ok with abstract, intro and comp details for now

Manuscript/CBD.bib

@@ -1,13 +1,26 @@
 %% This BibTeX bibliography file was created using BibDesk.
 %% https://bibdesk.sourceforge.io/
 
-%% Created for Pierre-Francois Loos at 2022-03-28 22:40:05 +0200 
+%% Created for Pierre-Francois Loos at 2022-03-30 15:46:27 +0200 
 
 
 %% Saved with string encoding Unicode (UTF-8) 
 
 
 
+@article{Christiansen_1996b,
+	author = {Ove Christiansen and Henrik Koch and Poul J{\o}rgensen},
+	date-added = {2022-03-30 15:45:28 +0200},
+	date-modified = {2022-03-30 15:45:28 +0200},
+	doi = {10.1063/1.472007},
+	eprint = {http://dx.doi.org/10.1063/1.472007},
+	journal = {J. Chem. Phys.},
+	pages = {1451--1459},
+	title = {Perturbative Triple Excitation Corrections to Coupled Cluster Singles and Doubles Excitation Energies},
+	volume = {105},
+	year = {1996},
+	bdsk-url-1 = {http://dx.doi.org/10.1063/1.472007}}
+
 @article{Matthews_2020,
 	author = {Matthews,Devin A. and Cheng,Lan and Harding,Michael E. and Lipparini,Filippo and Stopkowicz,Stella and Jagau,Thomas-C. and Szalay,P{\'e}ter G. and Gauss,J{\"u}rgen and Stanton,John F.},
 	date-added = {2022-03-28 21:47:45 +0200},
@@ -499,9 +512,9 @@
 	bdsk-url-1 = {https://doi.org/10.1063/1.3656734}}
 
 @article{Rowe_1968,
-	author = {ROWE, D. J.},
+	author = {Rowe, D. J.},
 	date-added = {2022-03-23 11:48:22 +0100},
-	date-modified = {2022-03-23 11:48:22 +0100},
+	date-modified = {2022-03-30 15:46:26 +0200},
 	doi = {10.1103/RevModPhys.40.153},
 	journal = {Rev. Mod. Phys.},
 	numpages = {0},

Manuscript/CBD.tex

@@ -207,7 +207,7 @@
 %D4h states
 %\newcommand{\oneBoneg}{$1{}^1B_{1g}$} same label as the D2h state
 \newcommand{\Atwog}{$1{}^3A_{2g}$}
-\newcommand{\Aoneg}{$2{}^1A_{1g}$}
+\newcommand{\Aoneg}{$1{}^1A_{1g}$}
 \newcommand{\Btwog}{$1{}^1B_{2g}$}
 
 \begin{document}	
@@ -233,7 +233,7 @@
 
 \begin{abstract}
 The cyclobutadiene molecule is a well-known playground for theoretical chemists and is particularly suitable to test ground- and excited-state methods.
-Indeed, due to its high spatial symmetry, especially at the $D_{4h}$ square geometry but also at the $D_{2h}$ rectangular structure, the ground and excited states of cyclobutadiene exhibit multi-configurational characters and single-reference methods, such as adiabatic time-dependent density-functional theory (TD-DFT) or equation-of-motion coupled cluster (EOM-CC), are notoriously known to struggle in such situations. 
+Indeed, due to its high spatial symmetry, especially at the $D_{4h}$ square geometry but also in the $D_{2h}$ rectangular arrangement, the ground and excited states of cyclobutadiene exhibit multi-configurational characters and single-reference methods, such as adiabatic time-dependent density-functional theory (TD-DFT) or equation-of-motion coupled cluster (EOM-CC), are notoriously known to struggle in such situations. 
 In this work, using a large panel of methods and basis sets, we provide an extensive computational study of the autoisomerization barrier (defined as the difference between the square and rectangular ground-state energies) and the vertical excitation energies of the $D_{2h}$ and $D_{4h}$ arrangements. 
 In particular, selected configuration interaction (SCI), multi-reference perturbation theory (CASSCF, CASPT2, and NEVPT2), and coupled cluster (CCSD, CC3, CCSDT, CC4, and CCSDTQ) calculations are performed. 
 The spin-flip formalism, which is known to provide a correct description of states with multi-configurational character, is also tested within TD-DFT (where numerous exchange-correlation functionals are considered) and the algebraic diagrammatic construction [ADC(2)-s, ADC(2)-x, and ADC(3)]. 
@@ -248,7 +248,7 @@ A theoretical best estimate is defined for the autoisomerization barrier and for
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 Despite the fact that excited states are involved in ubiquitous processes such as photochemistry, \cite{Bernardi_1990,Bernardi_1996,Boggio-Pasqua_2007,Klessinger_1995,Olivucci_2010,Robb_2007,VanderLugt_1969} catalysis or in solar cell technology, \cite{Delgado_2010} none of the currently existing methods has shown to provide accurate excitation energies in all scenarios due to the complexity of the process, the size of the systems, environment effects and many other possible factors.
-Indeed, each computational model has its own theoretical and/or technical limitations and the number of possible chemical scenario is so vast that the design of new excited-state methodologies is still a very active field of theoretical quantum chemistry.\cite{Roos_1996,Piecuch_2002,Dreuw_2005,Krylov_2006,Sneskov_2012,Gonzales_2012,Laurent_2013,Adamo_2013,Dreuw_2015,Ghosh_2018,Blase_2020,Loos_2020a,Hait_2021,Zobel_2021}
+Indeed, each computational model has its own theoretical and/or technical limitations and the number of possible chemical scenarios is so vast that the design of new excited-state methodologies is still a very active field of theoretical quantum chemistry.\cite{Roos_1996,Piecuch_2002,Dreuw_2005,Krylov_2006,Sneskov_2012,Gonzales_2012,Laurent_2013,Adamo_2013,Dreuw_2015,Ghosh_2018,Blase_2020,Loos_2020a,Hait_2021,Zobel_2021}
 
 Speaking of difficult tasks, the cyclobutadiene (CBD) molecule has been a real challenge for experimental and theoretical chemistry for many decades. \cite{Bally_1980} Due to its antiaromaticity \cite{Minkin_1994} and large angular strain, \cite{Baeyer_1885} CBD presents a high reactivity which made its synthesis a particularly difficult exercise. 
 The simple H\"uckel molecular orbital theory (wrongly) predicts a triplet ground state at the {\Dfour} square geometry, with two singly-occupied frontier orbitals that are degenerate by symmetry (Hund's rule), while state-of-the-art \textit{ab initio} methods (correctly) predict an open-shell singlet ground state.
@@ -264,13 +264,13 @@ Thus, the autoisomerization barrier (AB) is defined as the difference between th
 The energy of this barrier is estimated, experimentally, in the range of 1.6-10 \kcalmol, \cite{Whitman_1982} while previous multi-reference calculations yield an energy barrier in the range of 6-7 \kcalmol. \cite{Eckert-Maksic_2006}
 %All these specificities of CBD make it a real playground for excited-states methods. 
 
-The lowest-energy excited states of CBD in both geometries are represented in Fig.~\ref{fig:CBD}, where we have reported the {\oneAg} and {\tBoneg} states for the rectangular geometry and {\sBoneg} and {\Atwog} for the square one. 
+The lowest-energy excited states of CBD in both geometries are represented in Fig.~\ref{fig:CBD}, where we have reported the {\oneAg} and {\tBoneg} states for the rectangular geometry and the {\sBoneg} and {\Atwog} states for the square one. 
 Due to the energy scale, the higher-energy states ({\sBoneg} and {\twoAg} for {\Dtwo} and {\Aoneg} and {\Btwog} for {\Dfour}) are not shown. 
 Interestingly, the {\twoAg} and {\Aoneg} states have a strong contribution from doubly-excited configurations and these so-called double excitations \cite{Loos_2019} are known to be an absolute nightmare for adiabatic time-dependent density-functional theory (TD-DFT) \cite{Runge_1984,Casida_1995,Tozer_2000,Maitra_2004,Cave_2004,Levine_2006,Elliott_2011,Maitra_2012,Maitra_2017} and even for state-of-the-art methods like the approximate third-order coupled-cluster (CC3) \cite{Christiansen_1995,Koch_1997} or equation-of-motion coupled-cluster with singles, doubles, and triples (EOM-CCSDT). \cite{Kucharski_1991,Kallay_2004,Hirata_2000,Hirata_2004}
 
 In order to tackle the problem of multi-configurational character and double excitations, we have explored several routes. 
 The most evident way is to rely on multi-configurational methods, which are naturally designed to address such scenarios.
-Among these methods, one can mention the complete active space self-consistent field (CASSCF) method, \cite{Roos_1996} its second-order perturbatively-corrected variant (CASPT2) \cite{Andersson_1990,Andersson_1992,Roos_1995a} and the second-order $n$-electron valence state perturbation theory (NEVPT2) formalism. \cite{Angeli_2001,Angeli_2001a,Angeli_2002}
+Among these methods, one can mention the complete-active-space self-consistent field (CASSCF) method, \cite{Roos_1996} its second-order perturbatively-corrected variant (CASPT2) \cite{Andersson_1990,Andersson_1992,Roos_1995a} and the second-order $n$-electron valence state perturbation theory (NEVPT2) formalism. \cite{Angeli_2001,Angeli_2001a,Angeli_2002}
 The exponential scaling of the computational cost (with respect to the size of the active space) associated with these methods is the principal limitation to their applicability to large molecules. 
 
 Another way to deal with double excitations and multi-reference situations is to use high level truncation of the equation-of-motion (EOM) formalism \cite{Rowe_1968,Stanton_1993} of coupled-cluster (CC) theory. \cite{Kucharski_1991,Kallay_2003,Kallay_2004,Hirata_2000,Hirata_2004}
@@ -278,26 +278,27 @@ However, to provide a correct description of these situations, one have to take
 Again, due to the scaling of CC methods with the number of basis functions, their applicability is limited to small molecules.
 Although multi-reference CC methods have been specifically designed for such purposes, \cite{Jeziorski_1981,Mahapatra_1998,Mahapatra_1999,Lyakh_2012,Kohn_2013} they are computationally demanding and still far from being black-box.
 
-In this context, an interesting alternative to multi-configurational and CC methods is provided by selected CI (SCI) methods, \cite{Bender_1969,Whitten_1969,Huron_1973,Giner_2013,Evangelista_2014,Giner_2015,Caffarel_2016b,Holmes_2016,Tubman_2016,Liu_2016,Ohtsuka_2017,Zimmerman_2017,Coe_2018,Garniron_2018} which have proven, recently, to be able to provide near full CI (FCI) energies for small molecules for both ground- and excited-state energies. \cite{Caffarel_2014,Caffarel_2016a,Scemama_2016,Holmes_2017,Li_2018,Scemama_2018,Scemama_2018b,Li_2020,Loos_2018a,Chien_2018,Loos_2019,Loos_2020b,Loos_2020c,Loos_2020e,Garniron_2019,Eriksen_2020,Yao_2020,Williams_2020,Veril_2021,Loos_2021,Damour_2021}
+In this context, an interesting alternative to multi-configurational and CC methods is provided by selected configuration interaction (SCI) methods, \cite{Bender_1969,Whitten_1969,Huron_1973,Giner_2013,Evangelista_2014,Giner_2015,Caffarel_2016b,Holmes_2016,Tubman_2016,Liu_2016,Ohtsuka_2017,Zimmerman_2017,Coe_2018,Garniron_2018} which have proven, recently, to be able to provide near full CI (FCI) energies for small molecules for both ground- and excited-state energies. \cite{Caffarel_2014,Caffarel_2016a,Scemama_2016,Holmes_2017,Li_2018,Scemama_2018,Scemama_2018b,Li_2020,Loos_2018a,Chien_2018,Loos_2019,Loos_2020b,Loos_2020c,Loos_2020e,Garniron_2019,Eriksen_2020,Yao_2020,Williams_2020,Veril_2021,Loos_2021,Damour_2021}
 For example, the \textit{Configuration Interaction using a Perturbative Selection made Iteratively} (CIPSI) method limits the exponential increase of the size of the CI expansion by retaining the most energetically relevant determinants only, using a second-order energetic criterion to select perturbatively determinants in the FCI space. \cite{Huron_1973,Giner_2013,Giner_2015,Garniron_2017,Garniron_2018,Garniron_2019}
 Nonetheless, SCI methods remain very expensive and can be applied to a limited number of situations.
 
 Finally, another option to deal with these chemical scenarios is to rely on the cheaper spin-flip formalism, established by Krylov in 2001, \cite{Krylov_2001a,Krylov_2001b,Krylov_2002,Casanova_2020} where one accesses the ground and doubly-excited states via a single (spin-flip) excitation and deexcitation from the lowest triplet state, respectively.
-Obviously, spin-flip methods have their own flaws, especially spin contamination (\ie, an artificial mixing of electronic states with different spin multiplicities) due principally to spin incompleteness in the spin-flip expansion but also to the spin contamination of the reference configuration. \cite{Casanova_2020} 
+Obviously, spin-flip methods have their own flaws, especially spin contamination (\ie, the artificial mixing of electronic states with different spin multiplicities) due principally to spin incompleteness in the spin-flip expansion but also to the spin contamination of the reference configuration. \cite{Casanova_2020} 
 One can address part of this issue by increasing the excitation order or by complementing the spin-incomplete configuration set with the missing configurations, \cite{Sears_2003,Casanova_2008,Huix-Rotllant_2010,Li_2010,Li_2011a,Li_2011b,Zhang_2015,Lee_2018}
 both solutions being associated with an additional computational cost.
 
-In the present work, we investigate the accuracy of a large panel of computational methods on the autoisomerization barrier and the low-lying excited states of CBD at the {\Dtwo} and {\Dfour} ground-state geometries. 
+In the present work, we investigate the accuracy of each family of computational methods mentioned above on the autoisomerization barrier and the low-lying excited states of CBD at the {\Dtwo} and {\Dfour} ground-state geometries. 
 Computational details are reported in Section \ref{sec:compmet}.
 % for SCI (Subsection \ref{sec:SCI}), EOM-CC (Subsection \ref{sec:CC}), multi-configurational (Subsection \ref{sec:Multi}) and spin-flip (Subsection \ref{sec:sf}) methods. 
 Section \ref{sec:res} is devoted to the discussion of our results. 
 %First, we consider the autoisomerization barrier (Subsection \ref{sec:auto}) and then the excited states (Subsection \ref{sec:states}) of the CBD molecule for both geometries. 
+Finally, our conclusions are drawn in Section \ref{sec:conclusion}.
 
 %%% FIGURE 1 %%%
 \begin{figure}
 	\includegraphics[width=0.8\linewidth]{figure1.pdf}
-	\caption{Pictorial representation of the ground and excited states of CBD and its properties under investigation. 
+	\caption{Pictorial representation of the ground and excited states of CBD and the properties under investigation. 
-	The singlet ground-state (S) and triplet (T) properties are represented in black and red, respectively.
+	The singlet ground state (S) and triplet (T) properties are represented in black and red, respectively.
 	The autoisomerization barrier (AB) is also represented.}
 \label{fig:CBD}
 \end{figure}
@@ -312,7 +313,7 @@ Section \ref{sec:res} is devoted to the discussion of our results.
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \subsection{Selected configuration interaction calculations}
 \label{sec:SCI}
-For the SCI calculations, we rely on the CIPSI algorithm which is implemented in QUANTUM PACKAGE. \cite{Garniron_2019} 
+For the SCI calculations, we rely on the CIPSI algorithm implemented in QUANTUM PACKAGE, \cite{Garniron_2019} which iteratively select determinants in the FCI space.
 To treat electronic states on equal footing, we use a state-averaged formalism where the ground and excited states are expanded with the same set of determinants but with different CI coefficients. 
 Note that the determinant selection for these states are performed simultaneously via the protocol described in Refs.~\onlinecite{Scemama_2019,Garniron_2019}.
 
@@ -328,19 +329,17 @@ This type of extrapolation procedures is now routine in SCI methods as well as o
 \subsection{Coupled-cluster calculations}
 \label{sec:CC}
 Coupled-cluster theory provides a hierarchy of methods that yields increasingly accurate ground state energies by ramping up the maximum excitation degree of the cluster operator: \cite{Cizek_1966,Paldus_1972,Crawford_2000,Piecuch_2002b,Bartlett_2007,Shavitt_2009} 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_1991a,Kucharski_1992} etc.
-As mentioned above, CC theory can be extended to excited states via the EOM formalism, \cite{Rowe_1968,Stanton_1993} where one projects out the similarity-transformed Hamiltonian in a basis of excited determinants yielding the following systematically improvable family of methods for neutral excited states: EOM-CCSD, EOM-CCSDT, EOM-CCSDTQ, etc.\cite{Noga_1987,Koch_1990,Kucharski_1991,Stanton_1993,Christiansen_1998b,Kucharski_2001,Kowalski_2001,Kallay_2003,Kallay_2004,Hirata_2000,Hirata_2004}
+As mentioned above, CC theory can be extended to excited states via the EOM formalism, \cite{Rowe_1968,Stanton_1993} where one diagonalizes the similarity-transformed Hamiltonian in a CI basis of excited determinants yielding the following systematically improvable family of methods for neutral excited states: EOM-CCSD, EOM-CCSDT, EOM-CCSDTQ, etc.\cite{Noga_1987a,Koch_1990b,Kucharski_1991,Stanton_1993,Christiansen_1998,Kucharski_2001,Kowalski_2001,Kallay_2003,Kallay_2004,Hirata_2000,Hirata_2004}
 In the following, we will omit the prefix EOM for the sake of conciseness.
 Alternatively to the ``complete'' CC models, one can also employed the CC2, \cite{Christiansen_1995,Hattig_2000} CC3 \cite{Christiansen_1995,Koch_1995} and CC4 \cite{Kallay_2005,Matthews_2020,Loos_2021} methods which can be seen as cheaper approximations of CCSD, CCSDT, and CCSDTQ by skipping the most expensive terms and avoiding the storage of higher-excitations amplitudes.
 
 Here, we have performed CC calculations using various codes. 
-Typically, CCSD, CCSDT, and CCSDTQ as well as CC3 and CC4 calculations are achieved with CFOUR. \cite{Matthews_2020}
+Typically, CCSD, CCSDT, and CCSDTQ as well as CC3 and CC4 calculations are achieved with CFOUR, \cite{Matthews_2020} with which only singlet excited states can be computed.
-In some case, we have also computed CC3 energies and properties with DALTON.\cite{Aidas_2014} 
+In some case, we have also computed (singlet and triplet) excitation energies and properties (such as the percentage of single excitations involved in a given transition, namely $\%T_1$) at the CC3 and CCSDT levels with DALTON.\cite{Aidas_2014} 
 
-\alert{To avoid having to perform multi-reference CC calculations and because one cannot perform high-level CC calculations in the restricted open-shell formalism, it is worth mentioning that, for the {\Dfour} arrangement, we have considered the lowest \textit{closed-shell} singlet state of {\Aoneg} symmetry as reference. 
+To avoid having to perform multi-reference CC calculations and because one cannot perform high-level CC calculations in the restricted open-shell or unrestricted formalisms, it is worth mentioning that, for the {\Dfour} arrangement, we have considered the lowest \textit{closed-shell} singlet state {\Aoneg} as reference. 
-The open-shell ground state {\sBoneg} and the  {\Btwog} states are obtained as deexcitations.}
+Hence, the open-shell ground state {\sBoneg} and the {\Btwog} state appear as a deexcitation and an excitation, respectively.
-
+With respect to {\Aoneg}, {\sBoneg} has a dominant double excitation character, while {\Btwog} have a dominant single excitation character, hence their contrasting convergence behaviors with respect to the order of the CC expansion (see below).
-%The lowest-energy excited states of CBD in both geometries are represented in Fig.~\ref{fig:CBD}, where we have reported the {\oneAg} and {\tBoneg} states for the rectangular geometry and {\sBoneg} and {\Atwog} for the square one. 
-%Due to the energy scale, the higher-energy states ({\sBoneg} and {\twoAg} for {\Dtwo} and {\Aoneg} and {\Btwog} for {\Dfour}) are not shown. 
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
@@ -348,13 +347,13 @@ The open-shell ground state {\sBoneg} and the  {\Btwog} states are obtained as d
 \subsection{Multi-configurational calculations}
 \label{sec:Multi}
 State-averaged CASSCF (SA-CASSCF) calculations are performed with MOLPRO. \cite{Werner_2020} 
-For each excited state, a set of state-averaged orbitals is computed by taking into account the the excited state of interest as well as the ground state (even if it has a different symmetry).
+For each excited state, a set of state-averaged orbitals is computed by taking into account the excited state of interest as well as the ground state (even if it has a different symmetry).
 Two active spaces have been considered: (i) a minimal (4e,4o) active space including valence $\pi$ orbitals, and (ii) an extended (12e,12o) active space where we have additionally included the $\sigma_\text{CC}$ and $\sigma_\text{CC}^*$ orbitals.
 For ionic states, like the {\sBoneg} state of CBD, it is particularly important to take into account the $\sigma$-$\pi$ coupling.
 
-On top of this CASSCF treatment, CASPT2 calculations are performed within the RS2 contraction scheme, while the NEVPT2 energies are computed within both partially contracted (PC) and strongly contracted (SC) scheme. \cite{Angeli_2001,Angeli_2001a,Angeli_2002} 
+On top of this CASSCF treatment, CASPT2 calculations are performed within the RS2 contraction scheme, while the NEVPT2 energies are computed within both the partially contracted (PC) and strongly contracted (SC) schemes. \cite{Angeli_2001,Angeli_2001a,Angeli_2002} 
-In order to avoid the intruders state problem, a real-valued level shift of \SI{0.3}{\hartree} is set in CASPT2, \cite{Roos_1995b,Roos_1996} with an additional ionization-potential-electron-affinity (IPEA) shift of \SI{0.3}{\hartree} to avoid systematic overestimation of the vertical excitation energies. \cite{Ghigo_2004,Schapiro_2013,Zobel_2017,Sarkar_2022}
 Note that PC-NEVPT2 is theoretically more accurate than SC-NEVPT2 due to the larger number of external configurations and greater flexibility. 
+In order to avoid the intruders state problem in CASPT2, a real-valued level shift of \SI{0.3}{\hartree} is set, \cite{Roos_1995b,Roos_1996} with an additional ionization-potential-electron-affinity (IPEA) shift of \SI{0.3}{\hartree} to avoid systematic overestimation of the vertical excitation energies. \cite{Ghigo_2004,Schapiro_2013,Zobel_2017,Sarkar_2022}
 For the sake of comparison, in some cases, we have also performed multi-reference CI (MRCI) calculations.
 All these calculations are also carried out with MOLPRO. \cite{Werner_2020} 
 %and extended multistate (XMS) CASPT2 calculations are also performed in the cas of strong mixing between states with same spin and spatial symmetries. 
@@ -364,7 +363,7 @@ All these calculations are also carried out with MOLPRO. \cite{Werner_2020}
 \subsection{Spin-flip calculations}
 \label{sec:sf}
 Within the spin-flip formalism, one considers the lowest triplet state as reference instead of the singlet ground state.
-Ground-state energies are then computed as the sum of the triplet reference state and the corresponding deexcitation energy.
+Ground-state energies are then computed as sums of the triplet reference state energy and the corresponding deexcitation energy.
 Likewise, excitation energies with respect to the singlet ground state are computed as differences of excitation energies with respect to the reference triplet state.
 
 Nowadays, spin-flip techniques are broadly accessible thanks to intensive developments in the electronic structure community (see Ref.~\onlinecite{Casanova_2020} and references therein).
@@ -376,7 +375,7 @@ We have also carried out spin-flip calculations within the TD-DFT framework (SF-
 The B3LYP, \cite{Becke_1988b,Lee_1988a,Becke_1993b} PBE0 \cite{Adamo_1999a,Ernzerhof_1999} and BH\&HLYP hybrid functionals are considered, which contain 20\%, 25\%, 50\% of exact exchange, respectively. 
 These calculations are labeled as SF-TD-BLYP, SF-TD-B3LYP, SF-TD-PBE0, and SF-TD-BH\&HLYP in the following.
 Additionally, we have also computed SF-TD-DFT excitation energies using range-separated hybrid functionals: CAM-B3LYP,\cite{Yanai_2004a} LC-$\omega$PBE08, \cite{Weintraub_2009a} and $\omega$B97X-V. \cite{Mardirossian_2014}
-The main difference between these range-separated functionals is their amount of exact exchange at long-range: 75$\%$ for CAM-B3LYP and 100$\%$ for LC-$\omega$PBE08 and $\omega$B97X-V. 
+The main difference between these range-separated functionals is their amount of exact exchange at long range: 75$\%$ for CAM-B3LYP and 100$\%$ for LC-$\omega$PBE08 and $\omega$B97X-V. 
 Finally, the hybrid meta-GGA functional M06-2X \cite{Zhao_2008} and the range-separated hybrid meta-GGA functional M11 \cite{Peverati_2011} are also employed.
 Note that all SF-TD-DFT calculations are done within the Tamm-Dancoff approximation. \cite{Hirata_1999}
 %EOM-SF-CCSD and EOM-SF-CC(2,3) are also performed with Q-CHEM 5.2.1. 
@@ -417,7 +416,7 @@ For example,
 and so on.
 If neither CC4, nor CCSDT are feasible, then we rely on NEVPT2(12,12). 
 The procedure for each extrapolated value is explicitly mentioned as a footnote.
-Note that, due to error bar inherently linked to the CIPSI calculations (see Subsection \ref{sec:SCI}), these are mostly used as a safety net to further check the convergence of the CCSDTQ values.
+Note that, due to error bar inherently linked to the CIPSI calculations (see Subsection \ref{sec:SCI}), these are mostly used as an additional safety net to further check the convergence of the CCSDTQ estimates.
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%