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@@ -131,7 +131,7 @@ Indeed, selected CI (SCI) methods, \cite{Booth_2009,Giner_2013,Evangelista_2014,
 We refer the interested reader to Refs.~\onlinecite{Loos_2020a,Eriksen_2021} for recent reviews.
 SCI methods are based on a well-known fact: amongst the very large number of determinants contained in the FCI space, only a relative small fraction of them significantly contributes to the energy.
 Accordingly, the SCI+PT2 family of methods performs a sparse exploration of the FCI space by selecting iteratively only the most energetically relevant determinants of the variational space and supplementing it with a second-order perturbative correction (PT2). \cite{Huron_1973,Garniron_2017,Sharma_2017,Garniron_2018,Garniron_2019}
-Although the formal scaling of such algorithms remain exponential, the prefactor is greatly reduced which explains their current attractiveness in the electronic structure community and much wider applicability than their standard FCI parent.
+Although the formal scaling of such algorithms remains exponential, the prefactor is greatly reduced which explains their current attractiveness in the electronic structure community and much wider applicability than their standard FCI parent.
 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}
@@ -139,8 +139,8 @@ which popularity originates from its black-box nature, size-extensivity, and rel
 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}
-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,Marie_2021}
+MP4, \cite{Krishnan_1980} MP5, \cite{Kucharski_1989} and MP6 \cite{He_1996a,He_1996b} which scale 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, slowly convergent, or divergent behavior that limits 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,Marie_2021}
 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.
 
@@ -196,13 +196,14 @@ The sets $\cI_k$ and $\cA_k$ define, at the $k$th iteration, the internal and ex
 In the selection step, the perturbers corresponding to the largest $\abs*{e_{\alpha}^{(k)}}$ values are then added to the variational space at iteration $k+1$.
 In our implementation, the size of the variational space is roughly doubled at each iteration.
 Hereafter, we label these iterations over $k$ which consist in enlarging the variational space as \textit{macroiterations}.
-In practice, $\Evar^{(k)}$ is computed by diagonalizing the $\Ndet^{(k)} \times \Ndet^{(k)}$ CI matrix with elements $\mel{I}{\hH}{J}$ via Davidson's algorithm. \cite{Davidson_1975}
+In practice, $\Evar^{(k)}$ is the lowest eigenvalue of the $\Ndet^{(k)} \times \Ndet^{(k)}$ CI matrix with elements $\mel{I}{\hH}{J}$ obtained via Davidson's algorithm. \cite{Davidson_1975}
 The magnitude of $\EPT^{(k)}$ provides, at iteration $k$, a qualitative idea of the ``distance'' to the FCI limit. \cite{Garniron_2018}
 We then linearly extrapolate, using large variational wave functions, the CIPSI energy to $\EPT = 0$ (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.
-Most of the technology presented here has been borrowed from complete-active-space self-consistent-field (CASSCF) methods \cite{Werner_1980,Werner_1985,Sun_2017,Kreplin_2019,Kreplin_2020} but one of the strength of SCI methods is that one does not need to select an active space and to classify orbitals as active, inactive, and virtual orbitals.
+\toto{
+Most of the technology presented here has been borrowed from complete-active-space self-consistent-field (CASSCF) methods \cite{Werner_1980,Werner_1985,Sun_2017,Kreplin_2019,Kreplin_2020} but one of the strength of SCI methods is that one does not need to select an active space and to classify orbitals as active, inactive, and virtual orbitals.}
 Here, we detail our orbital optimization procedure within the CIPSI algorithm and we assume that the variational wave function is normalized, \ie, $\braket*{\Psivar}{\Psivar} = 1$.
 
 As stated in Sec.~\ref{sec:intro}, $\Evar$ depends on both the CI coefficients $\{ c_I \}_{1 \le I \le \Ndet}$ [see Eq.~\eqref{eq:Psivar}] but also on the orbital rotation parameters $\{\kappa_{pq}\}_{1 \le p,q \le \Norb}$.
@@ -307,19 +308,19 @@ This popular variant defines a region where the quadratic approximation \eqref{e
 By introducing a Lagrange multiplier $\lambda$ to control the trust-region size, one replaces Eq.~\eqref{eq:kappa_newton} by $\bk = - (\bH + \lambda \bI)^{-1} \cdot \bg$.
 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 cancel the step when the energy rises ensure the convergence of the algorithm.
-More details can be found in Ref.~\onlinecite{Nocedal_1999}.
+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}.
 
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 \section{Computational details}
 \label{sec:compdet}
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-The geometries of the twelve systems considered in the present study have been all obtained at the CC3/aug-cc-pVTZ level of theory and have been 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 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 QUANTUM PACKAGE. \cite{Garniron_2019}
+The geometries of the twelve systems considered in the present study were all obtained at the \toto{frozen-core?} 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}
+The MP2, MP3, MP4, CC2, CC3, CC4, CCSD, CCSDT, and CCSDTQ calculations were performed with CFOUR, \cite{Matthews_2020} while the CCSD(T) 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).
-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}
+Here, we employ the renormalized version of the PT2 correction which was recently implemented and tested for a more efficient extrapolation to the FCI limit thanks to a partial resummation of the higher orders 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).
 
@@ -333,14 +334,14 @@ Using optimized orbitals has the undeniable advantage to produce, for a given va
 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.
+\toto{Note 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.
 
 The present calculations have been performed on the AMD partition of GENCI's Irene supercomputer. 
 Each Irene's AMD node is a dual-socket AMD Rome (EPYC) CPU at 2.60 GHz with 256GiB of RAM, with a total of 64 physical cores per socket. 
 These nodes are connected via Infiniband HDR100. 
-In total, the present calculations have required around 3,000,000 core hours.
+In total, the present calculations have required around 3~million core hours.
 
 All the data (geometries, energies, etc) and supplementary material associated with the present manuscript are openly available in Zenodo at \href{http://doi.org/XX.XXXX/zenodo.XXXXXXX}{http://doi.org/XX.XXXX/zenodo.XXXXXXX}.
 
@@ -705,7 +706,7 @@ As a final remark, we would like to mention that even if the two families of CC
 
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 \begin{acknowledgements}
-This work was performed using HPC resources from GENCI-TGCC (2021-gen1738) and from CALMIP (Toulouse) under allocation 2021-18005, and was also partially supported by the European Centre of Excellence in Exascale Computing TREX--Targeting Real Chemical Accuracy at the Exascale.
+This work was performed using HPC resources from GENCI-TGCC (2021-gen1738) and from CALMIP (Toulouse) under allocation 2021-18005, and was also supported by the European Centre of Excellence in Exascale Computing TREX - Targeting Real Chemical Accuracy at the Exascale. This project has received funding from the European Union's Horizon 2020 - Research and Innovation program - under grant agreement no. 95216.
 This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No.~863481 and 952165).
 \end{acknowledgements}
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