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%% This BibTeX bibliography file was created using BibDesk.
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%% http://bibdesk.sourceforge.net/
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%% Created for Pierre-Francois Loos at 2021-07-20 16:08:44 +0200
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%% Created for Pierre-Francois Loos at 2021-07-21 08:17:37 +0200
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%% Saved with string encoding Unicode (UTF-8)
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@article{Magoulas_2021,
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author = {Magoulas, Ilias and Gururangan, Karthik and Piecuch, Piotr and Deustua, J. Emiliano and Shen, Jun},
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date-added = {2021-07-21 08:17:09 +0200},
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date-modified = {2021-07-21 08:17:24 +0200},
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doi = {10.1021/acs.jctc.1c00181},
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journal = {J. Chem. Theory Comput.},
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number = {7},
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pages = {4006-4027},
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title = {Is Externally Corrected Coupled Cluster Always Better Than the Underlying Truncated Configuration Interaction?},
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volume = {17},
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year = {2021},
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Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.1c00181}}
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@article{Xu_2020,
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author = {Xu, Enhua and Uejima, Motoyuki and Ten-no, Seiichiro L.},
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date-added = {2021-07-21 08:15:48 +0200},
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date-modified = {2021-07-21 08:16:01 +0200},
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doi = {10.1021/acs.jpclett.0c03084},
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journal = {J. Phys. Chem. Lett.},
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number = {22},
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pages = {9775-9780},
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title = {Towards Near-Exact Solutions of Molecular Electronic Structure: Full Coupled-Cluster Reduction with a Second-Order Perturbative Correction},
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volume = {11},
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year = {2020},
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Bdsk-Url-1 = {https://doi.org/10.1021/acs.jpclett.0c03084}}
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@article{Bozkaya_2011,
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author = {Bozkaya,U{\u g}ur and Turney,Justin M. and Yamaguchi,Yukio and Schaefer,Henry F. and Sherrill,C. David},
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date-added = {2021-07-20 16:08:28 +0200},
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@ -346,16 +372,6 @@
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year = {2017},
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Bdsk-Url-1 = {https://doi.org/10.1007/s10910-016-0688-6}}
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@misc{Magoulas_2021,
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archiveprefix = {arXiv},
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author = {Ilias Magoulas and Karthik Gururangan and Piotr Piecuch and J. Emiliano Deustua and Jun Shen},
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date-added = {2021-06-18 05:40:59 +0200},
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date-modified = {2021-06-18 05:41:08 +0200},
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eprint = {2102.10143},
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primaryclass = {physics.chem-ph},
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title = {Is Externally Corrected Coupled Cluster Always Better than the Underlying Truncated Configuration Interaction?},
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year = {2021}}
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@article{Lee_2021,
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author = {Lee, Seunghoon and Zhai, Huanchen and Sharma, Sandeep and Umrigar, C. J. and Chan, Garnet Kin-Lic},
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date-added = {2021-06-18 05:39:07 +0200},
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@ -1573,16 +1589,6 @@
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year = {2017},
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Bdsk-Url-1 = {http://dx.doi.org/10.1063/1.4977727}}
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@misc{Xu_2020,
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archiveprefix = {arXiv},
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author = {Enhua Xu and Motoyuki Uejima and Seiichiro L. Ten-no},
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date-added = {2020-10-10 13:54:02 +0200},
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date-modified = {2020-10-10 13:54:09 +0200},
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eprint = {2010.01850},
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primaryclass = {physics.chem-ph},
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title = {Towards near-exact solutions of molecular electronic structure: Full coupled-cluster reduction with a second-order perturbative correction},
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year = {2020}}
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@inbook{Caffarel_2016b,
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author = {Caffarel, Michel and Applencourt, Thomas and Giner, Emmanuel and Scemama, Anthony},
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booktitle = {Recent Progress in Quantum Monte Carlo},
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@ -49,6 +49,7 @@
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\newcommand{\cP}{\mathcal{P}}
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% energies
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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{\EPT}{E_\text{PT2}}
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@ -159,6 +160,13 @@ The performance of the ground-state gold standard CCSD(T) is also investigated.
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\end{figure*}
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%%% %%% %%%
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The present manuscript is organized as follows.
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In Sec.~\ref{sec:compdet}, computational details concerning geometries, basis sets, and methods are reported.
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Section \ref{sec:OO-CIPSI} provides theoretical details about the CIPSI algorithm and the orbital optimization procedure that we have employed here.
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In Sec.~\ref{sec:res}, we report our reference FCI correlation energies for the five-membered and six-membered cyclic molecules obtained thanks to extrapolated orbital-optimized CIPSI calculations (Sec.~\ref{sec:cipsi_res}).
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These reference correlation energies are then used to benchmark and study the convergence properties of various perturbative and CC methods (Sec.~\ref{sec:mpcc_res}).
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Finally, we draw our conclusions in Sec.~\ref{sec:ccl}.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Computational details}
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\label{sec:compdet}
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@ -361,7 +369,8 @@ More details can be found in Ref.~\onlinecite{Nocedal_1999}.
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\includegraphics[width=0.24\textwidth]{Triazine_EvsPT2}
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\caption{$\Delta \Evar$ as a function of $\EPT$ for the twelve cyclic molecules represented in Fig.~\ref{fig:mol}.
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Two sets of orbitals are considered: natural orbitals (NOs, in red) and optimized orbitals (OOs, in blue).
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The four-point linear fit using the four largest variational wave functions for each set is depicted as a dashed black line.
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The four-point weighted linear fit using the four largest variational wave functions for each set is depicted as a dashed black line.
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The weights are taken as the inverse square of the perturbative corrections.
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The CCSDTQ correlation energy is also represented as a thick black line.
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\label{fig:vsEPT2}}
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\end{figure*}
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@ -369,7 +378,7 @@ More details can be found in Ref.~\onlinecite{Nocedal_1999}.
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\begin{squeezetable}
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\begin{table*}
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\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.
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For the CIPSI estimates of the correlation energy, the fitting error associated with the 4-point linear fit is reported in parenthesis.
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For the CIPSI estimates of the correlation energy, the fitting error associated with the four-point linear fit is reported in parenthesis.
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\label{tab:Tab5-VDZ}}
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\begin{ruledtabular}
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\begin{tabular}{lcccccccccc}
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@ -404,7 +413,7 @@ More details can be found in Ref.~\onlinecite{Nocedal_1999}.
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\begin{squeezetable}
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\begin{table*}
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\caption{Total energy $E$ (in \SI{}{\hartree}) and correlation energy $\Delta E$ (in \SI{}{\milli\hartree}) for the frozen-core ground state of six-membered rings in the cc-pVDZ basis set.
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For the CIPSI estimates of the correlation energy, the fitting error associated with the 4-point linear fit is reported in parenthesis.
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For the CIPSI estimates of the correlation energy, the fitting error associated with the four-point linear fit is reported in parenthesis.
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\label{tab:Tab6-VDZ}}
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\begin{ruledtabular}
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\begin{tabular}{lcccccccccccccc}
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@ -440,12 +449,15 @@ More details can be found in Ref.~\onlinecite{Nocedal_1999}.
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\begin{squeezetable}
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\begin{table}
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\caption{
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Extrapolated correlation energies $\Delta \Evar$ (in \SI{}{\milli\hartree}) 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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The weights are taken as the inverse square of the perturbative corrections.
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For a $n$-point fit, the $n$ largest the 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}{lccc}
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Molecule & Number & \mc{2}{c}{Fitting parameter} \\
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Molecule & Number of & \mc{2}{c}{Fitting parameters} \\
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\cline{3-4}
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& of points & $\Delta \Evar$ & Standard error \\
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& fitting points & $\Delta \Evar$ & Fitting error \\
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\hline
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Cyclopentadiene & 3 & $-739.295$ & $0.199$ \\
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& 4 & $-739.309$ & $0.088$ \\
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@ -540,15 +552,19 @@ More details can be found in Ref.~\onlinecite{Nocedal_1999}.
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\label{sec:cipsi_res}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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We first study the convergence of the variational energy as a function of the number of determinants.
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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}.
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As one can see, the use of optimized orbitals greatly facilitate the convergence towards the FCI limit.
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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 $\EPT$ as well as its 4-point linear fit using the four largest variational wave functions.
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In both cases, the CCSDTQ correlation energy is also represented for comparison purposes.
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We first study the convergence of the CIPSI correlation energy $\Delta \Evar = \Evar - \EHF$ as a function of the number of determinants.
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Our motivation here is to generate FCI-quality reference correlation energies for the twelve cyclic molecules represented in Fig.~\ref{fig:mol} in order to benchmark, in a second time, the performance and convergence properties of various mainstream MP and CC methods (see Sec.~\ref{sec:mpcc_res}).
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For the natural and optimized orbital sets, we report, in Fig.~\ref{fig:vsNdet}, the evolution of the variational correlation energy $\Delta \Evar$ and its perturbatively corrected value $\Delta \Evar + \EPT$ with respect to the number of determinants for each cyclic molecule.
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As compared to natural orbitals (solid red lines), one can see that, for a given number of determinants, the use of optimized orbitals greatly lowers $\Delta \Evar$ (solid blue lines).
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Adding the perturbative correction $\EPT$ yields similar curves for both sets of orbitals (dashed lines).
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This indicates that, for a given number of determinants, $\EPT$ (which 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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The four-point weighted linear fit using the four largest variational wave functions are also represented (dashed black lines), while the CCSDTQ correlation energy (solid black line) is also reported for comparison purposes.
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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, hence facilitating the extrapolation procedure to the FCI limit.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Benchmark of CC and MP methods}
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\label{sec:cc_mp_res}
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\label{sec:mpcc_res}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{figure*}
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@ -576,6 +592,7 @@ In both cases, the CCSDTQ correlation energy is also represented for comparison
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%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Conclusion}
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\label{sec:ccl}
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%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%
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