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 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. A theoretical best estimate is defined for the automerization barrier and for each vertical transition energy.

I. INTRODUCTION

Despite the fact that excited states are involved in ubiquitous processes such as photochemistry, [1][2][3][4][START_REF] Olivucci | Computational Photochemistry[END_REF][START_REF] Robb | A Computational Strategy for Organic Photochemistry[END_REF][START_REF] Van Der Lugt | [END_REF] catalysis, 8 and solar cells, 9 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. [START_REF] Roos | Advances in Chemical Physics[END_REF][START_REF] Piecuch | [END_REF][12][13][14][15][16][17][18][19][20][21][22][23] Speaking of difficult tasks, the cyclobutadiene (CBD) molecule has been a real challenge for both experimental and theoretical chemistry for many decades. 24 Due to its antiaromaticity [START_REF] Minkin | Aromaticity and Antiaromaticity: Electronic and Structural Aspects[END_REF] and large angular strain, [START_REF] Baeyer | [END_REF] CBD presents a high reactivity making its synthesis a particularly difficult exercise. In the D 4h symmetry, the simple Hückel 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 ab initio methods correctly predict an open-shell singlet ground state. This degeneracy is lifted by the so-called pseudo Jahn-Teller effect, i.e., by a descent in symmetry (from D 4h to D 2h 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 27 and others. [28][29][30] In the D 2h symmetry, the 1 1 A g ground state has a weak multiconfigurational character with well-separated frontier orbitals that can be described by single-reference methods. However, in the D 4h symmetry, the 1 1 B 1g 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. a) Electronic mail: emonino@irsamc.ups-tlse.fr b) Electronic mail: loos@irsamc.ups-tlse.fr Of course, single-reference methods are naturally unable to describe such situations. Interestingly, the 1 1 B 1g ground state of the square arrangement is a transition state in the automerization reaction between the two rectangular structures (see Fig. 1), while the lowest triplet state, 1 3 A 2g , is a minimum on the triplet potential energy surface in the D 4h 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 1.6-10 kcal mol -1 , 31 while previous state-of-the-art ab initio calculations yield values in the 7-9 kcal mol -1 range. [32][33][34][35] The lowest-energy excited states of CBD in both symmetries are represented in Fig. 1, where we have reported the 1 1 A g and 1 3 B 1g states for the rectangular geometry and the 1 1 B 1g and 1 3 A 2g states for the square one. Due to the energy scale, the higher-energy states (1 1 B 1g and 2 1 A g for D 2h and 1 1 A 1g and 1 1 B 2g for D 4h ) are not shown. Interestingly, the 2 1 A g and 1 1 A 1g states have a strong contribution from doubly-excited configurations and these so-called double excitations 36 are known to be inaccessible with adiabatic time-dependent density-functional theory (TD-DFT) [37][START_REF] Casida | Recent Advances in Density Functional Methods[END_REF][39][40][41][42][43][START_REF] Maitra | Fundamentals of Time-Dependent Density Functional Theory[END_REF][START_REF] Maitra | [END_REF] and remain torturous for state-of-the-art methods like equation-of-motion third-order coupled-cluster (EOM-CC3) 46,47 or even coupled-cluster with singles, doubles, and triples (EOM-CCSDT). [48][49][50][51] 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 multiconfigurational methods, which are naturally designed to address such scenarios. Among these methods, one can mention the complete-active-space self-consistent field (CASSCF) method, [START_REF] Roos | Advances in Chemical Physics[END_REF] its second-order perturbatively-corrected variant (CASPT2) [52][53][START_REF] Roos | Quantum Mechanical Electronic Structure Calculations with Chemical Accuracy[END_REF] and the second-order n-electron valence state perturbation theory (NEVPT2) formalism. [START_REF] Angeli | [END_REF][56][57] Another way to deal with double excitations and multireference situations is to use high level truncation of the EOM formalism 58,59 of CC theory. [48][49][50][51]60 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. 36,[61][62][63] Although multi-reference CC methods have been designed, [64][65][66][67][68] they are computationally demanding and remain far from being black-box.

In this context, an interesting alternative to multiconfigurational and CC methods is provided by selected configuration interaction (SCI) methods, [69][70][71][72][73][74][START_REF] Caffarel | Recent Progress in Quantum Monte Carlo[END_REF][START_REF] Holmes | [END_REF][77][78][79][80][81][82] which are able to provide near full CI (FCI) ground-and excited-state energies of small molecules. 36,62,63,[83][84][85][86][87][88][89][90][91][92][93][94][95][96][97][98][99][100] For example, the 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. 71,72,74,82,94,101 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, [102][103][104][105] 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 (i.e., 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. 105 One can address part of this issue by increasing the excitation order or by complementing the spin-incomplete configuration set with the missing configurations. [106][107][108][109][110][111][112][113] 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 D 2h and D 4h ground-state geometries. Computational details are reported in Sec. II. Section III is devoted to the discussion of our results. Finally, our conclusions are drawn in Sec. IV.

II. COMPUTATIONAL DETAILS

A. Selected configuration interaction calculations

For the SCI calculations, we rely on the CIPSI algorithm implemented in QUANTUM PACKAGE, 94 which iteratively select determinants in the FCI space. To treat electronic states on an 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. 94 and 114. 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 var and a second-order perturbative correction E 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, i.e., E FCI ≈ E var + E PT2 . It is possible to estimate more precisely the FCI energy via an extrapolation procedure,

S T ≈ ≈ 1 1 A g 1 3 B 1g 1 1 A g 1 3 B 1g 1 1 B 1g 1 3 A 2g D 2h D 2h D 4h D 4h AB FIG.
1. 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 colored in black and red, respectively. The automerization barrier (AB) is also represented.

where the variational energy is extrapolated to E PT2 = 0. 86 Excitation energies are then computed as differences of extrapolated total energies. 36,62,63,91,92 Additionally, an error bar can be provided thanks to a recent method based on Gaussian random variables that is described in Ref. 98. This type of extrapolation procedures is now routine in SCI and similar techniques. 93,95,115

B. Coupled-cluster calculations

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: [START_REF] Piecuch | [END_REF][116][117][118][119][120] CC with singles and doubles (CCSD), 116,121 CC with singles, doubles, and triples (CCSDT), 122,123 CC with singles, doubles, triples, and quadruples (CCSDTQ), 124-126 etc. As mentioned above, CC theory can be extended to excited states via the EOM formalism, 58,59 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: [48][49][50][51]59,60,122,[127][128][129][130] EOM-CCSD, EOM-CCSDT, EOM-CCSDTQ, etc. In the fol-lowing, we will omit the prefix EOM for the sake of conciseness. Alternatively to the "complete" CC models, one can also employ the CC2, 46,131 CC3, 46,132 and CC4 99,133,134 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, 134 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 135 and at the CCSDT level with MRCC. 136 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 D 4h arrangement, we have considered the lowest closed-shell singlet state 1 1 A 1g as reference. Hence, the open-shell ground state, 1 1 B 1g , and the 1 1 B 2g state appear as a de-excitation and an excitation, respectively. With respect to 1 1 A 1g , 1 1 B 1g has a dominant double excitation character, while 1 1 B 2g has a dominant single excitation character, hence their contrasting convergence behaviors with respect to the order of the CC expansion (see below).

C. Multi-configurational calculations

State-averaged CASSCF (SA-CASSCF) calculations are performed for vertical transition energies, whereas state-specific CASSCF is used for computing the automerization barrier. 137 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 the valence π orbitals, and (ii) an extended (12e,12o) active space where we have additionally included the σ CC and σ * CC orbitals. For ionic excited states, like the 1 1 B 1g state of CBD, it is particularly important to take into account the σ-π coupling. [138][139][140] 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. [START_REF] Angeli | [END_REF][56][57] Note that PC-NEVPT2 is theoretically more accurate than SC-NEVPT2 due to the larger number of external configurations and greater flexibility. In order to avoid the intruder state problem in CASPT2, a real-valued level shift of 0.3 E h is set, [START_REF] Roos | Advances in Chemical Physics[END_REF]141 with an additional ionization-potential-electron-affinity (IPEA) shift of 0.25 E h to avoid systematic underestimation of the vertical excitation energies. [142][143][144][145] All these calculations are carried out with MOLPRO. 137

D. Spin-flip calculations

Within the spin-flip formalism, one considers the lowest triplet state as reference instead of the singlet ground state. Ground-state energies are then computed as sums of the triplet reference state energy and the corresponding de-excitation energy. Likewise, excitation energies with respect to the singlet ground state are computed as differences of excitation energies with respect to the reference triplet state.

Nowadays, spin-flip techniques are broadly accessible thanks to intensive developments in the electronic structure community (see Ref. 105 and references therein). Here, we explore the spin-flip version 146 of the algebraic-diagrammatic construction 147 (ADC) using the standard and extended secondorder ADC schemes, SF-ADC(2)-s 18,148 and SF-ADC(2)-x, 18 as well as its third-order version, SF-ADC(3). 18,149,150 These calculations are performed using Q-CHEM 5.2.1. 151 The spinflip version of our recently proposed composite approach, namely SF-ADC(2.5), 152 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), 153 with the same Q-CHEM 5.2.1 code. 151 The B3LYP, 154-156 PBE0 157,158 and BH&HLYP global hybrid GGA functionals are considered, which contain 20%, 25%, 50% of exact exchange, respectively. These calculations are labeled as 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 rangeseparated hybrid (RSH) functionals: CAM-B3LYP (19% of short-range exact exchange and 65% at long range), 159 LC-ωPBE08 (0% of short-range exact exchange and 100% at long range), 160 and ωB97X-V (16.7% of short-range exact exchange and 100% at long range). 161 Finally, the hybrid meta-GGA functional M06-2X (54% of exact exchange) 162 and the RSH meta-GGA functional M11 (42.8% of short-range exact exchange and 100% at long range) 163 are also employed. Note that all SF-TD-DFT calculations are done within the Tamm-Dancoff approximation. 164 Although there also exist spin-flip extension of EOM-CC methods, 102,165-168 they are not considered here.

E. Theoretical best estimates

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. 169 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 augcc-pVTZ basis. 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:

∆ ẼCCSDTQ aug-cc-pVTZ = ∆E CCSDTQ aug-cc-pVDZ + ∆E CC4 aug-cc-pVTZ -∆E CC4 aug-cc-pVDZ , (1)
while, when only CCSDTQ/6-31G+(d) values are available, we further extrapolate the CCSDTQ/aug-cc-pVDZ value as follows:

∆ ẼCCSDTQ aug-cc-pVDZ = ∆E CCSDTQ 6-31G+(d) + ∆E CC4 aug-cc-pVDZ -∆E CC4 6-31G+(d) .
(2) If we lack the CC4 data, we can follow the same philosophy and rely on CCSDT (for single excitations) or NEVPT2 (for double excitations). For example,

∆ ẼCC4 aug-cc-pVTZ = ∆E CC4 aug-cc-pVDZ + ∆E CCSDT aug-cc-pVTZ -∆E CCSDT aug-cc-pVDZ , (3)
and so on. If neither CC4, nor CCSDT are feasible, then we rely on NEVPT2 (12,12). The procedures applied for each extrapolated value are explicitly mentioned as footnote in the tables. Note that, due to error bar inherently linked to the CIPSI calculations (see Sec. II A), these are mostly used as an additional safety net to further check the convergence of the CCSDTQ estimates.

Additional tables gathering these TBEs as well as literature data for the automerization barrier and the vertical excitation energies can be found in the supporting information.

III. RESULTS AND DISCUSSION

A. Geometries

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. Therefore, we rely on CASPT2(12,12)/aug-cc-pVTZ for both the D 2h and D 4h ground-state structures. (Note that these optimizations are done without IPEA shift but with a level shift and a state-specific reference CASSCF wave function.) 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 1 1 A g state in the rectangular (D 2h ) arrangement and the restricted open-shell (RO) version of CCSD(T)/aug-cc-pVTZ to obtain the equilibrium geometry of the 1 3 A 2g state in the square (D 4h ) arrangement. These two geometries are the lowest-energy equilibrium structure of their respective spin manifold (see Fig. 1). The cartesian coordinates of these geometries are provided in the supporting information. Table I 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 0.01 Å only.

B. Automerization barrier

The results concerning the automerization barrier are reported in Table II for various basis sets and shown in Fig. 2 for the aug-cc-pVTZ basis. Our TBE with this basis set is 8.93 kcal mol -1 , which is in excellent agreement with previous studies [32][33][34][35] (see supporting information).

First, one can see large variations of the energy barrier at the SF-TD-DFT level, with differences as large as 10 kcal mol -1 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 approximately 50% of shortrange exact exchange (e.g., BH&HLYP, M06-2X, and M11) perform significantly better than the functionals having a small fraction of short-range exact exchange (e.g., B3LYP, PBE0, CAM-B3LYP, ωB97X-V, and LC-ωPBE08). However, they are still off by 1-4 kcal mol -1 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. 170 With the augmented double-ζ basis, the SF-TD-DFT results are basically converged to sub-kcal mol -1 accuracy, which is a drastic improvement compared to wave function approaches where this type of convergence is reached with the augmented triple-ζ basis only.

For the SF-ADC family of methods, the energy differences are much smaller with a maximum deviation of 2 kcal mol -1 between different versions. In particular, we observe that SF-ADC(2)-s and SF-ADC(3), which respectively scale as O(N 5 ) and O(N 6 ) (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 a Value obtained using CCSDT/aug-cc-pVTZ corrected by the difference between CC3/aug-cc-pVQZ and CC3/aug-cc-pVTZ. b Value obtained using CC4/aug-cc-pVDZ corrected by the difference between CCSDT/aug-cc-pVTZ and CCSDT/aug-cc-pVDZ. c Value obtained using CC4/aug-cc-pVTZ corrected by the difference between CCSDT/aug-cc-pVQZ and CCSDT/aug-cc-pVTZ. d Value obtained using CCSDTQ/6-31+G(d) corrected by the difference between CC4/aug-cc-pVDZ basis and CC4/6-31+G(d). e TBE value obtained using CCSDTQ/aug-cc-pVDZ corrected by the difference between CC4/aug-cc-pVTZ and CC4/aug-cc-pVDZ. f Value obtained using CCSDTQ/aug-cc-pVTZ corrected by the difference between CC4/aug-cc-pVQZ and CC4/aug-cc-pVTZ.

0.18 kcal mol -1 compared to the TBE/aug-cc-pVTZ basis reference value. Nonetheless, at a O(N 5 ) computational scaling, SF-ADC(2)-s is particularly accurate, even compared to highorder CC methods (see below). We note that SF-ADC(2)-x [which scales as O(N 6 )] 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). This behavior was previously reported by Dreuw's group. 18,150,171 Overall, even with the best exchange-correlation functional, SF-TD-DFT is clearly outperformed by the more expensive SF-ADC models.

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 1.5 kcal mol -1 for all bases. In this case, the NEVPT2 values are fairly accurate with differences below half a kcal mol -1 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 3 kcal mol -1 (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 minimal active space, with an energy difference of around 0.1-0.2 kcal mol -1 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 kcal mol -1 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: O(N 6 ), O(N 7 ), O(N 8 ), O(N 9 ), and O(N 10 ), respectively. 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.

C. Vertical excitation energies

D 2h rectangular geometry

Table III reports, at the D 2h rectangular equilibrium geometry of the 1 1 A g ground state, the vertical transition energies associated with the 1 3 B 1g , 1 1 B 1g , and 2 1 A g states obtained using the spin-flip formalism, while Table IV 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 a Value obtained using CC4/aug-cc-pVDZ corrected by the difference between CCSDT/aug-cc-pVTZ and CCSDT/aug-cc-pVDZ. b Value obtained using CC4/aug-cc-pVDZ corrected by the difference between PC-NEVPT2(12,12)/aug-cc-pVTZ and PC-NEVPT2(12,12)/aug-cc-pVDZ. c Value obtained using CCSDTQ/6-31+G(d) corrected by the difference between CC4/aug-cc-pVDZ and CC4/6-31+G(d). d TBE value obtained using CCSDTQ/aug-cc-pVDZ corrected by the difference between CC4/aug-cc-pVTZ and CC4/aug-cc-pVDZ.

1 3 B 1 g 1 1 B 1 g 2 1 A g 1 2 3 4 5 FIG.
3. Vertical excitation energies of the 1 3 B 1g , 1 1 B 1g , and 2 1 A g states at the D 2h rectangular equilibrium geometry of the 1 1 A g ground state using the aug-cc-pVTZ basis. See supporting information for the raw data.

theory is illustrated in Fig. 3. At the CC3/aug-cc-pVTZ level, the percentage of single excitation involved in the 1 3 B 1g , 1 1 B 1g , and 2 1 A g are 99%, 95%, and 1%, respectively. Therefore, the two formers are dominated by single excitations, while the latter state corresponds to a genuine double excitation.

First, let us discuss basis set effects at the SF-TD-DFT level (Table III). As expected, these are found to be small and the results are basically converged to the complete basis set limit with the triple-ζ basis, which is definitely not the case for the wave function methods. 172 Regarding now the accuracy of the vertical excitation energies, again, we see that, for 1 3 B 1g and 1 1 B 1g , the functionals with the largest amount of short-range exact exchange (e.g., BH&HLYP, M06-2X, and M11) are the most accurate. Functionals with a large share 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. 153 However, their overall accuracy remains average especially for the singlet states, 1 1 B 1g and 2 1 A g , with error of the order of 0.2-0.5 eV compared to the TBEs. The triplet state, 1 3 B 1g , is much better described with errors below 0.1 eV. Surprisingly, for the doubly-excited state, 2 1 A g , the hybrid functionals with a low percentage of exact exchange (B3LYP and PBE0) are the best performers with absolute errors below 0.05 eV. Note that, as evidenced by the data reported in supporting information, none of these states exhibit a strong spin contamination.

Second, we discuss the various SF-ADC schemes (Table III), i.e., 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 induces a small decrease in vertical excitation energies of 0.03 eV (0.06 eV) for the 1 3 B 1g (2 1 A g ) state, while the transition energy of the 1 1 B 1g state drops more significantly by about 0.2 eV. [The SF-ADC(2)x and SF-ADC(3) calculations with aug-cc-pVQZ were not feasible with our computational resources.] These basis set effects are fairly transferable to the other wave function methods that we have considered here. This further motivates the "pyramidal" extrapolation scheme that we have employed to produce the TBE values (see Sec. II E). Again, the extended version, SF-ADC(2)-x, does not seem to be relevant in the present context with much larger errors than the other schemes. Also, as reported previously, 152 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 2 1 A g where the error with respect to the TBE (0.140 eV) is larger than the SF-ADC(2)-s error (0.093 eV).

Let us now move to the discussion of the results obtained with standard wave function methods that are reported in Table IV. Regarding the multi-configurational calculations, the most striking result is the poor description of the 1 1 B 1g ionic state, especially with the (4e,4o) active space where CASSCF predicts this state higher in energy than the 2 1 A g 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 eV. This feature is characteristic of the inadequacy of the active space to model such a state. For the two other states, 1 3 B 1g and 2 1 A g , the errors at the CASPT2(4,4) and NEVPT2(4,4) levels are much smaller (below 0.1 eV). Using a larger active space re-solves 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 and doubly-excited states) although it is difficult to reach chemical accuracy (i.e., an error below 0.043 eV) on a systematic basis.

Finally, for the CC models (Table IV), the two states with a large %T 1 value, 1 3 B 1g and 1 1 B 1g , 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. 98 For the doubly-excited state, 2 1 A g , 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, as observed in an earlier work. 99 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.

D 4h square-planar geometry

In Table V, we report, at the D 4h square planar equilibrium geometry of the 1 3 A 2g state, the vertical transition energies associated with the 1 3 A 2g , 1 1 A 1g , and 1 1 B 2g states obtained using the spin-flip formalism, while Table VI 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. 4 for the aug-cc-pVTZ basis. Unfortunately, due to technical limitations, we could not compute %T 1 values associated with the 1 3 A 2g , 1 1 A 1g , and 1 1 B 2g excited states in the D 4h symmetry. However, it is clear from the inspection of the wave function that, with respect to the 1 1 B 1g ground state, 1 3 A 2g and 1 1 B 2g are dominated by single excitations, while 1 1 A 1g has a strong double excitation character.

As for the previous geometry we start by discussing the SF-TD-DFT results (Table V), and in particular the singlet-triplet gap, i.e., the energy difference between 1 1 B 1g and 1 3 A 2g . For all functionals, this gap is small (basically below 0.1 eV while the TBE value is 0.144 eV) but it is worth mentioning that B3LYP and PBE0 incorrectly deliver a negative singlet-triplet gap (hence a triplet ground state at this geometry). 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, 1 1 A 1g and 1 1 B 2g , 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 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 on the D 2h ground-state equilibrium structure and the automerization barrier, the functionals with a large fraction of short-range exact exchange yield more accurate results. Yet, the transition energy to 1 1 B 2g is off by half an eV compared to the TBE for BH&HLYP and M11, while the doubly-excited state is much closer to the reference value (errors of -0.251 and -0.312 eV for BH&HLYP and M11, respectively). With errors of -0.066, -0.097, and -0.247 eV for 1 3 A 2g , 1 1 A 1g , and 1 1 B 2g , M06-2X is the best performer here. Again, for all the excited states, the basis set effects are extremely small at the SF-TD-DFT level. We emphasize that the S 2 values reported in supporting information indicate again that there is no significant spin contamination in these excited states. Next, we discuss the various ADC schemes (Table V). Globally, we observe similar trends as those noted in Sec. III C 1. 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 (1 1 A 1g and 1 1 B 2g ), especially for the doubly-excited state 1 1 A 1g where it underestimates the vertical excitation energy by 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 1 1 A 1g . 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.

1 3 A 2 g 1 1 A 1 g
Let us turn to the multi-reference results (Table VI). 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. This is, of course, magnified with the (4e,4o) active space for which the second-order perturbative treatment is unable to provide a satisfying description due to the limited active space. In particular SC-NEVPT2(4,4)/aug-cc-pVTZ and PC-NEVPT2(4,4)/augcc-pVTZ underestimate the singlet-triplet gap by 0.072 and 0.097 eV and, more importantly, flip the ordering of 1 1 A 1g and 1 1 B 2g . Although 1 1 A 1g is not badly described, the excitation energy of the ionic state 1 1 B 2g is off by almost 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, e.g., an error up to -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 VI. As mentioned in Sec. II B, we remind the reader that these calculations are performed by considering the 1 1 A 1g state as reference, and that, therefore, 1 1 B 1g and 1 1 B 2g are obtained as a de-excitation and an excitation, respectively. Consequently, with respect to 1 1 A 1g , 1 1 B 1g has a dominant double excitation character, while 1 1 B 2g have a dominant single excitation character. This explains why one observes a slower convergence of the transition energies in the case of 1 1 B 1g as shown in Fig. 4. It is clear from the results of Table VI that, if one wants to reach high accuracy with such a computational strategy, it is mandatory to include quadruple excitations. Indeed, at the CCSDT/aug-cc-pVTZ level, the singlet-triplet gap is already very accurate (off by 0.005 eV only) while the excitation energies of the singlet states are still 0.131 and 0.688 eV away from their respective TBE. These deviations drop to 0.011 and -0.013 eV at the CC4/aug-cc-pVTZ level. As a final comment, we can note that the CCSDTQ-based TBEs and the CIPSI results are consistent if one takes into account the extrapolation error (see Sec. II A).

IV. CONCLUSIONS

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 (D 4h ) and rectangular (D 2h ) geometries, for which we have defined theoretical best estimates based on extrapolated CCSDTQ/augcc-pVTZ data.

The main take-home messages of the present work can be summarized as follows:

• Within the SF-TD-DFT framework, we advice to use a Value obtained using CC4/aug-cc-pVDZ corrected by the difference between CCSDT/aug-cc-pVTZ and CCSDT/aug-cc-pVDZ. b Value obtained using CC4/aug-cc-pVDZ corrected by the difference between PC-NEVPT2(12,12)/aug-cc-pVTZ and PC-NEVPT2(12,12)/aug-cc-pVDZ. c Value obtained using CCSDTQ/6-31+G(d) corrected by the difference between CCSDT/aug-cc-pVDZ and CCSDT/6-31+G(d). d Value obtained using CCSDTQ/6-31+G(d) corrected by the difference between CC4/aug-cc-pVDZ and CC4/6-31+G(d). e TBE value obtained using CCSDTQ/aug-cc-pVDZ corrected by the difference between CCSDT/aug-cc-pVTZ and CCSDT/aug-cc-pVDZ. f TBE value obtained using CCSDTQ/aug-cc-pVDZ corrected by the difference between CC4/aug-cc-pVTZ and CC4/aug-cc-pVDZ.

exchange-correlation (hybrids or range-separated hybrids) with a large fraction of short-range exact exchange. This has been shown to be clearly beneficial for the automerization barrier and the vertical excitation energies computed on both the D 2h and D 4h equilibrium geometries.

• At the SF-ADC level, we have found that, as expected, the extended scheme, SF-ADC(2)-x, systematically worsen the results compared to the cheaper standard version, SF-ADC(2)-s. Moreover, as previously reported, SF-ADC(2)-s and SF-ADC(3) have opposite error patterns which means that SF-ADC(2.5) emerges as an excellent compromise.

• For the D 4h square planar structure, a faithful energetic description of the excited states is harder to reach at the SF-TD-DFT level because of the strong multiconfigurational character. In such scenario, the SF-TD-DFT excitation energies can exhibit errors of the order of 1 eV compared to the TBEs. However, it was satisfying to see that the spin-flip version of ADC can lower these errors to 0.1-0.2 eV.

• 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.

• 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, CCS-DTQ and CIPSI yield equivalent excitation energies, hence confirming the outstanding accuracy of CCSDTQ in the context of molecular excited states.

10 FIG. 2 .

 102 FIG.2. Error (with respect to the TBE) in the automerization barrier (in kcal mol -1 ) of CBD at various levels of theory using the aug-cc-pVTZ basis. See supporting information for the total energies.

5 FIG. 4 .

 54 FIG.4. Vertical excitation energies (in eV) of the 3 A 2g , 1 1 A 1g , and 1 1 B 2g states at the D 4h square-planar equilibrium geometry of the 1 3 A 2g state using the aug-cc-pVTZ basis. See supporting information for the raw data.

TABLE I .

 I Optimized geometries associated with several states of CBD computed with various levels of theory. Bond lengths are in Å and angles (∠) are in degree.

	State	Method	C --C	C -C C -H ∠ H-C--C
	D 2h (1 1 A g )	CASPT2(12,12)/aug-cc-pVTZ a 1.354 1.566 1.077 CC3/aug-cc-pVTZ a 1.344 1.565 1.076	134.99 135.08
		CCSD(T)/cc-pVTZ b	1.343 1.566 1.074	135.09
	D 4h (1 1 B 1g ) CASPT2(12,12)/aug-cc-pVTZ a 1.449 1.449 1.076	135.00
	D 4h (1 3 A 2g ) CASPT2(12,12)/aug-cc-pVTZ a 1.445 1.445 1.076 RO-CCSD(T)/aug-cc-pVTZ a 1.439 1.439 1.075	135.00 135.00
		RO-CCSD(T)/cc-pVTZ b	1.439 1.439 1.073	135.00

a This work. b From Ref. 166.

TABLE II .

 II Automerization barrier (in kcal mol -1 ) of CBD computed with various computational methods and basis sets. The values in square parenthesis have been obtained by extrapolation via the procedure described in the corresponding footnote. The TBE/aug-cc-pVTZ value is highlighted in bold.

			Basis sets
	Method	6-31+G(d) aug-cc-pVDZ aug-cc-pVTZ aug-cc-pVQZ
	SF-TD-B3LYP	18.59	18.64	19.34	19.34
	SF-TD-PBE0	17.18	17.19	17.88	17.88
	SF-TD-BH&HLYP	11.90	12.02	12.72	12.73
	SF-TD-M06-2X	9.32	9.62	10.35	10.37
	SF-TD-CAM-B3LYP	18.05	18.10	18.83	18.83
	SF-TD-ωB97X-V	18.26	18.24	18.94	18.92
	SF-TD-LC-ωPBE08	19.05	18.98	19.74	19.71
	SF-TD-M11	11.03	10.25	11.22	11.12
	SF-ADC(2)-s	6.69	6.98	8.63
	SF-ADC(2)-x	8.63	8.96	10.37
	SF-ADC(2.5)	7.36	7.76	9.11
	SF-ADC(3)	8.03	8.54	9.58
	CASSCF(4,4)	6.17	6.59	7.38	7.41
	CASPT2(4,4)	6.56	6.87	7.77	7.93
	SC-NEVPT2(4,4)	7.95	8.31	9.23	9.42
	PC-NEVPT2(4,4)	7.95	8.33	9.24	9.41
	CASSCF(12,12)	10.19	10.75	11.59	11.62
	CASPT2(12,12)	7.24	7.53	8.51	8.71
	SC-NEVPT2(12,12)	7.10	7.32	8.29	8.51
	PC-NEVPT2(12,12)	7.12	7.33	8.28	8.49
	CCSD	8.31	8.80	9.88	10.10
	CC3	6.59	6.89	7.88	8.06
	CCSDT	7.26	7.64	8.68	[8.86] a
	CC4	7.40	7.78	[8.82] b	[9.00] c
	CCSDTQ	7.51	[7.89] d	[8.93] e	[9.11] f

TABLE III .

 III Spin-flip TD-DFT and ADC vertical excitation energies (with respect to the singlet 1 1 A g ground state) of the 1 3 B 1g , 1 1 B 1g , and 2 1 A g states of CBD at the D 2h rectangular equilibrium geometry of the 1 1 A g ground state.

			Excitation energies (eV)
	Method	Basis	1 3 B 1g	1 1 B 1g	2 1 A g
	SF-TD-B3LYP	6-31+G(d)	1.706	2.211	3.993
		aug-cc-pVDZ	1.706	2.204	3.992
		aug-cc-pVTZ	1.703	2.199	3.988
		aug-cc-pVQZ	1.703	2.199	3.989
	SF-TD-PBE0	6-31+G(d)	1.687	2.314	4.089
		aug-cc-pVDZ	1.684	2.301	4.085
		aug-cc-pVTZ	1.682	2.296	4.081
		aug-cc-pVQZ	1.682	2.296	4.079
	SF-TD-BH&HLYP	6-31+G(d)	1.552	2.779	4.428
		aug-cc-pVDZ	1.546	2.744	4.422
		aug-cc-pVTZ	1.540	2.732	4.492
		aug-cc-pVQZ	1.540	2.732	4.415
	SF-TD-M06-2X	6-31+G(d)	1.477	2.835	4.378
		aug-cc-pVDZ	1.467	2.785	4.360
		aug-cc-pVTZ	1.462	2.771	4.357
		aug-cc-pVQZ	1.458	2.771	4.352
	SF-TD-CAM-B3LYP	6-31+G(d)	1.750	2.337	4.140
		aug-cc-pVDZ	1.745	2.323	4.140
		aug-cc-pVTZ	1.742	2.318	4.138
		aug-cc-pVQZ	1.743	2.319	4.138
	SF-TD-ωB97X-V	6-31+G(d)	1.810	2.377	4.220
		aug-cc-pVDZ	1.800	2.356	4.217
		aug-cc-pVTZ	1.797	2.351	4.213
		aug-cc-pVQZ	1.797	2.351	4.213
	SF-TD-LC-ωPBE08	6-31+G(d)	1.917	2.445	4.353
		aug-cc-pVDZ	1.897	2.415	4.346
		aug-cc-pVTZ	1.897	2.415	4.348
		aug-cc-pVQZ	1.897	2.415	4.348
	SF-TD-M11	6-31+G(d)	1.566	2.687	4.292
		aug-cc-pVDZ	1.546	2.640	4.267
		aug-cc-pVTZ	1.559	2.651	4.300
		aug-cc-pVQZ	1.557	2.650	4.299
	SF-ADC(2)-s	6-31+G(d)	1.577	3.303	4.196
		aug-cc-pVDZ	1.513	3.116	4.114
		aug-cc-pVTZ	1.531	3.099	4.131
		aug-cc-pVQZ	1.544	3.101	4.140
	SF-ADC(2)-x	6-31+G(d)	1.557	3.232	3.728
		aug-cc-pVDZ	1.524	3.039	3.681
		aug-cc-pVTZ	1.539	3.031	3.703
	SF-ADC(2.5)	6-31+G(d)	1.496	3.328	4.219
		aug-cc-pVDZ	1.468	3.148	4.161
		aug-cc-pVTZ	1.475	3.131	4.178
	SF-ADC(3)	6-31+G(d)	1.435	3.352	4.242
		aug-cc-pVDZ	1.422	3.180	4.208
		aug-cc-pVTZ	1.419	3.162	4.224

TABLE IV .

 IV Vertical excitation energies (with respect to the 1 1 A g ground state) of the 1 3 B 1g , 1 1 B 1g , and 2 1 A g states of CBD at the D 2h rectangular equilibrium geometry of the 1 1 A g ground state. The values in square parenthesis have been obtained by extrapolation via the procedure described in the corresponding footnote. The TBE/augcc-pVTZ values are highlighted in bold.

			Excitation energies (eV)
	Method	Basis	1 3 B 1g	1 1 B 1g	2 1 A g
	CASSCF(4,4)	6-31+G(d)	1.662	4.657	4.439
		aug-cc-pVDZ	1.672	4.563	4.448
		aug-cc-pVTZ	1.670	4.546	4.441
		aug-cc-pVQZ	1.671	4.549	4.440
	CASPT2(4,4)	6-31+G(d)	1.440	3.162	4.115
		aug-cc-pVDZ	1.414	2.971	4.068
		aug-cc-pVTZ	1.412	2.923	4.072
		aug-cc-pVQZ	1.417	2.911	4.081
	SC-NEVPT2(4,4)	6-31+G(d)	1.407	2.707	4.145
		aug-cc-pVDZ	1.381	2.479	4.109
		aug-cc-pVTZ	1.379	2.422	4.108
		aug-cc-pVQZ	1.384	2.408	4.116
	PC-NEVPT2(4,4)	6-31+G(d)	1.409	2.652	4.120
		aug-cc-pVDZ	1.384	2.424	4.084
		aug-cc-pVTZ	1.382	2.368	4.083
		aug-cc-pVQZ	1.387	2.353	4.091
	CASSCF(12,12)	6-31+G(d)	1.675	3.924	4.220
		aug-cc-pVDZ	1.685	3.856	4.221
		aug-cc-pVTZ	1.686	3.844	4.217
		aug-cc-pVQZ	1.687	3.846	4.216
	CASPT2(12,12)	6-31+G(d)	1.508	3.407	4.099
		aug-cc-pVDZ	1.489	3.256	4.044
		aug-cc-pVTZ	1.480	3.183	4.043
		aug-cc-pVQZ	1.482	3.163	4.047
	SC-NEVPT2(12,12) 6-31+G(d)	1.522	3.409	4.130
		aug-cc-pVDZ	1.511	3.266	4.093
		aug-cc-pVTZ	1.501	3.188	4.086
		aug-cc-pVQZ	1.503	3.167	4.088
	PC-NEVPT2(12,12) 6-31+G(d)	1.487	3.296	4.103
		aug-cc-pVDZ	1.472	3.141	4.064
		aug-cc-pVTZ	1.462	3.063	4.056
		aug-cc-pVQZ	1.464	3.043	4.059
	CCSD	6-31+G(d)	1.346	3.422
		aug-cc-pVDZ	1.319	3.226
		aug-cc-pVTZ	1.317	3.192
		aug-cc-pVQZ	1.323	3.187
	CC3	6-31+G(d)	1.420	3.341	4.658
		aug-cc-pVDZ	1.396	3.158	4.711
		aug-cc-pVTZ	1.402	3.119	4.777
		aug-cc-pVQZ	1.409	3.113	4.774
	CCSDT	6-31+G(d)	1.442	3.357	4.311
		aug-cc-pVDZ	1.411	3.175	4.327
		aug-cc-pVTZ	1.411	3.139	4.429
	CC4	6-31+G(d)		3.343	4.067
		aug-cc-pVDZ		3.164	4.040
		aug-cc-pVTZ		[3.128] a	[4.032] b
	CCSDTQ	6-31+G(d)		3.340	4.073
		aug-cc-pVDZ		[3.161] c	[4.046] c
		aug-cc-pVTZ		[3.125] d	[4.038] d
	CIPSI	6-31+G(d)	1.486 ± 0.005 3.348 ± 0.024 4.084 ± 0.012
		aug-cc-pVDZ 1.458 ± 0.009 3.187 ± 0.035	4.04 ± 0.04
		aug-cc-pVTZ 1.461 ± 0.030 3.142 ± 0.035	4.03 ± 0.09

TABLE V .

 V Spin-flip TD-DFT and ADC vertical excitation energies (with respect to the singlet 1 1 B 1g ground state) of the 1 3 A 2g , 1 1 A 1g , and 1 1 B 2g states of CBD at the D 4h square-planar equilibrium geometry of the 1 3 A 2g state.

			Excitation energies (eV)
	Method	Basis	1 3 A 2g	1 1 A 1g	1 1 B 2g
	SF-TD-B3LYP	6-31+G(d)	-0.016	0.487	0.542
		aug-cc-pVDZ	-0.019	0.477	0.536
		aug-cc-pVTZ	-0.020	0.472	0.533
		aug-cc-pVQZ	-0.020	0.473	0.533
	SF-TD-PBE0	6-31+G(d)	-0.012	0.618	0.689
		aug-cc-pVDZ	-0.016	0.602	0.680
		aug-cc-pVTZ	-0.019	0.597	0.677
		aug-cc-pVQZ	-0.018	0.597	0.677
	SF-TD-BH&HLYP	6-31+G(d)	0.064	1.305	1.458
		aug-cc-pVDZ	0.051	1.260	1.437
		aug-cc-pVTZ	0.045	1.249	1.431
		aug-cc-pVQZ	0.046	1.250	1.432
	SF-TD-M06-2X	6-31+G(d)	0.102	1.476	1.640
		aug-cc-pVDZ	0.086	1.419	1.611
		aug-cc-pVTZ	0.078	1.403	1.602
		aug-cc-pVQZ	0.079	1.408	1.607
	SF-TD-CAM-B3LYP	6-31+G(d)	0.021	0.603	0.672
		aug-cc-pVDZ	0.012	0.585	0.666
		aug-cc-pVTZ	0.010	0.580	0.664
		aug-cc-pVQZ	0.010	0.580	0.664
	SF-TD-ωB97X-V	6-31+G(d)	0.040	0.600	0.670
		aug-cc-pVDZ	0.029	0.576	0.664
		aug-cc-pVTZ	0.026	0.572	0.662
		aug-cc-pVQZ	0.026	0.572	0.662
	SF-TD-LC-ωPBE08	6-31+G(d)	0.078	0.593	0.663
		aug-cc-pVDZ	0.060	0.563	0.659
		aug-cc-pVTZ	0.058	0.561	0.658
		aug-cc-pVQZ	0.058	0.561	0.659
	SF-TD-M11	6-31+G(d)	0.102	1.236	1.374
		aug-cc-pVDZ	0.087	1.196	1.362
		aug-cc-pVTZ	0.081	1.188	1.359
		aug-cc-pVQZ	0.080	1.185	1.357
	SF-ADC(2)-s	6-31+G(d)	0.345	1.760	2.096
		aug-cc-pVDZ	0.269	1.656	1.894
		aug-cc-pVTZ	0.256	1.612	1.844
	SF-ADC(2)-x	6-31+G(d)	0.264	1.181	1.972
		aug-cc-pVDZ	0.216	1.107	1.760
		aug-cc-pVTZ	0.212	1.091	1.731
	SF-ADC(2.5)	6-31+G(d)	0.234	1.705	2.087
		aug-cc-pVDZ	0.179	1.614	1.886
		aug-cc-pVTZ	0.168	1.594	1.849
	SF-ADC(3)	6-31+G(d)	0.123	1.650	2.078
		aug-cc-pVDZ	0.088	1.571	1.878
		aug-cc-pVTZ	0.079	1.575	1.853

TABLE VI .

 VI Vertical excitation energies (with respect to the 1 1 B 1g ground state) of the 1 3 A 2g , 1 1 A 1g , and 1 1 B 2g states of CBD at the D 4h square-planar equilibrium geometry of the 1 3 A 2g state. The values in square brackets have been obtained by extrapolation via the procedure described in the corresponding footnote. The TBE/aug-cc-pVTZ values are highlighted in bold.

			Excitation energies (eV)
	Method	Basis	1 3 A 2g	1 1 A 1g	1 1 B 2g
	CASSCF(4,4)	6-31+G(d)	0.447	2.257	3.549
		aug-cc-pVDZ	0.438	2.240	3.443
		aug-cc-pVTZ	0.434	2.234	3.424
		aug-cc-pVQZ	0.435	2.235	3.427
	CASPT2(4,4)	6-31+G(d)	0.176	1.588	1.899
		aug-cc-pVDZ	0.137	1.540	1.708
		aug-cc-pVTZ	0.128	1.506	1.635
		aug-cc-pVQZ	0.128	1.498	1.612
	SC-NEVPT2(4,4)	6-31+G(d)	0.083	1.520	1.380
		aug-cc-pVDZ	0.037	1.465	1.140
		aug-cc-pVTZ	0.024	1.428	1.055
		aug-cc-pVQZ	0.024	1.420	1.030
	PC-NEVPT2(4,4)	6-31+G(d)	0.085	1.496	1.329
		aug-cc-pVDZ	0.039	1.440	1.088
		aug-cc-pVTZ	0.026	1.403	1.003
		aug-cc-pVQZ	0.026	1.395	0.977
	CASSCF(12,12)	6-31+G(d)	0.386	1.974	2.736
		aug-cc-pVDZ	0.374	1.947	2.649
		aug-cc-pVTZ	0.370	1.943	2.634
		aug-cc-pVQZ	0.371	1.945	2.637
	CASPT2(12,12)	6-31+G(d)	0.235	1.635	2.170
		aug-cc-pVDZ	0.203	1.588	2.015
		aug-cc-pVTZ	0.183	1.538	1.926
		aug-cc-pVQZ	0.179	1.522	1.898
	SC-NEVPT2(12,12) 6-31+G(d)	0.218	1.644	2.143
		aug-cc-pVDZ	0.189	1.600	1.991
		aug-cc-pVTZ	0.165	1.546	1.892
		aug-cc-pVQZ	0.160	1.529	1.862
	PC-NEVPT2(12,12) 6-31+G(d)	0.189	1.579	2.020
		aug-cc-pVDZ	0.156	1.530	1.854
		aug-cc-pVTZ	0.131	1.476	1.756
		aug-cc-pVQZ	0.126	1.460	1.727
	CCSD	6-31+G(d)	0.148	1.788
		aug-cc-pVDZ	0.100	1.650
		aug-cc-pVTZ	0.085	1.600
		aug-cc-pVQZ	0.084	1.588
	CC3	6-31+G(d)		1.809	2.836
		aug-cc-pVDZ		1.695	2.646
		aug-cc-pVTZ		1.662	2.720
	CCSDT	6-31+G(d)	0.210	1.751	2.565
		aug-cc-pVDZ	0.165	1.659	2.450
		aug-cc-pVTZ	0.149	1.631	2.537
	CC4	6-31+G(d)		1.604	2.121
		aug-cc-pVDZ		1.539	1.934
		aug-cc-pVTZ		[1.511] a	[1.836] b
	CCSDTQ	6-31+G(d)	0.205	1.593	2.134
		aug-cc-pVDZ	[0.160] c	[1.528] d	[1.947] d
		aug-cc-pVTZ	[0.144] e	[1.500] f	[1.849] f
	CIPSI	6-31+G(d)	0.201 ± 0.003 1.602 ± 0.007	2.13 ± 0.04
		aug-cc-pVDZ 0.157 ± 0.003 1.587 ± 0.005 2.102 ± 0.027
		aug-cc-pVTZ	0.17 ± 0.03	1.63 ± 0.05

ACKNOWLEDGMENTS

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 used the HPC ressources from CALMIP (Toulouse) under allocation 2022-18005 and from the CCIPL center (Nantes).

SUPPORTING INFORMATION AVAILABLE

Included in the supporting information are the raw data, additional calculations and geometries, and the Cartesian coordinates of the various optimized geometries.