Manuscript corrections

Manuscript/CBD.tex

@@ -86,7 +86,7 @@
 \begin{abstract}
 \section*{Abstract}	
 Cyclobutadiene 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 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.
+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 \alert{standard} adiabatic time-dependent density-functional theory (TD-DFT) or \alert{standard} 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 automerization barrier (defined as the difference between the square and rectangular ground-state energies) and the vertical excitation energies at $D_{2h}$ and $D_{4h}$ equilibrium structures. 
 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 qualitatively correct description of these diradical states, is also tested within TD-DFT (combined with numerous exchange-correlation functionals) and the algebraic diagrammatic construction [ADC(2)-s, ADC(2)-x, and ADC(3)] schemes. 
@@ -112,17 +112,17 @@ This was confirmed by several experimental studies by Pettis and co-workers \cit
 In the {\Dtwo} symmetry, the {\oneAg} ground state has a weak multi-configurational character with well-separated frontier orbitals that can be described by single-reference methods. 
 However, in the {\Dfour} symmetry, the {\sBoneg} ground state is a diradical that has two degenerate singly occupied frontier orbitals.
 Therefore, one must take into account, at least, two electronic configurations to properly model this multi-configurational scenario. 
-Of course, single-reference methods are naturally unable to describe such situations. 
+Of course, \alert{standard} single-reference methods are naturally unable to describe such situations. 
 Interestingly, the {\sBoneg} ground state of the square arrangement is a transition state in the automerization reaction between the two rectangular structures (see Fig.~\ref{fig:CBD}), while the lowest triplet state,  {\Atwog}, is a minimum on the triplet potential energy surface in the {\Dfour} arrangement. 
 The automerization barrier (AB) is thus defined as the difference between the square and rectangular ground-state energies.
 The energy of this barrier is estimated, experimentally, in the range of \SIrange{1.6}{10}{\kcalmol}, \cite{Whitman_1982} while previous state-of-the-art \textit{ab initio} calculations yield values in the \SIrange{7}{9}{\kcalmol} range. \cite{Eckert-Maksic_2006,Li_2009,Shen_2012,Zhang_2019}
 
 The lowest-energy excited states of CBD in both symmetries 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 inaccessible with 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 remain torturous for state-of-the-art methods like equation-of-motion third-order coupled-cluster (EOM-CC3) \cite{Christiansen_1995,Koch_1997} or even coupled-cluster with singles, doubles, and triples (EOM-CCSDT). \cite{Kucharski_1991,Kallay_2004,Hirata_2000,Hirata_2004}
+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 inaccessible with \alert{standard} 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 remain torturous for state-of-the-art methods like \alert{standard} equation-of-motion third-order coupled-cluster (EOM-CC3) \cite{Christiansen_1995,Koch_1997} or even 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 approaches. 
-The most evident way is to rely on multi-configurational methods, which are naturally designed to address such scenarios.
+The most evident way is to rely on \alert{multi-reference} 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}
 %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. 
 
@@ -131,12 +131,12 @@ However, to provide a correct description of these situations, one has to take i
 %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 designed, \cite{Jeziorski_1981,Mahapatra_1998,Mahapatra_1999,Lyakh_2012,Kohn_2013} they are computationally demanding and remain far from being black-box.
 
-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 are able to provide near full CI (FCI) ground- and excited-state energies of small molecules. \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 \alert{multi-reference} 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 are able to provide near full CI (FCI) ground- and excited-state energies of small molecules. \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) de-excitation and excitation from the lowest triplet state, respectively.
+Finally, another option to deal with these chemical scenarios is to rely on the 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) de-excitation and excitation from the lowest triplet state, respectively.
-Obviously, spin-flip methods have their own flaws, especially spin contamination (\ie, the artificial mixing of electronic states with different spin multiplicities) due not only to the spin incompleteness in the spin-flip expansion but also to the potential spin contamination of the reference configuration. \cite{Casanova_2020} 
+\alert{One drawback of spin-flip methods is spin contamination} (\ie, the artificial mixing of electronic states with different spin multiplicities) due not only to the spin incompleteness in the spin-flip expansion but also to the potential 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 increased computational cost. 
 
@@ -193,7 +193,7 @@ With respect to {\Aoneg}, {\sBoneg} has a dominant double excitation character,
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsection{Multi-configurational calculations}
+\subsection{\alert{Multi-reference} calculations}
 \label{sec:Multi}
 State-averaged CASSCF (SA-CASSCF) calculations are performed for vertical transition energies, whereas state-specific CASSCF is used for computing the automerization barrier. \cite{Werner_2020} 
 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).
@@ -628,7 +628,7 @@ Again, the extended version, SF-ADC(2)-x, does not seem to be relevant in the pr
 Also, as reported previously, \cite{Loos_2020d} SF-ADC(2)-s and SF-ADC(3) have mirror error patterns making SF-ADC(2.5) particularly accurate except for the doubly-excited state {\twoAg} where the error with respect to the TBE (\SI{0.140}{\eV}) is larger than the SF-ADC(2)-s error (\SI{0.093}{\eV}).
 
 Let us now move to the discussion of the results obtained with standard wave function methods that are reported in Table \ref{tab:D2h}.
-Regarding the multi-configurational calculations, the most striking result is the poor description of the {\sBoneg} ionic state, especially with the (4e,4o) active space where CASSCF predicts this state higher in energy than the {\twoAg} state. 
+Regarding the \alert{multi-reference} calculations, the most striking result is the poor description of the {\sBoneg} ionic state, especially with the (4e,4o) active space where CASSCF predicts this state higher in energy than the {\twoAg} state. 
 Of course, the PT2 correction is able to correct the state ordering problem but cannot provide quantitative excitation energies due to the poor zeroth-order treatment.
 Another ripple effect of the unreliability of the reference wave function is the large difference between CASPT2 and NEVPT2 that differ by half an \si{\eV}. 
 This feature is characteristic of the inadequacy of the active space to model such a state.
@@ -835,7 +835,7 @@ This is, of course, magnified with the (4e,4o) active space for which the second
 In particular SC-NEVPT2(4,4)/aug-cc-pVTZ and PC-NEVPT2(4,4)/aug-cc-pVTZ underestimate the singlet-triplet gap by \SI{0.072}{} and \SI{0.097}{\eV} and, more importantly, flip the ordering of {\Aoneg} and {\Btwog}.
 Although {\Aoneg} is not badly described, the excitation energy of the ionic state {\Btwog} is off by almost \SI{1}{\eV}.
 Thanks to the IPEA shift in CASPT2(4,4), the singlet-triplet gap is accurate and the state ordering remains correct but the ionic state is still far from being well described.
-The (12e,12o) active space significantly alleviates these effects, and, as usual now, the agreement between CASPT2 and NEVPT2 is very much improved for each state, though the accuracy of multi-configurational approaches remains questionable for the ionic state with, \emph{e.g.,} an error up to \SI{-0.093}{\eV} at the PC-NEVPT2(12,12)/aug-cc-pVTZ level.
+The (12e,12o) active space significantly alleviates these effects, and, as usual now, the agreement between CASPT2 and NEVPT2 is very much improved for each state, though the accuracy of \alert{multi-reference} approaches remains questionable for the ionic state with, \emph{e.g.,} an error up to \SI{-0.093}{\eV} at the PC-NEVPT2(12,12)/aug-cc-pVTZ level.
 
 Finally, let us analyze the excitation energies computed with various CC models that are gathered in Table \ref{tab:D4h}.
 As mentioned in Sec.~\ref{sec:CC}, we remind the reader that these calculations are performed by considering the {\Aoneg} state as reference, and that, therefore, 
@@ -867,7 +867,7 @@ Moreover, as previously reported, SF-ADC(2)-s and SF-ADC(3) have opposite error
 In such scenario, the SF-TD-DFT excitation energies can exhibit errors of the order of \SI{1}{\eV} compared to the TBEs. 
 However, it was satisfying to see that the spin-flip version of ADC can lower these errors to \SIrange{0.1}{0.2}{\eV}.
 
-\item Concerning the multi-configurational methods, we have found that while NEVPT2 and CASPT2 can provide different excitation energies for the small (4e,4o) active space, the results become highly similar when the larger (12e,12o) active space is considered. 
+\item Concerning the \alert{multi-reference} methods, we have found that while NEVPT2 and CASPT2 can provide different excitation energies for the small (4e,4o) active space, the results become highly similar when the larger (12e,12o) active space is considered. 
 From a more general perspective, a significant difference between NEVPT2 and CASPT2 is usually not a good omen and can be seen as a clear warning sign that the active space is too small or poorly chosen.
 The ionic states remain a struggle for both CASPT2 and NEVPT2, even with the (12e,12o) active space.