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 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 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 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)].
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 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_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 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.
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 (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}
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 has two singly occupied frontier orbitals that are degenerate.
The singlet ground state, {\sBoneg}, 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.
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 an energy barrier in the range of \SIrange{7}{9}{\kcalmol}. \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.
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}
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}
However, to provide a correct description of these situations, one have 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}
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 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, 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 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.
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}.
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, $E_\text{FCI}\approx E_\text{var}+ E_\text{PT2}$.
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 via a recent method based on Gaussian random variables as described in Ref.~\onlinecite{Veril_2021}.
This type of extrapolation procedures is now routine in SCI methods as well as other state-of-the-art techniques. \cite{Eriksen_2020,Loos_2020e,Eriksen_2021}
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 diagonalizes the similarity-transformed Hamiltonian in a CI basis of excited determinants yielding the following systematically improvable family of methods for neutral excited states:\cite{Noga_1987a,Koch_1990b,Kucharski_1991,Stanton_1993,Christiansen_1998,Kucharski_2001,Kowalski_2001,Kallay_2003,Kallay_2004,Hirata_2000,Hirata_2004} EOM-CCSD, EOM-CCSDT, EOM-CCSDTQ, etc.
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} with which only singlet excited states can be computed.
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 CCSDT level with MRCC. \cite{mrcc}
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.
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).
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. \cite{Davidson_1996,Angeli_2009,BenAmor_2020}
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 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}
%and extended multistate (XMS) CASPT2 calculations are also performed in the cas of strong mixing between states with same spin and spatial symmetries.
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).
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), where one simply averages the SF-ADC(2)-s and SF-ADC(3) energies, is also tested in the following. \cite{Loos_2020d}
We have also carried out spin-flip calculations within the TD-DFT framework (SF-TD-DFT), \cite{Shao_2003} and these are also performed with Q-CHEM 5.2.1. \cite{qchem}
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 (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.
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 autoisomerisation barrier and the vertical excitation energies), we provide a theoretical best estimate (TBE) established in the aug-cc-pVTZ basis.
In most cases, these TBEs are defined using extrapolated CCSDTQ/aug-cc-pVTZ values or, in a single occasion, NEVPT2(12,12).
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.
For example, when CC4/aug-cc-pVTZ and CCSDTQ/aug-cc-pVDZ data are available, we proceed via the following basis set extrapolation:
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.
Tables gathering these TBEs as well as literature data for the automerization barrier and the vertical excitation energies can be found in the {\SupInf}.
Two different sets of geometries obtained with different levels of theory are considered for the automerization barrier and the excited states of the CBD molecule.
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.
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}).
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.
The results concerning the automerization barrier are reported in Table \ref{tab:auto_standard} for various basis sets and shown in Fig.~\ref{fig:AB} for the aug-cc-pVTZ basis.
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 large 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).
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}
With the augmented double-$\zeta$ basis, the SF-TD-DFT results are basically converged to sub-\kcalmol accuracy, which is a drastic improvement compared to wave function approaches where this type of convergence is reached with the augmented triple-$\zeta$ basis.
In particular, we observe that SF-ADC(2)-s and SF-ADC(3), which scale as $\order*{N^5}$ and $\order*{N^6}$ respectively (where $N$ is the number of basis functions), under- and overestimate the automerization barrier, making SF-ADC(2.5) a good compromise with an error of only \SI{0.18}{\kcalmol} with the aug-cc-pVTZ basis.
We note that SF-ADC(2)-x is probably not worth its extra cost [as compared to SF-ADC(2)-s] as it overestimates the energy barrier even more than SF-ADC(3).
Overall, even with the best exchange-correlation functional, SF-TD-DFT is clearly outperformed by the more expensive SF-ADC methods.
Concerning the multi-reference approaches with the minimal (4e,4o) active space, the TBEs are bracketed by the CASPT2 and NEVPT2 values that differ by approximately \SI{1.5}{\kcalmol} for all bases.
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).
Thought, the deviations between CASPT2(12,12) and NEVPT2(12,12) are much smaller 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 is also linked to 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.
Spin-flip TD-DFT and ADC vertical excitation energies (with respect to the singlet {\oneAg} ground state) of the {\tBoneg}, {\sBoneg}, and {\twoAg} states of CBD at the {\Dtwo} rectangular equilibrium geometry of the {\oneAg} ground state.}
Vertical excitation energies (with respect to the {\oneAg} ground state) of the {\tBoneg}, {\sBoneg}, and {\twoAg} states of CBD at the {\Dtwo} rectangular equilibrium geometry of the {\oneAg} ground state.
\caption{Vertical excitation energies of the {\tBoneg}, {\sBoneg}, and {\twoAg} states at the {\Dtwo} rectangular equilibrium geometry of the {\oneAg} ground state using the aug-cc-pVTZ basis.
Table \ref{tab:sf_D2h} reports, at the {\Dtwo} rectangular equilibrium geometry of the {\oneAg} ground state, the vertical transition energies associated with the {\tBoneg}, {\sBoneg}, and {\twoAg} states obtained using the spin-flip formalism, while Table \ref{tab:D2h} gathers the same quantities obtained with the multi-reference, CC, and CIPSI methods.
Considering the aug-cc-pVTZ basis, the evolution of the vertical excitation energies with respect to the level of theory is illustrated in Fig.~\ref{fig:D2h}.
At the CC3/aug-cc-pVTZ level, the percentage of single excitation involved in the {\tBoneg}, {\sBoneg}, and {\twoAg} are 99\%, 95\%, and 1\%, respectively.
First, let us discuss basis set effects at the SF-TD-DFT level.
As expected, these are found to be small and the results are basically converged to the basis set limit with the triple-$\zeta$ basis, which is definitely not the case for the wave function methods. \cite{Giner_2019}
Regarding now the accuracy of the vertical excitation energies, again, we clearly see that, for each transition, the functionals with the largest amount of short-range exact exchange (\eg, BH\&HLYP, M06-2X, and M11) are the most accurate.
Functionals with large fraction of exact exchange are known to perform best in the SF-TD-DFT framework as the Hartree-Fock exchange term is the only non-vanishing term in the spin-flip block. \cite{Shao_2003}
However, their overall accuracy remains average especially for the singlet states, {\sBoneg} and {\twoAg}, with error of the order of \SIrange{0.2}{0.5}{\eV} compared to the TBEs.
%First, we discuss the SF-TD-DFT values with hybrid functionals.
%For the B3LYP functional we can see that the energy differences for each state and throughout all bases are small with the largest one for the {\tBoneg} state with \SI{0.012}{\eV}.
%The same observation can be done for the PBE0 and BH\&HLYP functionals, we can also observe that adding exact exchange to the functional (20\% of exact exchange for B3LYP, 25\% for PBE0 and 50\% for BH\&HLYP increase the energy difference between the {\sBoneg} and the {\tBoneg} states.
%Put another way, increasing the amount of exact exchange in the functional reduces the energy difference between the {\tBoneg} and the {\oneAg} states.
%We can also notice that the vertical energies of the different states do not vary in the same way when adding exact exchange, for instance the energy variation for the {\tBoneg} state from PBE0 to BH\&HLYP is around \SI{0.1}{\eV} whereas for the {\sBoneg} and the {\twoAg} states this energy variation is about \SIrange{0.4}{0.5}{\eV} and \SI{0.34}{eV} respectively.
%For the RSH functionals we can not make the same observation, indeed we can see that for the CAM-B3LYP functional we have that the energy difference between the {\sBoneg} and the {\tBoneg} states is larger than for the $\omega$B97X-V and LC-$\omega$PBE08 functionals despite the fact that the latter ones have a larger amount of exact exchange.
%However we can observe that we have a small energy difference for each state throughout the bases and for each RSH functional.
%The M06-2X functional is an hybrid meta-GGA functional and contains 54\% of Hartree-Fock exchange, we can compare the energies with the BH\&HLYP functional and we can see that the energy differences are small with around \SIrange{0.03}{0.08}{\eV}.
%We can notice that the upper bound of \SI{0.08}{\eV} in the energy differences is due to the {\tBoneg} state.
%The M11 vertical energies are close to the BH\&HLYP ones for the triplet and the first singlet excited states with \SIrange{0.01}{0.02}{\eV} and \SIrange{0.08}{0.10}{\eV} of energy difference, respectively.
%For the {\twoAg} state the M11 energies are closer to the $\omega$B97X-V ones with \SIrange{0.05}{0.09}{\eV} of energy difference.
Second, we discuss the various SF-ADC schemes, \ie, SF-ADC(2)-s, SF-ADC(2)-x, and SF-ADC(3).
At the SF-ADC(2)-s level, going from the smallest 6-31+G(d) basis to the largest aug-cc-pVQZ basis, we see a small decrease in vertical excitation energies of about \SI{0.03}{\eV} for the {\tBoneg} state and around \SI{0.06}{\eV} for {\twoAg} state, while the transition energy of the {\sBoneg} state drops more significantly by about \SI{0.2}{\eV}.
Also, as reported previously, \cite{Loos_2020d} SF-ADC(2)-s and SF-ADC(3) have mirror error patterns which makes SF-ADC(2.5) particularly accurate with a maximum error of \SI{0.029}{\eV} for the doubly-excited state {\twoAg}.
Let us now move to the discussion of the results concerning 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.
Of course, the PT2 correction is able to correct the root-flipping 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.
For the two other states, {\tBoneg} and {\twoAg}, the errors at the CASPT2(4,4) and NEVPT2(4,4) levels are much smaller and typically below \SI{0.1}{\eV}.
Using a larger active 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, 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.
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 the present case.
The CCSDTQ excitation energies (which are used to define the TBEs) are systematically within the error bar of the CIPSI calculations, which confirms the outstanding performance of CC methods including quadruple excitations in the context of excited states.
Spin-flip TD-DFT and ADC vertical excitation energies (with respect to the singlet {\sBoneg} ground state) of the {\Atwog}, {\Aoneg}, and {\Btwog} states of CBD at the {\Dfour} square-planar equilibrium geometry of the {\Atwog} state.}
Vertical excitation energies (with respect to the {\sBoneg} ground state) of the {\Atwog}, {\Aoneg}, and {\Btwog} states of CBD at the {\Dfour} square-planar equilibrium geometry of the {\Atwog} state.
\caption{Vertical excitation energies (in \si{\eV}) of the {\Atwog}, {\Aoneg}, and {\Btwog} states at the {\Dfour} square-planar equilibrium geometry of the {\Atwog} state using the aug-cc-pVTZ basis.
In Table \ref{tab:sf_D4h}, we report, at the {\Dfour} square planar equilibrium geometry of the {\Atwog} state, the vertical transition energies associated with the {\Atwog}, {\Aoneg}, and {\Btwog} states obtained using the spin-flip formalism, while Table \ref{tab:D4h} gathers the same quantities obtained with the multi-reference, CC, and CIPSI methods.
The vertical excitation energies computed at various levels of theory are depicted in Fig.~\ref{fig:D4h} for the aug-cc-pVTZ basis.
Unfortunately, due to technical limitations, we could not compute $\%T_1$ values associated with the {\Atwog}, {\Aoneg}, and {\Btwog} excited states in the {\Dfour} symmetry.
However, it is clear from the inspection of the wave function that, with respect to the {\sBoneg} ground state, {\Atwog} and {\Btwog} are dominated by single excitations, while {\Aoneg} is strongly dominated by double excitations.
As for the previous geometry we start by discussing the SF-TD-DFT results, and in particular the singlet-triplet gap, \ie, the energy difference between {\sBoneg} and {\Atwog}.
For all functionals, this gap is small (basically below \SI{0.1}{\eV} while the TBE value is \SI{0.144}{\eV}) but it is worth mentioning that B3LYP and PBE0 predict a negative singlet-triplet gap (hence a triplet ground state).
Increasing the 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 short-range exact exchange.
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} equilibrium structure and the automerization barrier, functionals with a large fraction of short-range exact exchange deliver 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.
Then we discuss the various ADC scheme results, note that we were not able to obtain the vertical energies with the aug-cc-pVQZ basis due to computational resources.
For the ADC(2)-s scheme we can see that the energy difference for the triplet state are smaller than for the two other states, indeed we have an energy difference of \SI{0.09}{\eV} for the triplet state whereas we have \SI{0.15}{\eV} and \SI{0.25}{eV} for the {\Aoneg} and {\Btwog} states, again when considering all bases.
The energy difference for each state and through the bases are similar for the two other ADC schemes.
We can notice a large variation of the vertical energies for the {\Aoneg} state between ADC(2)-s and ADC(2)-x with around \SIrange{0.52}{0.58}{\eV} through all bases.
The ADC(3) vertical energies are very similar to the ADC(2) ones for the {\Btwog} state with an energy difference of \SIrange{0.01}{0.02}{\eV} for all bases, whereas we have an energy difference of \SIrange{0.04}{0.11}{\eV} and \SIrange{0.17}{0.22}{\eV} for the {\Aoneg} and {\Btwog} states, respectively.
Now we look at Table \ref{tab:D4h} for vertical energies obtained with standard methods, we have first the various CC methods, for all the CC methods we were only able to reach the aug-cc-pVTZ basis.
%For the CCSD method we cannot obtain the vertical energy for the $1\,{}^1B_{2g}$ state, we can notice a small energy variation through the bases for the triplet state with around 0.06 eV. The energy variation is larger for the $2\,{}^1A_{1g}$ state with 0.20 eV.
Note that for CC we started from a restricted Hartree-Fock (RHF) reference and in that case the ground state is the {\Aoneg} state, then the {\sBoneg} state is the single deexcitation and the {\Btwog} state is the double excitation from our ground state.
For the CC3 method we do not have the vertical energies for the triplet state {\Atwog}.
Considering all bases for the {\Aoneg}and {\Btwog} states we have an energy difference of about \SI{0.15}{\eV} and \SI{0.12}{\eV}, respectively.
The CCSDT energies are close to the CC3 ones for the {\Aoneg} state with an energy difference of around \SIrange{0.03}{0.06}{\eV} considering all bases.
For the{\Btwog} state the energy difference between the CC3 and the CCSDT values is larger with \SIrange{0.18}{0.27}{\eV}.
We can make a similar observation between the CC4 and the CCSDTQ values, for the {\Aoneg} state we have an energy difference of about \SI{0.01}{\eV} and this time we have smaller energy difference for the {\Btwog} with \SI{0.01}{\eV}.
Then we discuss the multi-reference results and this time we were able to reach the aug-cc-pVQZ basis.
Again we consider two different active spaces, four electrons in four orbitals and twelve electrons in twelve orbitals.
If we compare the CASSCF(4,4) and CASPT2(4,4) values we can see large energy differences for the {\Aoneg} and {\Btwog} states, for the {\Aoneg} state we have an energy difference of about \SIrange{0.67}{0.74}{\eV} and \SIrange{1.65}{1.81}{\eV} for the {\Btwog} state.
The energy difference is smaller for the triplet state with \SIrange{0.27}{0.31}{\eV}, we can notice that all the CASSCF(4,4) vertical energies are higher than the CASPT2(4,4) ones.
Then we have the NEVPT2(4,4) methods (SC-NEVPT2(4,4) and PC-NEVPT2(4,4)) for which the vertical energies are quite similar, however we can notice that we have a different ordering of the {\Aoneg} and {\Btwog} states with {\Btwog} higher in energy than {\Aoneg} for the two NEVPT2(4,4) methods.
Then we have the results for the same methods but with a larger active space.
For the CASSCF(12,12) we have similar values for the triplet state with energy difference of about \SI{0.06}{\eV} for all bases but larger energy difference for the {\Aoneg} state with around \SIrange{0.28}{0.29}{\eV} and \SIrange{0.79}{0.81}{\eV} for the {\Btwog} state.
Note that from CASSCF(4,4) to CASSCF(12,12) every vertical energy is smaller.
We can make the contrary observation for the CASPT2 method where from CASPT2(4,4) to CASPT2(12,12) every vertical energy is larger.
For the CASPT2(12,12) method we have similar values for the triplet state and for the {\Aoneg} state, considering all bases, with an energy difference of around \SIrange{0.05}{0.06}{\eV} and \SIrange{0.02}{0.05}{\eV} respectively.
The energy difference is larger for the {\Btwog} state with about \SIrange{0.27}{0.29}{\eV}.
Next we have the NEVPT2 methods, the first observation that we can make is that by increasing the size of the active space we retrieve the right ordering of the {\Aoneg} and {\Btwog} states.
Again, every vertical energy is higher in the NEVPT2(12,12) case than for the NEVPT2(4,4) one.
Figure \ref{fig:D4h} shows the vertical energies of the studied excited states described in Tables \ref{tab:sf_D4h} and \ref{tab:D4h}.
We see that all methods exhibit larger variations of the vertical energies due to the high symmetry of the {\Dfour} structure.
For the triplet state most of the methods used are able to give a vertical energy close to the TBE one, we can although see that CASSCF, with the two different active spaces, show larger energy error than most of the DFT functionals used.
Then for the, strongly multi-configurational character, {\Aoneg} state we have a good description by the CC and multi-reference methods with the largest active space, except for CASSCF.
The SF-ADC(2) and SF-ADC(3) schemes are also able to provide a good description of the {\Aoneg} state and even for the {\Btwog} we see that SF-ADC(2)-x give a worst result than SF-ADC(2).
Multi-configurational methods with the smallest active space do not demonstrate a good description of the two singlet excited states especially for the CASSCF method and for the the NEVPT2 methods that give the vertical energy of the {\Btwog} state below the {\Aoneg} one.
Note that CASPT2 improve a lot the description of all the states compared to CASSCF.
The various TD-DFT functionals are not able to describe correctly the two singlet excited states.
In the present study, we have benchmarked a larger number of computational methods on the automerization barrier and the vertical energies of the cyclobutadiene (CBD) molecule in its square ({\Dfour}) and rectangular ({\Dtwo}) arrangements, for which we have defined theoretical best estimates (TBEs) based on extrapolated CCSDTQ/aug-cc-pVTZ data.
%Otherwise we got the CCSDTQ/aug-cc-pVTZ values by correcting the CCSDTQ/aug-cc-pVDZ values by the difference between CC4/aug-cc-pVTZ and CC4/aug-cc-pVDZ (Eq.~\eqref{eq:aug-cc-pVTZ}) and we obtain the CCSDTQ/aug-cc-pVDZ values by correcting the CCSDTQ/6-31G+(d) values by the difference between CC4/aug-cc-pVDZ and CC4/6-31G+(d) (Eq.~\eqref{eq:aug-cc-pVDZ}).
%When the CC4/aug-cc-pVTZ values were not obtained we corrected the CC4/aug-cc-pVDZ values by the difference between CCSDT/aug-cc-pVTZ and CCSDT/aug-cc-pVDZ to obtain them (Eq.~\eqref{eq:CC4_aug-cc-pVTZ}).
%If the CC4 values have not been obtained then we used the same scheme that we just described but by using the CCSDT values.
%If neither the CC4 and CCSDTQ values were not available then we used the NEVPT2(12,12)/aug-cc-pVTZ values.
\titou{Within the SF-TD-DFT framework, we advice the use of exchange-correlation (hybrids or range-separated hybrids) with a large fraction of short-range exact exchange.
This has been shown to be beneficial for the automerization barrier and the vertical excitation energies in the {\Dtwo} and {\Dfour} arrangements.}
\titou{At the SF-ADC level, we have found that the extended scheme SF-ADC(2)-x is not good, while SF-ADC(2)-s and SF-ADC(3) have opposite behavior which means that SF-ADC(2.5) is really good.}
In order to provide a benchmark of the automerization barrier and vertical energies we used coupled-cluster (CC) methods with doubles (CCSD), with triples (CCSDT and CC3) and with quadruples (CCSTQ and CC4).
Due to the presence of multi-configurational states we used multi-reference methods (CASSCF, CASPT2 and NEVPT2) with two active spaces ((4,4) and (12,12)).
We also used spin-flip (SF-) within two frameworks, in TD-DFT with various global and range-separated hybrids functionals, and in ADC with the ADC(2)-s, ADC(2)-x and ADC(3) schemes.
The CC methods provide good results for the AB and vertical energies, however in the case of multi-configurational states CC with only triples is not sufficient and we have to include the quadruples to correctly describe these states.
multi-configurational methods also provide very solid results for the largest active space with second order correction (CASPT2 and NEVPT2).
With SF-TD-DFT the quality of the results are, of course, dependent on the functional but for the doubly excited states we have solid results.
In SF-ADC we have very good results compared to the TBEs even for the doubly excited states, nevertheless the ADC(2)-x scheme give almost systematically worse results than the ADC(2)-s ones and using the ADC(3) scheme does not always provide better values.
The description of the excited states of the {\Dtwo} structure give rise to good agreement between the single reference and multi-configurational methods due to the large $\%T_1$ percentage of the first two excited states.
When this percentage is much smaller as in the case of the doubly excited state {\twoAg} the spin-flip methods show very good results within the ADC framework but even at the TD-DFT level spin-flip display good results compared to the TBE value.
As said in the discussion, for the {\Dfour} geometry, the description of excited states is harder because of the strong multi-configurational character where SF-TD-DFT can present more than 1 eV of error compared to the TBE.
However, SF-ADC can show error of around \SIrange{0.1}{0.2}{\eV} which can be better than the multi-configurational methods results.
EM, AS, and PFL acknowledge funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No.~863481).
This work was performed using HPC ressources from CALMIP (Toulouse) under allocation 2022-18005.}
