Extremely Long C–C Bonds Predicted beyond 2.0 Å

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 D CC =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 the 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?(ii) What are the special features of these extremely long single bonds?
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 a diamino-o-carborane 2 following on the heels of others at 1.8 Å 3 , at 1.77 Å 4 , and around 1.7 Å somewhat earlier. 5,6,7,8,9 Theexample of a carborane contains a C-C distance between two six-coordinated carbons as long as 1.93 Å 10 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 as 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 1.8-2.0Å 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 why the longest two-electron single bond between two carbon atoms must break at about 2.0 Å. Alvarez has 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. 14On the theory 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 approximately 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. 16It is interesting that as Kubo et al. 1 noted, based on careful temperature-dependent XRD analysis, 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 have argued that this limit should be about 1.8 Å, slightly longer than suggested previously by Zavitsas 18 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 sp 3 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.The first question (i) can be addressed in a relatively straightforward manner by investigating the equilibrium geometries of 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 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. 20otwithstanding, 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 y 0 index of 0.128 in the ground state of this molecule.Scheme 2. Two VB structures of 1A (covalent) and 1Adr (diradical).Note the C 2v symmetry for both VB structures of the isolated molecule, 1A, as found in the crystal structure. 1 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 results.Scheme 3. Isomerization reaction involving twisting of the "wings" of some of the molecules discussed.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 D 12 , using accessible parameters in addition to the equilibrium bond distance (R e ), 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 studies 1,3,20 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 (ΔE ST ), which becomes small as the bond approaches dissociation. 22fore 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 D vdW =3.40 Å. 23 The shortest of these observed by XRD was for a dimer of tetracyanoethylene anion radical (TCNE -2 ) 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 PLY 2 , and a twoelectron 4-center (2e/4c) bond for (TCNE -2 ) 2 , should not be compared with the long single bonds in molecules shown in Scheme 1, or their analogs discussed here.
A further distinction relates to fluxional bonding.While the focus in this work is on very long equilibrium bond distances (R e ), 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 R XRD as long as 1.8 Å. 25,26,27 In the crystal structure of dimers of phenalenyl derivatives the observed R XRD =2.153 Å 28 is the result of large amplitude fluxional bonding, not to be confused by equilibrium bond distances. 29,30  target region of C-C bonds discussed in this paper is indicated by a green rectangle in Figure 1, which is larger than the typical stretched single C-C bond and shorter than pancake bonds.

Methods
Full geometry optimizations yielding the equilibrium stretched C-C bond length (R e ) have been performed with UB3LYP level of density functional theory (DFT) with 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 cluster 32 (MR-AQCC) calculations were run for ethane.Several descriptors were used to characterize the diradicaloid character of a molecule.The y 0 parameter as a descriptor of diradicaloid character was calculated according to the formula 33 y NOON where NOON LU is the natural orbital occupation number (NOON) for the lowest unoccupied orbital. 34In addition to the y 0 parameter, Fractional Orbital Density 35,36 (FOD) calculations B3LYP/6-311G+(d,p), with the recommended electronic Fermi temperature of T e =9000 K, were completed as another measure of diradicaloid character.Note that all FOD computations refer to the restricted DFT.The FOD analysis provides a quick measure of diradical character through N FOD , a parameter obtained by spatial integration of the FOD.To further probe the accuracy of the FOD analysis as a measure of 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. 37,38This 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, N U , in the relaxed dissociation of ethane.The N U values were obtained according to the nonlinear formula of Head-Gordon 39 as where n i is the occupation of the i th natural orbital (NO) and the sum is over all NOs.
these results, N FOD 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. 40,41The MR-AQCC calculations were performed with COLUMBUS. 42,43 e 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 this formula: Here  and  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. 44Similar approaches have been used for other weakly bonded systems. 45,46 where the triplet geometry was also optimized, they obtained a BDE of 113 kJ/mol (27 kcal/mol) 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  refers to the energy difference between a non-twisted and twisted conformer, as seen in Scheme 3. The BDE obtained in this manner is the following: The values for  obtained in this manner can be strongly affected by the differences in the strain energies of the two isomers and therefore turn out to be less useful than  . 13C NMR calculations were run for selected target molecules, whereby their 13 C 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 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 second column the "wings" of these butterfly shaped molecules.2. Key for target molecules within this study.Each target molecule corresponds to a number identifying the "body" and a letter for the "wings" possibly with additional substituents.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 maximum ring size of their "wings".Similarly, molecules with a number code 2 and 3 are related to molecules with the number code 1, except each number corresponds to different a "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 electrondonating 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 on their letter code A "wings".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 R e values in the range of 1.6-2.2Å, 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 R e 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. 3,5,6,7 Asa 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 of where the bond exists, but rather a distortion of the geometry of this molecule by slight twisting of its "wings", adopting a C 2 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, y 0 , and relatively low N FOD .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 appears to be pancakelike interactions between carbons between the two "wings", so a pancake bonding model 23 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 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 eclipsed 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 D 12 distances significantly elongated, surpassing many sterically hindered molecules.For example, 2E has an equilibrium D 12 of 2.228 Å which is longer than all 1A and 2A sterically hindered molecules with D 12 distances ranging from 2.048 to 2.192 Å.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 both had the effect of increasing D 12 .However, the effect on D 12 was more pronounced when shortening the body to 2A where D 12 increased by ~0.05 Å while lengthening the body marginally increased D 12 by less than 0.001 Å.However, because the body of all these Kubo-like molecules does not play a direct role in the bonding of D 12 , 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 D 12 .In contrast, the effect on D 12 due to variations 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 D 12 increases from 2.099 Å to 2.183 Å to 2.228 Å.These large increases in D 12 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 D 12 .
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 D 12 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, when adding electron-withdrawing groups to "bodies" of the target molecules, D 12 increased.Specifically, for the cyano substituted molecule (10A), a large increase in bond length was observed to 2.213 Å.
For all the target molecules in this study, correlations were constructed comparing bond length to a variety of parameters, including WBI, BDE ST , and N FOD .The respective WBI values correlate very well with D 12 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.Bond dissociation energies are physically well-defined quantities compared to bond orders which are not. 48However, as outlined in the methods section, a direct computation of the BDE in the presented cases is not possible.First, we display the  values in Figure 5 that can be used as surrogates of the BDE as per equation (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  computed in this manner becomes smaller than 10 kcal/mol at ~2.3 Å, which should be considered as the long limit of extremely stretched C-C bonds.However, molecule 10A with the computed R e = 2.213 Å still displays a significant  of 21.1 kcal/mol putting it on par with other very weak covalent bonds, such as the elongated bond (1.68 Å) present in 1,2di(9-anthryl)benzene. 49 Thus molecules on this long limit of extremely stretched C-C bonds still display qualities of bonding character.While there is no absolute cutoff for N FOD 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 non-twisted (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 2.1-2.3Å that are of interest due to their low N FOD values.These molecules are all derivatives of 2A, which interestingly all have longer D 12 distances than their corresponding 1A derivative counterparts.It is likely that these 2A derivatives have lower N FOD values because of their simplified "bodies"-which are less conjugated than 1A derivativesand 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 7. FOD density plots for 2A, 1A, and 2E calculated using B3LYP/def2-TZVP model chemistry (T e =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 N FOD 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 N FOD 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 N FOD is calculated by the integration of FOD over all space, N FOD 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 N FOD value between those of 2A and 2E.This hypothesis is confirmed by the N FOD values listed in Table 3, where for this series of molecules the values increase as follows: 1.37 e, 2.14 e, 3.18 e, 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 D 12 in 2E is too short for pancake bonding, the interaction between C1 and C2 must be covalent in nature.In contrast, the contacts D 21,23 and D 22,24 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 N FOD is to be expected.
For a selected group of molecules listed in Table 4 two minima were found: one with a non-twisted C 2v structure and another twisted C 2 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 C 2 conformer was lower in energy to the nontwisted C 2v conformer.Since Kubo et al. 1 determined that the higher-energy non-twisted 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 non-twisted 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 non-twisted C 2v isomer (2A) and a twisted C 2 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 lower than that of non-twisted 2A.Since 2A readily twists into its twisted conformer with slight distortions of D 12 , D 12 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  20-3-1-21 and  20-3-1-22 change little with an increase in D 12 .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 C 2v 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  orbitals that make up this elongated  bond, and thus rapidly breaking the central bond and the two above mentioned torsions differ.Then as D 12 increases with each point after the initial twisting at around 2.25Å up until around 2.50 Å, the molecule relaxes to its 2Atw conformer.In contrast, in the high symmetry pathway, the C 2v 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 C 2v symmetry where the central bond breaks without twisting by continual elongation of D 12 .At around 2.50 Å, however, the molecule twists into the 2Atw molecule, indicated by the black arrows in Figures 8 and 9.The misalignment of the -orbitals after twisting can be seen in Figure 10 for 2A.As the "wings" twist, these orbitals no longer overlap well, and thus the central D 12 bond breaks.Along the high symmetry pathway, with increasing D 12 , there is less orbital overlap up until 2.50 Å where the bond fully breaks losing electron sharing between the two carbons involved in the central bond.In the low symmetry reaction pathway, the "wings" twist breaking the C 2v symmetry and misaligning the  orbitals as early as 2.30 Å.As such, in the low symmetry pathway, the central bond breaks much earlier.It is expected that 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, 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 1.37 e to 2.48 e, in N FOD as 2A adopts a twisted conformation.While N FOD gradually increases with increasing D 12 , there is a sharp increase starting at around 2.25 Å as the molecule twists.This indicates that 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 spike in N FOD indicates the presence of a covalent C-C bond for 2A before any twisting takes place.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.
a There is only a twisted minimum 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  correlations where no C-C WBI or bond dissociation energy is expected.Furthermore, low WBI and high diradical character suggest each twisted molecule is in its diradical state without the presence of a D 12 bond.It should also be noted that the  values are positive for all molecules where both a twisted and non-twisted conformer was present, indicating that the twisted triplet (diradical) conformation is lower in energy than the non-twisted singlet molecule.Since the twisted isomer was found to always be 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 non-twisted conformer.In fact, Kubo et al. suggested 1 that 1A adopts the untwisted C 2v 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 non-twisted 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 non-twisted 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 D 12 .Dimer geometry calculations were completed for 2A, one of the simplest target molecules, to investigate non-bonding crystal packing effects, as described by Kubo et al. 1 Figure 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 non-twisted 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 non-twisted conformation, there are also non-bonding interactions that further stabilize the 2A dimer.In the non-twisted 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 non-bonding vdW interaction energy.In fact, this interaction energy for the 2A dimer (-23.6 kcal/mol) is almost twice as large as for the twisted dimer (-12.8 kcal/mol).When considering this large stabilizing energy for the non-twisted dimer, 2A would likely remain in its non-twisted 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 Kubo's observation that crystal packing effects stabilize 1A in its bonded, non-twisted form.While in-depth analysis for the isomerization reactions and packing was not completed for all the target molecules, these preliminary findings suggest that for all the molecules that exhibit lower-energy twisted isomers, the non-twisted conformation would be preferred in their crystal structure.The stabilization resulting from nonbonding vdW's interactions appears to be significant enough to favor the non-twisted bonded conformation of these target molecules. 13C NMR spectroscopy is a sensitive tool to explore the hybridization and environment of carbon atoms.Figure 14 displays the computed 13 C NMR chemical shifts for 2A in the bonded (C 2v ) and twisted diradicaloid (C 2 ) 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, S5, and 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 13 C NMR spectroscopy offers a tool to monitor these extremely elongated C-C single bonds.

Scheme 1 .
Scheme 1. Selected experimentally characterized examples of very long C-C single bonds.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.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 where even longer bond lengths, and (ii) whether these elongated bonds still display main characteristics of a C-C single chemical bond.

Figure 1 .
Figure 1.Schematic representation of a hypothetical histogram of unusual carbon-carbon bonds and contacts (D CC ).Molecules discussed in this work are encroaching on the "forbidden zone" 14 from the left

Figure 2
shows the MR-AQCC dissociation curve and the evolution of N U with increasing C-C distance.For comparison the FT-RDFT/B3LYP dissociation curve and N FOD 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 N FOD values are well described with the FT-RDFT method with values ~2 e in the dissociation region.This behavior coupled with the observation that N FOD values correlate well with N U values obtained at the MR-AQCC level indicates that the fractional occupation is well reproduced by the FT-RDFT method.Based on

Figure 2 .
Figure 2. Potential energy curves (in relation to the minimum geometry) for relaxed displacement along the C-C bond in ethane and N U values calculated with the MR-AQCC(PPMC)/6-311+G(d,p) and N FOD FT-RDFT/B3LYP/6-311G+(d,p) methods.

Figure 3 .
Figure 3. a) HOMO and 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 D 21,23 and D 22,24 in Scheme 4.

Figure 4 .
Figure 4. Correlation between D 12 and the WBI for molecules with extremely long covalent single C-C bonds.For selected data points, their corresponding molecules are shown.The non-filled blue data point refers to 2E.

Figure 5 .
Figure 5. Correlation between D 12 and  for molecules with extremely long covalent single C-C bonds.The data points for molecules 10A and 10D are indicated by arrows.The non-filled blue data point refers to 2E.

Figure 6 .
Figure 6.Correlation between D 12 and the N FOD for molecules with extremely long covalent single C-C bonds.The non-filled blue data point refers to 2E, see text.

Figure 6
Figure6illustrates the positive linear correlation between N FOD and D 12 .While there is no absolute cutoff for N FOD 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 non-twisted (in blue) and twisted (in orange) molecules and are consistent with those in Figures4 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 2.1-2.3Å that are of interest due to their low N FOD values.These molecules are all derivatives of 2A, which interestingly all have longer D 12 distances than their corresponding 1A derivative counterparts.It is likely that these 2A derivatives have lower N FOD values because of their simplified "bodies"-which are less conjugated than 1A derivativesand thus minimize delocalization of radical electrons.This can be confirmed by visualizing the FOD densities of 1A and 2A as seen in Figure7.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 8 .
Figure 8. Isomerization reaction torsional coordinates along a D 12 relaxed scan of 2A comparing torsions α and β and D 12 .The red bonds in the insert indicate the two disrotatory axes of torsion.

Figure 9 .
Figure 9. Isomerization reaction coordinate plot of 2A comparing energy at each step to D 12 .The point at D 12 =2.5 Å is the longest at which a C 2v structure could be optimized.For longer D 12 values the computations converge to the lower energy twisted C 2 structure as indicated by the black arrow.

Figure 11
depicts the HOMO molecular orbitals of 2A along the high and low symmetry pathways from D 12 =2.25 Å to 2.50 Å.

Figure 10 .
Figure 10.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  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 D 12 .

Figure 11 .
Figure 11.High and low symmetry alpha/beta HOMO molecular orbitals of 2A along 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.

Figure 13 .
Figure 13.Optimized dimer structures of 2A and 2Atw.The closest C-C distances between the wings of each dimer are displayed.

Table 1 .
List of target molecules in their non-twisted configurations investigated in this study.

Table 3 .
Key results for non-twisted target molecules from computational modeling calculated at the UB3LYP-GD3/6-311+G(d,p) level of theory.