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. 2023 May 11;127(20):4440–4454. doi: 10.1021/acs.jpca.3c01209

Extremely Long C–C Bonds Predicted beyond 2.0 Å

Eero J J Korpela †, Jhonatas R Carvalho ‡, Hans Lischka ‡, Miklos Kertesz †,*
PMCID: PMC10950299  PMID: 37166124

Abstract

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A number of conjugated molecules are designed with extremely long single C–C bonds beyond 2.0 Å. Some of the investigated molecules are based on analogues to the recently discovered molecule by Kubo et al. These bonds are analyzed by a variety of indices in addition to their equilibrium bond length including the Wiberg bond index, bond dissociation energy (BDE), and measures of diradicaloid character. All unrestricted DFT calculations indicate no diradical character supported by high-level multireference calculations. Finally, NFOD was computed through fractional orbital density (FOD) calculations and used to compare relative differences of diradicaloid character across twisted molecules without central C–C bonding and those with extremely elongated C–C bonds using a comparison with the C–C bond breaking in ethane. No example of direct C–C bonds beyond 2.4 Å are seen in the computational modeling; however, extremely stretched C–C bonds in the vicinity of 2.2 Å are predicted to be achievable with a BDE of 15–25 kcal mol–1.

Introduction

A recent result by Kubo et al.1 showed the presence of a chemical bond between two carbon atoms at DCC = 2.042 Å, as identified by X-ray diffraction (XRD) in a highly strained environment. This remarkable finding was realized by two perpendicularly facing fluorenyl rings in tris(9-fluorenylidene)methane, 1A, a kind of butterfly shape with the two “wings” being joined at the “body” illustrated in Scheme 1 together with selected examples of extremely long C–C single bonds. The purpose of this work is to explore variations on this molecule computationally by looking for two questions: (i) Is it possible to obtain molecular structures with even longer single bonds? and (ii) What are the special features of these extremely long single bonds?

Scheme 1. Selected Experimentally Characterized Examples of Very Long C–C Single Bonds.

Scheme 1

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Numbers in parenthesis following the first author’s name and year are in Å and indicate the length of the long bond shown by a red dashed line.

In this article, we place this discovery by Kubo et al. in the context of the historical progression of longer and longer single bonds obtained in several laboratories over the years, all of which displayed bond lengths as long as the recent 1.93 Å value for diamino-o-carborane2 following on the heels of others at 1.8,3 1.77,4 and around 1.7 Å somewhat earlier.5−9 The example of a carborane contains a C–C distance between two six-coordinated carbons as long as 1.93 Å10 and another with 1.99 Å,11 with a Wiberg bond index of 0.33. Mandal and Datta describe carborenes with C–C bonds as long as 2.01 Å.12 During these series of discoveries, the limit of the longest C–C single bond has been gradually pushed to larger and larger values. Ishigaki et al. reasoned a few years ago that molecular examples with C–C bonds longer than 1.8–2.0 Å should be forthcoming.3Scheme 1 displays some of these molecules with unusually long C–C bonds in organic molecules. Organic ligands in transition-metal complexes occasionally also display very long single C–C bonds, e.g., by Han et al.13 at 1.87(2) Å.

There are no unambiguous theoretical reasons as to why the longest two-electron single bond between two carbon atoms must break at about 2.0 Å. Alvarez surveyed the periodic table, searching for improved van der Waals radii and for the presence or absence of a “van der Waals gap” in the distribution of contact distances in the CSD and finds one for the carbon atoms bound to an oxygen atom.14 On the theoretical side, based on atoms-in-molecules and electron localization function computations, Isea argued that C–C single bonds should still show key characteristics of sigma bonds up to ∼2.0 Å, but not beyond 2.0 Å.15 Based on the analysis of a large database, Lobato et al. arrived at a similar conclusion recently.16 Based on careful temperature-dependent XRD analysis, Kubo et al.1 interestingly noted that the intrinsic distance of the long C–C bond in 1 is somewhat shorter than 2.042 Å, close to ∼1.98 Å due to crystal packing effects. Cho et al.17 argued that this limit should be about 1.8 Å, slightly longer than suggested previously by Zavitsas18 and Schreiner et al.8 based on the dependency of the binding energy as a function of the long C–C bond distance between two sp3 carbon atoms connected to adamantanes or alkanes. They estimated that at about 1.8 Å, the C–C bond dissociation energy (BDE) becomes very small or zero in the series of highly crowded adamantanes. While the accurate assessment of the BDE is challenging, its value is of importance in the presented discussions as we evaluate its approximate value with a singlet–triplet energy gap. Overall, based on the history of the problem, any C–C bond distance longer than 1.8 Å should be considered unusual and worthy of analysis.

Based on the discovery by Kubo et al. and previous cases of very long C–C single bonds, we have continued to ask the questions whether (i) examples can be found with even longer bond lengths, and (ii) whether these elongated bonds still display main characteristics of a C–C single chemical bond.

The first question (i) can be addressed in a relatively straightforward manner by investigating the equilibrium geometries of the proposed molecules with computational methods that are sufficiently reliable in predicting geometries and relative energies. The second question (ii) is more subtle. There are a number of physical parameters that can be used to characterize and compare the strengths of chemical bonds; none of them are perfect, especially when applied to weak bonds. Another complication is that in many weak and long single bonds, steric repulsion plays a significant role,19 as if the effect would be primarily due to bond stretching. In many of these cases, the separation of these opposing effects, bond formation, and steric repulsion leading to bond stretching, is to some degree elusive and arbitrary.

Scheme 2 indicates an intriguing feature of the long C–C bond in 1: a simple VB argument would indicate some, possibly strong, diradicaloid character, as is the case with highly stretched bonds.20 Notwithstanding, Kubo et al.1 convincingly argue based on CASSCF(6,6)/6-311G(d) computations that molecule 1A has a very small diradicaloid character as measured by the y0 index of 0.128 in the ground state of this molecule.

Scheme 2. Two VB Structures of 1A (Covalent) and 1Adr (Diradical).

Scheme 2

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Note the C2v symmetry for both VB structures of the isolated molecule, 1A, as found in the crystal structure.1

An additional complication in investigating extremely long and therefore relatively weak covalent single C–C bonds is the possibility that the bond can break, resulting in a diradical isomer. This possibility is present, for example, for 2A in the form of a twisting deformation, as illustrated in Scheme 3. It will be interesting to explore these deformations, the energetics of these isomerization reactions, and how to prevent them should they lead to a lower-energy twisted diradical, which in fact turns out to be the case in more than one of the presented molecules.

Scheme 3. Isomerization Reaction Involving Twisting of the “Wings” of Some of the Molecules Discussed.

Scheme 3

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Red arrows indicate the conrotatory twists. 2Atw is a structural isomer that has a local minimum on the computed potential energy surface.

In the following, we will characterize the very long covalent single C–C bonds, identified as D12, using accessible parameters in addition to the equilibrium bond distance (Re), including the Wiberg bond index (WBI),21 and the bond dissociation energy (BDE). In addition, we will be interrogating these weak bonds by their diradical character, which serves to indicate a measure of the degree of dissociation and the degree of electron pairing in the bond. The discussion of the diradicaloid character of extremely stretched bonds has been a common theme in most studies1,3,19 as a way to describe how far along the dissociation a particular stretched bond may be. We generally found a low level of diradicaloid character for bond distances up to even 2.0 Å. A further measure of the strength of the covalent bonds investigated is provided by the singlet–triplet energy difference (ΔEST), which becomes small as the bond approaches dissociation.22

Before enumerating the methodology and turning to the results, one comment on terminology can be helpful to avoid a possible misunderstanding. There is a category of weak C–C bonds, typically binding radicals together that are characterized by multicenter electron sharing, the prototypical example being the pairing of phenalenyl (PLY) dimers. The C···C contact distances in these so-called pancake bonds are shorter than twice the van der Waals radius of carbon at DvdW = 3.40 Å.23 The shortest of these observed by XRD was for a dimer of tetracyanoethylene anion radical (TCNE–)2 at 2.801 Å.24 However, these pancake bonds, due to their multicenter nature, e.g., a two-electron 12-center (2e/12c) bond for PLY2 and a two-electron 4-center (2e/4c) bond for (TCNE–)2, should not be compared with the long single bonds in molecules shown in Scheme 1 or their analogues discussed here.

A further distinction relates to fluxional bonding. While the focus in this work is on very long equilibrium bond distances (Re), XRD data may indicate extremely long C–C bond distances that correspond in fact to an average of a bond distance distribution due to fluxional bonding, as they may occur for example in bisnorcaradienes with RXRD as long as 1.8 Å.25−27 In the crystal structure of dimers of phenalenyl derivatives, the observed RXRD = 2.153 Å28 is the result of large-amplitude fluxional bonding, not to be confused by equilibrium bond distances.29,30

The target region of C–C bonds discussed in this paper is indicated by a green rectangle in Figure 1, which encompasses bonds that are longer than the typical stretched single C–C bonds and shorter than pancake bonds.

Figure 1.

Figure 1

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Schematic representation of a hypothetical histogram of unusual carbon–carbon bonds and contacts (DCC). Molecules discussed in this work are encroaching on the “forbidden zone”14 from the left and correspond to bond distances represented by the green rectangle. Blue points and lines indicate only the rapid increase of the relative numbers of such contact distances on the left and right, and hence no specific vertical scale is indicated emphasizing the relative scarcity of extremely stretched bonds and pancake bonds.

Methods

Full geometry optimizations yielding the equilibrium stretched C–C bond length (Re) have been performed with UB3LYP level of density functional theory (DFT) with the empirical dispersion term included in the total energy using the GD3 parametrization,31 where U indicates the spin-unrestricted version. The 6-311+G(d,p) basis set was used except where noted otherwise. Each local minimum or transition structure (TS) was confirmed by zero or one imaginary frequency, respectively. To investigate the diradicaloid character of the electronic structure of molecules, UB3LYP/6-311+G(d,p) calculations were run for all molecules, while higher-level multireference-averaged quadratic coupled cluster32 (MR-AQCC) calculations were run for ethane. Several descriptors were used to characterize the diradicaloid character of a molecule. The y0 parameter as a descriptor of the diradicaloid character was calculated according to the formula33

graphic file with name jp3c01209_m001.jpg 1

where NOONLU is the natural orbital occupation number (NOON) for the lowest unoccupied orbital.34 In addition to the y0 parameter, fractional orbital density35,36 (FOD) calculations B3LYP/6-311G+(d,p), with the recommended electronic Fermi temperature of Te = 9000 K, were completed as another measure of the diradicaloid character. Note that all FOD computations refer to the restricted DFT as FT-RDFT. The FOD analysis provides a quick measure of diradical character through NFOD, a parameter obtained by spatial integration of the FOD. To further probe the accuracy of the FOD analysis as a measure of the diradical character, MR-AQCC/6-311G+(d,p) calculations were performed for the dissociation of ethane. The reference wavefunction, which was also used for initial multiconfiguration self-consistent field calculations, was constructed within a general valence bond (GVB) perfect-pairing multiconfigurational (PPMC) approach.37,38 This wavefunction is of direct-product form where electron pairs are assigned to pairs of active orbitals whose occupancies are determined variationally. Only singlet coupling of all electron pairs were allowed. These MR-AQCC calculations were used to obtain the potential energy curve and the number of effective unpaired electrons, NU, in the relaxed dissociation of ethane. The NU values were obtained according to the nonlinear formula of Head-Gordon39 as eq 2.

graphic file with name jp3c01209_m002.jpg 2

where ni is the occupation of the ith natural orbital (NO) and the sum is over all NOs. Figure 2 shows the MR-AQCC dissociation curve and the evolution of NU with increasing C–C distance. For comparison, the FT-RDFT/B3LYP dissociation curve and NFOD values calculated with the FT-RDFT/B3LYP/6-311+G(d,p) method are also shown. Both methods produce almost identical potential energy curves. Similarly, the NFOD values are well described with the FT-RDFT method with values ∼ 2e in the dissociation region. This behavior coupled with the observation that NFOD values correlate well with NU values obtained at the MR-AQCC level indicates that the fractional occupation is well reproduced by the FT-RDFT method. Based on these results, NFOD was used to compare the diradical character across all molecules included in the study. The Gaussian 16 program was used in most of this work. For the FOD calculations, the ORCA 5.0 program was used.40,41 The MR-AQCC calculations were performed with COLUMBUS.42,43

Figure 2.

Figure 2

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Potential energy curves (relative to the minimum geometry) for relaxed displacement along the C–C bond in ethane and NU values calculated with the MR-AQCC(PPMC)/6-311+G(d,p) method and NFOD values with the FT-RDFT/B3LYP/6-311G+(d,p) methods.

The evaluation of the bond strength via BDE is essential. Unfortunately, in the systems under consideration, a simple dissociation of the highly stretched C–C bond is not possible due to the complex topology of these molecules that engender various strains and tethering. Consequently, we have employed two indirect approaches to estimate the BDE of the long C–C bonds. Here, we summarize these computation protocols and their justifications. It needs to be noted that the separation of strain and other relaxation from the intrinsic BDE is not trivial and is by definition model-dependent. Nevertheless, we expect that useful trends will emerge from these data and their comparisons.

  • (1)
    Estimation of BDE by considering the vertical transition from the singlet ground state to the triplet excited state, according to the formula
    graphic file with name jp3c01209_m003.jpg 3
    Here, ES and ET are the singlet ground state and lowest triplet state energies, respectively, computed by the spin-unrestricted formalism.

    The approximation of the BDE in this manner goes back to the analysis of single-bond dissociation by Michl.44 Similar approaches have been used for other weakly bonded systems.45,46 Kubo et al. estimated the respective BDE for 1A to be 138 kJ mol–1 (33 kcal mol–1). In a relaxed geometry version of the same method where the triplet geometry was also optimized, they obtained a BDE of 113 kJ mol–1 (27 kcal mol–1) for 1A, noting that due to relaxation, this measure includes the release of some of the angle strain seen in 1A.1

  • (2)
    A second approach relies on the possible presence on the potential energy surface (PES) of an isomeric structure without the weak bond in question. Such structures may be present in some cases, and not in others. In the cases where these structures are present, the BDEisomers refers to the energy difference between a nontwisted and twisted conformer, as seen in Scheme 3. The BDE obtained in this manner is as following
    graphic file with name jp3c01209_m004.jpg 4)
    The values for BDEisomers obtained in this manner can be strongly affected by the differences in the strain energies of the two isomers and therefore turned out to be less useful than BDEST. In Table 3 bonded refers to untwisted, and unbonded to twisted conformations.

Table 3. Key Results for Nontwisted Target Molecules from Computational Modeling Calculated at the UB3LYP-GD3/6-311+G(d,p) Level of Theory.

molecule Re = D12 (Å) WBI BDEST (kcal mol–1) NFODa [e]
1A 2.048 0.437 32.4 1.76
1A(CN:4,12) 2.071 0.409 28.2 1.98
1A(F:4) 2.071 0.431 28.7 1.82
1A(Me:4) 2.151 0.354 19.8 2.03
1A(Me:5) 2.052 0.433 20.9 1.79
1A(Me:6) 2.068 0.421 30.2 1.81
1A(Me:7) 2.073 0.417 29.7 1.82
1A(Me:4,5) 2.118 0.379 24.9 1.94
1A(Me:4,6) 2.165 0.341 18.3 2.08
1A(Me:4,7) 2.157 0.347 19.3 2.06
1A(Me:4,8) 2.152 0.342 20.0 2.05
1A(Me:4,9) 2.153 0.352 19.6 2.05
1A(Me:4,10) 2.137 0.363 22.0 2.01
1A(Me:4,11) 2.097 0.405 27.7 1.89
1A(Me:4,12) 2.191 0.321 15.4 2.16
1A(Me:4,19) 2.117 0.390 25.5 1.93
1A(OH:5,10,13,18) 2.070 0.416 32.3 1.99
1B 1.641 0.801 63.7 1.46
1C 1.625 0.765 68.9 0.88
2A 2.099 0.388 27.8 1.37
2A(F:4,12) 2.107 0.343 24.7 1.43
2A(Me:4) 2.144 0.400 22.7 1.50
2A(Me:4,12) 2.192 0.399 17.1 1.65
2A(Me:4,11,12,19) 2.180 0.379 21.2 1.59
2A(Br:4,11,12,19) 2.170 0.368 20.1 1.78
2B 1.639 0.802 62.9 1.00
2C 1.636 0.763 67.2 0.43
2D 2.183 0.312 23.1 2.14
2E 2.228 0.247 23.4 3.18
3A 2.049 0.436 32.3 2.13
5F 1.636 0.868 83.1 1.02
6F 1.664 0.883 76.8 1.06
7F 1.667 0.852 55.4 1.24
8F 1.597 0.900 72.8 1.15
9F 1.736 0.801 73.9 1.06
10A 2.213 0.344 21.1 1.73
10A(Me:4) 2.231 0.294 17.6 1.83
10D 2.254 0.305 21.0 2.40
11A 2.079 0.397 25.9 1.35
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a

NFOD calculations computed with FT-RB3LYP/def2-TZVP (Te = 9000 K) level of theory.

13C NMR calculations were run for selected target molecules, whereby their 13C NMR chemical shifts were computed by the GIAO-UB3LYP-GD3/6-311+G(d,p) method.47 These structures were optimized using the same level of theory. TSM was used as the reference computed also using GIAO-UB3LYP-GD3/6-311+G(d,p).

Molecular Design

For all of the target molecules of this work, the two carbon atoms in question have three other carbon atoms attached to them in addition to the long bond being investigated. In this sense, they are analogues of the molecules shown in Scheme 1. All target molecules in this study can be seen in Table 1. Moreover, the names of each molecule are defined using Table 2, where the first column represents the “body” and the second column the “wings” of these butterfly-shaped molecules.

Table 1. List of Target Molecules in Their Nontwisted Configurations Investigated in This Study.

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Table 2. Key for Target Molecules within This Studya,b.

graphic file with name jp3c01209_0021.jpg

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a

Each target molecule corresponds to a number identifying the “body” and a letter for the “wings” possibly with additional substituents.

b

The red highlighted bonds indicate the linking units for the “wings” and “body”. E.g., 1A corresponds to the molecule in Scheme 2, and 2A corresponds to the one in Scheme 3.

Molecules with letter codes B–E are related to and derived from those with a letter code A. Their distinguishing feature is the number of rings and the maximum ring size of their “wings”. Similarly, molecules with number codes 2 and 3 are related to molecules with the number code 1, except that each number corresponds to a different “body”, as seen in Table 2. Molecules 5F–8F are related to and derived from Ishigaki’s molecules, one of which is seen in Scheme 1. Molecule 9F is related to the Toda molecule in Scheme 1. Finally, molecules 10A and 11A are derivatives of 2A with electron-withdrawing or electron-donating groups on all available sites of the molecule’s “body”. For some of these molecules, there are other derivatives with various substituents on their “body” and “wings” that were included as target molecules. Scheme 4 shows atomic numbering used in this work.

Scheme 4. Numbering Used to Identify Target Molecules with Additional Substituents Added to Their Letter Code A “Wings”.

Scheme 4

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E.g., 1A(Me:4) corresponds to molecule 1A with a methyl substituent at C4.

Results and Discussion

Key results of the computational modeling are presented in Table 3. The results consistently indicate extremely long covalent single C–C bonds with equilibrium bond distance Re values in the range of 1.6–2.2 Å, some of which are remarkably long, placing them in the unusual category within the green rectangle in Figure 1. The molecules that stand out having the longest Re values will be discussed.

While geometric parameters of typical C–C covalent bonds are nearly constant depending on orbital hybridization, throughout the literature, there are several molecules such as those presented in Scheme 1 that rely on steric effects to distort these typically stable geometric parameters under highly strained conditions.3,5−7 As a result, in this study, steric effects are a core strategy used to probe the limits of covalent single C–C bonds. One of the longest observed bonds in this study that utilizes steric hinderance as its primary mode of elongation is 1A(Me:4,12) with an equilibrium bond distance of 2.191 Å. With methyl groups positioned at carbons 1 and 9, both fluorene “wings” are forced to separate from one another due to steric repulsion. This separation is not a simple elongation of the bond along the axis where the bond exists, but rather a distortion of the geometry of this molecule by slight twisting of its “wings”, adopting a C2 geometry. This twisting is similar to that shown in Scheme 3; however, the molecule does not fully adopt a twisted conformer without an elongated central carbon bond, as confirmed by the zero diradical character, y0, and relatively low NFOD. Instead, this molecule twists slightly, which elongates the bond, to lower the van der Waals repulsion between the methyl substituents and hydrogen atoms on the opposing fluorenyl “wing”. For most of the other molecules that have substituents on their “wings” and exhibit elongated bonds, a similar reasoning of steric hindrance can be used.

As explained above, the elongated bonds of the target molecules are too short to fall in the category of pancake bonds. However, for molecules with the large “wings” D and E, there appear to be pancake-like interactions between carbons of the two “wings”, and hence a pancake bonding model22 can be used to understand the attractive interaction between the “wings” in these systems. One such through space bonding interaction is indicated by the in-phase orbitals between the two “wings” seen in Figure 3 for the highest occupied molecular orbital (HOMO) of 2E. The interaction is labeled between two carbons at a length of 2.922 Å, which is within the typical range of pancake bonding. If such pancake bonding would not be present, this short contact distance would imply large steric repulsion. A geometric consequence of this pancake-like interaction is reflected in the optimized geometries for these types of molecules with extended macrocycle “wings”. Unlike for the fluorenyl “winged” molecules, these larger macrocycle molecules converged to energy minima where their “wings” almost completely eclipse one another. This eclipsed conformation shortens the distance between wings, allowing for pancake-like bonding interactions. While this study did not focus on these interactions, it still should be noted that molecules with “wings” D and E have D12 distances significantly elongated, surpassing many sterically hindered molecules. For example, 2E has an equilibrium D12 of 2.228 Å, which is longer than all 1A and 2A sterically hindered molecules with D12 distances ranging from 2.048 to 2.192 Å.

Figure 3.

Figure 3

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(a) HOMO and lowest unoccupied molecular orbital (LUMO) of 2E calculated using the B3LYP-GD3/6-311+G(d,p) method. Purple and red surfaces represent the relative signs of the orbital coefficients drawn at the 0.03 e au–3 level. (b) Alternative view of HOMO of 2E drawn at a 0.01 e au–3 level. The equilibrium distances indicated at 2.922 Å correspond to D21,23 and D22,24 in Scheme 4.

The effects of extending and shortening the “body” and “wings” of 1A were investigated by two series of molecules: 1A, 2A, 3A and 2A, 2D, 2E. In the first series 1A–3A, the fluorenyl body was shortened in 2A to a methylene group and lengthened in 3A to a 12H-dibenzo[b,h]fluorene group. The effect of altering the body of 1A is unclear since both lengthening and shortening the body had an effect of increasing D12. However, the effect on D12 was more pronounced when shortening the body to 2A where D12 increased by ∼0.05 Å, while lengthening the body marginally increased D12 by less than 0.001 Å. However, because the body of all of these Kubo-like molecules does not play a direct role in the bonding of D12, as seen for example in the two frontier MOs of 2E in Figure 3, it was expected that modifications of the “body” of 1A would have little impact on D12. In contrast, the effect on D12 due to variations in the “wings” was far more pronounced. Going from 2A to 2D to 2E, the fluorenyl “wings” were extended on both sides by one benzene group, resulting in significant D12 increases from 2.099 to 2.183 to 2.228 Å. These large increases in D12 can be explained by the increase in pancake bonding as a result of larger macrocycle conjugated systems. Unlike in the first series where the “body” was systematically changed, the “wings” of the Kubo-like molecules play a large role in bonding. Specifically, C1 and C2, both part of the “wing” macrocycles, are the two carbons involved in D12.

The inductive effect via electron-donating and electron-withdrawing substituents was also explored as a way to stabilize extremely elongated bonds. This effect was tested by adding electron-withdrawing and electron-donating groups to either the “body” or “wings” of 2A. When adding electron-withdrawing and electron-donating groups to the “wings” of any molecule under study, steric effects dominated the observed response to bond length. For example, adding halogens or methyl groups to the “wings” resulted in partial twisting, a result of the confined space between wings leading to the elongation of the D12 bond. In contrast, the effects of adding electron-withdrawing and electron-donating groups to the body of the target molecules were less clear. A decrease was seen in the bond length with the addition of electron-donating methyl groups to the body of 2A, 11A. However, D12 increased when adding electron-withdrawing groups to “bodies” of the target molecules. Specifically, for the cyano-substituted molecule (10A), a large increase in bond length was observed to 2.213 Å.

For all of the target molecules in this study, correlations were constructed comparing the bond length to a variety of parameters, including WBI, BDEST, and NFOD. The respective WBI values correlate very well with D12, as illustrated in Figure 4. These data indicate that several molecules in the dataset have significant bond orders with bond distances larger than 2.0 Å with WBI values of 0.3 and larger for bond distances of up to ∼2.25 Å, which are significantly longer than that of the Kubo molecule. However, the nearly linear correlation indicates that no C–C WBI is expected beyond around 2.5 Å. It should be noted that molecules beyond 2.4 Å exhibit twisted conformers with significant diradical character and low WBI, indicating the absence of a bond.

Figure 4.

Figure 4

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Correlation between D12 and the WBI for molecules with extremely long covalent single C–C bonds. For selected data points, their corresponding molecules are shown. The nonfilled blue data point refers to 2E.

Bond dissociation energies are physically well-defined quantities compared to bond orders, which are not.48 However, as outlined in the Methods section, a direct computation of the BDE in the presented cases is not possible. First, we display the BDEST values in Figure 5 that can be used as surrogates of the BDE as per eq 3. The trends are similar to that seen in Figure 4 for the WBI except that the linear trendline indicates a shorter limit where the extremely stretched C–C bonding diminishes to the absence of any bonding at ∼2.45 Å. The strength of the BDEST computed in this manner becomes smaller than 10 kcal mol–1 at ∼2.3 Å, which should be considered as the long limit of extremely stretched C–C bonds. However, molecule 10A with the computed Re= 2.213 Å still displays a significant BDEST of 21.1 kcal mol–1 putting it on par with other very weak covalent bonds, such as the elongated bond (1.68 Å) present in 1,2-di(9-anthryl)benzene.49 Thus, molecules on this long limit of extremely stretched C–C bonds still display qualities of bonding character.

Figure 5.

Figure 5

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Correlation between D12 and BDEST for molecules with extremely long covalent single C–C bonds. The data points for molecules 10A and 10D are indicated by arrows. The nonfilled blue data point refers to 2E.

Figure 6 illustrates the positive linear correlation between NFOD and D12. While there is no absolute cutoff for NFOD that indicates the presence or absence of a C–C bond, it can be used as a relative measure of diradical character, which increases as a covalent bonding weakens. These data indicate a large difference in the diradical character between the nontwisted (in blue) and twisted (in orange) molecules and are consistent with those in Figures 4 and 5. These data also support the presence of covalent bonding up to around 2.3 Å. There is a region of data points below the trendline from 2.1 to 2.3 Å that are of interest due to their low NFOD values. These molecules are all derivatives of 2A, all of which interestingly have longer D12 distances than their corresponding 1A derivative counterparts. It is likely that these 2A derivatives have lower NFOD values because of their simplified “bodies”—which are less conjugated than 1A derivatives—and thus minimize delocalization of radical electrons. This can be confirmed by visualizing the FOD densities of 1A and 2A as seen in Figure 7. This figure reveals no FOD density on the simplified “body” of 2A, as compared to some FOD density on the larger, conjugated “body” of 1A.

Figure 6.

Figure 6

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Correlation between D12 and NFOD for molecules with extremely long covalent single C–C bonds. The nonfilled blue data point refers to 2E, see text.

Figure 7.

Figure 7

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FOD density plots for 2A, 1A, and 2E calculated using B3LYP/def2-TZVP model chemistry (Te = 9000 K). FOD surfaces are drawn at a 0.005 e au–3 level.

There also seems to be an outlier in Figure 6 with an unusually high NFOD value that refers to molecule 2E, as indicated by the empty blue circle. Looking at 2E’s FOD density plot in Figure 7 reveals a potential reason for its high NFOD value. Compared to the FOD plots of both 1A and 2A, it is clear that the radical electrons are significantly more delocalized across the large macrocyle “wings” of 2E. Since NFOD is calculated by the integration of FOD over all space, NFOD is expected to increase when radical electrons are delocalized over a larger region. Using this reasoning, it would be expected that molecule 2D, with larger “wings” than 2A but smaller than those of 2E, would have an NFOD value between those of 2A and 2E. This hypothesis is confirmed by the NFOD values listed in Table 3, where for this series of molecules, the values increase as follows: 1.37e, 2.14e, 3.18e, for 2A, 2D, and 2E, respectively. As a result of its high FOD value and molecular orbitals seen in Figure 3, the interaction between the two wings of molecule 2E can be described as two pancake-bonded radicals. Since D12 in 2E is too short for pancake bonding, the interaction between C1 and C2 must be covalent in nature. In contrast, the contacts D21,23 and D22,24 are too long for covalent bonding but within the range for pancake bonding. All of this is to say, 2E is unlike the rest of the presented molecules in that there is a mix of covalent and pancake bonding, so an increase in its NFOD is to be expected.

For a selected group of molecules listed in Table 4, two minima were found: one with a nontwisted C2v structure and another with a twisted C2 structure. While the existence of two isomers was not confirmed for all molecules, we expect that two geometric minima should be present for most of the molecules presented in Table 1. In all confirmed cases, however, the twisted C2 conformer was lower in energy to the nontwisted C2v conformer. Since Kubo et al.1 determined that the higher-energy nontwisted conformer of 1A was present in the crystal structure, a potential energy scan (PES) was completed to understand the reaction coordinate of such isomerization reactions and why the crystal structure revealed the presence of a higher-energy nontwisted isomer. In this work, a relaxed potential energy scan was performed on the simplest molecule in our database, 2A, to investigate the isomerization reaction pathway between 2A and 2Atw. As illustrated in Scheme 3, 2A has a nontwisted C2v isomer (2A) and a twisted C2 isomer (2Atw). Similar to 1A, the twisted 2Atw structure was lower in energy. More specifically, 2Atw’s ground state energy was 5.59 kcal mol–1 lower than that of nontwisted 2A. Since 2A readily twists into its twisted conformer with slight distortions of D12, D12 was frozen at each point of the scan to obtain intermediary points along the PES. Figure 8 shows the isomerization reaction pathway in terms of the molecule’s geometry. In the first portion of the figure, as indicated by the blue points below 2.25 Å, torsions α = 20–3–1–21 and β = 20–3–1–22 change little with an increase in D12. In this region before 2.25 Å, there is no twisting of the “wings”. Instead, the central bond weakens through bond elongation while conserving its C2v geometry. At around 2.25 Å, there appear to be two distinct pathways through which the central bond of 2A breaks: high- and low-symmetry pathways. The relative energies of each point along these pathways are depicted in Figure 9. In the low-symmetry pathway, the molecule begins to twist at around 2.25 Å, misaligning the π orbitals that make up this elongated π-bond, and thus rapidly breaking the central bond, and the two abovementioned torsions differ. Then, as D12 increases with each point after the initial twisting at around 2.25 Å up until around 2.50 Å, the molecule relaxes to its 2Atw conformer. In contrast, in the high-symmetry pathway, the C2v geometry is preserved, indicated by the blue points that extend past 2.25 Å. Instead of twisting earlier at around 2.25 Å, the molecule preserves its C2v symmetry where the central bond breaks without twisting by continual elongation of D12. At around 2.50 Å, however, the molecule twists into the 2Atw molecule, indicated by the black arrows in Figures 8 and 9.

Table 4. Physical Parameters of Molecules that Exhibit a Lower-Energy Twisted Conformer Relative to the Untwisted Conformer at the UB3LYP-GD3/6-311+G(d,p) Level of Theory.

molecule Re (Å) WBI ΔBDEisomers (kcal mol–1) ΔBDEST (kcal mol–1) diradical character (y0)
1Atw 2.466 0.147 3.63 –2.76 0.895
1A(Br:4)tw 2.479 0.017 a –2.02 0.963
2Atw 2.510 0.018 5.59 –1.48 0.736
2A(Me:4,11,12,19)tw 2.604 0.016 7.14 –0.924 0.614
2A(Br:4,11,12,19)-tw 2.610 0.029 6.54 –0.678 0.550
2Dtw 2.510 0.022 3.15 –1.29 0.824
10Atw 2.481 0.107 7.21 –1.00 0.438
10A(Me:4,12)tw 2.490 0.014 a –1.19 0.555
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a

There is only a twisted minimum.

Figure 8.

Figure 8

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Isomerization reaction torsional coordinates along D12 in a relaxed scan of 2A comparing torsions α and β and D12. The red bonds in the inset indicate the two disrotatory axes of torsion.

Figure 9.

Figure 9

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Isomerization reaction coordinate plot of 2A comparing energy as a function of D12. The point at D12 = 2.5 Å is the longest at which a C2v structure could be optimized. For longer D12 values, the computations converge to the lower-energy twisted C2 structure as indicated by the black arrow.

The misalignment of π-orbitals after twisting can be seen in Figure 10 for 2Atw. As the “wings” twist, these orbitals no longer overlap well, and thus the central D12 bond breaks. Figure 11 depicts the HOMO molecular orbitals of 2A along the high- and low-symmetry pathways from D12 = 2.25–2.50 Å. Along the high-symmetry pathway, with increasing D12, there is less orbital overlap up until 2.50 Å where the bond breaks, losing most electron sharing between the two carbons involved in the central bond. In the low-symmetry reaction pathway the “wings” twist, breaking the C2v symmetry and misaligning the π orbitals as early as 2.30 Å. As such, in the low-symmetry pathway, the central D12 bond breaks much earlier.

Figure 10.

Figure 10

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HOMO of 2A and 2Atw calculated using the UB3LYP-GD3/6-311+G(d,p) method drawn at the 0.03 e au–3 level. The π orbitals involved in the central C–C bonding overlap to form a bond in 2A; however, these orbitals are not aligned for perfect overlap in 2Atw, preventing orbital overlap and sharing of electrons along D12.

Figure 11.

Figure 11

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High- and low-symmetry α/β HOMO molecular orbitals of 2A along the PES scan. Molecular orbitals were calculated using the UB3LYP-GD3/6-311+G(d,p) method drawn at the 0.03 e au–3 level.

The high-symmetry pathway is less likely for the isomerization of 2A. This is because each blue point past 2.25 Å is a high-energy conformer that can with any slight deformation adopt a twisted conformation. As seen in Figure 9, the activation energy of the isomerization PES is surprisingly low. It had been hypothesized that a large activation energy for such isomerization reactions was the key reason why 1A had adopted a higher-energy conformation in its crystal structure. However, since the activation energy is less than 1 kcal mol–1, there must be other effects that restrict molecule 1A from adopting its lower-energy twisted conformer in its crystal structure.

FOD calculations were run on this PES to investigate the diradical character of each conformer. Figure 12 reveals a large relative increase, from 1.37e to 2.48e, in NFOD as 2A adopts a twisted conformation. While NFOD gradually increases with increasing D12, there is a sharp increase starting at around 2.25 Å, as the molecule twists. This indicates that the diradical character significantly increases as the “wings” of 2A twist and the central bond breaks. After the bond has broken, further increases in bond length from 2.30 to 2.65 Å have little effect on the diradical character. This sudden increase in NFOD indicates the presence of a covalent C–C bond for 2A before any twisting takes place.

Figure 12.

Figure 12

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Isomerization reaction scan of NFOD for 2A at B3LYP/def2-TZVP (Te = 9000 K) level of theory.

The isomerization reaction involving twisting of the “wings”, as depicted in Scheme 3, was investigated for selected molecules seen in Table 4. All equilibrium bond distances were near ∼2.5 Å, which is the limit predicted by both WBI and BDEST correlations where near zero C–C WBI or bond dissociation energy values are expected. Furthermore, low WBI and high diradical character suggest each twisted molecule is in its diradical state without the presence of a D12 bond. It should also be noted that the ΔBDEisomers values are positive for all molecules where both a twisted and a nontwisted conformer were present, indicating that the twisted singlet (diradical) conformation is lower in energy than the nontwisted singlet molecule. Since the twisted isomer was found to be always lower in energy and the isomerization reaction of 2A revealed a small activation energy, crystal packing effects were investigated as a possible stabilizing effect for the higher-energy nontwisted conformer. In fact, Kubo et al. suggested1 that 1A adopts the untwisted C2v conformation as a result of crystal packing effects where the two fluorenyl rings face each other in a perpendicular configuration.1 It should be noted that for molecules 1A(Br:4)tw and 10A(Me:4,12)tw, no nontwisted conformer was found. For these molecules, the steric repulsions due to the sizes of the halogen atom or two methyl groups were too large for nontwisted energy minima to exist. To minimize steric hinderances, these molecules are forced to adopt their twisted conformation. This means that there is a limit to the size and number of substituents one can place on the “wings” of the target molecules to further elongate the equilibrium D12.

Dimer geometry calculations were completed for 2A, one of the simplest target molecules, to investigate nonbonding crystal packing effects, as described by Kubo et al.1Figure 13 illustrates the packing for the dimers of 2A and 2Atw. For the 2A dimer, the “wings” of the neighboring monomer appear to lock each molecule of the dimer in its nontwisted form. Unlike in the monomer, where there is space for the “wings” to twist, as a dimer, this space is taken up by the opposing molecule, restricting the twisting isomerization reaction from taking place. While steric repulsions likely play a significant role in stabilizing the 2A dimer in the nontwisted conformation, there are also nonbonding interactions that further stabilize the 2A dimer. In the nontwisted dimer, the “wings” of each molecule are more closely packed and overlap more compared to the 2Atw dimer (Figure 13). This close packing results in a larger nonbonding vdW interaction energy. In fact, this interaction energy for the 2A dimer (−23.6 kcal mol–1) is almost twice as large as for the twisted dimer (−12.8 kcal mol–1). When considering this large stabilizing energy for the nontwisted dimer, 2A would likely remain in its nontwisted form in its crystal structure despite the twisted monomer being a lower-energy conformation and the low activation energy of the isomerization reaction. This finding supports the observation in ref (1) that crystal packing effects stabilize 1A in its bonded, nontwisted form. While indepth analysis for the isomerization reactions and packing was not completed for all of the target molecules, these preliminary findings suggest that for all of the molecules that exhibit lower-energy twisted isomers, the nontwisted conformation would be preferred in their crystal structure. The stabilization resulting from nonbonding vdW’s interactions appears to be significant enough to favor the nontwisted bonded conformation of these target molecules.

Figure 13.

Figure 13

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Optimized dimer structures of 2A and 2Atw. The closest C–C distances between the wings of each dimer are displayed.

13C NMR spectroscopy is a sensitive tool to explore the hybridization and environment of carbon atoms. Figure 14 displays the computed 13C NMR chemical shifts for 2A in the bonded (C2v) and twisted diradicaloid (C2) conformation. According to the calculation, the peak around 100 ppm corresponds to the chemical shifts of C1 and C2 in the bonded conformation, while this peak moves by ∼50 ppm to a much higher value when the bond is broken (2Atw). A similar major increase in chemical shift is seen for pairs 1A/1Atw, 2D/2DTw, and 10A/10Atw as shown in Figures S4–S6, respectively. Figure S7 shows the development of a similar shift by almost 100 ppm as the single bond is gradually broken in ethane. It appears that 13C NMR spectroscopy offers a tool to monitor these extremely elongated C–C single bonds.

Figure 14.

Figure 14

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Theoretically predicted 13C chemical shifts of (a) 2A and (b) 2Atw calculated by the GIAO-B3LYP-GD3/6-311+G(d,p) method. The structures were optimized at the UB3LYP-GD3/6-311+G(d,p) level of theory and converged to a C2v symmetry for 2A and C2 symmetry for 2Atw. TSM was also computed at GIAO-B3LYP-GD3/6-311+G(d,p) and used as the reference.

Conclusions

Generally, it is assumed that a “forbidden zone” exists, separating the extremely elongated single C–C bond distances from the shortest of the intermolecular pancake bonds as illustrated in Figure 1. The discovery by Kubo et al. has reduced this forbidden range by increasing the lower limit to about 2.04 Å. The presented analysis of a wide-ranging selection of molecules and molecular models predicts that the upper limit of extremely stretched C–C single bonds should be revised to about 2.2 Å. The trends described in this work show that the strengths of the extremely stretched C–C bonds decrease nearly linearly with increasing bond length, parallel with the decrease of the computed WBI values. Importantly, no diradical character was exhibited in the designed target molecules with computed equilibrium bond lengths exceeding 2.0 Å. Only in molecules that adopt a twisted configuration was the bond broken, creating a diradical. Finally, we highlight that 13C NMR chemical shift values depend sensitively on the length of the extremely elongated C–C bond, potentially providing a tool for their characterization.1

Acknowledgments

Support by the U.S. National Science Foundation for this research (M.K.: Grant No. CHE-2107820; H.L.: Grant No. CHE-2107820) is gratefully acknowledged. M.K. and E.J.J.K. are grateful for ample computer time provided by the High-Performance Computer Center, Georgetown University. H.L. and J.C. are grateful for ample computer time provided by the High-Performance Computer Center of the Texas Tech University.

Supporting Information Available

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.jpca.3c01209.

  • Figures of atomic numbering, predicted Raman and 13C NMR spectra, and details of DFT calculations (PDF)

The authors declare no competing financial interest.

Supplementary Material

jp3c01209_si_001.pdf (432.6KB, pdf)

References

  1. Kubo T.; Suga Y.; Hashizume D.; Suzuki H.; Miyamoto T.; Okamoto H.; Kishi R.; Nakano M. Long carbon–carbon bonding beyond 2 Å in tris(9-fluorenylidene)methane. J. Am. Chem. Soc. 2021, 143, 14360–14366. 10.1021/jacs.1c07431. [DOI] [PubMed] [Google Scholar]
  2. Li J.; Pang R.; Li Z.; Lai G.; Xiao X.-Q.; Müller T. Exceptionally Long C–C Single Bonds in Diamino-o-Carborane as Induced by Negative Hyperconjugation. Angew. Chem., Int. Ed. 2019, 58, 1397–1401. 10.1002/anie.201812555. [DOI] [PubMed] [Google Scholar]
  3. Ishigaki Y.; Shimajiri T.; Takeda T.; Katoono R.; Suzuki T. Longest C–C Single Bond among Neutral Hydrocarbons with a Bond Length beyond 1.8 Å. Chem 2018, 4, 795–806. 10.1016/j.chempr.2018.01.011. [DOI] [Google Scholar]
  4. Kawai H.; Takeda T.; Fujiwara K.; Inabe T.; Suzuki T. Exceptionally Large Difference in Bond Length among Conformational Isomorphs of a Hexaphenylethane Derivative with a Dispiropyracene Skeleton. Cryst. Growth Des. 2005, 5, 2256–2260. 10.1021/cg050064r. [DOI] [Google Scholar]
  5. Toda F.; Tanaka K.; Stein Z.; Goldberg I. Extremely Long C-C Bonds in Strained 1,1,2,2-Tetraphenylcyclobutaarenes: 3,8-Dichloro-1,1,2,2-tetraphenylcyclobuta[b]naphthalene, C36H24Cl2, and 3,6,9,10-Tetrachloro-4,5-dimethyl-1,1,2,2,7,7,8,8-octaphenyldicyclobuta[b,h]phenanthrene Toluene Solvate, C68H46Cl4.1.5C7H8. Acta Crystallogr., Sect. C: Cryst. Struct. Commun. 1996, 52, 177–180. 10.1107/S0108270195011097ACCESS. [DOI] [Google Scholar]
  6. Kammermeier S.; Jones P. G.; Herges R. [2+2] Cycloaddition products of tetradehydrodianthracene: experimental and theoretical proof of extraordinary long C-C single bonds. Angew. Chem., Int. Ed. 1997, 36, 1757–1760. 10.1002/anie.199717571. [DOI] [Google Scholar]
  7. Tanaka K.; Takamoto N.; Tezuka Y.; Kato M.; Toda F. Preparation and Structural Study of Naphtho- and Anthrocyclobutene Derivatives Which Have Extremely Long C–C Bonds. Tetrahedron 2001, 57, 3761–3767. 10.1016/S0040-4020(01)00249-6. [DOI] [Google Scholar]
  8. Schreiner P. R.; Chemish L. V.; Gunchenko P. A.; Tikbonchuk E. Y.; Hausmann H.; Serafin M.; Schlecht S.; Dahl J. E. P.; Carlson R. M. K.; Fokin A. A. Overcoming lability of extremely long alkane carbon–carbon bonds through dispersion forces. Nature 2011, 477, 308–311. 10.1038/nature10367. [DOI] [PubMed] [Google Scholar]
  9. Grimme S.; Schreiner P. R. Steric crowding can stabilize a labile molecule: solving the hexaphenylethane riddle. Angew. Chem., Int. Ed. 2011, 50, 12639–12642. 10.1002/anie.201103615. [DOI] [PubMed] [Google Scholar]
  10. Li J.; Pang R.; Li Z.; Lai G.; Xiao X.-Q.; Müller T. Exceptionally Long C–C Single Bonds in Diamino-o-Carborane as Induced by Negative Hyperconjugation. Angew. Chem., Int. Ed. 2019, 58, 1397–1401. 10.1002/anie.201812555. [DOI] [PubMed] [Google Scholar]
  11. Wu Y.; Zhang J.; Xie Z. How long a C–C bond can be? An example of extraordinary long C–C single bond in 1, 2-diarylamino-o-carborane. Chin. Chem. Lett. 2019, 30, 1530–1532. 10.1016/j.cclet.2019.04.034. [DOI] [Google Scholar]
  12. Mandal N.; Datta A. Molecular designs for expanding the limits of ultralong C–C bonds and ultrashort H···H non-bonded contacts. Chem. Commun. 2020, 56, 15377–15386. 10.1039/D0CC06690G. [DOI] [PubMed] [Google Scholar]
  13. Han Y. F.; Lin Y. J.; Jia W. G.; Wang G. L.; Jin G. X. Template-controlled topochemical photodimerization based on “organometallic macrocycles” through single-crystal to single-crystal transformation. Chem. Commun. 2008, 1807–1809. 10.1039/b717554j. [DOI] [PubMed] [Google Scholar]
  14. Alvarez S. A cartography of the van der Waals territories. Dalton Trans. 2013, 42, 8617–8636. 10.1039/c3dt50599e. [DOI] [PubMed] [Google Scholar]
  15. Isea R. What is the maximum stretching for C–C single bond?. J. Mol. Struct.: THEOCHEM 2001, 540, 131–138. 10.1016/S0166-1280(00)00721-1. [DOI] [Google Scholar]
  16. Lobato A.; Salvadó M. A.; Recio J. M.; Taravillo M.; Baonza V. G. Highs and Lows of Bond Lengths: Is There Any Limit?. Angew. Chem. 2021, 133, 17165–17173. 10.1002/ange.202102967. [DOI] [PMC free article] [PubMed] [Google Scholar]
  17. Cho D.; Ikabata Y.; Yoshikawa T.; Lee J. Y.; Nakai H. Theoretical study of extremely long yet stable carbon–carbon bonds: effect of attractive C···H interactions and small radical stabilization of diamondoids. Bull. Chem. Soc. Jpn. 2015, 88, 1636–1641. 10.1246/bcsj.20150264. [DOI] [Google Scholar]
  18. Zavitsas A. A. The relation between bond lengths and dissociation energies of carbon–carbon bonds. J. Phys. Chem. A 2003, 107, 897–898. 10.1021/jp0269367. [DOI] [Google Scholar]
  19. Suzuki T.; Takeda T.; Kawai H.; Fujiwara K. Ultralong C–C bonds in hexaphenylethane derivatives. Pure Appl. Chem. 2008, 80, 547–553. 10.1351/pac200880030547. [DOI] [Google Scholar]
  20. Yost S. R.; Head-Gordon M. J. Efficient implementation of NOCI-MP2 using the resolution of the identity approximation with application to charged dimers and long C–C bonds in ethane derivatives. Chem. Theory Comput. 2018, 14, 4791–4805. 10.1021/acs.jctc.8b00697. [DOI] [PubMed] [Google Scholar]
  21. Wiberg K. B. Application of the Pople-Santry-Segal CNDO Method to the Cyclopropylcarbinyl and Cyclobutyl Cation and to Bicyclobutane. Tetrahedron 1968, 24, 1083–1096. 10.1016/0040-4020(68)88057-3. [DOI] [Google Scholar]
  22. Tian Y.-H.; Kertesz M. Is There a Lower Limit to the CC Bonding Distances in Neutral Radical Pi-Dimers? The Case of Phenalenyl Derivatives. J. Am. Chem. Soc. 2010, 132, 10648–10649. 10.1021/ja103396h. [DOI] [PubMed] [Google Scholar]
  23. Preuss K. E. Pancake bonds: π-Stacked dimers of organic and light-atom radicals. Polyhedron 2014, 79, 1–15. 10.1016/j.poly.2014.04.005. [DOI] [Google Scholar]
  24. Piccoli P. M. B.; Schultz A. J.; Sparkes H. A.; Howard J. A.; Arif A. M.; Dawe L. N.; Miller J. S. [MeNC5H5]2[TCNE]2 (TCNE = tetracyanoethylene). Single crystal X-ray and neutron diffraction characterization of an exceptionally long 2.8 Å C–C bond. CrystEngComm 2009, 11, 686–690. 10.1039/b816817b. [DOI] [Google Scholar]
  25. Bianchi R.; Pilati T.; Simonetta M. The Influence of Substituents on the Equilbrium Bisnorcaradiene ⇄ [10]Annulene. The Crystal and Molecular Structure of 11-Methyltricyclo[4,4,1,01,6]-undeca-2,4,7,9-tetraene-11-carbonitrile. Acta Crystallogr., Sect. B: Struct. Crystallogr. Cryst. Chem. 1978, B34, 2157–2162. 10.1107/S0567740878007621. [DOI] [Google Scholar]
  26. Dorn H. C.; Yannoni C. S.; Limbach H.-H.; Vogel E. Evidence for a Nonclassical Structure of a 1,6- Methano[10]annulene: A Cryogenic 13C CPMAS NMR Study of the 11,11-Dimethyl Derivative. J. Phys. Chem. A 1994, 98, 11628–11629. 10.1021/j100096a002. [DOI] [Google Scholar]
  27. Humason A.; Zou W.; Cremer D. 11, 11-Dimethyl-1, 6-methano [10] annulene-- An Annulene with an Ultralong CC Bond or a Fluxional Molecule?. J. Phys. Chem. A 2015, 119, 1666–1682. 10.1021/jp5082966. [DOI] [PubMed] [Google Scholar]
  28. Morita Y.; Aoki T.; Fukui K.; Nakazawa S.; Tamaki K.; Suzuki S.; Fuyuhiro A.; Yamamoto K.; Sato K.; Shiomi D.; et al. A New Trend in Phenalenyl Chemistry: A Persistent Neutral Radical, 2,5,8-Tri-tert-butyl-1,3-diazaphenalenyl, and the Excited Triplet State of the Gable syn-Dimer in the Crystal of Column Motif. Angew. Chem. 2002, 114, 1871–1874. . [DOI] [PubMed] [Google Scholar]
  29. Morita Y.; Suzuki S.; Fukui K.; Nakazawa S.; Kitagawa H.; Kishida H.; Okamoto H.; Naito A.; Sekine A.; Ohashi Y.; et al. Thermochromism in an organic crystal based on the coexistence of σ-and π-dimers. Nat. Mater. 2008, 7, 48–51. 10.1038/nmat2067. [DOI] [PubMed] [Google Scholar]
  30. Tian Y.-H.; Huang J. S.; Kertesz M. Fluxional σ-Bonds of 2,5,8-Tri-tert-butyl-1,3-diazaphenalenyl Dimers: Stepwise [3,3], [5,5] and [7,7] Sigmatropic Rearrangements via π-Dimer Intermediates. Phys. Chem. Chem. Phys. 2010, 12, 5084–5093. 10.1039/b925259b. [DOI] [PubMed] [Google Scholar]
  31. Grimme S.; Antony J.; Ehrlich S.; Krieg H. A consistent and accurate ab initio parameterization of density functional dispersion correction (DFT-D) for the 94 elements H-Pu. J. Chem. Phys. 2010, 132, 154104 10.1063/1.3382344. [DOI] [PubMed] [Google Scholar]
  32. Szalay P. G.; Bartlett R. J. Multi-reference averaged quadratic coupled-cluster method: a size-extensive modification of multi-reference CI. Chem. Phys. Lett. 1993, 214, 481–8. 10.1016/0009-2614(93)85670-J. [DOI] [Google Scholar]
  33. Doehnert D.; Koutecky J. Occupation Numbers of Natural Orbitals as a Criterion for Biradical Character. Different Kinds of Biradicals. J. Am. Chem. Soc. 1980, 102, 1789–1796. 10.1021/ja00526a005. [DOI] [Google Scholar]
  34. There are other definitions of diradicaloid character, see; Stuyver T.; Chen B.; Zeng T.; Geerlings P.; de Proft F.; Hoffmann R. Do Diradicals Behave Like Radicals?. Chem. Rev. 2019, 119, 11291–11351. 10.1021/acs.chemrev.9b00260. [DOI] [PubMed] [Google Scholar]
  35. Grimme S.; Hansen A. A Practicable Real-Space Measure and Visualization of Static Electron-Correlation Effects. Angew. Chem., Int. Ed. 2015, 54, 12308–12313. 10.1002/anie.201501887. [DOI] [PubMed] [Google Scholar]
  36. Bauer C. A.; Hansen A.; Grimme S. The Fractional Occupation Number Weighted Density as a Versatile Analysis Tool for Molecules with a Complicated Electronic Structure. Chem. - Eur. J. 2017, 23, 6150. 10.1002/chem.201604682. [DOI] [PubMed] [Google Scholar]
  37. Harding L. B.; Goddard W. A. III Generalized valence bond description of the low-lying states of formaldehyde. J. Am. Chem. Soc. 1975, 97, 6293–6299. 10.1021/ja00855a001. [DOI] [Google Scholar]
  38. Shepard R.The Multiconfiguration Self-Consistent Field Method. In Advances in Chemical Physics; MLawley K. P., Ed.; Ab Initio Methods in Quantum Chemistry-II; Wiley: New York, 1987; Vol. 69, pp 63–200. [Google Scholar]
  39. Head-Gordon M. Characterizing unpaired electrons from the one-particle density matrix. Chem. Phys. Lett 2003, 372, 508–511. 10.1016/S0009-2614(03)00422-6. [DOI] [Google Scholar]
  40. Frisch M. J.; Trucks G. W.; Schlegel H. B.; Scuseria G. E.; Robb M. A.; Cheeseman J. R.; Scalmani G.; Barone V.; Petersson G. A.; Nakatsuji H.. et al. Gaussian 16, Revision A.03; Gaussian, Inc.: Wallingford CT, 2016.
  41. Neese F.; Wennmohs F.; Becker U.; Riplinger C. The ORCA quantum chemistry program package. J. Chem. Phys. 2020, 152, 224108 10.1063/5.0004608. [DOI] [PubMed] [Google Scholar]
  42. Lischka H.; Müller T.; Szalay P. G.; Shavitt I.; Pitzer R. M.; Shepard R. Wiley Interdiscip. Rev.: Comput. Mol. Sci. 2011, 1, 191. [Google Scholar]
  43. Lischka H.; Shepard R.; Müller T.; Szalay P. G.; Pitzer R. M.; Aquino A. J.; Araújo do Nascimento M. M.; Barbatti M.; Belcher L. T.; Blaudeau J. P.; et al. The generality of the GUGA MRCI approach in COLUMBUS for treating complex quantum chemistry. J. Chem. Phys. 2020, 152, 134110 10.1063/1.5144267. [DOI] [PubMed] [Google Scholar]
  44. Michl J. Relationship of bonding to electronic spectra. Comments. Acc. Chem. Res. 1990, 23, 127–128. 10.1021/ar00173a001. [DOI] [Google Scholar]
  45. Mota F.; Miller J. S.; Novoa J. J. Comparative Analysis of the Multicenter, Long Bond in [TCNE]·– and Phenalenyl Radical Dimers: A Unified Description of Multicenter, Long Bonds. J. Am. Chem. Soc. 2009, 131, 7699–7707. 10.1021/ja9002298. [DOI] [PubMed] [Google Scholar]
  46. Cui Z.-h.; Lischka H.; Beneberu H. Z.; Kertesz M. Rotational barrier in phenalenyl neutral radical dimer: Separating pancake and van der Waals interactions. J. Am. Chem. Soc. 2014, 136, 5539–5542. 10.1021/ja412862n. [DOI] [PubMed] [Google Scholar]
  47. Wolinski K.; Hilton J. F.; Pulay P. Efficient Implementation of the Gauge-Independent Atomic Orbital Method for NMR Chemical Shift Calculations. J. Am. Chem. Soc. 1990, 112, 8251–60. 10.1021/ja00179a005. [DOI] [Google Scholar]
  48. Mayer I. Bond order and valence indices: A personal account. J. Comput. Chem. 2007, 28, 204–221. 10.1002/jcc.20494. [DOI] [PubMed] [Google Scholar]
  49. Nishiuchi T.; Uno S.; Hirao Y.; Kubo T. Intramolecular Interaction, Photoisomerization, and Mechanical C–C Bond Dissociation of 1,2-Di(9-anthryl)benzene and Its Photoisomer: A Fundamental Moiety of Anthracene-Based π-Cluster Molecules. J. Org. Chem. 2016, 81, 2106–2112. 10.1021/acs.joc.6b00134. [DOI] [PubMed] [Google Scholar]

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