Multireference Averaged Quadratic Coupled Cluster (MR-AQCC) Study of the Geometries and Energies for ortho-, meta- and para-Benzyne

The diradical benzyne isomers are excellent prototypes for evaluating the ability of an electronic structure method to describe static and dynamic correlation. The benzyne isomers are also interesting molecules with which to study the fundamentals of through-space and through-bond diradical coupling that is important in so many electronic device applications. In the current study, we utilize the multireference methods MC-SCF, MR-CISD, MR-CISD+Q, and MR-AQCC with an (8,8) complete active space that includes the σ, σ*, π and π* orbitals, to characterize the electronic structure of ortho-, meta- and para-benzyne. We also determine the adiabatic and vertical singlet–triplet splittings for these isomers. MR-AQCC and MR-CISD+Q produced energy gaps in good agreement with previously obtained experimental values. Geometries, orbital energies and unpaired electron densities show significant through-space coupling in the o- and m-benzynes, while p-benzyne shows through-bond coupling, explaining the dramatically different singlet–triplet gaps between the three isomers.


■ INTRODUCTION
The benzyne diradicals have been the subject of many experimental and computational studies due to their fundamental nature and unusual bonding, as well as their range of potential applications.−38 In addition, recent studies show the benzynes as building blocks for larger systems such as graphene. 39,40The three isomers of benzyne, ortho (o)-, meta (m)-, and para (p)-benzyne (Figure 1), are the subjects of this study.
Through-space and through-bond interactions between the radical electrons lead to a stabilized singlet state which falls below the triplet state in energy. 41,42The ability of the diradical to abstract hydrogen atoms from proximate sources such as DNA is believed to originate in the triplet state and therefore the singlet−triplet (S−T) energy gap is a measure of the hydrogen abstraction reactivity of the diradical. 43−62 The open-shell nature of these molecules makes for interesting electron interactions that are challenging to describe accurately.−72 Early approaches using unrestricted Hartree−Fock methods proved to be insufficient in capturing the full extent of electron coupling interactions while density functional theory (DFT) and coupled-cluster methods were shown to be better. 25,29,44,45,55,59However, it is not clear if a density-based approach captures the physical nature of these diradicals, and therefore it cannot be known a priori whether DFT is an adequate method. 55,73or many diradicals, the difficulty in obtaining a proper theoretical characterization is due to the multiconfigurational nature of the molecules.If the diradical lobes are degenerate more than one electron configuration may be needed to fully capture the physical behavior of the system.For instance, in pbenzyne the dehydro lobes on C1 and C4 can combine to produce a σ and σ* orbital (Figure 2).If the σ and σ* orbitals are degenerate or nearly degenerate, then the configurations (σ) 2 , (σ*) 2 and (σ) 1 (σ*) 1 need to be included in the reference wave function, i.e. this is a multiconfigurational molecule that needs a multireference wave function (MR) for proper characterization.In addition to the MR problem, polyradicals often have a high density of low lying electronic states, and spin contamination may occur where low lying states of different spin mix with the state being characterized.
Electronic structure methods have been developed to treat the multiconfigurational nature of diradicals.−96 The state specific Mukherjee multireference coupled cluster singles and doubles (Mk-CCSD) method developed by Evangelista et al. 44,66,67 is another example of a multireference coupled-cluster method.−101 Most recently, DFT approaches have been applied to the MR problem with the development of the state-averaged multiconfiguration pairdensity functional theory, 102 the renormalized singles plus the   The Journal of Physical Chemistry A particle−particle Tamm-Dancoff approximation within the density functional approximation, 103 and configurational interaction based on constrained DFT. 104As a general alternative to these methods traditional multireference (MR) configuration interaction (MR-CI) and the related MR-AQCC method 105,106 allow flexible choices of wave functions and application to virtually any desired spin multiplicity. 107A major advantage of MR-CI and MR-AQCC is their variational nature in which the methodological treatment can be systematically improved, and analytic energy gradients are easily accessible.
An important consequence of the different radical locations in the three benzyne isomers is that they have different energy gaps between the lowest lying singlet state and triplet state.Wenthold et al. 56 showed via photoelectron spectroscopy that o-benzyne has the largest S−T gap (37.5 kcal/mol).Radical electron coupling was attributed mainly to through-space interactions due to the proximity of the radicals. 8,41,42In comparison, p-benzyne has the smallest S−T gap (3.8 kcal/ mol) attributed mainly to through-bond interactions. 8,41,42,56lark and Davidson 108 analyzed p-benzyne derivatives via a CASSCF/cc-pVDZ treatment, and identified many with exceptionally small singlet−triplet splitting energies, and even a few with a triplet ground state.m-benzyne, with a S−T gap of 21.0 kcal/mol, falls in the middle of the other two isomers. 56A valence bond study performed by Wei et al. in 2009 suggested that stabilization of the singlet state in m-benzyne due to through-bond interactions is slightly (10%) more than through-space interactions. 109npaired electrons can interact via spin polarization.In addition to through-space and through-bond interactions, Crawford et al. 110 have used this phenomenon to explain the singlet stabilization in p-benzyne.A representation of the spin polarization for the singlet and triplet states of all three benzyne isomers is shown in Figure 3. Spin polarization effects are governed by the intraatomic Hund's rule as well as the electron coupling in bond pairs. 111These rules dictate that electrons in orthogonal orbitals on the same atom be spin aligned (α−α or β−β interactions) while bonding electrons shared between atoms must be spin paired (α−β interactions).In o-and p-benzyne, there are two stabilizing β−β interactions on the radical centers in the singlet state while there is one β−β interaction and one destabilizing α−β interaction in the triplet state.In m-benzyne, however, there are two β−β interactions in the triplet state rather than the singlet state.Thus, spin polarization plays a role in singlet stabilization for oand p-benzyne but not m-benzyne.
4][45][46][47][48][49][50][51]53,54,61,108,112 Fewer studies have been done on the triplet state, and include an o-benzyne SCF geometry optimization by Scheiner et al., 53 a study by Slipchenko and Krylov using a Spin-Flip-DFT approach on all three isomers, 54 a study by Debbert and Cramer using various theoretical approaches, 30,61 and an analysis of the three isomers by Clark and Davidson using the CASSCF method. 57 A topc of special interest is the accurate determination of the singlet state structure of m-benzyne and the geometry implications of the radical carbon interactions.Kraka et al. 47 showed that restricted density functional methods tended to characterize singlet m-benzyne as a bicyclic structure with a bond between the C1 and C3 radical carbons while unrestricted DFT methods as well as coupled-cluster methods show a monocyclic structure.Winker and Sander 113 also performed geometry optimizations at various levels of theory and confirmed the monocyclic structure of singlet m-benzyne.Thus, like S−T gaps, the C1−C3 distance has become an indicator of how well a method captures the multiconfigurational nature of this particular diradical.
Our work focuses on characterizing the energetics and geometries of the singlet and triplet states of o-, m-, and pbenzyne using a highly correlated, multireference approach.Multiconfigurational self-consistent field (MCSCF) is a method that can be used for molecules with low-lying ground and excited states.To overcome the static correlation problem brought on by the open-shelled nature of diradicals, various multireference methods, as already introduced above, can be employed.The multireference configuration interaction with singles and doubles (MR-CISD) method 114 can be used along with the state specific multireference average quadratic coupled cluster (MR-AQCC) method. 105,106,115The MR-AQCC method is of special interest since it allows a balanced description of quasi-degenerate configurations in the reference wave function and of dynamic electron correlation including size-extensivity corrections.These methods were used in a previous study by Wang et al. 55 to study the vertical gaps and other properties of p-benzyne and proved to properly capture the physical nature of the benzyne diradicals.Recently, MR-AQCC has also been used for the calculation of acenes and periacenes. 39,116The biradical character of zethrenes, 117 pyrazine 118 and anthracene 119 have been studied in this way.Also, the antiferromagnetic coupling in non-Kekulédimethylenepolycyclobutadienes, taking into account spin multiplicities up to septets, has been investigated. 120anauer and Koḧn 60 performed an analysis of singlet− triplet splitting energies of the benzynes using MR-CCSD(T) with correlation-consistent double and triple-ζ basis sets (cc-pVDZ/TZ).However, Wang et al. demonstrated that the CAS(2,2) active space used by Hanauer and Koḧn was inadequate for the characterization of p-benzyne due to interactions with the π orbitals.Wei et al. 109 in 2009 demonstrated that the inadequacy of the CAS(2,2) active space for m-benzyne is mostly due to through-bond interactions.In this work, we use a CAS (8,8) active space that includes all of the π orbitals for all the benzyne isomers in order to capture these effects.
To the best of our knowledge, this is the first MR-AQCC analysis of the geometries and energies of all of the benzyne isomers.MR-AQCC is a genuine multireference method allowing flexible incorporation of quasi-degenerate configurations into the reference space to account for static electron correlation.It stands out from other quantum methods as it not only includes dynamic electron correlation using single and double excitations into the virtual space, but also includes coherent size-consistency contributions and is free of spin contamination.

■ METHODS
Initial geometry optimizations on the triplet states of o-, m-, and p-benzyne were performed using the UB3LYP method with a 6-31G** basis set in Q-Chem. 121A single point calculation was then performed using the restricted open-shell Hartree−Fock (ROHF) method with the cc-pVDZ basis set.The molecular orbitals from these calculations were characterized based on D 2h (p-benzyne) and C 2v (o-and m-benzyne) symmetry and used to determine the orbital space for subsequent calculations.Orbitals were identified as either The Journal of Physical Chemistry A doubly occupied (DOCC) or part of the complete active space (CAS).The CAS was used to construct all possible configuration state functions for any particular state symmetry.MCSCF, MR-CI and MR-AQCC geometry optimizations were performed with a CAS (8,8) active space.This included two radical electrons occupying the σ and σ* orbitals on the didehydrocarbons and six electrons occupying the pi and pi* orbitals.The symmetries of these orbitals are characterized and shown in Table 1. Figure 4 shows the eight CAS orbitals for the three benzyne isomers.
The lowest lying singlet and triplet states were calculated independently for each isomer.In o-and m-benzyne, the lowest lying singlet states have 1 A 1 symmetry, while the lowest lying triplet states have 3 B 2 symmetry.For p-benzyne, the lowest

The Journal of Physical Chemistry A
lying singlet state has 1 A g symmetry, and the lowest lying triplet state has 3 B 3u symmetry.
The MCSCF wave functions were used to provide the MOs for the MR-CISD and MR-AQCC calculations.Electrons in six core orbitals (1−3a 1 and 1−3b 2 for o-benzyne, 1−4a 1 and 1− 2b 2 for m-benzyne, 1−2a g , 1−2b 1u , 1b 2u , 1b 3g for p-benzyne) were frozen.The problem of size-extensivity associated with configuration interaction calculations was addressed a posteriori with the Davidson correction using the Pople method (MR-CISD+Q). 122lthough there have been analyses on the complete active orbital set of p-benzyne, 108,110 to the best of our knowledge, the orbitals for o-and m-benzyne have not been previously reported.The MCSCF orbital energies for the CAS will be discussed in the Results section.
Both o-and m-benzyne were optimized with C 2v geometry with the yz plane serving as the molecular plane and the z-axis serving as the C 2 rotation axis.para-benzyne was optimized with D 2h geometry in the yz molecular plane.All methods were used in combination with the correlation consistent cc-pVDZ and cc-pVTZ basis sets developed by Dunning. 123,124The geometries optimized from the MR-AQCC/cc-pVTZ calculation were used to perform single point energy calculations at the MCSCF, MR-CISD, and MR-AQCC levels to obtain vertical S−T excitation energies.The calculations were performed using the COLUMBUS 7.0 program 125−128 with the DALTON atomic orbital integral package. 129he effective unpaired electron densities (UED) 130,131 and Mulliken populations were determined for the MR-AQCC/cc-pVTZ singlet and triplet state geometries.A nonlinear model as suggested by Head-Gordon 132 was used to obtain the UED so that the contribution from the nearly occupied and unoccupied natural orbitals is reduced. 39,131RESULTS AND DISCUSSION Energy Gaps and Electron Configurations.The adiabatic S−T gaps at each level of theory, along with the dominant electron configurations from the MR-AQCC calculations for the singlet and triplet state of each isomer, are shown in Tables 2−4.For the singlet states, we find that the weight (c 2 ) of the π

The Journal of Physical Chemistry A
percent.This confirms that the CAS (8,8) reference space used in this study is capturing well the open shell character.This provides a balanced basis for calculating the total electron correlation energy involving single and double excitations into the virtual orbital space derived from the CAS(8,8) reference configurations.As expected, all of the isomeric triplets are single configurational(π 1 2 π 2 2 π 3 2 σ 1 (σ*) 1 ).The weighting of the second configuration in the singlet state correlates with (a) the decrease in stability of the singlet (vide infra; Figure 5), (b) a decrease in the S−T gap (vide infra; Figure 5) and (c) a narrowing of the σ and σ* orbital energy gap (vide infra; Table 6) as the radical electrons move further apart in the o-, m-and p-benzyne isomers.The multiconfigurational wave function that results serves as affirmation for our multireference approach to the problem.
The S−T gap is an indication of the extent of radical interaction, and having a singlet lie lower in energy than a triplet attests to singlet state stabilization.o-benzyne, having radical centers that are closest to each other, has the largest gap, where in contrast p-benzyne, having radical centers that are farthest apart, has the smallest gap. Figure 5 shows the relative energies of the benzyne isomers.It is evident that the singlet states are destabilized as the distance between radical centers increase from o-to p-benzyne.Given that the experimental heats of formation are known for the benzyne isomers, we can use these values (107.3 ± 3.5, 121.9 ± 3.1, and 138.0 ± 1.0 kcal/mol for ortho-, meta-, and para-benzyne, respectively) 36,37,56,133,134 to see that our AQCC/cc-pVTZ relative energies for the singlets are quite accurate (Figure 5).For instance, using our AQCC/TZ geometry optimized singlets, we calculate the difference in energy for ortho− meta, meta−para and ortho-para as 12.6, 17.6, and 30.2 kcal/ mol.These can be compared to the relative experimental energy differences of 14.6, 16.1, and 30.7 kcal/mol.We can also combine the experimental absolute heats of formation with the calculated singlet−triplet gaps to arrive at the energies of the isomeric triplets (Figure S1).This approach suggests that the triplet isomers are all relatively close in energy, i.e. 141.8, 142.9, and 144.8 kcal/mol for para, meta and orthobenzyne.This is perhaps not surprising that the triplet relative energies would be more isoenergetic than the singlet energies where through-space and through-bond coupling play a more discriminating role in the stability of the various isomers.
As the level of theory increases, and dynamical electron configuration is included, the calculated adiabatic gap approaches the experimental value.The good agreement between our CISD + Q and AQCC results suggests an accurate size-extensivity correction.In p-benzyne, the MR-AQCC overestimates the S−T gap.
The AQCC/TZ singlet state geometries were used to compute vertical S−T gaps (Tables 2−4).The gaps are higher than the adiabatic gaps, which is to be expected.The magnitude of the change reflects the importance of geometry relaxation and the need to use a computational method that captures the multiconfigurational nature of the isomers.The difference between the vertical and adiabatic gap for p-benzyne (Table 4) is significantly smaller than for o-and m-benzyne (Tables 2 and 3, respectively).This is due, in part, to the similar geometry for the singlet and triplet state of p-benzyne whereas in o-and m-benzynes the geometries differ significantly (vide infra).
Table 5 compares our ΔE(ST) results to splittings obtained with other methods as reported in the literature.Judging by the wave functions shown in Tables 2−4, o-benzyne displays the least diradical character of any of the benzynes, and therefore previous methods that had difficulty characterizing the p-and m-benzynes generally perform well for the o-benzyne isomer.Spin-flip-optimized orbital coupled-cluster doubles (SF-OD) is the most accurate in this case, with the MR-CI and MR-AQCC methods also showing good agreement with experiment, especially for o-and m-benzyne.

The Journal of Physical Chemistry A
energies (Table 6) provide further explanation for the variation in isomeric singlet state energies and subsequent singlet− triplet gaps (Tables 2−4).A visualization of these orbitals can be seen in Figure 4.As the radical electrons move further apart in space, the two-configurational nature of the isomer increases, i.e. the ratio of the configuration weights π 1 2 π 2 2 π 3 2 σ 2 : π 1 2 π 2 2 π 3 2 (σ*) 2 is 65:4 for o-benzyne; 62:7 for mbenzyne and 49:20 for p-benzyne.As expected, in the singlet state of both o-and m-benzyne, the π and σ orbitals are occupied and thus have a negative energy, while the π* and σ* orbitals are positive reflecting the decreased probability of being occupied.In the triplet state of o-and m-benzyne, there is electron occupation in the π, σ and σ* orbitals.In the pbenzyne singlet state, the antisymmetric σ* radical orbital falls lower than the symmetric σ orbital.This is indicative of through bond coupling as described by Hoffmann et al. 42 and Crawford et al. 110 Through bond coupling results from the mixing of unpaired/nonbonded σ electrons with the σ paired electrons in the ring.In para-benzyne, the radical electrons are separated by three intervening σ bonds and the mixing symmetry of this arrangement stabilizes the antisymmetric σ* orbital.This causes the σ* orbital to lie lower than the σ orbital in para-benzyne but not the other isomers. 41,42In the pbenzyne singlet, the energies for the σ and σ* orbitals are both negative reflecting the increased two-configurational nature of this isomer.This leads to a destabilization of this singlet isomer and subsequent narrowing of the singlet−triplet gap (Figure 5).
An analysis of the NO populations of the isomeric MR-AQCC/TZ singlets confirms that σ orbitals not corresponding to the radical electrons are all either close to doubly occupied or unoccupied (with an NO population below a threshold of 0.03e).This confirms that σ orbitals other than those in Figure 2 do not contribute to the open shell character of these diradical isomers.The electron correlation of these orbitals and their through-bond interactions are thus well described by the single and double excitations into the virtual orbital space.
Geometries.The AQCC/TZ geometry optimized structures are shown in Figure 7. Several differences in the singlet and triplet geometries provide evidence for the radical electron interaction present in the benzyne isomers.In general, the geometry of the triplet states are much closer to that of benzene. 50,55For o-benzyne, the shortening of the C1−C2 bond in the singlet state geometry is apparent: at the MR-AQCC/cc-pVTZ level, the C1−C2 bond is 1.26 Å while in the triplet state the C1−C2 bond is 1.40.The singlet state is much more distorted relative to benzene, and less symmetrical, due to the coupling between the didehydrocarbon atoms.The shortening of the C1−C2 bond in the singlet state geometry results in a distortion of the bond angles, with an increase in the C1−C2−C3 angle (involved in bond shortening) and a decrease in the C2−C3−C4 angle.The nature of the C1−C2 bond in o-benzyne has been discussed in previous studies. 46,50,51,53,57These prior works all report the C1−C2 bond to be between 1.25 and 1.26 Å, which is in agreement with our results.However, different interpretations are given as to whether the length of this bond constitutes a full triple bond.
The singlet and triplet geometries of m-benzyne (Figure 7) show a similar trend whereby the singlet is significantly more distorted and less symmetrical than the triplet.In the singlet state geometry, the through-space interaction between the didehydro carbon atoms pinches together the C1−C3 distance, even though these atoms are not directly bonded together.This effect of radical electron interaction can be seen in the angle distortion in the C1−C2−C3 angle.Compared to the angle of 115.2°degrees for the triplet state, the singlet state has an angle of 99.0°.The monocyclic/bicyclic structure of mbenzyne has become a standard benchmark test for the ability of a computational quantum method to properly describe a multiconfigurational molecular system.Higher level multireference approaches describe m-benzyne as a monocyclic structure rather than a bicyclic structure containing a C1−C3 bond. 44,47Table 7 shows a comparison of our C1−C3 bond length results at the cc-pVTZ level with previously published distances.It is apparent that the multireference approach taken in this study accurately characterizes the structure of mbenzyne.
Important distances and angles of p-benzyne are also presented in Figure 7.The difference between the singlet and triplet geometries of p-benzyne is much smaller when compared to the same differences for o-and m-benzyne.There is no significant change in the bond distances, and the angles of the radical carbons only change by a few degrees.

The Journal of Physical Chemistry A
The MC-SCF optimized MOs (Figure 4) and AQCC geometries (Figure 7) provide insight into the underlying differences between the singlet and triplet states that lead to quite different isomeric energy splittings.For instance, the proximity of the unpaired electrons in o-benzyne allows strong through-space coupling, resulting in a molecule with a fairly closed-shell nature.One may even draw a resonance structure for this isomer containing a triple bond (Figure 6).This interaction significantly stabilizes the singlet while destabilizing the triplet (ΔE(ST) (AQCC/TZ) = 38.21kcal/mol).For comparison, closed shell pentacene has a UED close to 1 and a T1 energy of ∼23 kcal/mol, suggesting that o-benzyne may be more closed-shell than pentacene. 136For m-benzyne, the ground state singlet is quite distorted, likely due to significant through space interaction between the unpaired electrons, leading to the unusual resonance structure shown in Figure 6.Again we see significant stabilization of the singlet (but not as much as for o-benzyne; Figure 5) likely due to through-space and through-bond coupling, leading to a (ΔE(ST) (AQCC/ TZ) = 21.76 kcal/mol). 109For p-benzyne, through space interaction is unlikely but through bond coupling stabilizes the singlet below the triplet 41,42 (ΔE(ST) (AQCC/TZ) = 5.99 kcal/mol).
UED.To better gauge the open-shell nature of these isomers, and to compare their electron density distributions, Mulliken electron populations, the number of effectively unpaired electrons (N U ) and the UED were determined using the MR-AQCC/cc-pVTZ wave functions.The UED are visualized in Figure 8.For calibration, for closed shell molecules such as benzene, we expect to see Mulliken values that correlate roughly with the number of electrons "assigned" to particular atoms, i.e. ∼6 and ∼0.8 for C and H, respectively.The UED plots provide a qualitative description of the amount of interaction between the unpaired electrons as well as their delocalization throughout the molecule.N U values quantify these effects.The total number of unpaired electrons (N U ) and the unpaired electron density 137 were determined using eq 1 via the nonlinear formula established by Head-Gordon, 131 which involves summing over all NO occupations where n i is the occupation of the ith NO and M is the number of NOs.The UED values and plots provide good estimations of the relative difference in unpaired electron density between the singlet and triplet states.
For o-benzyne, the total unpaired electron density is 0.457 and 2.285 for the singlet and triplet states, respectively, demonstrating that there is significantly more electron interaction in the singlet state, leading to less UED and consequent stabilization of the singlet state.The unpaired electron density is distributed mostly among the carbon atoms, with very little in the hydrogen atoms in both the singlet and triplet state.In the singlet state for o-benzyne, the diradical carbons 1 and 2 (0.120 electrons) have twice the contribution compared with carbons 3 and 6 (0.053) or carbons 4 and 5 (0.046).In the triplet state, however, the diradical atoms (0.916) have approximately ten times as much UED as the other carbon atoms (0.102, 0.092).In contrast to the singlet state, the UED on the diradical carbons for the triplet state is much closer to one, indicative of very little interaction between the radical electrons.
m-benzyne displays similar density patterns as o-benzyne.For instance, in m-benzyne, the radical-containing carbon atoms have 0.184 unpaired electrons in the singlet state as compared to 0.881 in the triplet state.The total UED is 0.626 for the singlet state and 2.280 for the triplet state.The singlet state total UED is slightly for m-benzyne as compared to obenzyne, while the triplet state total UED is about the same for both isomers.The higher UED for the singlet state of mbenzyne, relative to o-benzyne, suggests that there is less radical electron interaction in the singlet state of m-benzyne, leading to a smaller adiabatic and vertical gap compared with obenzyne.The plots in Figure 8 show the unpaired electron density in blue, from which we can see that the UED is significantly higher in the triplet state of m-benzyne.
p-benzyne, in contrast, shows a relative unpaired electron density of 0.566 for the radical carbons in the singlet state and 0.862 for the radical carbons in the triplet state.The difference The Journal of Physical Chemistry A between the UED of the two states is less than the difference seen in either o-or m-benzyne.Similarly, the total UED is 1.604 for the singlet state and 2.270 for the triplet state.Although the UED in the triplet remains close to 2 as for the other benzyne isomers, the UED in the singlet state increases significantly.Once again, the location of the radicals affects the extent of their interaction.The fact that p-benzyne has a higher UED in the singlet state is a confirmation that the smaller S−T gap, relative to o-and m-benzyne, is due to the singlet state not being as stabilized by radical interactions.

The Journal of Physical Chemistry A
Additional computational details including a graphical

Figure 3 .
Figure 3. Spin polarization for the singlet and triplet states of the benzyne isomers.Electrons are represented according to the intraatomic Hund's rule and the pair coupling principle.

Figure 4 .
Figure 4. Active space molecular orbitals for the benzyne isomers optimized at the MCSCF/cc-pVTZ level of theory.

Figure 5 .
Figure 5. Relative energies of the benzyne isomers from AQCC/cc-pVTZ geometry optimizations.Singlet ground state relative energies are in shown in blue while triplet state relative energies are in orange.

Figure 6 .
Figure 6.Resonance structures for o-and m-benzyne showing through-space coupling that leads to these diradicals having a fairly closed-shell nature.

■
CONCLUSIONSWe have characterized the geometries, and the adiabatic and vertical S−T gaps, for the lowest two states of o-, m-, and pbenzyne using multireference methods MC-SCF, MR-CISD, MR-CISD+Q and MR-AQCC.An (8,8) CAS was used that included the σ, σ*, π and π* orbitals.Comparison with previously published data shows that these methods are capturing the various open-shelled nature of the different isomers and produce energies that are in good agreement with experiment.The distance between the unpaired electrons

Figure 7 .
Figure 7. Geometry optimized structures of the benzyne isomers.Distances (Å) and angles (deg) are shown at the MCSCF (red), MRCI (green), and MR-AQCC (blue) levels of theory with the cc-pVTZ basis set.

Table 1 .
CAS Molecular Orbital Symmetries for the Benzyne Isomers

Table 5 .
Comparison of Computed and Measured ST Splittings for the Benzyne Isomers a

Table 6 .
Orbital Energies (au) of the Lowest Lying Singlet and Triplet for the benzynes Calculated at the MCSCF Level with a cc-pVTZ Basis Set Distances calculated as part of this study are shown in bold. a