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Manuscript/CBD.bib
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@ -1,13 +1,52 @@
%% This BibTeX bibliography file was created using BibDesk.
%% https://bibdesk.sourceforge.io/
%% Created for Pierre-Francois Loos at 2022-04-07 21:30:45 +0200
%% Created for Pierre-Francois Loos at 2022-04-07 21:53:18 +0200
%% Saved with string encoding Unicode (UTF-8)
@article{Dutta_2013,
author = {Dutta,Achintya Kumar and Pal,Sourav and Ghosh,Debashree},
date-added = {2022-04-07 21:53:18 +0200},
date-modified = {2022-04-07 21:53:18 +0200},
doi = {10.1063/1.4821936},
journal = {J. Chem. Phys.},
number = {12},
pages = {124116},
title = {Perturbative approximations to single and double spin flip equation of motion coupled cluster singles doubles methods},
volume = {139},
year = {2013},
bdsk-url-1 = {https://doi.org/10.1063/1.4821936}}
@article{Casanova_2009a,
author = {Casanova,David and Slipchenko,Lyudmila V. and Krylov,Anna I. and Head-Gordon,Martin},
date-added = {2022-04-07 21:52:59 +0200},
date-modified = {2022-04-07 21:52:59 +0200},
doi = {10.1063/1.3066652},
journal = {J. Chem. Phys.},
number = {4},
pages = {044103},
title = {Double spin-flip approach within equation-of-motion coupled cluster and configuration interaction formalisms: Theory, implementation, and examples},
volume = {130},
year = {2009},
bdsk-url-1 = {https://doi.org/10.1063/1.3066652}}
@article{Casanova_2009b,
abstract = {A new formulation of the spin-flip (SF) method is presented. The electronic wave function is specified by the definition of an active space and through α-to-β excitations from a Hartree--Fock reference. The method belongs to the restricted active space (RAS) family{,} where the CI expansion is restricted by classifying the molecular orbitals in three subspaces. Properties such as spin completeness{,} variationality{,} size consistency{,} size intensivity{,} and orbital invariance are discussed. The implementation and applications use a particular truncation of the wave function{,} with the inclusion of hole and particle contributions such that for fixed active space size{,} the number of amplitudes is linear in molecular size. This approach is used to investigate single and double bond-breaking{,} the singlet--triplet gap of linear acenes{,} electronic transitions in three Ni(ii) octahedral complexes{,} the low-lying states of the 2{,}5-didehydrometaxylylene (DDMX) tetraradical and the ground state multiplicity of 28 non-Kekul{\'e} structures. The results suggest that this approach can provide a quite well balanced description of nearly degenerate electronic states at moderate computational cost.},
author = {Casanova, David and Head-Gordon, Martin},
date-added = {2022-04-07 21:52:59 +0200},
date-modified = {2022-04-07 21:52:59 +0200},
doi = {10.1039/B911513G},
journal = {Phys. Chem. Chem. Phys.},
pages = {9779-9790},
title = {Restricted active space spin-flip configuration interaction approach: theory{,} implementation and examples},
volume = {11},
year = {2009},
bdsk-url-1 = {http://dx.doi.org/10.1039/B911513G}}
@article{Wormit_2014,
author = {Michael Wormit and Dirk R. Rehn and Philipp H.P. Harbach and Jan Wenzel and Caroline M. Krauter and Evgeny Epifanovsky and Andreas Dreuw},
date-added = {2022-04-07 21:30:28 +0200},

66
Manuscript/CBD.tex
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@ -84,8 +84,8 @@
\affiliation{\LCPQ}
\begin{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 for struggling for cyclobutadiene.
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.
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 of both the $D_{2h}$ 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 states with multi-configurational character, 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.
@ -102,7 +102,8 @@ A theoretical best estimate is defined for the automerization barrier and for ea
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, and solar cells, \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, the impact of the environment, and many other factors.
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 remains a very active field of theoretical quantum chemistry.\cite{Roos_1996,Piecuch_2002b,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 both 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 making its synthesis a particularly difficult exercise.
Speaking of difficult tasks, the cyclobutadiene (CBD) molecule has been a real challenge for both 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 making its synthesis a particularly difficult exercise.
In the {\Dfour} symmetry, the simple H\"uckel molecular orbital theory wrongly predicts a triplet ground state (Hund's rule) with two singly-occupied frontier orbitals that are degenerate by symmetry, while state-of-the-art \textit{ab initio} methods correctly predict an open-shell singlet ground state.
This degeneracy is lifted by the so-called Jahn-Teller effect, \ie, by a descent in symmetry (from {\Dfour} to {\Dtwo} point group) via a geometrical distortion of the molecule, leading to a closed-shell singlet ground state in the rectangular geometry (see below).
This was confirmed by several experimental studies by Pettis and co-workers \cite{Reeves_1969} and others. \cite{Irngartinger_1983,Ermer_1983,Kreile_1986}
@ -111,19 +112,18 @@ In the {\Dtwo} symmetry, the {\oneAg} ground state has a weak multi-configuratio
However, in the {\Dfour} symmetry, the {\sBoneg} ground state has two singly occupied frontier orbitals that are degenerate.
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.
Interestingly singlet {\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 {\Atwog} triplet state is a minimum on the triplet potential energy surface in the {\Dfour} arrangement.
Interestingly singlet {\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 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 beyond reach 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 remain a nightmare for state-of-the-art methods like the equation-of-motion third-order coupled-cluster (EOM-CC3) \cite{Christiansen_1995,Koch_1997} or even the coupled-cluster with singles, doubles, and triples (EOM-CCSDT) methods. \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 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}
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.
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.
%\hl{Deux choses: 1. la derniere phrase est vraie mais ne s'applique pas au CBD, pas utile je trouve 2. Martial a essaye CASPT3 ?}
Another way to deal with double excitations and multi-reference situations is to use high level truncation of the EOM formalism \cite{Rowe_1968,Stanton_1993} of CC theory. \cite{Kucharski_1991,Kallay_2003,Kallay_2004,Hirata_2000,Hirata_2004}
However, to provide a correct description of these situations, one has to take into account, at the very least, contributions from the triple excitations in the CC expansion. \cite{Watson_2012,Loos_2018a,Loos_2019,Loos_2020b}
@ -133,13 +133,11 @@ Although multi-reference CC methods have been designed, \cite{Jeziorski_1981,Mah
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}
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.
%\hl{idem: OK, on a compris, c'est bonbon tout cela et a la fin vous ferez B3LYP comme tout le monde :-) :-) }
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.
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 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.
%\hl{mentionner que SF surtout avec TD, un peu avec ADC et CCSD mais peu avec CCSDTQ ? pas sur, j'y pense en lisant}
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.
In the present work, we define highly-accurate reference values and investigate the accuracy of each family of computational methods mentioned above on the automerization barrier and the low-lying excited states of CBD at the {\Dtwo} and {\Dfour} ground-state geometries.
Computational details are reported in Sec.~\ref{sec:compmet}.
@ -150,7 +148,7 @@ Finally, our conclusions are drawn in Sec.~\ref{sec:conclusion}.
\begin{figure}
\includegraphics[width=\linewidth]{figure1}
\caption{Pictorial representation of the ground and lowest 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 colored in black and red, respectively.
The automerization barrier (AB) is also represented.}
\label{fig:CBD}
\end{figure}
@ -168,8 +166,8 @@ For the SCI calculations, we rely on the CIPSI algorithm implemented in QUANTUM
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}.
For a given size of the variational wave function, the CIPSI energy is the sum of two terms: the variational energy obtained by diagonalization of the CI matrix in the reference space $E_\text{var}$ and a second-order perturbative correction $E_\text{PT2}$ which estimates the contribution of the external determinants that are not included in the variational space at a given iteration.
The sum of these two energies is, for large enough wave functions, an estimate of the FCI energy, \ie, $E_\text{FCI} \approx E_\text{var} + E_\text{PT2}$.
For a given size of the variational wave function and for each electronic state, the CIPSI energy is the sum of two terms: the variational energy obtained by diagonalization of the CI matrix in the reference space $E_\text{var}$ and a second-order perturbative correction $E_\text{PT2}$ which estimates the contribution of the external determinants that are not included in the variational space at a given iteration.
The sum of these two energies is, for large enough wave functions, an estimate of the FCI energy of a given state, \ie, $E_\text{FCI} \approx E_\text{var} + E_\text{PT2}$.
It is possible to estimate more precisely the FCI energy via an extrapolation procedure, where the variational energy is extrapolated to $E_\text{PT2} = 0$. \cite{Holmes_2017}
Excitation energies are then computed as differences of extrapolated total energies. \cite{Chien_2018,Loos_2018a,Loos_2019,Loos_2020b,Loos_2020c}
Additionally, an error bar can be provided thanks to a recent method based on Gaussian random variables that is described in Ref.~\onlinecite{Veril_2021}.
@ -185,7 +183,7 @@ In the following, we will omit the prefix EOM for the sake of conciseness.
Alternatively to the ``complete'' CC models, one can also employ 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 high-order 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} with which only singlet excited states can be computed but for CCSD.
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 (except for CCSD).
In some cases, 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 level with DALTON \cite{Aidas_2014} and at the CCSDT level with MRCC. \cite{mrcc}
To avoid having to perform multi-reference CC calculations or 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.
@ -219,7 +217,7 @@ Likewise, excitation energies with respect to the singlet ground state are compu
Nowadays, spin-flip techniques are broadly accessible thanks to intensive developments in the electronic structure community (see Ref.~\onlinecite{Casanova_2020} and references therein).
Here, we explore the spin-flip version \cite{Lefrancois_2015} of the algebraic-diagrammatic construction \cite{Schirmer_1982} (ADC) using the standard and extended second-order ADC schemes, SF-ADC(2)-s \cite{Trofimov_1997,Dreuw_2015} and SF-ADC(2)-x, \cite{Dreuw_2015} as well as its third-order version, SF-ADC(3). \cite{Dreuw_2015,Trofimov_2002,Harbach_2014}
These calculations are performed using Q-CHEM 5.2.1. \cite{qchem}
The spin-flip version of our recently proposed composite approach, namely SF-ADC(2.5), \cite{Loos_2020d} where one simply averages the SF-ADC(2)-s and SF-ADC(3) energies, is also tested in the following.
The spin-flip version of our recently proposed composite approach, namely SF-ADC(2.5), \cite{Loos_2020d} where one simply averages the SF-ADC(2)-s and SF-ADC(3) energies, is also tested in the following.
We have also carried out spin-flip calculations within the TD-DFT framework (SF-TD-DFT), \cite{Shao_2003} with the same Q-CHEM 5.2.1 code. \cite{qchem}
The B3LYP, \cite{Becke_1988b,Lee_1988a,Becke_1993b} PBE0 \cite{Adamo_1999a,Ernzerhof_1999} and BH\&HLYP global hybrid GGA functionals are considered, which contain 20\%, 25\%, 50\% of exact exchange, respectively.
@ -227,6 +225,8 @@ These calculations are labeled as SF-TD-BLYP, SF-TD-B3LYP, SF-TD-PBE0, and SF-TD
Additionally, we have also computed SF-TD-DFT excitation energies using range-separated hybrid (RSH) functionals: CAM-B3LYP (19\% of short-range exact exchange and 65\% at long range), \cite{Yanai_2004a} LC-$\omega$PBE08 (0\% of short-range exact exchange and 100\% at long range), \cite{Weintraub_2009a} and $\omega$B97X-V (16.7\% of short-range exact exchange and 100\% at long range). \cite{Mardirossian_2014}
Finally, the hybrid meta-GGA functional M06-2X (54\% of exact exchange) \cite{Zhao_2008} and the RSH meta-GGA functional M11 (42.8\% of short-range exact exchange and 100\% at long range) \cite{Peverati_2011} are also employed.
Note that all SF-TD-DFT calculations are done within the Tamm-Dancoff approximation. \cite{Hirata_1999}
Although there also exist spin-flip extension of EOM-CC methods, \cite{Krylov_2001a,Levchenko_2004,Manohar_2008,Casanova_2009a,Dutta_2013} they are not considered here.
%EOM-SF-CCSD and EOM-SF-CC(2,3) are also performed with Q-CHEM 5.2.1.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -234,12 +234,13 @@ Note that all SF-TD-DFT calculations are done within the Tamm-Dancoff approximat
\subsection{Theoretical best estimates}
\label{sec:TBE}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
When technically possible, each level of theory is tested with four Gaussian basis sets, namely, 6-31+G(d) and aug-cc-pVXZ with X $=$ D, T, and Q. \cite{Dunning_1989}
When technically possible, each level of theory is tested with four gaussian basis sets, namely, 6-31+G(d) and aug-cc-pVXZ with X $=$ D, T, and Q. \cite{Dunning_1989}
This helps us to assess the convergence of each property with respect to the size of the basis set.
More importantly, for each studied quantity (i.e., the automerization barrier and the vertical excitation energies), we provide a theoretical best estimate (TBE) established in the aug-cc-pVTZ basis.
These TBEs are defined using extrapolated CCSDTQ/aug-cc-pVTZ values but, in a single occasion, in which the NEVPT2(12,12) value is used.
These TBEs are defined using extrapolated CCSDTQ/aug-cc-pVTZ values except in a single occasion where the NEVPT2(12,12) value is used.
The extrapolation of the CCSDTQ/aug-cc-pVTZ values is done via a ``pyramidal'' scheme, where we employ systematically the most accurate level of theory and the largest basis set available.
The viability of this scheme lies on the transferability of basis set effects within wave function methods (see below).
For example, when CC4/aug-cc-pVTZ and CCSDTQ/aug-cc-pVDZ data are available, we proceed via the following basis set extrapolation:
\begin{equation}
\label{eq:aug-cc-pVTZ}
@ -283,11 +284,11 @@ Two different sets of geometries obtained with different levels of theory are co
First, because the automerization barrier is obtained as a difference of energies computed at distinct geometries, it is paramount to obtain these at the same level of theory.
However, due to the fact that the ground state of the square arrangement is a transition state of singlet open-shell nature, it is technically difficult to optimize the geometry with high-order CC methods.
Therefore, we rely on CASPT2(12,12)/aug-cc-pVTZ for both the {\Dtwo} and {\Dfour} ground-state structures.
(Note that these optimizations are done without IPEA shift.)
\titou{(Note that these optimizations are done without IPEA shift.)}
Second, because the vertical transition energies are computed for a particular equilibrium geometry, we can afford to use different methods for the rectangular and square structures.
Hence, we rely on CC3/aug-cc-pVTZ to compute the equilibrium geometry of the {\oneAg} state in the rectangular ({\Dtwo}) arrangement and the restricted open-shell (RO) version of CCSD(T)/aug-cc-pVTZ to obtain the equilibrium geometry of the {\Atwog} state in the square ({\Dfour}) arrangement.
These two geometries are the lowest-energy equilibrium structure of their respective spin manifold (see Fig.~\ref{fig:CBD}).
The Cartesian coordinates of these geometries are provided in the {\SupInf}.
The cartesian coordinates of these geometries are provided in the {\SupInf}.
Table \ref{tab:geometries} reports the key geometrical parameters obtained at these levels of theory as well as previous geometries computed by Manohar and Krylov at the CCSD(T)/cc-pVTZ level.
One notes globally satisfying agreement between the tested methods with variations of the order of \SI{0.01}{\angstrom} only.
%================================================
@ -392,7 +393,7 @@ Our TBE with this basis set is 8.93 \kcalmol.
First, one can see large variations of the energy barrier at the SF-TD-DFT level, with differences as large as \SI{10}{\kcalmol} between the different functionals for a given basis set.
Nonetheless, it is clear that the performance of a given functional is directly linked to the amount of exact exchange at short range.
Indeed, hybrid functionals with a ca. 50\%\ fraction of short-range exact exchange (\eg, BH\&HLYP, M06-2X, and M11) perform significantly better than the functionals having a small fraction of short-range exact exchange (\eg, B3LYP, PBE0, CAM-B3LYP, $\omega$B97X-V, and LC-$\omega$PBE08).
Indeed, hybrid functionals with a ca.~50\%\ fraction of short-range exact exchange (\eg, BH\&HLYP, M06-2X, and M11) perform significantly better than the functionals having a small fraction of short-range exact exchange (\eg, B3LYP, PBE0, CAM-B3LYP, $\omega$B97X-V, and LC-$\omega$PBE08).
However, they are still off by \SIrange{1}{4}{\kcalmol} from the TBE reference value, the most accurate result being obtained with M06-2X.
For the RSH functionals, the automerization barrier is much less sensitive to the amount of longe-range exact exchange.
Another important feature of SF-TD-DFT is the fast convergence of the energy barrier with the size of the basis set. \cite{Loos_2019d}
@ -409,13 +410,12 @@ Concerning the multi-reference approaches with the minimal (4e,4o) active space,
In this case, the NEVPT2 values are fairly accurate with differences below half a \si{\kcalmol} compared to the TBEs.
The CASSCF results predict an even lower barrier than CASPT2 due to the well known lack of dynamical correlation at the CASSCF level.
For the larger (12e,12o) active space, we see larger differences of the order of \SI{3}{\kcalmol} through all the bases between CASSCF and the second-order variants (CASPT2 and NEVPT2).
However, the deviations between CASPT2(12,12) and NEVPT2(12,12) are much smaller than with the compact active space, with an energy difference of around \SIrange{0.1}{0.2}{\kcalmol} for all bases, CASPT2 being slightly more accurate than NEVPT2 in this case.
However, the deviations between CASPT2(12,12) and NEVPT2(12,12) are much smaller than with the minimal active space, with an energy difference of around \SIrange{0.1}{0.2}{\kcalmol} for all bases, CASPT2 being slightly more accurate than NEVPT2 in this case.
For each basis set, both CASPT2(12,12) and NEVPT2(12,12) are less than a \si{\kcalmol} away from the TBEs.
For the two active spaces that we have considered here, the PC- and SC-NEVPT2 schemes provide nearly identical barriers independently of the size of the one-electron basis.
Finally, for the CC family of methods, we observe the usual systematic improvement following the series CCSD $<$ CC3 $<$ CCSDT $<$ CC4 $<$ CCSDTQ, which parallels their increase in computational cost: $\order*{N^6}$, $\order*{N^7}$, $\order*{N^8}$, $\order*{N^9}$, and $\order*{N^{10}}$, respectively.
Note that the introduction of the triple excitations is clearly mandatory to have an accuracy beyond SF-TD-DFT, while it is also clear that the iterative triples and quadruples can be included approximately via the CC3 and CC4 methods, respectively.
\hl{Pas sur de comprendre, entre CC3 et CCSDT les differences sont pas si small que cela, donc que vlz vs dire...}
Note that the introduction of the triple excitations is clearly mandatory to have an accuracy beyond SF-TD-DFT, and we observe that CCSDT is definitely an improvement over its cheaper, approximated version, CC3.
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@ -627,7 +627,8 @@ This feature is characteristic of the inadequacy of the active space to model su
For the two other states, {\tBoneg} and {\twoAg}, the errors at the CASPT2(4,4) and NEVPT2(4,4) levels are much smaller (below \SI{0.1}{\eV}).
Using a larger active space resolves most of these issues: CASSCF predicts the correct state ordering (though the ionic state is still badly described in term of energetics), CASPT2 and NEVPT2 excitation energies are much closer, and their accuracy is often improved (especially for the triplet state) although it is difficult to reach chemical accuracy (\ie, an error below \SI{0.043}{\eV}) on a systematic basis.
Finally, for the CC models (Table \ref{tab:D2h}), the two states with a large $\%T_1$ value, {\tBoneg} and {\sBoneg}, are already extremely accurate at the CC3 level, and systematically improved by CCSDT and CC4, which is in line with the results of the QUEST database. \cite{Veril_2021}
Finally, for the CC models (Table \ref{tab:D2h}), the two states with a large $\%T_1$ value, {\tBoneg} and {\sBoneg}, are already extremely accurate at the CC3 level, and systematically improved by CCSDT and CC4.
This trend is in line with the observations made on the QUEST database. \cite{Veril_2021}
For the doubly-excited state, {\twoAg}, the convergence of the CC expansion is much slower but it is worth pointing out that the inclusion of approximate quadruples via CC4 is particularly effective, in line with an earlier work. \cite{Loos_2021}
The CCSDTQ excitation energies (which are used to define the TBEs) are systematically within the error bar of the CIPSI extrapolations, which confirms the outstanding performance of CC methods that include quadruple excitations in the context of excited states.
@ -804,17 +805,17 @@ For all functionals, this gap is small (basically below \SI{0.1}{\eV} while the
Increasing the fraction of exact exchange in hybrids or relying on RSHs (even with a small amount of short-range exact exchange) allows to recover a positive gap and a singlet ground state.
At the SF-TD-DFT level, the energy gap between the two singlet excited states, {\Aoneg} and {\Btwog}, is particularly small and grows moderately with the amount of exact exchange at short range.
The influence of the exact exchange on the singlet energies is quite significant with an energy difference of the order of \SI{1}{\eV} between the functional with the smallest amount of exact exchange (B3LYP) and the functional with the largest amount (M06-2X).
As for the excitation energies computed at the {\Dtwo} ground-state equilibrium structure and the automerization barrier, the functionals with a large fraction of short-range exact exchange yield much more accurate results.
As for the excitation energies computed on the {\Dtwo} ground-state equilibrium structure and the automerization barrier, the functionals with a large fraction of short-range exact exchange yield much more accurate results.
Yet, the transition energy to {\Btwog} is off by more than half an \si{\eV} compared to the TBE, while the doubly-excited state is much closer to the reference value (errors of \SI{-0.251}{}, \SI{-0.097}{}, and \SI{-0.312}{\eV} for BH\&HLYP, M06-2X, and M11, respectively).
Again, for all the excited states, the basis set effects are extremely small at the SF-TD-DFT level.
We underline that the $\expval*{S^2}$ values reported in {\SupInf} indicate again that there is no significant spin contamination in these excited states.
We emphasize that the $\expval*{S^2}$ values reported in {\SupInf} indicate again that there is no significant spin contamination in these excited states.
Next, we discuss the various ADC schemes (Table \ref{tab:sf_D4h}).%DJ: inutile, redite: where we were not able to compute the vertical energies with the aug-cc-pVQZ basis due to our limited computational resources.
Next, we discuss the various ADC schemes (Table \ref{tab:sf_D4h}).
Globally, we observe similar trends as those noted in Sec.~\ref{sec:D2h}.
Concerning the singlet-triplet gap, each scheme predicts it to be positive.
Although it provides a decent singlet-triplet gap value, SF-ADC(2)-x seems to particularly struggle with the singlet excited states ({\Aoneg} and {\Btwog}), especially for the doubly-excited state {\Aoneg} where it underestimates the vertical excitation energy by \SI{0.4}{\eV}.
Again, averaging the SF-ADC(2)-s and SF-ADC(3) transition energies is beneficial in most cases at the exception of {\Aoneg}.
Although the basis set effects are larger than at the SF-TD-DFT level, they remain quite moderate at the SF-ADC level, and this holds for all wave function method in general.
Although the basis set effects are larger than at the SF-TD-DFT level, they remain quite moderate at the SF-ADC level, and this holds for wave function methods in general.
Let us turn to the multi-reference results (Table \ref{tab:D4h}).
For both active spaces, expectedly, CASSCF does not provide a quantitive energetic description, although it is worth mentioning that the right state ordering is preserved.
@ -841,7 +842,7 @@ As a final comment, we can note that the CCSDTQ-based TBEs and the CIPSI results
In the present study, we have benchmarked a larger number of computational methods on the automerization barrier and the vertical excitation energies of cyclobutadiene in its square ({\Dfour}) and rectangular ({\Dtwo}) geometries, for which we have defined theoretical best estimates based on extrapolated CCSDTQ/aug-cc-pVTZ data.
The main take-home messages of the present work are
The main take-home messages of the present work can be summarized as follows:
\begin{itemize}
\item Within the SF-TD-DFT framework, we advice to use exchange-correlation (hybrids or range-separated hybrids) with a large fraction of short-range exact exchange.
@ -856,11 +857,10 @@ However, it was satisfying to see that the spin-flip version of ADC can lower th
\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.
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.
\item The ionic states remain a struggle for both CASPT2 and NEVPT2, even with the (12e,12o) active space.
\item In the context of CC methods, although the inclusion of triple excitations (via CC3 or CCSDT) yields very satisfactory results in most cases, the inclusion of quadruples excitation (via CC4 or CCSDTQ) is mandatory to reach high accuracy (especially in the case of doubly-excited states).\hl{la aussi: accuracy. This is as expected especially true for doubly excited states ?)}
We also point out that, considering the error bar related to the CIPSI extrapolation procedure, CCSDTQ and CIPSI yield equivalent excitation energies, hence confirming the outstanding accuracy of CCSDTQ in the context of molecular excited states.
\item In the context of CC methods, although the inclusion of triple excitations (via CC3 or CCSDT) yields very satisfactory results in most cases, the inclusion of quadruples excitation (via CC4 or CCSDTQ) is mandatory to reach high accuracy (especially in the case of doubly-excited states).
Finally, we point out that, considering the error bar related to the CIPSI extrapolation procedure, CCSDTQ and CIPSI yield equivalent excitation energies, hence confirming the outstanding accuracy of CCSDTQ in the context of molecular excited states.
\end{itemize}