Fabris corrections

Manuscript/Ec.tex

@@ -94,7 +94,7 @@
 
 % Abstract
 \begin{abstract}
-In the continuity of our recent work on the benzene molecule [\href{https://doi.org/10.1063/5.0027617}{J.~Chem.~Phys.~\textbf{153}, 176101 (2020)}], itself motivated by the blind challenge of Eriksen \textit{et al.} [\href{https://doi.org/10.1021/acs.jpclett.0c02621}{J.~Phys.~Chem.~Lett.~\textbf{11}, 8922 (2020)}] on the same system, we report accurate full configuration interaction (FCI) frozen-core correlation energy estimates for twelve five- and six-membered ring molecules (cyclopentadiene, furan, imidazole, pyrrole, thiophene, benzene, pyrazine, pyridazine, pyridine, pyrimidine, tetrazine, and triazine) in the standard correlation-consistent double-$\zeta$ Dunning basis set (cc-pVDZ).
+In the continuity of our recent work on the benzene molecule [\href{https://doi.org/10.1063/5.0027617}{J.~Chem.~Phys.~\textbf{153}, 176101 (2020)}], itself motivated by the blind challenge of Eriksen \textit{et al.} [\href{https://doi.org/10.1021/acs.jpclett.0c02621}{J.~Phys.~Chem.~Lett.~\textbf{11}, 8922 (2020)}] on the same system, we report accurate full configuration interaction (FCI) frozen-core correlation energy estimates for twelve five- and six-membered ring molecules (cyclopentadiene, furan, imidazole, pyrrole, thiophene, benzene, pyrazine, pyridazine, pyridine, pyrimidine, s-tetrazine, and s-triazine) in the standard correlation-consistent double-$\zeta$ Dunning basis set (cc-pVDZ).
 Our FCI correlation energy estimates, with estimated error below 1 millihartree, are based on energetically optimized-orbital selected configuration interaction (SCI) calculations performed with the \textit{Configuration Interaction using a Perturbative Selection made Iteratively} (CIPSI) algorithm.
 Having at our disposal these accurate reference energies, the respective performance and convergence properties of several popular and widely-used families of quantum chemistry methods are investigated.
 In particular, we study the convergence properties of i) the M{\o}ller-Plesset perturbation series up to fifth-order (MP2, MP3, MP4, and MP5), ii) the iterative approximate single-reference coupled-cluster series CC2, CC3, and CC4, and iii) the single-reference coupled-cluster series CCSD, CCSDT, and CCSDTQ.
@@ -146,7 +146,7 @@ Again, MP perturbation theory and CC methods can be coupled.
 The CCSD(T) method, \cite{Raghavachari_1989} where one includes iteratively the single and double excitations and perturbatively (from MP4 and partially MP5) the triple excitations, known as the ``gold-standard'' of quantum chemistry for weakly correlated systems thanks to its excellent accuracy/cost ratio, is probably the most iconic example of such coupling.
 
 Motivated by the recent blind test of Eriksen \textit{et al.}\cite{Eriksen_2020}~reporting the performance of a large panel of emerging electronic structure methods [the many-body expansion FCI (MBE-FCI), \cite{Eriksen_2017,Eriksen_2018,Eriksen_2019a,Eriksen_2019b} adaptive sampling CI (ASCI), \cite{Tubman_2016,Tubman_2018,Tubman_2020} iterative CI (iCI), \cite{Liu_2014,Liu_2016,Lei_2017,Zhang_2020} semistochastic heat-bath CI (SHCI), \cite{Holmes_2016,Holmes_2017,Sharma_2017} the full coupled-cluster reduction (FCCR), \cite{Xu_2018,Xu_2020} density-matrix renormalization group (DMRG), \cite{White_1992,White_1993,Chan_2011} adaptive-shift FCI quantum Monte Carlo (AS-FCIQMC), \cite{Booth_2009,Cleland_2010,Ghanem_2019} and cluster-analysis-driven FCIQMC (CAD-FCIQMC) \cite{Deustua_2017,Deustua_2018}] on the non-relativistic frozen-core correlation energy of the benzene molecule in the standard correlation-consistent double-$\zeta$ Dunning basis set (cc-pVDZ), some of us have recently investigated the performance of the SCI method known as \textit{Configuration Interaction using a Perturbative Selection made Iteratively} (CIPSI). \cite{Huron_1973,Giner_2013,Giner_2015,Garniron_2018,Garniron_2019} on the very same system \cite{Loos_2020e} [see also Ref.~\onlinecite{Lee_2020} for a study of the performance of phaseless auxiliary-field quantum Monte Carlo (ph-AFQMC) \cite{Motta_2018}].
-In the continuity of this recent work, we report here a significant extension by estimating the (frozen-core) FCI/cc-pVDZ correlation energy of twelve cyclic molecules (cyclopentadiene, furan, imidazole, pyrrole, thiophene, benzene, pyrazine, pyridazine, pyridine, pyrimidine, tetrazine, and triazine) with the help of CIPSI employing energetically-optimized orbitals at the same level of theory. \cite{Yao_2020,Yao_2021}
+In the continuity of this recent work, we report here a significant extension by estimating the (frozen-core) FCI/cc-pVDZ correlation energy of twelve cyclic molecules (cyclopentadiene, furan, imidazole, pyrrole, thiophene, benzene, pyrazine, pyridazine, pyridine, pyrimidine, s-tetrazine, and s-triazine) with the help of CIPSI employing energetically-optimized orbitals at the same level of theory. \cite{Yao_2020,Yao_2021}
 These systems are depicted in Fig.~\ref{fig:mol}.
 This set of molecular systems corresponds to Hilbert spaces with sizes ranging from $10^{29}$ to $10^{36}$.
 In addition to CIPSI, the performance and convergence properties of several series of methods are investigated.
@@ -155,7 +155,8 @@ The performance of the ground-state gold standard CCSD(T) as well as the complet
 
 %%% FIG 1 %%%
 \begin{figure*}
-	\includegraphics[width=\linewidth]{mol}
+	\includegraphics[width=0.8\linewidth]{ring5}
+	\includegraphics[width=\linewidth]{ring6}
 	\caption{
 	Five-membered rings (top) and six-membered rings (bottom) considered in this study.
 	\label{fig:mol}}
@@ -299,10 +300,6 @@ Note that a tight convergence is not critical here as a new set of microiteratio
 This procedure might sound computationally expensive but one has to realize that the microiterations are usually performed only for relatively compact variational spaces.
 Therefore, the computational bottleneck remains the diagonalization of the CI matrix for very large variational spaces.
 
-%\begin{equation}
-%	\Evar =  \sum_{pq} h_p^q \gamma_p^q + \frac{1}{2} \sum_{pqrs} v_{pq}^{rs} \Gamma_{pq}^{rs},
-%\end{equation}
-
 To enhance the convergence of the microiteration process, we employ an adaptation of the Newton-Raphson method known as ``trust region''. \cite{Nocedal_1999}
 This popular variant defines a region where the quadratic approximation \eqref{eq:EvarTaylor} is an adequate representation of the objective energy function \eqref{eq:Evar_c_k} and it evolves during the optimization process in order to preserve the adequacy via a constraint on the step size preventing it from overstepping, \ie, $\norm{\bk} \leq \Delta$, where $\Delta$ is the trust radius.
 By introducing a Lagrange multiplier $\lambda$ to control the trust-region size, one replaces Eq.~\eqref{eq:kappa_newton} by $\bk = - (\bH + \lambda \bI)^{-1} \cdot \bg$.
@@ -368,7 +365,7 @@ All the data (geometries, energies, etc) and supplementary material associated w
 	\includegraphics[width=0.24\textwidth]{Triazine_EvsNdet}
 	\caption{$\Delta \Evar$ (solid) and $\Delta \Evar + \EPT$ (dashed) as functions of the number of determinants $\Ndet$ in the variational space 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 CCSDTQ correlation energy is represented as a thick black line.
+	The FCI estimate of the correlation energy is represented as a thick black line.
 	\label{fig:vsNdet}}
 \end{figure*}
 %%% %%% %%%
@@ -393,7 +390,7 @@ All the data (geometries, energies, etc) and supplementary material associated w
 	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 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.
+	The FCI estimate of the correlation energy is represented as a thick black line.
 	\label{fig:vsEPT2}}
 \end{figure*}
 %%% %%% %%%
@@ -445,7 +442,7 @@ All the data (geometries, energies, etc) and supplementary material associated w
 	\label{tab:Tab6-VDZ}}
 	\begin{ruledtabular}
 	\begin{tabular}{lcccccccccccccc}
-				&	\mc{2}{c}{Benzene}	&	\mc{2}{c}{Pyrazine}	&	\mc{2}{c}{Pyridazine}	&	\mc{2}{c}{Pyridine}	&	\mc{2}{c}{Pyrimidine}	&	\mc{2}{c}{Tetrazine}	&	\mc{2}{c}{Triazine}	\\
+				&	\mc{2}{c}{Benzene}	&	\mc{2}{c}{Pyrazine}	&	\mc{2}{c}{Pyridazine}	&	\mc{2}{c}{Pyridine}	&	\mc{2}{c}{Pyrimidine}	&	\mc{2}{c}{s-Tetrazine}	&	\mc{2}{c}{s-Triazine}	\\
 					\cline{2-3}	\cline{4-5}	\cline{6-7} \cline{8-9} \cline{10-11} \cline{12-13} \cline{14-15}
 	Method		&	$E$	&	$\Delta E$	&	$E$	&	$\Delta E$	&	$E$	&	$\Delta E$	&	$E$	&	$\Delta E$
 				&	$E$	&	$\Delta E$	&	$E$	&	$\Delta E$	&	$E$	&	$\Delta E$	\\
@@ -475,92 +472,92 @@ All the data (geometries, energies, etc) and supplementary material associated w
 \end{squeezetable}
 %%% %%% %%%
 
-
 %%% TABLE III %%%
 \begin{squeezetable}
 \begin{table}
 	\caption{
 	Extrapolated correlation energies $\Delta \Eextrap$ (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.
+	Two sets of orbitals are considered: natural orbitals and optimized orbitals.
 	The weights are taken as the inverse square of the perturbative corrections.
 	For a $m$-point fit, the $m$ largest variational wave functions are used.
 	\label{tab:fit}}
 	\begin{ruledtabular}
-	\begin{tabular}{lccc}
-	Molecule	&	Number of		&	\mc{2}{c}{Fitting parameters}	\\
-									\cline{3-4}
-				&	 fitting points	&	$\Delta \Eextrap$	&	Fitting error	\\
+	\begin{tabular}{lccccc}
+	Molecule	&	Number of		&	\mc{2}{c}{Natural orbitals}	&	\mc{2}{c}{Optimized orbitals}	\\
+									\cline{3-4}\cline{5-6}
+				&	 fitting points	&	$\Delta \Eextrap$	&	Fitting error	&	$\Delta \Eextrap$	&	Fitting error	\\
 	\hline
-	Cyclopentadiene	&	3	&	$-739.295$	&	$0.199$	\\
-					&	4	&	$-739.309$	&	$0.088$	\\
-					&	\bf5	&	$\bf-739.230$	&	$\bf0.074$	\\
-					&	6	&	$-739.304$	&	$0.072$	\\
-					&	7	&	$-739.292$	&	$0.055$	\\
+	Cyclopentadiene	&	3	&	$-740.639$	&	$0.273$	&	$-739.295$	&	$0.199$		\\
+					&	4	&	$-740.243$	&	$0.306$	&	$-739.309$	&	$0.088$		\\
+					&\bf5	&	$-740.047$	&	$0.242$	&	$\bf-739.230$&	$\bf0.074$	\\
+					&	6	&	$-739.952$	&	$0.187$	&	$-739.304$	&	$0.072$		\\
+					&	7	&	$-739.761$	&	$0.204$	&	$-739.292$	&	$0.055$		\\
 	\hline
-	Furan			&	3	&	$-767.790$	&	$0.064$	\\
-					&	4	&	$-768.104$	&	$0.196$	\\
-					&	\bf5	&	$\bf-768.194$	&	$\bf0.135$	\\
-					&	6	&	$-768.060$	&	$0.131$	\\
-					&	7	&	$-768.086$	&	$0.101$	\\
+	Furan			&	3	&	$-766.090$	&	$0.729$	&	$-767.790$	&	$0.064$		\\
+					&	4	&	$-766.445$	&	$0.459$	&	$-768.104$	&	$0.196$		\\
+					&\bf5	&	$-766.582$	&	$0.318$	&	$\bf-768.194$	&$\bf0.135$	\\
+					&	6	&	$-766.366$	&	$0.288$	&	$-768.060$	&	$0.131$		\\
+					&	7	&	$-766.507$	&	$0.254$	&	$-768.086$	&	$0.101$		\\
 	\hline
-	Imidazole		&	3	&	$-778.295$	&	$0.356$	\\
-					&	4	&	$-778.270$	&	$0.150$	\\
-					&	\bf5	&	$\bf-778.178$	&	$\bf0.105$	\\
-					&	6	&	$-778.174$	&	$0.072$	\\
-					&	7	&	$-778.051$	&	$0.099$	\\
+	Imidazole		&	3	&	$-778.148$	&	$2.197$	&	$-778.295$	&	$0.356$		\\
+					&	4	&	$-777.436$	&	$1.107$	&	$-778.270$	&	$0.150$		\\
+					&\bf5	&	$-776.300$	&	$0.996$	&	$\bf-778.178$	&$\bf0.105$	\\
+					&	6	&	$-776.104$	&	$0.712$	&	$-778.174$	&	$0.072$		\\
+					&	7	&	$-776.098$	&	$0.541$	&	$-778.051$	&	$0.099$		\\
 	\hline
-	Pyrrole			&	3	&	$-758.650$	&	$0.321$	\\
-					&	4	&	$-758.389$	&	$0.174$	\\
-					&	\bf5	&	$\bf-758.460$	&	$\bf0.110$	\\
-					&	6	&	$-758.352$	&	$0.100$	\\
-					&	7	&	$-758.347$	&	$0.075$	\\
+	Pyrrole			&	3	&	$-758.309$	&	$0.447$	&	$-758.650$	&	$0.321$		\\
+					&	4	&	$-758.749$	&	$0.393$	&	$-758.389$	&	$0.174$		\\
+					&\bf5	&	$-758.405$	&	$0.359$	&	$\bf-758.460$	&$\bf0.110$	\\
+					&	6	&	$-758.136$	&	$0.334$	&	$-758.352$	&	$0.100$		\\
+					&	7	&	$-757.990$	&	$0.283$	&	$-758.347$	&	$0.075$		\\
 	\hline
-	Thiophene		&	3	&	$-728.744$	&	$0.691$	\\
-					&	4	&	$-729.052$	&	$0.331$	\\
-					&	\bf5	&	$\bf-728.948$	&	$\bf0.203$	\\
-					&	6	&	$-728.987$	&	$0.140$	\\
-					&	7	&	$-729.067$	&	$0.117$	\\
+	Thiophene		&	3	&	$-728.054$	&	$0.134$	&	$-728.744$	&	$0.691$		\\
+					&	4	&	$-728.240$	&	$0.139$	&	$-729.052$	&	$0.331$		\\
+					&\bf5	&	$-728.243$	&	$0.087$	&	$\bf-728.948$	&$\bf0.203$	\\
+					&	6	&	$-728.242$	&	$0.062$	&	$-728.987$	&	$0.140$		\\
+					&	7	&	$-728.420$	&	$0.144$	&	$-729.067$	&	$0.117$		\\
 	\hline
-	Benzene			&	3	&	$-862.325$	&	$0.279$	\\
-					&	4	&	$-863.024$	&	$0.424$	\\
-					&	\bf5	&	$\bf-862.890$	&	$\bf0.266$	\\
-					&	6	&	$-862.360$	&	$0.383$	\\
-					&	7	&	$-862.083$	&	$0.339$	\\
+	Benzene			&	3	&	&		&	$-862.325$	&	$0.279$		\\
+					&	4	&	&		&	$-863.024$	&	$0.424$		\\
+					&\bf5	&	&		&	$\bf-862.890$	&$\bf0.266$	\\
+					&	6	&	&		&	$-862.360$	&	$0.383$		\\
+					&	7	&	&		&	$-862.083$	&	$0.339$		\\
 	\hline
-	Pyrazine		&	3	&	$-904.867$	&	$1.420$	\\
-					&	4	&	$-904.588$	&	$0.650$	\\
-					&	\bf5	&	$\bf-904.550$	&	$\bf0.385$	\\
-					&	6	&	$-903.982$	&	$0.439$	\\
-					&	7	&	$-903.746$	&	$0.359$	\\
+	Pyrazine		&	3	&	$-904.148$	&	$0.035$	&	$-904.867$	&	$1.420$		\\
+					&	4	&	$-904.726$	&	$0.377$	&	$-904.588$	&	$0.650$		\\
+					&\bf5	&	$-904.274$	&	$0.383$	&	$\bf-904.550$	&$\bf0.385$	\\
+					&	6	&	$-903.980$	&	$0.341$	&	$-903.982$	&	$0.439$		\\
+					&	7	&	$-903.621$	&	$0.370$	&	$-903.746$	&	$0.359$		\\
 	\hline
-	Pyridazine		&	3	&	$-909.292$	&	$0.024$	\\
-					&	4	&	$-908.808$	&	$0.230$	\\
-					&	\bf5	&	$\bf-908.820$	&	$\bf0.133$	\\
-					&	6	&	$-908.342$	&	$0.303$	\\
-					&	7	&	$-908.368$	&	$0.224$	\\
+	Pyridazine		&	3	&	$-910.856$	&	$3.053$	&	$-909.292$	&	$0.024$		\\
+					&	4	&	$-908.222$	&	$1.834$	&	$-908.808$	&	$0.230$		\\
+					&\bf5	&	$-909.282$	&	$1.191$	&	$\bf-908.820$	&$\bf0.133$	\\
+					&	6	&	$-912.566$	&	$1.727$	&	$-908.342$	&	$0.303$		\\
+					&	7	&	$-910.694$	&	$2.210$	&	$-908.368$	&	$0.224$		\\
 	\hline
-	Pyridine		&	3	&	$-883.363$	&	$0.047$	\\
-					&	4	&	$-883.413$	&	$0.029$	\\
-					&	\bf5	&	$\bf-882.700$	&	$\bf0.405$	\\
-					&	6	&	$-882.361$	&	$0.341$	\\
-					&	7	&	$-882.023$	&	$0.330$	\\
+	Pyridine		&	3	&	&		&	$-883.363$	&	$0.047$		\\
+					&	4	&	&		&	$-883.413$	&	$0.029$		\\
+					&\bf5	&	&		&	$\bf-882.700$	&$\bf0.405$	\\
+					&	6	&	&		&	$-882.361$	&	$0.341$		\\
+					&	7	&	&		&	$-882.023$	&	$0.330$		\\
 	\hline
-	Pyrimidine		&	3	&	$-900.817$	&	$0.726$	\\
-					&	4	&	$-900.383$	&	$0.356$	\\
-					&	\bf5	&	$\bf-900.496$	&	$\bf0.214$	\\
-					&	6	&	$-900.698$	&	$0.190$	\\
-					&	7	&	$-900.464$	&	$0.206$	\\
+	Pyrimidine		&	3	&	$-900.386$	&	$1.884$	&	$-900.817$	&	$0.726$		\\
+					&	4	&	$-901.441$	&	$0.991$	&	$-900.383$	&	$0.356$		\\
+					&\bf5	&	$-900.354$	&	$0.865$	&	$\bf-900.496$	&$\bf0.214$	\\
+					&	6	&	$-900.240$	&	$0.594$	&	$-900.698$	&	$0.190$		\\
+					&	7	&	$-899.689$	&	$0.565$	&	$-900.464$	&	$0.206$		\\
 	\hline
-	Tetrazine		&	3	&	$-957.559$	&	$0.246$	\\
-					&	4	&	$-957.299$	&	$0.160$	\\
-					&	\bf5	&	$\bf-957.869$	&	$\bf0.349$	\\
-					&	6	&	$-957.744$	&	$0.247$	\\
-					&	7	&	$-957.709$	&	$0.183$	\\
+	s-Tetrazine		&	3	&	&		&	$-957.559$	&	$0.246$		\\
+					&	4	&	&		&	$-957.299$	&	$0.160$		\\
+					&\bf5	&	&		&	$\bf-957.869$	&$\bf0.349$	\\
+					&	6	&	&		&	$-957.744$	&	$0.247$		\\
+					&	7	&	&		&	$-957.709$	&	$0.183$		\\
 	\hline
-	Triazine		&	3	&	$-919.596$	&	$0.105$	\\
-					&	4	&	$-918.457$	&	$0.538$	\\
-					&	\bf5	&	$\bf-918.355$	&	$\bf0.312$	\\
-					&	6	&	$-918.206$	&	$0.226$	\\
-					&	7	&	$-917.876$	&	$0.267$	\\
+	s-Triazine		&	3	&	$-917.221$	&	$0.693$	&	$-919.596$	&	$0.105$		\\
+					&	4	&	$-918.723$	&	$0.913$	&	$-918.457$	&	$0.538$		\\
+					&\bf5	&	$-917.402$	&	$0.956$	&	$\bf-918.355$	&$\bf0.312$	\\
+					&	6	&	$-916.517$	&	$0.862$	&	$-918.206$	&	$0.226$		\\
+					&	7	&	$-916.544$	&	$0.643$	&	$-917.876$	&	$0.267$		\\
 	\end{tabular}
 	\end{ruledtabular}
 \end{table}
@@ -580,14 +577,14 @@ Adding the perturbative correction $\EPT$ yields similar curves for both sets of
 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.
 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}.
+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}.
 
 %%% FIG 4 %%%
 \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 green), and optimized orbitals (OOs, in blue).
-	The CCSDTQ correlation energy is represented as a thick black line.
+	The FCI estimate of the correlation energy is represented as a thick black line.
 	\label{fig:BenzenevsNdet}}
 \end{figure}
 %%% %%% %%%
@@ -606,6 +603,12 @@ Although we cannot provide a mathematically rigorous error bar, the data provide
 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}.
 
+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 the variation in extrapolated values and fitting error is much larger with the natural set.
+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.
+
+
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \subsection{Benchmark of CC and MP methods}
 \label{sec:mpcc_res}
@@ -630,12 +633,12 @@ Note that it is pleasing to see that, although different geometries are consider
 	\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 for reference.
+	The FCI estimate of the correlation energy is represented as a black line.
 	\label{fig:MPCC}}
 \end{figure*}
 %%% %%% %%%
 
-%%% TABLE III %%%
+%%% TABLE IV %%%
 \begin{squeezetable}
 \begin{table}
 	\caption{
@@ -683,7 +686,7 @@ 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 weakly-correlated 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 s-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.