$ & $-908.342$ & $0.303$ \\
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& 7 & $-910.694$ & $2.210$ & $-908.368$ & $0.224$ \\
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Pyridazine & 3 & $-887.410$ & $-910.856$ & $23.446$ & $3.053$ & $-895.565$ & $-909.292$ & $13.726$ & $0.024$ \\
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& 4 & $-887.410$ & $-908.222$ & $20.811$ & $1.834$ & $-895.565$ & $-908.808$ & $13.243$ & $0.230$ \\
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&\bf5 & $-887.410$ & $-909.282$ & $21.871$ & $1.191$ &$\bf-895.565$ &$\bf-908.820$ &$\bf13.255$ &$\bf0.133$ \\
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& 6 & $-887.410$ & $-912.566$ & $25.156$ & $1.727$ & $-895.565$ & $-908.342$ & $12.777$ & $0.303$ \\
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& 7 & $-887.410$ & $-910.694$ & $23.283$ & $2.210$ & $-895.565$ & $-908.368$ & $12.802$ & $0.224$ \\
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\hline
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Pyridine & 3 & $-883.025$ & $3.919$ & $-883.363$ & $0.047$ \\
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& 4 & $-883.862$ & $1.869$ & $-883.413$ & $0.029$ \\
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&\bf5 & $-881.664$ & $1.760$ & $\bf-882.700$ &$\bf0.405$ \\
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& 6 & $-880.422$ & $1.456$ & $-882.361$ & $0.341$ \\
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& 7 & $-880.191$ & $1.084$ & $-882.023$ & $0.330$ \\
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Pyridine & 3 & $-861.424$ & $-883.025$ & $21.601$ & $3.919$ & $-868.803$ & $-883.363$ & $14.560$ & $0.047$ \\
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& 4 & $-861.424$ & $-883.862$ & $22.438$ & $1.869$ & $-868.803$ & $-883.413$ & $14.610$ & $0.029$ \\
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&\bf5 & $-861.424$ & $-881.664$ & $20.240$ & $1.760$ &$\bf-868.803$ &$\bf-882.700$ &$\bf13.897$ &$\bf0.405$ \\
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& 6 & $-861.424$ & $-880.422$ & $18.998$ & $1.456$ & $-868.803$ & $-882.361$ & $13.558$ & $0.341$ \\
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& 7 & $-861.424$ & $-880.191$ & $18.768$ & $1.084$ & $-868.803$ & $-882.023$ & $13.221$ & $0.330$ \\
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\hline
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Pyrimidine & 3 & $-900.386$ & $1.884$ & $-900.817$ & $0.726$ \\
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& 4 & $-901.441$ & $0.991$ & $-900.383$ & $0.356$ \\
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&\bf5 & $-900.354$ & $0.865$ & $\bf-900.496$ &$\bf0.214$ \\
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& 6 & $-900.240$ & $0.594$ & $-900.698$ & $0.190$ \\
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& 7 & $-899.689$ & $0.565$ & $-900.464$ & $0.206$ \\
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Pyrimidine & 3 & $-879.958$ & $-900.386$ & $20.428$ & $1.884$ & $-887.009$ & $-900.817$ & $13.808$ & $0.726$ \\
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& 4 & $-879.958$ & $-901.441$ & $21.483$ & $0.991$ & $-887.009$ & $-900.383$ & $13.374$ & $0.356$ \\
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&\bf5 & $-879.958$ & $-900.354$ & $20.396$ & $0.865$ &$\bf-887.009$ &$\bf-900.496$ &$\bf13.487$ &$\bf0.214$ \\
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& 6 & $-879.958$ & $-900.240$ & $20.283$ & $0.594$ & $-887.009$ & $-900.698$ & $13.689$ & $0.190$ \\
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& 7 & $-879.958$ & $-899.689$ & $19.732$ & $0.565$ & $-887.009$ & $-900.464$ & $13.455$ & $0.206$ \\
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\hline
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s-Tetrazine & 3 & $-958.736$ & $0.320$ & $-957.559$ & $0.246$ \\
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& 4 & $-958.727$ & $0.148$ & $-957.299$ & $0.160$ \\
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&\bf5 & $-958.500$ & $0.172$ & $\bf-957.869$ &$\bf0.349$ \\
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& 6 & $-958.162$ & $0.260$ & $-957.744$ & $0.247$ \\
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& 7 & $-958.161$ & $0.198$ & $-957.709$ & $0.183$ \\
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s-Tetrazine & 3 & $-942.162$ & $-958.736$ & $16.574$ & $0.320$ & $-944.077$ & $-957.559$ & $13.4815$ & $0.246$ \\
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& 4 & $-942.162$ & $-958.727$ & $16.564$ & $0.148$ & $-944.077$ & $-957.299$ & $13.2221$ & $0.160$ \\
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&\bf5 & $-942.162$ & $-958.500$ & $16.337$ & $0.172$ &$\bf-944.077$ &$\bf-957.869$ &$\bf13.7916$ &$\bf0.349$ \\
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& 6 & $-942.162$ & $-958.162$ & $16.000$ & $0.260$ & $-944.077$ & $-957.744$ & $13.6665$ & $0.247$ \\
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& 7 & $-942.162$ & $-958.161$ & $15.999$ & $0.198$ & $-944.077$ & $-957.709$ & $13.6319$ & $0.183$ \\
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\hline
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s-Triazine & 3 & $-917.221$ & $0.693$ & $-919.596$ & $0.105$ \\
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& 4 & $-918.723$ & $0.913$ & $-918.457$ & $0.538$ \\
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&\bf5 & $-917.402$ & $0.956$ & $\bf-918.355$ &$\bf0.312$ \\
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& 6 & $-916.517$ & $0.862$ & $-918.206$ & $0.226$ \\
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& 7 & $-916.544$ & $0.643$ & $-917.876$ & $0.267$ \\
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s-Triazine & 3 & $-898.283$ & $-917.221$ & $18.938$ & $0.693$ & $-905.180$ & $-919.596$ & $14.4152$ & $0.105$ \\
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& 4 & $-898.283$ & $-918.723$ & $20.440$ & $0.913$ & $-905.180$ & $-918.457$ & $13.2768$ & $0.538$ \\
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&\bf5 & $-898.283$ & $-917.402$ & $19.119$ & $0.956$ &$\bf-905.180$ &$\bf-918.355$ &$\bf13.1745$ &$\bf0.312$ \\
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& 6 & $-898.283$ & $-916.517$ & $18.233$ & $0.862$ & $-905.180$ & $-918.206$ & $13.0251$ & $0.226$ \\
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& 7 & $-898.283$ & $-916.544$ & $18.261$ & $0.643$ & $-905.180$ & $-917.876$ & $12.6956$ & $0.267$ \\
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\end{tabular}
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\end{ruledtabular}
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\end{table}
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\end{table*}
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\end{squeezetable}
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%%% %%% %%%
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@ -647,7 +651,7 @@ As compared to natural orbitals (solid red lines), one can see that, for a given
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Adding the perturbative correction $\EPT$ yields very similar curves for both sets of orbitals (dashed lines).
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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.
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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.
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From Fig.~\ref{fig:vsEPT2}, it is clear that the behavior of $\Delta \Evar$ is much more linear and produces smaller $\EPT$ values when optimized orbitals are selected, hence facilitating the extrapolation procedure to the FCI limit (see below).
|
||||
From Fig.~\ref{fig:vsEPT2}, \alert{it is clear one produces smaller $\EPT$ values when optimized orbitals are selected, 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 FCI estimate of the correlation energy (solid black line) is reported for reference in Figs.~\ref{fig:vsNdet} and \ref{fig:vsEPT2}.
|
||||
|
||||
Figure \ref{fig:BenzenevsNdet} compares the convergence of $\Delta \Evar$ for natural, localized, and optimized orbitals for benzene.
|
||||
|
|
@ -660,12 +664,14 @@ To this end, we have extrapolated the orbital-optimized variational CIPSI correl
|
|||
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 list 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.
|
||||
\alert{The extrapolation distance $\Delta \Edist$ defined as the difference between the final computed energy $\Delta \Efinal$ and $\Delta \Eextrap$ is also reported.}
|
||||
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 \alert{carry an error of the order of one millihartree}.
|
||||
Logically, the FCI estimates for the five-membered rings seem slightly more accurate than for the (larger) six-membered rings.
|
||||
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 (\SI{-862.9}{\milli\hartree}) is very close to the one reported in Ref.~\onlinecite{Loos_2020e} (\SI{-863.4}{\milli\hartree}).
|
||||
|
||||
Table \ref{tab:fit} does report extrapolated correlation energies and fitting errors for both natural and optimized orbitals.
|
||||
Again, the superiority of the latter set is clear as both the variation in extrapolated values and the fitting error are much larger with the natural set.
|
||||
\alert{Moreover, the extrapolation distance $\Delta \Edist$ is systematically decreases by several \si{\milli\hartree}.}
|
||||
Taking cyclopentadiene as an example, the extrapolated values vary by almost \SI{1}{\milli\hartree} with natural orbitals and less than \SI{0.1}{\milli\hartree} with the optimized set.
|
||||
The fitting errors follow the same trend.
|
||||
|
||||
|
|
@ -702,14 +708,17 @@ Importantly here, one notices that MP4 [which scales as $\order*{N^7}$] is syste
|
|||
\label{sec:ccl}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
Using the SCI algorithm named \textit{Configuration Interaction using a Perturbative Selection made Iteratively} (CIPSI), we have produced FCI-quality frozen-core correlation energies for twelve cyclic molecules (see Fig.~\ref{fig:mol}) in the correlation-consistent double-$\zeta$ Dunning basis set (cc-pVDZ).
|
||||
These estimates, which are likely accurate to a few tenths of a millihartree, have been obtained by extrapolating CIPSI energies to the FCI limit based on a set of orbitals obtained by minimizing the CIPSI variational energy.
|
||||
These estimates, which \alert{probably carry an error of the order of one millihartree}, have been obtained by extrapolating CIPSI energies to the FCI limit based on a set of orbitals obtained by minimizing the CIPSI variational energy.
|
||||
Using energetically optimized orbitals, one can reduce the size of the variational space by one order of magnitude for the same variational energy as compared to natural orbitals.
|
||||
|
||||
Thanks to these reference FCI energies, we have then benchmarked three families of popular electronic structure methods: i) the MP perturbation series up to fifth-order (MP2, MP3, MP4, and MP5), ii) the approximate CC series CC2, CC3, and CC4, and iii) the ``complete'' CC series CCSD, CCSDT, and CCSDTQ.
|
||||
With a $\order*{N^7}$ scaling, MP4 provides an interesting accuracy/cost ratio for this particular set of weakly correlated systems, while MP5 systematically worsen the perturbative estimates of the correlation energy.
|
||||
In addition, CC3 (where the triples are computed iteratively) outperforms the perturbative-triples CCSD(T) method with the same $\order*{N^7}$ scaling, its completely renormalized version CR-CC(2,3), as well as its more expensive parent, CCSDT.
|
||||
A similar trend is observed for the methods including quadruple excitations, where the $\order*{N^9}$ CC4 model has been shown to be slightly more accurate than CCSDTQ [which scales as $\order*{N^{10}}$], both methods providing correlation energies within \SI{2}{\milli\hartree} of the FCI limit.
|
||||
Of course, the present trends are only valid for this particular class of (weakly-correlated) molecules and it would be desirable to have a broader variety of systems in the future by including more challenging systems such as, for example, transition metal compounds.
|
||||
\alert{These observations slightly alter the method ranking provided in Sec.~\ref{sec:intro}.
|
||||
Of course, the present trends are only valid for this particular class of (weakly-correlated) molecules.
|
||||
For example, the performance of CC3 might decline for larger systems.}
|
||||
Thus, it would be desirable to have a broader variety of systems in the future by including more challenging systems such as, for example, transition metal compounds.
|
||||
Some work along this line is currently being performed.
|
||||
|
||||
As perspectives, we are currently investigating the performance of the present approach for excited states in order to expand the QUEST database of vertical excitation energies. \cite{Veril_2021}
|
||||
|
|
@ -728,8 +737,6 @@ This project has received funding from the European Research Council (ERC) under
|
|||
%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
The data that support the findings of this study are openly available in Zenodo at \url{http://doi.org/10.5281/zenodo.5150663}.
|
||||
|
||||
\clearpage
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\bibliography{Ec}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
|
|
|||
|
|
@ -44,20 +44,31 @@ I have a few comments in the following before publication.}
|
|||
CR-CC(2,3) would be similar to CCSD(T) but I am not sure if it is a size-extensive model.}
|
||||
\\
|
||||
\alert{
|
||||
We have mentioned this theoretical ranking at the end of the Introduction.
|
||||
CR-CC(2,3) is indeed a size-extensive model and its theoretical accuracy should be slightly better than CCSD(T).
|
||||
}
|
||||
|
||||
\item
|
||||
{It seems that the results of NO are more linear than those of OO in Fig 3. This contradicts the discussion in the right column of page 7.
|
||||
}
|
||||
\\
|
||||
\alert{
|
||||
\alert{Indeed, for relatively large PT2 corrections, the NO curves look more linear.
|
||||
However, if one considers the last four or five points, the OO curves are also very linear.
|
||||
The main point here is that OOs produce much smaller PT2 values than NOs.
|
||||
To avoid confusions, we have removed our comment on the linearity of the curves, mentioning only that OOs yield much smaller PT2 values then NOs.
|
||||
}
|
||||
|
||||
\item
|
||||
{It is claimed that the FCI estimates in Table III are likely accurate to less than 1mEh. I think this is too optimistic as some results in the table range by 1-2 mEh with different number of ftting points. I also think the extrapolation distances need to be listed for comparison to the error-bar.
|
||||
{It is claimed that the FCI estimates in Table III are likely accurate to less than 1mEh.
|
||||
I think this is too optimistic as some results in the table range by 1-2 mEh with different number of fitting points.
|
||||
I also think the extrapolation distances need to be listed for comparison to the error-bar.
|
||||
}
|
||||
\\
|
||||
\alert{
|
||||
We strongly believe that our correlation energy estimates are accurate to a few tenths of a millihartree.
|
||||
However, the error might be slightly larger for the six-membered rings.
|
||||
To be conservative, we have decided to state, in the revised manuscript, that the present estimates carry an error of the order of one millihartreee.
|
||||
Morevoer, we now list the extrapolation distances in Table III alongside other useful information and show that OOs systematically decrease the extrapolation distance by several millihartree as compared to NOs.
|
||||
}
|
||||
|
||||
\item
|
||||
|
|
@ -65,7 +76,10 @@ CR-CC(2,3) would be similar to CCSD(T) but I am not sure if it is a size-extensi
|
|||
}
|
||||
\\
|
||||
\alert{
|
||||
}
|
||||
We have mentioned these points in the concluding section of the manuscript.
|
||||
This is indeed not always the case.
|
||||
The performance of the MP series can be quite hard to predict.
|
||||
Moreover, the performance of CC3 may depend on the size of the systems considered and is known to work best for relatively small systems.}
|
||||
|
||||
\end{enumerate}
|
||||
|
||||
|
|
|
|||
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Reference in New Issue