Bonding in Singlet and Triplet Butalene - American Chemical Society

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Bonding in Singlet and Triplet Butalene: Insights from Spin-coupled Theory David L. Cooper, Peter B. Karadakov, and Brian J. Duke J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.5b00522 • Publication Date (Web): 17 Feb 2015 Downloaded from http://pubs.acs.org on February 18, 2015

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The Journal of Physical Chemistry

Bonding in Singlet and Triplet Butalene: Insights from Spin-Coupled Theory David L. Cooper,*,† Peter B. Karadakov‡ and Brian J. Duke§ †

Department of Chemistry, University of Liverpool, Liverpool, L69 7ZD, U.K.

‡

Department of Chemistry, University of York, Heslington, York, YO10 5DD, U.K.

§

Faculty of Pharmacy and Pharmaceutical Sciences, Monash University, 381 Royal Parade,

Parkville, VIC 3052, Australia ABSTRACT: Spin-coupled (SC, equivalent to full generalized valence bond) calculations for the 1Ag ground state of butalene at its optimal D2h planar geometry show that cross-ring Dewar-like modes of spin coupling are of comparable importance to the more usually considered Kekulé-like modes. There are marked similarities to the SC description of one of the isomers of benzo[1,2:4,5]dicyclobutadiene. A complication for both of these systems is the existence of SC solutions in which some of the orbitals resemble in- and out-of-phase combinations of semi-localized atom-centered orbitals. The lowest triplet state, for which a nonplanar C2v geometry is preferred, is somewhat more straightforward to analyze: the SC description of the 3B2 state is dominated by a very simple pattern of two π bonds and two well-localized triplet-coupled orbitals. Keywords: butalene; para-bonded structures; spin-coupled theory; full generalized valence bond.

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1. INTRODUCTION It is now well established that there are at least two local minima of D2h symmetry on the singlet ground state potential energy hypersurface of 1,4-dehydrobenzene. These consist of an open-ring form I, also known as p-benzyne, and a closed-ring form II, known as butalene (see Figure 1). There is experimental evidence for the trapping of both of these isomers as intermediates1−3 in various chemical reactions and there have been many theoretical studies, stretching back at least to the early 1950s.4 Somewhat more recently, the calculations of Warner and Jones5 placed I about 39 kcal/mol below II, with the barrier to ring opening from II to I being just 3.5−5.5 kcal/mol. Their estimate of the singlet-triplet gap in butalene is 33.5 kcal/mol. According to calculations due to Hess,6 the transition state between II and I actually corresponds to a ‘twisted’, non-planar structure with D2 symmetry. Not only have conventional representations of singlet butalene tended to assume the dominance of Kekulé-like structures but Sakai and Kita7 restricted themselves to such structures when applying their combination method of asymmetric Kekulé structures (CMAK) approach.8 On the other hand, a recent study of benzo[1,2:4,5]dicyclobutadiene,9 which also features two four-membered rings, has shown using spin-coupled (equivalent to full generalized valence bond) theory that para-bonded structures can be at least as important for describing one of the isomers as are any of the Kekulé-like structures. One purpose of the present work is to investigate whether the same is true for the singlet ground state of butalene. We also examine the bonding in the lowest triplet state, not only to discover how much it resembles or differs from that in the ground state, but also to identify the degree to which the triplet character is localized.

2. COMPUTATIONAL FRAMEWORK The π-space spin-coupled (SC) wave functions used here for butalene can be written in the following form:  17  6   ΨSC = Â ∏ σiα σiβ  ∏ πµ Θ  µ=1    i=1    

(1)

All of the doubly-occupied orthonormal ‘inactive’ σ orbitals and the nonorthogonal ‘active’ π orbitals were simultaneously fully optimized alongside the expansion coefficients of Θ, the normalized spin function for the active electrons, in the full spin space10 of five functions for a six-electron singlet or of nine functions for a six-electron triplet: 2 ACS Paragon Plus Environment

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Θ = ∑ ck Θk

(2)

k

We make some use in the present work of the traditional Rumer spin basis11,12 for which each canonical spin function Rk can be uniquely identified by listing its singlet-coupled pairs, for example: k ≡ (µ1–µ2,µ3–µ4,µ5–µ6)

(3)

The notation µi–µj in eq 3 indicates that the spins of the electrons occupying SC orbitals πµi and πµj are coupled to a singlet. We may assess the relative importance or weights Wk of the different Rumer modes of spin coupling in the normalized spin function for the active electrons, Θ, by means of the Chirgwin-Coulson13 definition: Wk = ∑ cl ck ∆lk

(4)

l

in which ∆ is the overlap matrix with elements ∆lk = 〈Rl | Rk〉. An alternative is to use so-called inverse-overlap weights,14 which quantify the unique contribution within a given wave function of a particular component: Wk = |ck|2 / (∆–1)kk

(5) 14,15

It is conventional to renormalize these Wk values so that they add to unity.

We also make some use in the present work of the Serber spin basis.16,17 Serber spin functions are formed through the successive coupling of singlet and triplet pairs according to the standard rules for the addition of angular momenta. The six-electron Serber spins functions for a given total spin multiplicity are conveniently represented in the following compact notation: k ≡ ((s12s34)S4;s56)

(5)

in which sµν takes the value 0 or 1 according to whether the spins of the electrons occupying SC orbitals πµ and πν are coupled to a singlet or triplet, respectively, and S4 is the total spin of the first four electrons. Serber spin functions are orthonormal and so their weights in the normalized spin function for the active electrons, Θ, are simply Wk = |ck|2. The values of the spin-coupling coefficients ck in eq 2 and thus of the weights Wk depend not only on the specific choice of full spin basis but also on the ordering of the different π orbitals, whereas the total SC wave function in eq 1 is of course invariant to such choices. It is straightforward to transform between these different representations of ΨSC using a specialized code for the symbolic generation and manipulation of spin eigenfunctions

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(SPINS).18 Additionally, as was done for benzo[1,2:4,5]dicyclobutadiene,9 we may also examine values of the spin correlation matrix elements Qµν, defined according to: Qµν = 〈Θ | ŝ(πµ) · ŝ(πν) | Θ〉

(6)

in which ŝ(πµ) is the one-electron spin operator associated with the electron occupying orbital πµ. Such values of Qµν, which range between −¾ (pure singlet) and +¼ (pure triplet), have the advantage of being independent of the choice of full spin basis but they do of course depend on the chosen orbital ordering. Some preliminary SC (or full GVB) calculations on butalene were carried out with the VB2000 code19,20 incorporated in GAMESS(US)21,22 but all of the SC calculations that are described here made use of the CASVB module23−26 within the general-purpose quantum chemistry program package MOLPRO.27,28 Many of these results were subsequently replicated using VB2000/GAMESS(US). Geometries for the singlet and triplet state were optimized at the RCCSD(T)/cc-pVTZ and ROHF-CCSD(T)/cc-pVTZ levels, respectively, using the CFOUR program package,29 and π-space CASSCF energies at these geometries were obtained using MOLPRO. The various pictorial representations of SC orbitals were produced from Virtual Reality Markup Language (VRML) files that were generated with MOLDEN.30

3. RESULTS AND DISCUSSION 3.1. Singlet ground state of butalene. The key geometric parameters from our geometry optimization of the singlet ground state (1Ag) of butalene in D2h symmetry at the RCCSD(T)/cc-pVTZ level are summarized in Figure 2 and the full set of coordinates is available in Section S-A in the Supporting Information. All of the frequencies are real. A key feature of the optimized geometry is the fairly large value of r1 (ca. 1.56 Å) for the long central bond, which is of course suggestive of predominantly single bond character. The shortest CC distances, ca. 1.40 Å, are for C1C2 and the symmetry-equivalent bonds, with C1C6 (and C3C4) being slightly longer, by ca. 0.06 Å. In exploratory calculations for butalene (1Ag) at this geometry using the trivial STO-3G basis set we found that it was possible to converge to two different spin-coupled (SC) solutions that are very close in energy (see Table 1). Very slightly higher in energy is a solution, which we label 'L', in which there are semi-localized orbitals on each of the carbon atoms (see top row of Figure 3). The largest symmetry-unique orbital overlaps are those between πC1 and πC2 (ca. 0.54) and between πC1 and πC6 (ca. 0.51), where we have labelled 4 ACS Paragon Plus Environment

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each orbital according to the center with which it is most associated (see Figure 2). As is to be anticipated from the long C2C5 bond length, the overlap between πC2 and πC5 is much smaller (ca. 0.21). The full sets of orbital overlaps, spin correlation matrices, and ChirgwinCoulson and inverse overlap weights in the Rumer basis are available in Table S1 in the Supporting Information. The canonical Rumer functions for the SC 'L' description of this system are shown in Figure 4. By analogy to the findings of a recent SC study of benzo[1,2:4,5]dicyclobutadiene,9 we expect it to be relatively unfavorable to have two π bonds in the same cyclobutadiene ring if they are in the C1C2 and C5C6 positions (or at C2C3 and C4C5 in the other ring). Accordingly, we find that the usually-considered Kekulé-like structures R1 and R4 are not overwhelmingly dominant, contrary to previous assumptions. With the orbitals ordered as πC1,πC2,πC3,πC4,πC5,πC6 around the outer ring, we find that R1 and R4 contribute 26% each to the spin function for the active electrons, with the corresponding Chirgwin-Coulson weight for the minority structure R5 being just 2%. Instead, there are also important contributions of 23% from each of the two cross-ring Dewar-like modes of spin coupling, R2 and R3. Analogous cross-ring modes were found to have similar importance for one of the isomers of benzo[1,2:4,5]dicyclobutadiene9 and for benzocyclobutadiene.31 The lower energy SC solution, which we label 'X', essentially features combinations of orbitals from the 'L' solution, which we may denote πC2,πC1+C3,πC1−C3,πC5,πC4+C6,πC4−C6 (see bottom row of Figure 3). Within the approximation that the orbitals πC1±C3 for the SC 'X' solution can be rewritten as (πC1 ±πC3)/√2 for the SC 'L' case, and that there is no change in the πC2 orbitals, we may rewrite the spatial part of the 'X' wave function in terms of 'L' orbitals. Doing this, we find that the pattern of spin coupling in the 'X' solution resembles that for the 'L' solution, i.e. we find again that the Chirgwin-Coulson weights of the four structures R1 to R4 are all fairly similar to one another, with the corresponding weight of R5 being very small. Various numerical results for the two solutions are available in Tables S1 and S2 in the Supporting Information. Whether it is the 'L' or 'X' solution that is lower in energy for this system is a consequence of a relatively fine balance between the different numbers of orbital and spin degrees of freedom. Inspection of the converged solutions shows that the 'X' case has one more spin degree of freedom than does 'L', but at the expense of one less orbital degree of freedom. (The simple procedure that was used to compare the patterns of spin coupling in the two solutions did of course ignore the different numbers of degrees of freedom.) It was also 5 ACS Paragon Plus Environment


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possible for benzo[1,2:4,5]dicyclobutadiene to obtain rival 'L' and 'X' solutions.9 We note that the existence of SC descriptions that feature in- and out-of-phase combinations of functions associated with symmetry-related atoms were initially considered indicators of antiaromatic or diradical character,32,33 but this interpretation was subsequently shown to be incorrect when a solution of this type could be obtained for the archetypal aromatic molecule benzene.34 In the special case of the H4 model system, McWeeny showed that the existence of SC 'X' solutions can be linked to invariances in the spin function for the active electrons against one or more subgroups of spin permutations, and emphasized the conceptual advantages of using instead SC 'L' solutions that are based on semi-localized atom-centered orbitals.35 Subsequent SC calculations on singlet butalene using the cc-pVTZ basis set converged to a SC 'X' solution with orbitals, shown in Figure 5, that are very reminiscent of those obtained in the exploratory calculations. This SC 'X' description with the cc-pVTZ basis set recovers ca. 91% of the electron correlation that is incorporated in the corresponding π-space CASSCF(6,6) wave function (see Table 2). Various additional numerical results, reported in Table S3 in the Supporting Information, show marked similarities to those from the exploratory calculations. When the spatial part of the wave function is expressed in terms of notional 'L' orbitals, ignoring the different numbers of degrees of freedom in the 'X' and 'L' cases, the spin function for the active electrons is again found to have comparable contributions from the Kekulé-like structures R1 and R4 and the cross-ring Dewar-like modes of spin coupling R2 and R3, with only a small contribution from R5. The close similarities between the spin functions for the active electrons in the STO-3G SC 'L', STO-3G SC 'X' and cc-pVTZ SC 'X' solutions can also be demonstrated by transforming to the Serber spin basis (see Table S4 in the Supporting Information). We were unfortunately unable to converge properly onto a higher-lying SC 'L' solution when using the cc-pVTZ basis set. This suggests that the existence of the SC 'L' solution for the 1Ag ground state of this molecule may be limited to certain basis sets. The main focus of this paper is of course the unusual SC descriptions of the singlet ground state of butalene and the comparison with the SC description of its lowest triplet state. Nonetheless, it does seem worthwhile to consider briefly also the electronic structure of the singlet ground state of the lower-lying open-ring form I (p-benzyne). As has been well documented, the latter is a relatively tricky system to treat accurately. Of particular relevance to the present work is the detailed analysis of Crawford et al.36 which showed why 6 ACS Paragon Plus Environment

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RHF-CCSD(T) cannot be considered reliable for this system. More recently, Jagau and Gauss37 used instead Mukherjee’s multireference coupled-cluster method (Mk-MRCC) which should avoid some of the usual problems for this system that have been identified with many other approaches, especially those based on RHF reference functions. Given the known problematic nature of the singlet ground state of p-benzyne, it could prove useful in future work to investigate multiconfigurational SC descriptions of p-benzyne that allow for a small number of additional structures in which the total numbers of active σ and π electrons are not fixed at 2 and 6, respectively. This is an important aspect that we intend to pursue in due course, comparing SC descriptions of o-, m- and p-benzyne. In the meantime, we summarize in Section S-D in the Supporting Information the results of standard, single-configuration SC calculations for the singlet ground state of p-benzyne that were carried out at the Mk-MRCC/cc-pCVTZ D2h 1Ag geometry reported by Jagau and Gauss.37 This SC wave function, which recovers more than 95% of the electron correlation that is incorporated in the corresponding CASSCF(8,8) description (see Table S5 in the Supporting Information), turns out to consist of well-localized σ orbitals at the two dehydro centers and a benzene-like description of the π system. The spins of the electrons occupying the two active σ orbitals are predominantly singlet coupled, with the degree of triplet character being less than 7%. We find for the corresponding SC description of the 3B2u state of p-benzyne, calculated at the Mk-MRCC/cc-pCVTZ D2h 3B2u geometry reported by Jagau and Gauss,37 that the main difference from the ground state is that the spins of these two σ electrons are predominantly triplet coupled, with the degree of singlet character being less than 4%. 3.2. Lowest triplet state of butalene. Exploratory calculations were carried out for triplet butalene

(3B2u), again using the

trivial STO-3G

basis

set and

adopting the

RCCSD(T)/cc-pVTZ geometry that was optimized for the ground state. We find in this case that a SC solution with semi-localized orbitals, SC 'L', lies nearly 8 millihartree lower than the corresponding SC 'X' solution, with the two sets of orbitals being rather similar to those for the singlet ground state. Pictures of these orbitals, as well as further numerical results, are available in Section S-E in the Supporting Information. In view of the energy difference between the two types of solution, we focused in our subsequent SC calculations on 'L' solutions. We find that the ROHF-CCSD(T)/cc-pVTZ vertical excitation energy from the singlet ground state to the triplet state is over 70 kcal/mol, whereas geometry optimization for D2h 7 ACS Paragon Plus Environment


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triplet butalene (3B2u) at this level of theory results in a somewhat different geometry at a much lower energy (see Table 3). The key geometric parameters for butalene (3B2u) are summarized in Figure 2 and the full set of coordinates is available in Section S-A in the Supporting Information. The most dramatic change relative to the ground state geometry is the shortening of the central bond to ca. 1.42 Å, accompanied by a reduction of r3 to ca. 1.40 Å and an increase of r2 to ca. 1.48 Å. However, frequency analysis shows this to be a first-order saddle point, with an imaginary mode that is suggestive of a distortion to a nonplanar C2v geometry (see later). Warner and Jones5 found a nonplanar triplet geometry with more bond alternation than for the ground state. Nonetheless, it is instructive to consider first the SC descriptions of the triplet state (3B2u) at the two D2h geometries. The SC calculations for the 3B2u state converge to SC 'L' solutions that consistently recover more than 95% of the electron correlation that is incorporated in the corresponding CASSCF calculations (see Table 4). The changes in the deformations of the orbitals towards nearest neighbors when going from the ground state geometry (see top row of Figure 6) to the optimized D2h triplet state geometry (see middle row of Figure 6) and in the overlaps between the orbitals (Table S9 in the Supporting Information) are consistent with the various changes in nuclear separations shown in Figure 2, as is to be expected. The canonical Rumer functions for this system, which are shown in Figure S3 in the Supporting Information, are not well adapted to the overall spatial symmetry of this system. Instead, it proves much more straightforward to analyze the mode of spin coupling by transforming instead to the Serber basis set, after reordering the orbitals as follows: πC1,πC6,πC3,πC4,πC2,πC5. We find that the largest single contribution is for the triplet spin function ((00)0;1)) (see eq 5): this corresponds to singlet coupling of the spins associated with πC1 and πC6, and similarly of those for πC3 and πC4, but triplet coupling of the spins associated with πC2 and πC5. The weight of this particular mode increases from 66% at the ground state geometry to 79% at the optimized D2h triplet state geometry. Taking account also of the contributions from the other Serber functions, or (equivalently) examining the relevant elements of the spin correlation matrices, we find that the overall degree of triplet coupling of the spins associated with πC2 and πC5 are 72% and 80%, respectively, at the ground state and optimized D2h triplet state geometries. It is clear that the triplet character is fairly localized in this state. Geometry optimization for C2v triplet butalene (3B2) at the ROHF-CCSD(T)/cc-pVTZ level results in a yet lower energy (see Table 3). This time, all of the frequencies are real. The key geometric parameters are summarized in Figure 7 and the full set of coordinates is 8 ACS Paragon Plus Environment

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available in Section S-A in the Supporting Information. The much reduced value of r3 (ca. 1.37 Å) is suggestive of significant double bond character for C1C6 and C3C4. Although the system is nonplanar, so that the σ and π labels used in eq 1 are no longer strictly applicable, it turned out to be straightforward to distinguish a σ-like framework from a valence π-like active space. The SC calculations at this geometry again converge onto a 'L' solution (see bottom row of Figure 6), this time recovering more than 98% of the electron correlation that is incorporated in the corresponding CASSCF wave function (see Table 4). Of these two wave functions, SC is much easier to interpret in valence bond terms, while sacrificing very little in terms of computational accuracy. We observe further increases in the weight of the dominant mode of spin coupling in the Serber basis and in the overall degree of triplet coupling of the spins associated with πC2 and πC5, which rises to 93%. In spite of the fact that the C2C5 distance (ca. 1.49 Å) is somewhat shorter than in the ground state, the triplet character is strongly localized in this part of the molecule. (The full sets of orbital overlaps, spin correlation matrices and weights in the Serber basis are available for all three geometries in Tables S9 and S10 in the Supporting Information.)

4. CONCLUSIONS We have confirmed that the D2h geometry for the 1Ag singlet ground state of butalene corresponds to a true minimum on the RCCSD(T)/cc-pVTZ potential energy hypersurface of 1,4-dehydrobenzene. Spin-coupled (SC) calculations for the 1Ag ground state at this geometry show that the two cross-ring Dewar-like modes of spin coupling R2 and R3 (see Figure 4) are of comparable importance in the spin function for the active electrons to the more usually considered Kekulé-like modes, R1 and R4. In keeping with the large value of r1 for the C2C5 distance (see Figure 2), spin-coupling mode R5 is relatively unimportant. Although it is based on a single product of orbitals, the SC wave function recovers more than 90% of the nondynamical correlation energy that is incorporated in the corresponding CASSCF wave function. There are marked similarities between the SC descriptions presented here and those for one of the isomers of benzo[1,2:4,5]dicyclobutadiene.9 An additional complication, for both systems, is the existence of SC 'X' solutions in which some of the orbitals are based on inand out-of-phase combinations of semi-localized atom-centered SC 'L' orbitals for symmetryrelated centers. However, we find for the 1Ag singlet ground state of butalene that when the orbital product for the SC 'X' solution is expanded in terms of notional SC 'L' orbitals, the 9 ACS Paragon Plus Environment


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spin function for the active electrons again has comparable contributions from the four structures R1 to R4, with the corresponding Chirgwin-Coulson weight of R5 being small. The SC description of the lowest triplet state of butalene is somewhat more straightforward, with the SC 'L' solutions being energetically preferred. Starting from the ground state geometry, ROHF-CCSD(T)/cc-pVTZ geometry optimization of a D2h structure for the 3B2u state resulted in a somewhat shorter C2C5 distance (see Figure 2). Nonetheless, the SC calculations show an increase in the corresponding degree of triplet character that is localized in C2C5. According to frequency analysis, this D2h structure is not a true minimum: an imaginary mode suggests distortion to C2v symmetry. Accordingly, geometry optimization at the ROHF-CCSD(T)/cc-pVTZ level produced a lower energy C2v structure (see Figure 7). This time, all of the frequencies were found to be real, so that this is now a true minimum. The SC description of the 3B2 state at this geometry recovers more than 98% of the nondynamical correlation energy that is incorporated in the corresponding CASSCF wave function and it is particular straightforward: it is dominated by C1C6 and C3C4 π bonds, with the triplet character being heavily localized in C2C5. The intricate interplay between the SC 'L' and 'X' solutions, as well as the somewhat unconventional balance between the contributions of different spin functions/structures for the singlet state, show that preconceptions and chemical intuition can sometimes be misleading: they are no substitute for an unbiased analysis that is based on fully-variational wave functions which are capable of accommodating a wide range of computational outcomes.

ASSOCIATED CONTENT Supporting Information Various additional numerical results and orbital depictions for singlet butalene, singlet p-benzyne and triplet butalene. This material is available free of charge via the Internet at http://pubs.acs.org.

AUTHOR INFORMATION Corresponding Author *E-mail: [email protected]. Tel: +44 151 794 3532. Notes The authors declare no competing financial interest.


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Table 1. Energies (in hartree) and percentages of CASSCF(6,6) correlation energy recovered for butalene (1Ag) using STO-3G at the RCCSD(T)/cc-pVTZ geometry. Method

Energy

%

RHF

−226.441513

0%

SC 'L'

−226.562763 94.3%

SC 'X'

−226.563473 94.8%

CASSCF(6,6) −226.570098

100%

Table 2. Energies (in hartree) and percentages of CASSCF(6,6) correlation energy recovered for butalene (1Ag) using cc-pVTZ at the RCCSD(T)/cc-pVTZ geometry. Method

Energy

%

RHF

−229.359607

0%

SC 'X'

−229.435333

91.2%

CASSCF(6,6)

−229.442650

100%

Table 3. CCSD(T)/cc-pVTZ energies (in hartree) at optimized geometries. State

Geometry

Method

Energy

∆E (kcal/mol)

RCCSD(T)

−230.475534

0.0

triplet

D2h singlet ROHF-CCSD(T) −230.361163

71.8

triplet

D2h triplet

ROHF-CCSD(T) −230.402773

45.7

triplet

C2v triplet

ROHF-CCSD(T) −230.427177

30.3

singlet D2h singlet



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Table 4. Energies (in hartree) and percentages of CASSCF(6,6) correlation energy recovered for triplet butalene using cc-pVTZ. Geometry

Method

Energy

%

D2h singlet

ROHF

−229.269756

0%

SC

−229.340054

97.2%

CASSCF(6,6)

−229.342115

100%

ROHF

−229.308823

0%

SC

−229.373522

96.3%

CASSCF(6,6)

−229.376031

100%

ROHF

−229.351147

0%

SC

−229.408519

98.2%

CASSCF(6,6)

−229.409582

100%

D2h triplet

C2v triplet


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Figure captions Figure 1. Open-ring and closed-ring D2h isomers of 1,4-dehydrobenzene. Figure 2. Atom (re)numbering and D2h geometries for singlet and triplet butalene. Figure 3. Symmetry-unique 1Ag SC orbitals (STO-3G basis): 'L' (top row) and 'X' (bottom row). Figure 4. Canonical Rumer structures for singlet butalene. Figure 5. Symmetry-unique SC orbitals for singlet butalene (cc-pVTZ basis). Figure 6. Symmetry-unique SC orbitals for triplet butalene (cc-pVTZ basis) at three geometries, as explained in the text. Figure 7. Atom (re)numbering and C2v geometry for triplet butalene.



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REFERENCES (1) Jones, R. R.; Bergman, R. G. p-Benzyne. Generation as an Intermediate in a Thermal Isomerization Reaction and Trapping Evidence for the 1,4-Benzenediyl Structure. J. Am. Chem. Soc. 1972, 94, 660–661. (2) Breslow, R.; Napierski, J.; Clarke, T. C. The Generation and Trapping of Butalene. J. Am. Chem. Soc. 1975, 97, 6275–6276. (3) Breslow, R.; Khanna, P. L. Further Evidence for the Formation of Butalene in the Reaction of 1-chloro-[2.2.0]bicyclohexa-2,5-dienes[Dewar Chlorobenzenes] with a Strong Base. Tetrahedron Lett. 1977, 3429–3432. (4) Roberts, J. D.; Streitwieser, A.; Regan, C. M. Small-Ring Compounds. X. Molecular Orbital Calculations of Properties of Some Small-Ring Hydrocarbons and Free Radicals. J. Am. Chem. Soc. 1952, 74, 4579–4582. (5) Warner, P. M.; Jones, G. B. Butalene and Related Compounds: Aromatic or Antiaromatic? J. Am. Chem. Soc. 2001, 123, 10322–10328. (6) Hess, B. A., Jr. Do Bicyclic Forms of m- and p-Benzyne Exist? Eur. J. Org. Chem. 2001, 2185–2189. (7) Sakai, S.; Kita, Y. Theoretical Studies of the Structures and Local Aromaticity of Conjugated Polycyclic Hydrocarbons using Three Aromatic Indices. Chem. Phys. Lett. 2013, 578, 49–53. (8) Sakai, S.; Udagawa, T.; Kita, Y. Theoretical Studies on the Structures and the Aromaticity for Condensed Cyclobutadienoids Series: The Combination of Kekulé Structures. J. Phys. Chem. A 2009, 113, 13964–13971. (9) Cooper, D. L.; Karadakov, P. B. Bonding in Benzodicyclobutadiene Isomers: Insights from Modern Valence Bond Theory. Mol. Phys. 2014, 112, 2840–2852. (10) Pauncz, R. The Symmetric Group in Quantum Chemistry; CRC Press: Boca Raton, FL, 1995. (11) Rumer, G. Zur Theorie der Spinvalenz. Göttinger Nachr. 1932, 3, 337–341. (12) Simonetta, M.; Gianinetti, E.; Vandoni, I. Valence‐Bond Theory for Simple Hydrocarbon Molecules, Radicals, and Ions. J. Chem. Phys. 1968, 48, 1579–1594. (13) Chirgwin, B. H.; Coulson, C. A. The Electronic Structure of Conjugated Systems. VI. Proc. R. Soc. London, A 1950, 201, 196–209. (14) Gallup, G. A.; Norbeck, J. M. Population Analyses of Valence-Bond Wave functions and BeH2. Chem. Phys. Lett. 1973, 21, 495–500. 14 ACS Paragon Plus Environment

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(15) Thorsteinsson, T.; Cooper, D. L. Nonorthogonal Weights of Modern VB Wave functions. Implementation and Applications within CASVB. J. Math. Chem. 1998, 23, 105– 126. (16) Serber, R. Extension of the Dirac Vector Model to Include Several Configurations. Phys. Rev. 1934, 45, 461–467. (17) Serber, R. The Solution of Problems Involving Permutation Degeneracy. J. Chem. Phys. 1934, 2, 697–710. (18) Karadakov, P. B.; Gerratt, J.; Cooper, D. L.; Raimondi, M. SPINS: A Program for Symbolic Generation and Transformation of Spin Eigenfunctions. Theor. Chim. Acta 1995, 90, 51–73. (19) Li, J.; McWeeny, R. VB2000: Pushing Valence Bond Theory to New Limits. Int. J. Quantum Chem. 2002, 89, 208–216. (20) Li, J.; Duke, B. J.; Klapötke, T. M.; McWeeny, R. Spin Density of Spin-free Valence Bond Wave Functions and its Implementation in VB2000. J. Theor. Comput. Chem. 2008, 7, 853–867. (21) Schmidt, M. W.; Baldridge, K. K.; Boatz, J. A.; Elbert, S. T.; Gordon, M. S.; Jensen, J. H.; Koseki, S.; Matsunaga, N.; Nguyen, K. A.; Su, S. J. et al. General Atomic and Molecular Electronic Structure System. J. Comput. Chem. 1993, 14, 1347–1363. (22) Gordon, M. S.; Schmidt, M. W. Advances in Electronic Structure Theory: GAMESS a Decade Later. In Theory and Applications of Computational Chemistry: The First Forty Years; Dykstra, C. E.; Frenking, G.; Kim, K. S.; Scuseria, G. E., Eds.; Elsevier: Amsterdam, 2005, pp. 1167–1189. (23) Thorsteinsson, T.; Cooper, D. L. Exact Transformations of CI spaces, VB Representations of CASSCF Wave functions and the Optimization of VB Wave functions. Theor. Chim. Acta 1996, 94, 233–245. (24) Cooper, D. L.; Thorsteinsson, T.; Gerratt, J. Fully-variational Optimization of Modern VB Wave Functions Using the CASVB Strategy. Int. J. Quantum Chem. 1997, 65, 439–451. (25) Cooper, D. L.; Thorsteinsson, T.; Gerratt, J. Modern VB Representations of CASSCF Wave Functions and the Fully-Variational Optimization of Modern VB Wave Functions Using the CASVB Strategy. Adv. Quantum Chem. 1999, 32, 51–67.



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(37) Jagau, T.-C.; Gauss, J. Ground and Excited State Geometries via Mukherjee’s Multireference Coupled-Cluster Method. Chem. Phys. 2012, 401, 73–87.



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Figure 1. Open-ring and closed-ring D2h isomers of 1,4-dehydrobenzene. 179x74mm (300 x 300 DPI)


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Figure 2. Atom (re)numbering and D2h geometries for singlet and triplet butalene. 185x75mm (300 x 300 DPI)



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Figure 3. Symmetry-unique 1Ag SC orbitals (STO-3G basis): 'L' (top row) and 'X' (bottom row). 260x142mm (300 x 300 DPI)


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Figure 4. Canonical Rumer structures for singlet butalene. 217x120mm (300 x 300 DPI)



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Figure 5. Symmetry-unique SC orbitals for singlet butalene (cc-pVTZ basis). 260x74mm (300 x 300 DPI)


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Figure 6. Symmetry-unique SC orbitals for triplet butalene (cc-pVTZ basis) at three geometries, as explained in the text. 176x212mm (300 x 300 DPI)



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Figure 7. Atom (re)numbering and C2v geometry for triplet butalene. 139x81mm (300 x 300 DPI)


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Graphical Abstract 155x114mm (300 x 300 DPI)


Bonding in Singlet and Triplet Butalene - American Chemical Society

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