saving work in 2nd part of results

Manuscript/Ec.tex

@@ -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 four-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 four 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}}
@@ -395,7 +395,7 @@ More details can be found in Ref.~\onlinecite{Nocedal_1999}.
 \begin{squeezetable}
 \begin{table*}
 	\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.
-	For the CIPSI estimates of the FCI correlation energy, the fitting error associated with the weighted four-point linear fit is reported in parenthesis.
+	For the CIPSI estimates of the FCI correlation energy, the fitting error associated with the weighted five-point linear fit is reported in parenthesis.
 	\label{tab:Tab5-VDZ}}
 	\begin{ruledtabular}
 	\begin{tabular}{lcccccccccc}
@@ -421,7 +421,7 @@ More details can be found in Ref.~\onlinecite{Nocedal_1999}.
 		CCSD(T)	&	$-193.5439$	&	$-735.6$	&	$-229.4073$	&	$-764.0$	&	$-225.6099$	&	$-774.5$	&	$-209.5836$	&	$-754.9$	&	$-552.0458$	&	$-724.8$
 \\
 		\hline
-		FCI	&					&	$-739.3(1)$			&					&	$-768.1(2)$		&					&	$-778.3(1)$				&					&	$-758.4(2)$			&			&	$-729.1(3)$\\
+		FCI	&					&	$-739.2(1)$			&					&	$-768.2(1)$		&					&	$-778.2(1)$				&					&	$-758.5(1)$			&			&	$-728.9(3)$\\
 	\end{tabular}
 	\end{ruledtabular}
 \end{table*}
@@ -432,7 +432,7 @@ More details can be found in Ref.~\onlinecite{Nocedal_1999}.
 \begin{squeezetable}
 \begin{table*}
 	\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.
-	For the CIPSI estimates of the FCI correlation energy, the fitting error associated with the weighted four-point linear fit is reported in parenthesis.
+	For the CIPSI estimates of the FCI correlation energy, the fitting error associated with the weighted five-point linear fit is reported in parenthesis.
 	\label{tab:Tab6-VDZ}}
 	\begin{ruledtabular}
 	\begin{tabular}{lcccccccccccccc}
@@ -458,7 +458,7 @@ More details can be found in Ref.~\onlinecite{Nocedal_1999}.
 		\hline
 		CCSD(T)	&	$-231.5798$	&	$-857.5$	&	$-263.6024$		&	$-899.4$	&	$-263.5740$	&	$-904.1$	&	$-247.5929$	&	$-877.7$	&	$-263.6099$	&	$-896.2$	&	$-295.5680$		&	$-952.2$	&	$-279.6305$		&	$-913.1$	\\
 		\hline
-		FCI	&				&	$-863.0(4)$		&				&	$-904.6(6)$		&				&	$-908.8(2)$		&				&	$-883.4(0)$		&				&	$-900.4(4)$		&		&	$-957.3(2)$		&		&	$-918.5(5)$\\
+		FCI	&				&	$-862.9(3)$		&				&	$-904.6(4)$		&				&	$-908.8(1)$		&				&	$-882.7(4)$		&				&	$-900.5(2)$		&		&	$-957.9(4)$		&		&	$-918.4(3)$\\
 	\end{tabular}
 	\end{ruledtabular}
 \end{table*}
@@ -482,87 +482,87 @@ More details can be found in Ref.~\onlinecite{Nocedal_1999}.
 	\hline
 	Cyclopentadiene	&	3	&	$-739.295$	&	$0.199$	\\
 					&	4	&	$-739.309$	&	$0.088$	\\
-					&	5	&	$-739.230$	&	$0.074$	\\
+					&	\bf5	&	$\bf-739.230$	&	$\bf0.074$	\\
 					&	6	&	$-739.304$	&	$0.072$	\\
 					&	7	&	$-739.292$	&	$0.055$	\\
-					&	8	&	$-739.119$	&	$0.131$	\\
+%					&	8	&	$-739.119$	&	$0.131$	\\
 	\hline
 	Furan			&	3	&	$-767.790$	&	$0.064$	\\
 					&	4	&	$-768.104$	&	$0.196$	\\
-					&	5	&	$-768.194$	&	$0.135$	\\
+					&	\bf5	&	$\bf-768.194$	&	$\bf0.135$	\\
 					&	6	&	$-768.060$	&	$0.131$	\\
 					&	7	&	$-768.086$	&	$0.101$	\\
-					&	8	&	$-767.904$	&	$0.154$	\\
+%					&	8	&	$-767.904$	&	$0.154$	\\
 	\hline
 	Imidazole		&	3	&	$-778.295$	&	$0.356$	\\
 					&	4	&	$-778.270$	&	$0.150$	\\
-					&	5	&	$-778.178$	&	$0.105$	\\
+					&	\bf5	&	$\bf-778.178$	&	$\bf0.105$	\\
 					&	6	&	$-778.174$	&	$0.072$	\\
 					&	7	&	$-778.051$	&	$0.099$	\\
-					&	8	&	$-777.992$	&	$0.089$	\\
+%					&	8	&	$-777.992$	&	$0.089$	\\
 	\hline
 	Pyrrole			&	3	&	$-758.650$	&	$0.321$	\\
 					&	4	&	$-758.389$	&	$0.174$	\\
-					&	5	&	$-758.460$	&	$0.110$	\\
+					&	\bf5	&	$\bf-758.460$	&	$\bf0.110$	\\
 					&	6	&	$-758.352$	&	$0.100$	\\
 					&	7	&	$-758.347$	&	$0.075$	\\
-					&	8	&	$-758.357$	&	$0.059$	\\
+%					&	8	&	$-758.357$	&	$0.059$	\\
 	\hline
 	Thiophene		&	3	&	$-728.744$	&	$0.691$	\\
 					&	4	&	$-729.052$	&	$0.331$	\\
-					&	5	&	$-728.948$	&	$0.203$	\\
+					&	\bf5	&	$\bf-728.948$	&	$\bf0.203$	\\
 					&	6	&	$-728.987$	&	$0.140$	\\
 					&	7	&	$-729.067$	&	$0.117$	\\
-					&	8	&	$-728.876$	&	$0.162$	\\
+%					&	8	&	$-728.876$	&	$0.162$	\\
 	\hline
 	Benzene			&	3	&	$-862.325$	&	$0.279$	\\
 					&	4	&	$-863.024$	&	$0.424$	\\
-					&	5	&	$-862.890$	&	$0.266$	\\
+					&	\bf5	&	$\bf-862.890$	&	$\bf0.266$	\\
 					&	6	&	$-862.360$	&	$0.383$	\\
 					&	7	&	$-862.083$	&	$0.339$	\\
-					&	8	&	$-861.711$	&	$0.370$	\\
+%					&	8	&	$-861.711$	&	$0.370$	\\
 	\hline
 	Pyrazine		&	3	&	$-904.867$	&	$1.420$	\\
 					&	4	&	$-904.588$	&	$0.650$	\\
-					&	5	&	$-904.550$	&	$0.385$	\\
+					&	\bf5	&	$\bf-904.550$	&	$\bf0.385$	\\
 					&	6	&	$-903.982$	&	$0.439$	\\
 					&	7	&	$-903.746$	&	$0.359$	\\
-					&	8	&	$-903.549$	&	$0.311$	\\
+%					&	8	&	$-903.549$	&	$0.311$	\\
 	\hline
 	Pyridazine		&	3	&	$-909.292$	&	$0.024$	\\
 					&	4	&	$-908.808$	&	$0.230$	\\
-					&	5	&	$-908.820$	&	$0.133$	\\
+					&	\bf5	&	$\bf-908.820$	&	$\bf0.133$	\\
 					&	6	&	$-908.342$	&	$0.303$	\\
 					&	7	&	$-908.368$	&	$0.224$	\\
-					&	8	&	$-908.229$	&	$0.198$	\\
+%					&	8	&	$-908.229$	&	$0.198$	\\
 	\hline
 	Pyridine		&	3	&	$-883.363$	&	$0.047$	\\
 					&	4	&	$-883.413$	&	$0.029$	\\
-					&	5	&	$-882.700$	&	$0.405$	\\
+					&	\bf5	&	$\bf-882.700$	&	$\bf0.405$	\\
 					&	6	&	$-882.361$	&	$0.341$	\\
 					&	7	&	$-882.023$	&	$0.330$	\\
-					&	8	&	$-881.732$	&	$0.322$	\\
+%					&	8	&	$-881.732$	&	$0.322$	\\
 	\hline
 	Pyrimidine		&	3	&	$-900.817$	&	$0.726$	\\
 					&	4	&	$-900.383$	&	$0.356$	\\
-					&	5	&	$-900.496$	&	$0.214$	\\
+					&	\bf5	&	$\bf-900.496$	&	$\bf0.214$	\\
 					&	6	&	$-900.698$	&	$0.190$	\\
 					&	7	&	$-900.464$	&	$0.206$	\\
-					&	8	&	$-900.226$	&	$0.227$	\\
+%					&	8	&	$-900.226$	&	$0.227$	\\
 	\hline
 	Tetrazine		&	3	&	$-957.559$	&	$0.246$	\\
 					&	4	&	$-957.299$	&	$0.160$	\\
-					&	5	&	$-957.869$	&	$0.349$	\\
+					&	\bf5	&	$\bf-957.869$	&	$\bf0.349$	\\
 					&	6	&	$-957.744$	&	$0.247$	\\
 					&	7	&	$-957.709$	&	$0.183$	\\
-					&	8	&	$-957.558$	&	$0.176$	\\
+%					&	8	&	$-957.558$	&	$0.176$	\\
 	\hline
 	Triazine		&	3	&	$-919.596$	&	$0.105$	\\
 					&	4	&	$-918.457$	&	$0.538$	\\
-					&	5	&	$-918.355$	&	$0.312$	\\
+					&	\bf5	&	$\bf-918.355$	&	$\bf0.312$	\\
 					&	6	&	$-918.206$	&	$0.226$	\\
 					&	7	&	$-917.876$	&	$0.267$	\\
-					&	8	&	$-917.533$	&	$0.308$	\\
+%					&	8	&	$-917.533$	&	$0.308$	\\
 	\end{tabular}
 	\end{ruledtabular}
 \end{table}
@@ -581,7 +581,7 @@ 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 four-point weighted linear fit using the four largest variational wave functions are also represented (dashed black lines). 
+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).
 
@@ -602,7 +602,7 @@ 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 four-point linear fit using the four largest variational wave functions (see Fig.~\ref{fig:vsEPT2}).
+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.
 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.
@@ -639,21 +639,60 @@ Note that it is pleasing to see that, although different geometries are consider
 \end{figure*}
 %%% %%% %%%
 
+%%% TABLE III %%%
+\begin{squeezetable}
+\begin{table}
+	\caption{
+	Mean absolute error (MAE), mean signed error (MSE), and minimum (Min) and maximum (Max) absolute errors (in \SI{}{\milli\hartree}) with respect to the FCI correlation energy for various methods. 
+	The formal computational scaling of each method is also reported.
+	\label{tab:stats}}
+	\begin{ruledtabular}
+	\begin{tabular}{lcdddd}
+	Method	&	Scaling			&	\tabc{MAE}	&	\tabc{MSE}	&	\tabc{Min}	&	\tabc{Max}	\\
+	\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	\\
+	MP4		&	$\order{N^7}$	&	2.1		&	2.0		&	4.7		&	0.7		\\
+	MP5		&	$\order{N^8}$	&	9.4		&	9.4		&	13.6	&	5.8		\\
+	\hline
+	CC2		&	$\order{N^5}$	&	58.9	&	58.9	&	73.5	&	48.9	\\
+	CC3		&	$\order{N^7}$	&	2.7		&	2.7		&	3.8		&	2.1		\\
+	CC4		&	$\order{N^9}$	&	1.5		&	1.5		&	2.3		&	0.8		\\
+	\hline
+	CCSD	&	$\order{N^6}$	&	39.4	&	39.4	&	48.8	&	32.0	\\
+	CCSDT	&	$\order{N^8}$	&	4.5		&	4.5		&	6.3		&	3.0		\\
+	CCSDTQ	&	$\order{N^{10}}$&	1.8		&	1.8		&	2.6		&	1.0		\\
+	\hline
+	CCSD(T)	&	$\order{N^7}$	&	4.5		&	4.5		&	5.7		&	3.6		\\
+	\end{tabular}
+	\end{ruledtabular}
+\end{table}
+\end{squeezetable}
+%%% %%% %%%
+
 Using the CIPSI estimates of the FCI correlation energy produced in Sec.~\ref{sec:cipsi_res}, we now study the performance and convergence properties of three series of methods: i) MP2, MP3, MP4, and MP5, ii) CC2, CC3, and CC4, and iii) CCSD, CCSDT, and CCSDTQ.
 Additionally, we also report CCSD(T) correlation energies.
 All these data are reported in Tables \ref{tab:Tab5-VDZ} and \ref{tab:Tab6-VDZ} for the five- and six-membered rings, respectively. 
 In Fig.~\ref{fig:MPCC}, we show, for each molecule, the convergence of the correlation energy for each series of methods as a function of the computational cost of the corresponding method.
 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, let us investigate 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}
+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.
+ 
+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 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 is thus systematically worse than MP4 for these 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 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.
+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.
-As observed in our recent study on excitation energies, \cite{Loos_2021} CC4 is an outstanding approximation to its CCSDTQ parent.
+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 here that CC3 and CC4 provide correlation energies only one or two millihartree different, which shows that CC3 is particularly effective for ground-state energetics.
+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.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Conclusion}