diff --git a/Manuscript/Ec.tex b/Manuscript/Ec.tex
index fffab0f..7e034a7 100644
--- a/Manuscript/Ec.tex
+++ b/Manuscript/Ec.tex
@@ -113,13 +113,13 @@ Loosely speaking, to make any method practical, three main approximations must b
 The first fundamental approximation, known as the Born-Oppenheimer (or clamped-nuclei) approximation, consists in assuming that the motion of nuclei and electrons are decoupled. \cite{Born_1927}
 The nuclei coordinates can then be treated as parameters in the electronic Hamiltonian.
 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.
+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_2002b,Bartlett_2007,Shavitt_2009} a reference Slater determinant $\PsiO$ [usually taken as the Hartree-Fock (HF) wave function] is multiplied by a wave operator defined as the exponentiated excitation operator $\hT = \sum_{k=1}^\Nel \hT_k$ (where $\Nel$ is the number of electrons).
 
-The truncation of $\hT$ allows to define a hierarchy of non-variational and size-extensive methods with improved accuracy:
+The truncation of $\hT$ allows to define a hierarchy of non-variational and size-extensive methods with increasing levels of 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 $\order*{\Norb^{6}}$, $\order*{\Norb^{8}}$, and $\order*{\Norb^{10}}$, respectively (where $\Norb$ denotes the number of orbitals).
-Parallel to the ``complete'' CC series presented above, an alternative family 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}
+Parallel to the ``complete'' CC series presented above, an alternative family of approximate iterative CC models has 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 $\order*{\Norb^{5}}$, $\order*{\Norb^{7}}$, 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,Chrayteh_2021,Sarkar_2021}
@@ -136,7 +136,7 @@ Although the formal scaling of such algorithms remains exponential, the prefacto
 Note that, very recently, several groups \cite{Aroeira_2021,Lee_2021,Magoulas_2021} have coupled CC and SCI methods via the externally-corrected CC methodology, \cite{Paldus_2017} showing promising performances for weakly and strongly correlated systems.
 
 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.
+whose 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 $\order*{\Norb^{5}}$] 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}
@@ -306,14 +306,14 @@ By introducing a Lagrange multiplier $\lambda$ to control the trust-region size,
 The addition of the level shift $\lambda \geq 0$ removes the negative eigenvalues and ensures the positive definiteness of the Hessian matrix by reducing the step size.
 By choosing the right value of $\lambda$, $\norm{\bk}$ is constrained into a hypersphere of radius $\Delta$ and is able to evolve from the Newton direction at $\lambda = 0$ to the steepest descent direction as $\lambda$ grows.
 The evolution of the trust radius during the optimization and the use of a condition to reject the step when the energy rises ensure the convergence of the algorithm.
-More detail can be found in Ref.~\onlinecite{Nocedal_1999}.
+More details can be found in Ref.~\onlinecite{Nocedal_1999}.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Computational details}
 \label{sec:compdet}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 The geometries of the twelve systems considered in the present study were all obtained at the CC3/aug-cc-pVTZ level of theory and were extracted from a previous study. \cite{Loos_2020a}
-Note that, for the sake of consistency, the geometry of benzene considered here is different from one of Ref.~\onlinecite{Loos_2020e} which was obtained at a lower level of theory [MP2/6-31G(d)]. \cite{Schreiber_2008}
+Note that, for the sake of consistency, the geometry of benzene considered here is different from the one of Ref.~\onlinecite{Loos_2020e} which was obtained 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 were performed with CFOUR, \cite{Matthews_2020} the CCSD(T) and CR-CC(2,3) calculations were made with GAMESS 2014R1, \cite{gamess} and MP5 calculations were computed with GAUSSIAN 09. \cite{g09}
 The CIPSI calculations were performed with QUANTUM PACKAGE. \cite{Garniron_2019}
 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).
@@ -325,13 +325,13 @@ Although the FCI energy has the enjoyable property of being independent of the s
 In the present study, we investigate, in particular, the convergence behavior of the CIPSI energy for two sets of orbitals: 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.
 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.
+Successive orbital optimizations are then performed, which consist in minimizing the variational CIPSI energy at each macroiteration 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.
 Using optimized orbitals has the undeniable advantage to produce, for a given variational energy, more compact CI expansions (see Sec.~\ref{sec:res}).
 For the benzene molecule, we also explore the use of localized orbitals (LOs) which are produced with 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. \cite{Loos_2020e}
 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 both localized and optimized orbitals do break the spatial symmetry.
+Note that both localized and optimized orbitals do break the spatial symmetry.
 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 fulfill this property as we aim for the lowest possible energy of a closed-shell singlet state.
 We have found that $\expval*{\Hat{S}^2}$ is, nonetheless, very close to zero for each system.
 
@@ -590,10 +590,10 @@ The five-point weighted linear fit using the five largest variational wave funct
 %%% %%% %%%
 
 Figure \ref{fig:BenzenevsNdet} compares the convergence of $\Delta \Evar$ for the natural, localized, and optimized sets of orbitals in the particular case of benzene.
-As mentioned in Sec.~\ref{sec:compdet}, although both the localized and optimized orbitals break the spatial symmetry to take advantage of the local nature of electron correlation, the latter set further improve on the use of former set.
+As mentioned in Sec.~\ref{sec:compdet}, although both the localized and optimized orbitals break the spatial symmetry to take advantage of the local nature of electron correlation, {\color{red} [I don't understand what is meant here:]} the latter set further improve on the use of former set.
 More quantitatively, optimized orbitals produce the same variational energy as localized orbitals with, roughly, a ten-fold reduction in the number of determinants.
 A similar improvement is observed going from natural to localized orbitals.
-Accordingly to these observations, all our FCI correlation energy estimates have been produced with the set of optimized orbitals.
+According 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 five-point linear fit using the five largest variational wave functions (see Fig.~\ref{fig:vsEPT2}).
 The fitting weights have been taken as the inverse square of the perturbative corrections.
@@ -647,7 +647,7 @@ The fitting errors follow the same trend.
 	\label{tab:stats}}
 	\begin{ruledtabular}
 	\begin{tabular}{lcdddd}
-	Method	&	Scaling			&	\tabc{MAE}	&	\tabc{MSE}	&	\tabc{Min}	&	\tabc{Max}	\\
+	Method	&	Scaling			&	\tabc{MAE}	&	\tabc{MSE}	&	\tabc{Max}	&	\tabc{Min}	\\
 	\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	\\
@@ -678,12 +678,12 @@ 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, 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}
+Unfortunately, CC with singles, doubles, triples, quadruples, and pentuples (CCSDTQP) calculations are out of reach here. \cite{Hirata_2000,Kallay_2001}
 As expected for the present set of weakly correlated systems, 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 (instead of iteratively) the triple excitations, while CCSD(T) and CR-CC(2,3) performs equally well.
+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 (instead of iteratively) the triple excitations, while CCSD(T) and CR-CC(2,3) perform equally well.
 
  
-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}
+Second, let us look into the series of MP approximations which is known, as mentioned in Sec.~\ref{sec:intro}, to potentially exhibit ``surprising'' behaviors 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 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.