saving work before OO section

Manuscript/Ec.bib

@@ -1,7 +1,7 @@
 %% This BibTeX bibliography file was created using BibDesk.
 %% http://bibdesk.sourceforge.net/
 
-%% Created for Pierre-Francois Loos at 2021-06-19 07:02:58 +0200 
+%% Created for Pierre-Francois Loos at 2021-07-02 10:54:28 +0200 
 
 
 %% Saved with string encoding Unicode (UTF-8) 

Manuscript/Ec.tex

@@ -22,13 +22,15 @@
 \newcommand{\tabc}[1]{\multicolumn{1}{c}{#1}}
 \newcommand{\QP}{\textsc{quantum package}}
 
+\newcommand{\Nel}{n}
+\newcommand{\Norb}{N}
 \newcommand{\Ndet}{N_\text{det}}
-\newcommand{\Norb}{N_\text{orb}}
 
 \newcommand{\br}{\boldsymbol{r}}
 \newcommand{\bc}{\boldsymbol{c}}
 \newcommand{\bR}{\boldsymbol{R}}
 \newcommand{\bX}{\boldsymbol{X}}
+\newcommand{\bH}{\boldsymbol{H}}
 
 \newcommand{\hH}{\Hat{H}}
 \newcommand{\hh}{\Hat{h}}
@@ -87,17 +89,17 @@ The nuclei coordinates can then be treated as parameters in the electronic Hamil
 The second central approximation which makes calculations feasible by a computer is the basis set approximation where one introduces a set of pre-defined basis functions to represent the many-electron wave function of the system.
 In most molecular calculations, a set of one-electron, atom-centered gaussian basis functions are introduced to expand the so-called one-electron molecular orbitals which are then used to build the many-electron Slater determinants.
 The third and most relevant approximation in the present context is the ansatz (or form) of the electronic wave function $\Psi$.
-For example, in configuration interaction (CI) methods, the wave function is expanded as a linear combination of Slater determinants, while in (single-reference) coupled-cluster (CC) theory, \cite{Cizek_1966,Paldus_1972,Crawford_2000,Piecuch_2002,Bartlett_2007,Shavitt_2009} a reference Slater determinant $\Psi_0$ [usually taken as the Hartree-Fock (HF) wave function] is multiplied by a wave operator defined as the exponentiated excitation operator $\Hat{T} = \sum_{k=1}^n \Hat{T}_k$ (where $n$ is the number of electrons).
+For example, in configuration interaction (CI) methods, the wave function is expanded as a linear combination of Slater determinants, while in (single-reference) coupled-cluster (CC) theory, \cite{Cizek_1966,Paldus_1972,Crawford_2000,Piecuch_2002,Bartlett_2007,Shavitt_2009} a reference Slater determinant $\Psi_0$ [usually taken as the Hartree-Fock (HF) wave function] is multiplied by a wave operator defined as the exponentiated excitation operator $\Hat{T} = \sum_{k=1}^\Nel \Hat{T}_k$ (where $\Nel$ is the number of electrons).
 
 The truncation of $\Hat{T}$ allows to define a hierarchy of non-variational and size-extensive methods with improved accuracy:
-CC with singles and doubles (CCSD), \cite{Cizek_1966,Purvis_1982} CC with singles, doubles, and triples (CCSDT), \cite{Noga_1987a,Scuseria_1988} CC with singles, doubles, triples, and quadruples (CCSDTQ), \cite{Oliphant_1991,Kucharski_1992} with corresponding computational scalings of $\order*{N^{6}}$, $\order*{N^{8}}$, and $\order*{N^{10}}$, respectively (where $N$ denotes the number of orbitals).
+CC with singles and doubles (CCSD), \cite{Cizek_1966,Purvis_1982} CC with singles, doubles, and triples (CCSDT), \cite{Noga_1987a,Scuseria_1988} CC with singles, doubles, triples, and quadruples (CCSDTQ), \cite{Oliphant_1991,Kucharski_1992} with corresponding computational scalings of $\order*{\Norb^{6}}$, $\order*{\Norb^{8}}$, and $\order*{\Norb^{10}}$, respectively (where $\Norb$ denotes the number of orbitals).
 Parallel to the ``complete'' CC series presented above, an alternative series of approximate iterative CC models have been developed by the Aarhus group in the context of CC response theory \cite{Christiansen_1998} where one skips the most expensive terms and avoids the storage of the higher-excitation amplitudes:  CC2, \cite{Christiansen_1995a} CC3, \cite{Christiansen_1995b,Koch_1997} and CC4 \cite{Kallay_2005,Matthews_2021} 
-These iterative methods scale as $\order*{N^{5}}$, $\order*{N^{7}}$, and $\order*{N^{9}}$, respectively, and can be seen as cheaper approximations of CCSD, CCSDT, and CCSDTQ.
+These iterative methods scale as $\order*{\Norb^{5}}$, $\order*{\Norb^{7}}$, and $\order*{\Norb^{9}}$, respectively, and can be seen as cheaper approximations of CCSD, CCSDT, and CCSDTQ.
 Coupled-cluster methods have been particularly successful at computing accurately various properties for small- and medium-sized molecules.
 \cite{Kallay_2003,Kallay_2004a,Gauss_2006,Kallay_2006,Gauss_2009}  
 
 A similar systematic truncation strategy can be applied to CI methods leading to the well-established family of methods known as CISD, CISDT, CISDTQ, \ldots~where one systematically increases the maximum excitation degree of the determinants taken into account. 
-Except for full CI (FCI) where all determinants from the Hilbert space (\ie, with excitation degree up to $N$) are considered, truncated CI methods are variational but lack size-consistency.
+Except for full CI (FCI) where all determinants from the Hilbert space (\ie, with excitation degree up to $\Nel$) are considered, truncated CI methods are variational but lack size-consistency.
 The non-variationality of truncated CC methods being less of an issue than the size-inconsistency of the truncated CI methods, the formers have naturally overshadowed the latters in the electronic structure landscape.
 However, a different strategy has recently made a come back in the context of CI methods. \cite{Bender_1969,Whitten_1969,Huron_1973}
 Indeed, selected CI (SCI) methods, \cite{Booth_2009,Giner_2013,Evangelista_2014,Giner_2015,Holmes_2016,Tubman_2016,Liu_2016,Ohtsuka_2017,Zimmerman_2017,Coe_2018,Garniron_2018} where one iteratively selects the energetically relevant determinants from the FCI space (usually) based on a perturbative criterion, has been recently shown to be highly successful in order to produce reference energies for ground and excited states in small- and medium-size molecules \cite{Holmes_2017,Li_2018,Li_2020,Loos_2018a,Chien_2018,Loos_2019,Loos_2020b,Loos_2020c,Loos_2020e,Garniron_2019,Eriksen_2020,Yao_2020,Veril_2021,Loos_2021} thanks to efficient deterministic, stochastic or hybrid algorithms well suited for massive parallelization. 
@@ -110,9 +112,9 @@ Note that, very recently, several groups \cite{Aroeira_2021,Lee_2021,Magoulas_20
 A rather different strategy in order to reach the holy grail FCI limit is to resort to M{\o}ller-Plesset (MP) perturbation theory, \cite{Moller_1934}
 which popularity originates from its black-box nature, size-extensivity, and relatively low computational scaling, making it easily applied to a broad range of molecular systems.
 Again, at least in theory, one can obtain the exact energy of the system by ramping up the degree of the perturbative series. \cite{Marie_2021} 
-The second-order M{\o}ller-Plesset (MP2) method \cite{Moller_1934} [which scales as $\order*{N^{5}}$] has been broadly adopted in quantum chemistry for several decades, and is now included in the increasingly popular double-hybrid functionals \cite{Grimme_2006} alongside exact HF exchange.
+The second-order M{\o}ller-Plesset (MP2) method \cite{Moller_1934} [which scales as $\order*{\Norb^{5}}$] has been broadly adopted in quantum chemistry for several decades, and is now included in the increasingly popular double-hybrid functionals \cite{Grimme_2006} alongside exact HF exchange.
 Its higher-order variants [MP3, \cite{Pople_1976}
-MP4, \cite{Krishnan_1980} MP5, \cite{Kucharski_1989} and MP6 \cite{He_1996a,He_1996b} which scales as $\order*{N^{6}}$, $\order*{N^{7}}$, $\order*{N^{8}}$, and $\order*{N^{9}}$ respectively] have been investigated much more scarcely.
+MP4, \cite{Krishnan_1980} MP5, \cite{Kucharski_1989} and MP6 \cite{He_1996a,He_1996b} which scales as $\order*{\Norb^{6}}$, $\order*{\Norb^{7}}$, $\order*{\Norb^{8}}$, and $\order*{\Norb^{9}}$ respectively] have been investigated much more scarcely.
 However, it is now widely recognised that the series of MP approximations might show erratic, slow, or divergent behavior that limit its applicability and systematic improvability. \cite{Laidig_1985,Knowles_1985,Handy_1985,Gill_1986,Laidig_1987,Nobes_1987,Gill_1988,Gill_1988a,Lepetit_1988,Malrieu_2003} 
 Again, MP perturbation theory and CC methods can be coupled. 
 The CCSD(T) method, \cite{Raghavachari_1989} known as the gold-standard of quantum chemistry for weakly correlated systems, is probably the most iconic example of such coupling.
@@ -132,8 +134,7 @@ The performance of the ground-state gold standard CCSD(T) is also investigated.
 	Five-membered rings (top) and six-membered rings (bottom) considered in this study.
 	\label{fig:mol}}
 \end{figure*}
-%%% FIG 1 %%%
-
+%%% %%% %%%
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Computational details}
@@ -142,33 +143,31 @@ The geometries of the twelve systems considered in the present study have been a
 Note that, for the sake of consistency, the geometry of benzene considered here is different from one of Ref.~\onlinecite{Loos_2020e} which has been computed at a lower level of theory [MP2/6-31G(d)]. \cite{Schreiber_2008}
 The MP2, MP3, MP4, CC2, CC3, CC4, CCSD, CCSDT, and CCSDTQ calculations have been performed with CFOUR, \cite{Matthews_2020} while the CCSD(T) and MP5 calculations have been computed with Gaussian 09. \cite{g09}
 The CIPSI calculations have been performed with {\QP}. \cite{Garniron_2019}
-\titou{The particularity of the current implementation is that the selection step and the PT2 correction are computed \textit{simultaneously} via a hybrid semistochastic algorithm \cite{Garniron_2017,Garniron_2019} (which explains the statistical error associated with the PT2 correction in the following).
-Moreover, a renormalized version of the PT2 correction (dubbed rPT2 below) has been recently implemented and tested for a more efficient extrapolation to the FCI limit thanks to a partial resummation of the higher-order of perturbation. \cite{Garniron_2019} 
-We refer the interested reader to Ref.~\onlinecite{Garniron_2019} where one can find all the details regarding the implementation of the rPT2 correction and the CIPSI algorithm.}
+In the current implementation, the selection step and the PT2 correction are computed simultaneously via a hybrid semistochastic algorithm. \cite{Garniron_2017,Garniron_2019} (which explains the statistical error associated with the PT2 correction in the following).
+Here, we employ the renormalized version of the PT2 correction which has been recently implemented and tested for a more efficient extrapolation to the FCI limit thanks to a partial resummation of the higher-order of perturbation. \cite{Garniron_2019} 
+We refer the interested reader to Ref.~\onlinecite{Garniron_2019} where one can find all the details regarding the implementation of the PT2 correction and the CIPSI algorithm.
 For all these calculations, we consider Dunning's correlation-consistent double-$\zeta$ basis (cc-pVDZ).
 
 Although the FCI energy has the enjoyable property of being independent of the set of one-electron orbitals used to construct the many-electron Slater determinants, as a truncated CI method, the convergence properties of CIPSI strongly dependent on this orbital choice.
-In the present study, we investigate the convergence behavior of the CIPSI energy for four distinct sets: natural orbitals (NOs), localized orbitals (LOs), and optimized orbitals (OOs).
+In the present study, we investigate the convergence behavior of the CIPSI energy for two sets of orbitals in particular: natural orbitals (NOs) and optimized orbitals (OOs).
 Following our usual procedure, \cite{Scemama_2018,Scemama_2018b,Scemama_2019,Loos_2018a,Loos_2019,Loos_2020a,Loos_2020b,Loos_2020c,Loos_2020e} we perform first a preliminary SCI calculation using HF orbitals in order to generate a SCI wave function with at least $10^7$ determinants.
-Then, natural orbitals (NOs) are computed based on this wave function, and subsequently localized orbitals.
-The Boys-Foster localization procedure \cite{Boys_1960} that we apply to the natural orbitals is performed in several orbital windows: \titou{i) core, ii) valence $\sigma$, iii) valence $\pi$, iv) valence $\pi^*$, v) valence $\sigma^*$, vi) the higher-lying $\sigma$ orbitals, and vii) the higher-lying $\pi$ orbitals.}
-Like Pipek-Mezey, \cite{Pipek_1989} this choice of orbital windows allows to preserve a strict $\sigma$-$\pi$ separation in planar systems like the ones considered here.
-Because they take advantage of the local character of electron correlation, localized orbitals have been shown to provide faster convergence towards the FCI limit compared to natural orbitals. \cite{Angeli_2003,Angeli_2009,BenAmor_2011,Suaud_2017,Chien_2018,Eriksen_2020,Loos_2020e} 
-Using these localized orbitals as starting point, we also perform successive orbital optimizations, which consist in minimizing the variational CIPSI energy at each iteration up to approximately $2 \times 10^5$ determinants.
+Natural orbitals are computed based on this wave function and they are used to perform a new CIPSI run.
+Successive orbital optimizations are then performed, which consist in minimizing the variational CIPSI energy at each iteration up to approximately $2 \times 10^5$ determinants.
 When convergence is achieved in terms of orbital optimization, as our ``production'' run, we perform a new CIPSI calculation from scratch using this set of optimized orbitals.
-As we shall see below, employing optimized orbitals has the advantage to produce a smoother and faster convergence of the SCI energy toward the FCI limit.
+In some cases, we also explore the use of localized orbitals (LOs) which are produced with the help of the Boys-Foster localization procedure \cite{Boys_1960} that we apply to the natural orbitals in several orbital windows in order to preserve a strict $\sigma$-$\pi$ separation in the planar systems considered here.
+Because they take advantage of the local character of electron correlation, localized orbitals have been shown to provide faster convergence towards the FCI limit compared to natural orbitals. \cite{Angeli_2003,Angeli_2009,BenAmor_2011,Suaud_2017,Chien_2018,Eriksen_2020,Loos_2020e} 
+As we shall see below, employing optimized orbitals has the advantage to produce an even smoother and faster convergence of the SCI energy toward the FCI limit.
 Note that, unlike excited-state calculations where it is important to enforce that the wave functions are eigenfunctions of the $\Hat{S}^2$ spin operator, \cite{Chilkuri_2021} the present wave functions do not fulfil this property as we aim for the lowest possible energy of a singlet state. 
 We have found that $\expval*{\Hat{S}^2}$ is, nonetheless, very close to zero for each system.
 
-
 %%%%%%%%%%%%%%%%%%%%%%%%%
 \section{CIPSI with optimized orbitals}
 %%%%%%%%%%%%%%%%%%%%%%%%%
 
-Here, we provide key details about the CIPSI method. 
+Here, we provide key details about the CIPSI method. \cite{Huron_1973,Garniron_2019}
 Note that we focus on the ground state but the present discussion can be easily extended to excited states. \cite{Scemama_2019,Veril_2021}
 
-At each iteration $k$, the total CIPSI energy $E_\text{CIPSI}^{(k)}$ is defined as the sum of the variational energy 
+At the $k$th iteration, the total CIPSI energy $E_\text{CIPSI}^{(k)}$ is defined as the sum of the variational energy 
 \begin{equation}
 	E_\text{var}^{(k)} = \frac{\mel*{\Psi_\text{var}^{(k)}}{\Hat{H}}{\Psi_\text{var}^{(k)}}}{\braket*{\Psi_\text{var}^{(k)}}{\Psi_\text{var}^{(k)}}}
 \end{equation} 
@@ -178,21 +177,27 @@ and a second-order perturbative correction
 	= \sum_{\alpha \in \mathcal{A}_k} e_{\alpha} 
 	= \sum_{\alpha \in \mathcal{A}_k} \frac{\mel*{\Psi_\text{var}^{(k)}}{\Hat{H}}{\alpha}}{E_\text{var}^{(k)} - \mel*{\alpha}{\Hat{H}}{\alpha}}
 \end{equation}
-where $\Psi_\text{var}^{(k)} = \sum_{I \in \mathcal{I}_k} c_I^{(k)} \ket*{I}$ is the variational wave function, $\mathcal{I}_k$ is the set of internal determinants $\ket*{I}$ and $\mathcal{A}_k$ is the set of external determinants $\ket*{\alpha}$ which do not belong to the variational space but are linked to it via a nonzero matrix element, \ie, $\mel*{\Psi_\text{var}^{(k)}}{\Hat{H}}{\alpha} \neq 0$.
+where $\hH$ is the (non-relativistic) electronic Hamiltonian, 
+\begin{equation}
+\label{eq:Psivar}
+	\Psi_\text{var}^{(k)} = \sum_{I \in \mathcal{I}_k} c_I^{(k)} \ket*{I}
+\end{equation}
+is the variational wave function, $\mathcal{I}_k$ is the set of internal determinants $\ket*{I}$ and $\mathcal{A}_k$ is the set of external determinants $\ket*{\alpha}$ which do not belong to the variational space but are linked to it via a nonzero matrix element, \ie, $\mel*{\Psi_\text{var}^{(k)}}{\Hat{H}}{\alpha} \neq 0$.
 The sets $\mathcal{I}_k$ and $\mathcal{A}_k$ define, at the $k$th iteration, the internal and external spaces, respectively.
-In practice, $E_\text{var}^{(k)}$ is computed by diagonalizing the CI matrix in the reference space and the magnitude of $E_\text{PT2}$ provides a qualitative idea of the ``distance'' to the FCI limit.
+In practice, $E_\text{var}^{(k)}$ is computed by diagonalizing the $\Ndet^{(k)} \times \Ndet^{(k)}$ CI matrix $\bH$ with elements $H_{IJ} = \mel{I}{\hH}{J}$ via Davidson's algorithm \cite{Davidson_1975} and the magnitude of $E_\text{PT2}$ provides a qualitative idea of the ``distance'' to the FCI limit. \cite{Garniron_2018}
 We then linearly extrapolate, using large variational space, the CIPSI energy to $E_\text{PT2} = 0$ (which effectively corresponds to the FCI limit). 
+Further details concerning the extrapolation procedure are provided below (see Sec.~\ref{sec:res}).
 
+Orbital optimization techniques at the SCI level are theoretically straightforward, but practically challenging. \cite{Yao_2020,Yao_2021}
+Here, we detail our orbital optimization procedure within the CIPSI algorithm and we assume that the variational wave function is normalized, \ie, $\braket*{\Psi_\text{var}}{\Psi_\text{var}} = 1$. 
+Then, the variational energy can be written as
+\begin{equation}
+	E_\text{var}(\bc,\bX) = \mel{\Psi_\text{var}}{e^{\hX} \hH e^{-\hX}}{\Psi_\text{var}},
+\end{equation}
+where $\bc$ gathers the CI coefficients, $\bX$ the orbital rotation parameters and $\hX$ is a one-electron anti-hermitian operator, which creates a rotation matrix when exponentiated, \ie, $\bR = e^{\bX}$. 
+From a more general point of view, the variational energy $E_\text{var}$ depends on both the coefficient $\{ c_I \}_{1 \le I \le \Ndet^{(k)}}$ [see Eq.~\eqref{eq:Psivar}] but also on the orbital rotation parameter $\{X_{pq}\}_{1 \le p,q \le \Norb}$.
 
-Orbital optimization techniques at the SCI level are theoretically straightforward, but practically challenging.
-Here, we detail our orbital optimization procedure with the CIPSI algorithm.
-From a more general point of view, the variational energy $E_\text{var}^{(k)}$ depends on both the coefficient $\{ c_I \}_{1 \le I \le \Ndet^{(k)}}$ but also on the orbital rotation parameter $\{X_{pq}\}_{1 \le \mu p,q \le \Norb}$.
-%such that the $p$th orbital is
-%\begin{equation}
-%	\phi_p(\br) = \sum_{\mu} C_{\mu p} \chi_{\mu}(\br)
-%\end{equation}
-%where $\chi_{\mu}(\br)$ is a basis function.
-The diagonalization of the CI matrix ensures that
+For a given set of orbitals, The diagonalization of the CI matrix ensures that
 \begin{equation}
 	\pdv{E_\text{var}(\bc,\bX)}{c_I} = 0,
 \end{equation}
@@ -200,16 +205,10 @@ but, a priori, we have
 \begin{equation}
 	\pdv{E_\text{var}(\bc,\bX)}{X_{pq}} \neq 0,
 \end{equation}
+Although one could use a second order method to minimize the corresponding energy, one has to realize that the size of the CI space is much larger than orbital space.
+It is therefore more appropriate to perform a minimization of the variational energy with respect to the orbital rotation parameters and then compute the new CI coefficients by re-diagonalizing the CI matrix. 
 
 
-Here, we assume that the variational wave function is normalized, \ie, $\braket*{\Psi_\text{var}}{\Psi_\text{var}} = 1$. 
-Then, the previous equation can be rewritten, 
-\begin{equation}
-	E(\bc,\bX) = \mel{\Psi_\text{var}}{e^{\hX} \hH e^{-\hX}}{\Psi_\text{var}},
-\end{equation}
-where $\bc$ gathers the CI coefficients, $\bX$ the orbital rotation parameters and $\hX$ is a one-electron anti-hermitian operator, which creates a rotation matrix when exponentiated, \ie, $\bR = e^{\bX}$. 
-The energy $E$ depends on the determinants and their number, thus the orbital optimization will not be the same for two different sets of determinants.
-
 To understand the concept of orbital rotation, we look at this operator $\bX$ in more details, 
 \begin{equation}
 	\hX = \sum_{p > q} \sum_{\sigma} X_{pq} (\hat{a}_{p \sigma}^{\dagger} \hat{a}_{q \sigma} - \hat{a}_{q \sigma}^{\dagger} \hat{a}_{p \sigma}),
@@ -262,18 +261,33 @@ And the Hessian of the energy with respect to the orbital rotation, $H_{pq,rs}$,
 with $\delta_{ij}$ is the Kronecker delta, $\mathcal{P}_{pq}$ the permutation operator $\mathcal{P}_{pq} = 1
 - (p \leftrightarrow q)$ and where $(p \leftrightarrow q)$ applied to an equation returns the same equation with the indices $p$ and $q$ swapped.
 
+Trust region method and Newton's method are very similar, they use a quadratical approximation of a real function, i.e.,
+Taylor expansion truncated at the second order. In the Newton method, the step is given by the minimizer of the
+quadratical approximation, contrary to the trust region method which gives the minimizer of the quadratical approximation
+in the trust region. The trust region defines a region where the quadratical approximation is a adequate representation
+of the real function and it evolves during the optimization process in order to preserve the adequacy. The constraint
+for the step size is the following, $\norm{\bm{X}_{k+1}} \leq \Delta_k$ with  $\Delta_k$ the trust radius. By putting the
+constraint with a Lagrange multiplier $\lambda$ and derivating the Lagrangian, the solution is 
+$\bm{X}_{k+1} = - (\bm{H_k} + \lambda \bm{I})^{-1} \cdot \bm{g}_k$. 
+The addition of a constant $\lambda \geq 0$ on the diagonal of the hessian removes the negative eigenvalues and
+reduce the size of the step since the calculation uses its inverse. By choosing the right $\lambda$ the step size is contraint
+into a hypersphere of radius $\Delta_k$. In addition, the evolution of $\Delta_k$ during the optimization and the use of
+a condition to cancel a step ensure the convergence of the algorithm.
+More details could be found in Numerical Optimization, Nocedal \& Wright (1999)
+
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Results and discussion}
+\label{sec:res}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 \begin{table*}
-	\caption{Total energy $E$ (in \Eh) and correlation energy $\Ec$ (in \mEh) for the frozen-core ground state of five-membered rings in the cc-pVDZ basis set.
+	\caption{Total energy $E$ (in \Eh) and correlation energy $\Delta E$ (in \mEh) for the frozen-core ground state of five-membered rings in the cc-pVDZ basis set.
 	\label{tab:Tab5-VDZ}}
 	\begin{ruledtabular}
 	\begin{tabular}{lcccccccccc}
 				&	\mc{2}{c}{Cyclopentadiene}	&	\mc{2}{c}{Furan}	&	\mc{2}{c}{Imidazole}	&	\mc{2}{c}{Pyrrole}	&	\mc{2}{c}{Thiophene}	\\
 					\cline{2-3}	\cline{4-5}	\cline{6-7} \cline{8-9} \cline{10-11}
-		Method	&	$E$&	$\Ec$	&	$E$	&	$\Ec$	&	$E$	&	$\Ec$	&	$E$	&	$\Ec$	&	$E$	&	$\Ec$ 	\\
+		Method	&	$E$&	$\Delta E$	&	$E$	&	$\Delta E$	&	$E$	&	$\Delta E$	&	$E$	&	$\Delta E$	&	$E$	&	$\Delta E$ 	\\
 		\hline
 		HF		&	$-192.8083$	&				&	$-228.6433$	&				&	$-224.8354$	&				&	$-208.8286$	&		&	-551.3210	&	\\
 		\hline
@@ -300,14 +314,14 @@ with $\delta_{ij}$ is the Kronecker delta, $\mathcal{P}_{pq}$ the permutation op
 
 \begin{squeezetable}
 \begin{table*}
-	\caption{Total energy $E$ (in \Eh) and correlation energy $\Ec$ (in \mEh) for the frozen-core ground state of six-membered rings in the cc-pVDZ basis set.
+	\caption{Total energy $E$ (in \Eh) and correlation energy $\Delta E$ (in \mEh) for the frozen-core ground state of six-membered rings in the cc-pVDZ basis set.
 	\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}	\\
 					\cline{2-3}	\cline{4-5}	\cline{6-7} \cline{8-9} \cline{10-11} \cline{12-13} \cline{14-15}
-	Method		&	$E$	&	$\Ec$	&	$E$	&	$\Ec$	&	$E$	&	$\Ec$	&	$E$	&	$\Ec$	
-				&	$E$	&	$\Ec$	&	$E$	&	$\Ec$	&	$E$	&	$\Ec$	\\
+	Method		&	$E$	&	$\Delta E$	&	$E$	&	$\Delta E$	&	$E$	&	$\Delta E$	&	$E$	&	$\Delta E$	
+				&	$E$	&	$\Delta E$	&	$E$	&	$\Delta E$	&	$E$	&	$\Delta E$	\\
 	\hline
 		HF		&	$-230.7222$	&			&	$-262.7030$	&			&	$-262.6699$	&			&	$-246.7152$	&			&	$-262.7137$	&			&	$-294.6157$		&		&	$-278.7173$	\\
 		\hline
@@ -326,47 +340,52 @@ with $\delta_{ij}$ is the Kronecker delta, $\mathcal{P}_{pq}$ the permutation op
 		\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
-		CIPSI	&				&	$-863.0$		&				&			&				&	$-908.8$		&				&	$-883.4$		&				&	$-900.4$		&		&	$-957.3$		&		&	$-918.5$\\
+		CIPSI	&				&	$-863.0$		&				&	$-904.6$		&				&	$-908.8$		&				&	$-883.4$		&				&	$-900.4$		&		&	$-957.3$		&		&	$-918.5$\\
 	\end{tabular}
 	\end{ruledtabular}
 \end{table*}
 \end{squeezetable}    
 
 \begin{figure*}
-%	\includegraphics[width=0.15\textwidth]{Cyclopentadiene_vs_Ndet}
-%	\includegraphics[width=0.15\textwidth]{Furan_vs_Ndet}
-%	\includegraphics[width=0.15\textwidth]{Imidazole_vs_Ndet}
-%	\includegraphics[width=0.15\textwidth]{Pyrrole_vs_Ndet}
-%	\includegraphics[width=0.15\textwidth]{Thiophene_vs_Ndet}
-%	\includegraphics[width=0.15\textwidth]{Benzene_vs_Ndet}
-%	\\
-%	\includegraphics[width=0.15\textwidth]{Pyrazine_vs_Ndet}
-%	\includegraphics[width=0.15\textwidth]{Pyridazine_vs_Ndet}
-%	\includegraphics[width=0.15\textwidth]{Pyridine_vs_Ndet}
-%	\includegraphics[width=0.15\textwidth]{Pyrimidine_vs_Ndet}
-%	\includegraphics[width=0.15\textwidth]{Tetrazine_vs_Ndet}
-%	\includegraphics[width=0.15\textwidth]{Triazine_vs_Ndet}
-	\caption{$E_\text{var}$ and $E_\text{var} + E_\text{PT2}$ as functions of the number of determinants $\Ndet$ in the variational space for the twelve cyclic molecules represented in Fig.~\ref{fig:mol}.
-		The CCSDTQ correlation energy is represented as a thick black line.
+	\includegraphics[width=0.24\textwidth]{Cyclopentadiene_EvsNdet}
+	\includegraphics[width=0.24\textwidth]{Furan_EvsNdet}
+	\includegraphics[width=0.24\textwidth]{Imidazole_EvsNdet}
+	\includegraphics[width=0.24\textwidth]{Pyrrole_EvsNdet}
+	\\
+	\includegraphics[width=0.24\textwidth]{Thiophene_EvsNdet}
+	\includegraphics[width=0.24\textwidth]{Benzene_EvsNdet}
+	\includegraphics[width=0.24\textwidth]{Pyrazine_EvsNdet}
+	\includegraphics[width=0.24\textwidth]{Pyridazine_EvsNdet}
+	\\
+	\includegraphics[width=0.24\textwidth]{Pyridine_EvsNdet}
+	\includegraphics[width=0.24\textwidth]{Pyrimidine_EvsNdet}
+	\includegraphics[width=0.24\textwidth]{Tetrazine_EvsNdet}
+	\includegraphics[width=0.24\textwidth]{Triazine_EvsNdet}
+	\caption{$\Delta E_\text{var}$ (solid) and $\Delta E_\text{var} + E_\text{PT2}$ (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.
 	\label{fig:vsNdet}}
 \end{figure*}
 
 \begin{figure*}
-%	\includegraphics[width=0.15\textwidth]{Cyclopentadiene_vs_EPT2}
-%	\includegraphics[width=0.15\textwidth]{Furan_vs_EPT2}
-%	\includegraphics[width=0.15\textwidth]{Imidazole_vs_EPT2}
-%	\includegraphics[width=0.15\textwidth]{Pyrrole_vs_EPT2}
-%	\includegraphics[width=0.15\textwidth]{Thiophene_vs_EPT2}
-%	\includegraphics[width=0.15\textwidth]{Benzene_vs_EPT2}
-%	\\
-%	\includegraphics[width=0.15\textwidth]{Pyrazine_vs_EPT2}
-%	\includegraphics[width=0.15\textwidth]{Pyridazine_vs_EPT2}
-%	\includegraphics[width=0.15\textwidth]{Pyridine_vs_EPT2}
-%	\includegraphics[width=0.15\textwidth]{Pyrimidine_vs_EPT2}
-%	\includegraphics[width=0.15\textwidth]{Tetrazine_vs_EPT2}
-%	\includegraphics[width=0.15\textwidth]{Triazine_vs_EPT2}
-	\caption{$E_\text{var}$ as a function of $E_\text{PT2}$ for the twelve cyclic molecules represented in Fig.~\ref{fig:mol}.
-	The CCSDTQ correlation energy is represented as a thick black line.
+	\includegraphics[width=0.24\textwidth]{Cyclopentadiene_EvsPT2}
+	\includegraphics[width=0.24\textwidth]{Furan_EvsPT2}
+	\includegraphics[width=0.24\textwidth]{Imidazole_EvsPT2}
+	\includegraphics[width=0.24\textwidth]{Pyrrole_EvsPT2}
+	\\
+	\includegraphics[width=0.24\textwidth]{Thiophene_EvsPT2}
+	\includegraphics[width=0.24\textwidth]{Benzene_EvsPT2}
+	\includegraphics[width=0.24\textwidth]{Pyrazine_EvsPT2}
+	\includegraphics[width=0.24\textwidth]{Pyridazine_EvsPT2}
+	\\
+	\includegraphics[width=0.24\textwidth]{Pyridine_EvsPT2}
+	\includegraphics[width=0.24\textwidth]{Pyrimidine_EvsPT2}
+	\includegraphics[width=0.24\textwidth]{Tetrazine_EvsPT2}
+	\includegraphics[width=0.24\textwidth]{Triazine_EvsPT2}
+	\caption{$\Delta E_\text{var}$ as a function of $E_\text{PT2}$ 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 linear fit using the four largest variational wave functions for each set is depicted as a dashed black line.
+	The CCSDTQ correlation energy is also represented as a thick black line.
 	\label{fig:vsNdet}}
 \end{figure*}