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. 2020 Dec 10;124(51):10849–10855. doi: 10.1021/acs.jpca.0c09692

Aromaticity of Even-Number Cyclo[n]carbons (n = 6–100)

Glib V Baryshnikov †,‡,*, Rashid R Valiev §,∥,*, Rinat T Nasibullin ⊥, Dage Sundholm ∥, Theo Kurten ∥, Hans Ågren †,#
PMCID: PMC7770816  PMID: 33301674

Abstract

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The recently synthesized cyclo[18]carbon molecule has been characterized in a number of studies by calculating electronic, spectroscopic, and mechanical properties. However, cyclo[18]carbon is only one member of the class of cyclo[n]carbons—standalone carbon allotrope representatives. Many of the larger members of this class of molecules have not been thoroughly investigated. In this work, we calculate the magnetically induced current density of cyclo[n]carbons in order to elucidate how electron delocalization and aromatic properties change with the size of the molecular ring (n), where n is an even number between 6 and 100. We find that the Hückel rules for aromaticity (4k + 2) and antiaromaticity (4k) become degenerate for large Cn rings (n > 50), which can be understood as a transition from a delocalized electronic structure to a nonaromatic structure with localized current density fluxes in the triple bonds. Actually, the calculations suggest that cyclo[n]carbons with n > 50 are nonaromatic cyclic polyalkynes. The influence of the amount of nonlocal exchange and the asymptotic behavior of the exchange–correlation potential of the employed density functionals on the strength of the magnetically induced ring current and the aromatic character of the large cyclo[n]carbons is also discussed.

Introduction

Since the recent synthesis of cyclo[18]carbon,1 a large number of studies of spectroscopic,2−4 aromatic,5−10 structural,5,10−14 mechanical,15 and electronic properties7,16−21 of cyclo[n]carbons have been reported. Recently, Anderson et al.(22) proposed a more efficient protocol for synthesizing cyclo[18]carbon. Thus, one can expect that also other cyclo[n]carbons will be synthesized in the near future. Cyclo[n]carbons absorb light in the near-UV region and have a number of low-lying excited states that can be reached by dipole-forbidden electronic transitions from the ground state, implying that the de-excitation from them is not expected to lead to any luminescence.2 Cyclo[n]carbons should on the other hand exhibit extraordinary reactivity because of the −C≡C triple bonds, implying that they may be used as precursors for graphene allotropes.23 It has also been suggested that cyclo[18]carbon can function as an electron acceptor24 and form metal complexes21 and catenanes.6 The reactivity of cyclo[n]carbons also depends on their degree of aromaticity, that is, to the extent of the delocalization of the π electrons around the ring. The aromaticity of cyclo[n]carbons has been studied in a number of works focusing on the double antiaromaticity/aromaticity of molecules with (4k) and (4k + 2) (k = 1–7) π electrons.5−10,25−27 The studies show that for small n values, the (4k + 2) cyclo[n]carbons are indeed doubly aromatic with two subsystems of the delocalized (4k + 2) π electrons. The cyclo[n]carbons were found to have delocalized π electrons in the molecular plane inside and outside the ring, which is denoted as “in” in Scheme 1. There is an independent delocalized π-electron system out of the molecular plane above and below the ring, denoted as “out” in Scheme 1. For cyclo[18]carbon, the two delocalized donut-like π-electron systems contain 18π electrons each.5,7,10,25 The two π-electron systems are not identical because the pin orbitals on the outside of the ring have a slightly smaller overlap than the pout orbitals because of the bent curvature of the ring.

Scheme 1. Electronic Structure of Cyclo[n]carbons (Left) and Formal Chemical Structure of Cyclo[18]carbon (Right).

Scheme 1

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Fowler et al. studied the double aromaticity of cyclo[n]carbons with n = 6–30 and the interplay between the πin and πout electron systems.25 They found that the aromatic properties of the two delocalized subsystems are independent, obeying the Hückel rule of aromaticity.25 The aromatic properties of cyclo[n]carbons have also been studied by calculating nucleus-independent chemical shift (NICS) values in the center of the molecular ring.9,25,28 Such NICS values are, however, not a very reliable aromaticity criterion because the radius of the ring increases with the increasing number of carbon atoms in the ring, making the ring center further distant from the delocalized electrons in the larger cyclo[n]carbons. The NICS values in the center of the ring therefore decrease when expanding the ring, even though the ring would sustain a ring current of the same strength as for a smaller ring.

Seenithurai and Chai29 recently reported a systematic density functional theory (DFT) study of the energy differences between the lowest singlet and triplet states and between the lowest singlet and quintet states of cyclo[n]carbons (n = 8–100). They also reported ionization potentials, electron affinities, fundamental gaps, occupation numbers of the frontier orbitals, and the relative energy with respect to the linear isomers. Their calculations yielded similar trends as obtained in earlier studies.8,25,28 For small cyclo[n]carbons with n < 32, there is a clear difference between the properties of the rings with (4k + 2) and (4k) carbon atoms, whereas for larger cyclo[n]carbons, the molecules with (4k + 2) and (4k) carbon atoms have similar properties. Even though they studied electronic properties that are related to aromaticity and electron delocalization, they did not explicitly discuss the Hückel aromaticity and antiaromaticity of the cyclo[n]carbons.

We recently reported calculations of aromatic properties of Hückel antiaromatic (4k) cyclo[n]carbons.30 The calculations of magnetically induced current densities showed that the ring current strength decreases with increasing n. For cyclo[44]carbon, the aromatic character switches from antiaromatic to aromatic, that is, the net ring current is paratropic for cyclo[40]carbon, while the ring current is unexpectedly diatropic for cyclo[44]carbon. The underlying reason for the switch from the antiaromatic to aromatic character was not elucidated.

In this work, we perform systematic studies of the electronic structure and the aromatic character of cyclo[n]carbons with n = 6–100. The calculations show that the Hückel antiaromatic (4k) and Hückel aromatic (4k + 2) cyclo[n]carbons become nonaromatic for n > 50. Thus, large cyclo[n]carbons can be considered as unsaturated carbon rings with localized single and triple bonds, which may be useful information for future synthesis of large cyclo[n]carbons.

Computational Details

The molecular structures of the cyclo[n]carbons (n = 6–100) were initially optimized at the DFT level using the M06-2X functional and the def2-TZVP basis set.31,32 It has previously been shown5,10 that only density functionals with a large amount of nonlocal Hartree–Fock exchange (>40%) yield the correct alternation of single and triple bonds in (4k + 2) cyclo[n]carbons starting from the C14 molecule, while the use of density functionals with no or a moderate amount of Hartree–Fock exchange results in the incorrect cumulenic structure for C14 and C18. Optimizations at the BHLYP/def2-TZVP32−34 and ωB97XD/def2-TZVP32,35 levels have been performed in order to investigate the unexpected oscillations in the strength of the magnetically induced ring current for cyclo[n]carbons in the range of n = 36–52 that were obtained at the M06-2X/def2-TZVP level. Calculations at the M06-L/def2-TZVP32,36 level (0% Hartree–Fock exchange) have also been carried out for the C36–C52 series in order to assess the role of Hartree–Fock exchange.

The applicability of the employed DFT methods for studies of large cyclo[n]carbons was checked by calculating the electronic structure of the singlet ground state of cyclo[n]carbons (n = 30, 40, 50, 60, 70, 80, 90, and 100) at the complete active space self-consistent field (CASSCF) level.37,38 For the (4k + 2) systems (C30, C50, C70, and C90), the (8, 8) active space was used (eight active electrons in eight active orbitals) and for the (4k) molecules (C40, C60, C80, and C100), the (12, 12) CASSCF scheme was employed. In the single-point CASSCF calculations, we used the 6-31G(d,p),39−41 6-311G(d,p),40−42 and cc-pVTZ43 basis sets and the BHLYP/def2-TZVP-optimized molecular structures because this level of theory gives the expected and most likely correct dependence of the strength of the magnetically induced ring current on the size of the ring.

The calculations showed that the studied molecules have a closed-shell singlet ground state dominated by a single-reference configuration. The CASSCF calculations were performed with the Firefly software.44,45 The GIMIC code46−48 was used for calculating the magnetically induced current densities and for integrating the ring current strengths. Gauge-independent magnetically induced current densities were calculated using density matrices obtained in the GIAO49 nuclear magnetic resonance (NMR) shielding calculations. The magnetic field was oriented along the z axis, perpendicular to the plane of the molecular ring. The strengths of the current densities were obtained by numerical integration of the current density passing a rectangular surface perpendicular to the molecule plane. The DFT calculations were performed with the Gaussian 16 software.50

Results and Discussion

Structure of Cyclo[n]carbons

The experimental polyalkyne structure of cyclo[18]carbon1,22 was obtained from the calculations at the DFT level when using functionals with a large amount of Hartree–Fock exchange.5,12 In this work, we have optimized the molecular structures using the M06-2X, BHLYP, and ωB97XD functionals, which yielded a polyalkyne structure for cyclo[18]carbon. The Cartesian coordinates of the molecular structures can be found in the Supporting Information. Cyclo[14]carbon is an intermediate between the cumulene structures of C6 and C10 and the polyalkyne-like structure of C18. The larger cyclo[n]carbons consisting of (4k + 2) carbon atoms have even more pronounced polyalkyne structures. The Peierls transition of the molecular structure has been reported to occur for cyclo[n]carbons with n = 14, 18, or 22 depending on the employed computational level.25,51,52 In our calculations using the M06-2X, BHLYP, and ωB97XD functionals, the Peierls transition occurs between C14 and C18 (Figure 1, Table S1). In addition to the strong bond length alternation (BLA) that appears in C18 and the larger cyclo[n]carbons (n = 22–100), the bond angle alternation (BAA) disappears for C18 and larger cyclo[n]carbons (Figure 1, Table S2).

Figure 1.

Figure 1

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BLA of cyclo[n]carbons as a function of n is shown in the left graph. The BAA of cyclo[n]carbons as a function of n is shown in the right graph. BLA and BAA have been calculated for the cyclo[n]carbons (n = 6–100) using the BHLYP, M06-2X, and ωB97XD functionals.

For (4k) cyclo[n]carbons, the BAA drastically changes from C12 (BAA = 30°) to C16 (BAA = 0.04°). All larger cyclo[n]carbons (after C16–C18), regardless of whether they are of the (4k) or (4k + 2) type, show in principle similar structures with significant BLA but without BAA (Tables S1 and S2). However, the BLA becomes saturated (the BLA difference between two nearest analogues is <0.001 Å) at C32–C34, which approximately matches the size of the cyclo[n]carbon where the magnetically induced current strength vanishes, showing the importance of BLA as a descriptor of the electron delocalization.

CASSCF Studies of Cyclo[n]carbons

In a recent paper, Seenithurai and Chai29 employed thermally assisted occupation DFT (TAO-DFT) calculations for investigating the ground electronic state of cyclo[n]carbons (n = 10–100). Based on the analysis of the occupation numbers of frontier molecular orbitals, they found that the polyradical nature increases with increasing size of the cyclo[n]carbon. In order to check this statement, we performed CASSCF calculations for each 10th cyclo[n]carbon starting from C30. The results are summarized in Table 1. The results of the CASSCF calculations are basis set independent and give occupation numbers of the active orbitals (x) that are close to 2.00 and 0.00, implying that the weight (w) of the closed-shell singlet determinant is also close to 1. Thus, we can conclude that all the studied cyclo[n]carbons possess a single-reference ground singlet electronic state in sheer contrast to the results obtained in TAO-DFT calculations.

Table 1. Occupation Number (x) of the Highest Molecular Orbital with an Occupation Number Near 2 Obtained in the CASSCF Calculation and the Weight (w) of Closed-Shell Determinant for the S0 State Using Different Basis Sets.

molecule 6-31G(d,p) 6-311G(d,p) cc-pvTZ
C30 x = 1.94; w = 0.95 x = 1.95; w = 0.94 x = 1.94; w = 0.94
C40 x = 1.94; w = 0.94 x = 1.94; w = 0.91 x = 1.94; w = 0.91
C50 x = 1.94; w = 0.94 x = 1.92; w = 0.94 x = 1.99; w = 0.99
C60 x = 1.94; w = 0.91 x = 1.92; w = 0.94 x = 1.99; w = 0.99
C70 x = 1.98; w = 0.96 x = 1.92; w = 0.94 x = 1.99; w = 0.99
C80 x = 1.94; w = 0.92 x = 1.92; w = 0.91 x = 1.99; w = 0.99
C90 x = 1.99; w = 0.99 x = 1.92; w = 0.90 x = 1.99; w = 0.99
C100 x = 1.99; w = 0.99 x = 1.91; w = 0.90 x = 2.00; w = 0.99
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Aromaticity of Cyclo[n]carbons Obtained with GIMIC (n = 6–34)

Applying the GIMIC approach to the (4k) and (4k + 2) cyclo[n]carbons (n = 6–100), one can analyze the distribution of the magnetically induced ring current. The ring current strength is a reliable indicator of the electron delocalization in cyclo[n]carbons (n = 6–100). Starting from the smallest Hückel aromatic (4k + 2) cyclo[n]carbon, namely, cyclo[6]carbon with k = 1, one can see that it sustains a net diatropic current of 14 nA T–1 at the M06-2X, BHLYP, and ωB97XD levels (Table S3), which means that C6 is aromatic. This agrees with several previous studies25,26 where C6 was found to be doubly aromatic because of the delocalization of two sextets of π electrons lying in and out of the molecular plane. Despite that the net current is diatropic for the C6 ring, a paratropic component can also be seen in the current density map (Figures S1 and S2).

The smallest Hückel antiaromatic (4k) cyclo[8]carbon (k = 2) considered here is clearly antiaromatic, sustaining a net ring current of −33.3 nA T–1 at the BHLYP level of theory. Ring current strengths of −42.3 and −35.7 nA T–1 were obtained at the M06-2X and ωB97XD levels, respectively (Table S3). The diatropic contribution to the ring current of C8 is negligibly small. The strength of the paratropic ring current decreases with n for the larger Hückel antiaromatic (4k) cyclo[n]carbons with k = 3–8. The ring current strength of cyclo[32]carbon is −4.6 nA T–1 at the BHLYP level. A similar value of −4.3 nA T–1 is obtained at the M06-2X level, while the ωB97XD calculations yield a ring current strength of 0.9 nA T–1. The topology of the current density shows that the current density tends to become more localized to the triple bonds with increasing k for the (4k) cyclo[n]carbons (k = 3–8). The diatropic contribution to the current density appears at the edge of the triple bonds, which means that the global electron delocalization is broken (Figure S2). The transition from delocalized to localized topology of the current density for the Hückel antiaromatic (4k) cyclo[n]carbons is clearly seen for C28, whose net paratropic ring current is predominantly delocalized over the ring, whereas the paratropic current density of C32 is predominantly localized to the triple bonds (Figure 2). A drastic transition is also seen at C28–C32, whose ring current strengths are −7.9/–4.6, −15.3/–4.3, and −5.4/–0.9 nA T–1 at the BHLYP, M06-2X, and ωB97XD levels, respectively (Table S3).

Figure 2.

Figure 2

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Magnetically induced current density maps for selected (4k + 2) (top) and (4k) (bottom) cyclo[n]carbons (n = 26, 28, 30, 32, 48, and 50) calculated at the BHLYP/def2-TZVP level. The red and blue arrows denote the circulation direction of the paratropic and diatropic magnetically induced ring currents, respectively.

The Hückel aromatic (4k + 2) cyclo[n]carbons behave similarly but with opposite signs, that is, the direction of the ring current. Judged from the ring current strength, the degree of aromaticity increases from C6 to C14 (Table S3), whereas starting from C18, the net diatropic ring current strength gradually declines with increasing n and becomes only 3.9 nA T–1 for C34 at the BHLYP level. Paratropic contributions appear at each single bond (i.e., between each pair of triple bonds, which are marked with pink dotted lines in Figure 2). The transition from globally delocalized to bond-localized topology of the current density is observed between C26 (delocalized) and C30 (localized). This transition is accompanied by a decrease in the net ring current strength from 10.8 to 6.6 nA T–1 at the BHLYP level. Thus, GIMIC calculations predict a gradual decline in the global electron delocalization with increasing n for the cyclo[n]carbons starting from (4k + 2) C26 and (4k) C28.

Aromaticity of Cyclo[n]carbons from the GIMIC Approach (n = 36–100)

The (4k) C48 and (4k + 2) C50 molecules have almost identical current density maps, as seen in Figure 2. Based on the GIMIC calculations, they and the larger cyclo[n]carbons are nonaromatic polyalkynes, which perfectly explains the previously observed saturation of electronic descriptors such as the ionization potential, electron affinity, singlet–triplet splitting, singlet–quintet splitting, and so forth of the larger cyclo[n]carbons.8,28,29

GIMIC calculations at the BHLYP level lead to a systematic decline in the ring current strength of the larger cyclo[n]carbons (n = 36–100). The ring current strengths calculated at the M06-2X level for the cyclo[n]carbons with n = 38–50 differ significantly from the expected trend obtained at the BHLYP level (Figure 3). The ring current strengths calculated in this range using the M06-2X functional are a strongly irregular function of n and not a regularly oscillating function as obtained at the BHLYP level (Figure 3), indicating some limitations of the M06-2X functional for calculating magnetically induced current densities, even though the BHLYP and M06-2X functionals have about the same amount of Hartree–Fock exchange. The two functionals yield similar molecular structures with practically the same BLA and BAA (Tables S1 and S2).

Figure 3.

Figure 3

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Strength of the magnetically induced current for C6–C44 calculated using different DFT functionals.

The multireference problem of the ground state of the cyclo[n]carbons could be ruled out based on CASSCF calculations as discussed above. A possible reason for the erroneous oscillations in the C38–C50 region obtained at the M06-2X level seems to be an unsuitable parametrization of the functional for calculations of magnetically induced current densities. In the previous GIMIC studies, we successfully used the M06-2X functional in calculations on aromatic and antiaromatic porphyrinoids.53−55 However, generally, the M06-2X functional exaggerates the diatropic contribution to the ring current,55,56 which we actually also observe for C60–C100. The strength of the ring current gradually increases from 0.7 nA T–1 for C44 to 2.4 nA T–1 for C100 (Figure 3).

The M06-2X functional was also criticized by Stoychev et al., who found that M06-2X was the worst functional for calculating NMR shielding constants among the employed ones.57 They found that the M06-L functional is much better aimed for such studies.57 We therefore performed current density calculations using the M06-L functional for the problematic C38–C50 molecules (Table S4) and found a similar asymptotic behavior as obtained with the BHLYP functional (the plot is inserted in Figure 3).

We also tested the ωB97XD functional and found a similar trend for C6–C44 as obtained at the BHLYP level. Calculations using the ωB97XD functional suggest that the ring current strength increases with n in the range of C44–C100. The ring current strength gradually increases from 0.6 nA T–1 for C44 to 6.8 nA T–1 for C100 (Figure 3). The unexpected increase in the ring current strength obtained with the ωB97XD functional is most likely an artifact, the reason of which has not been elucidated. For large cyclo[n]carbons (n > 44) that sustain localized single and triple bonds, the short-range Hartree–Fock exchange scheme may overestimate exchange interactions58 that might result in a monotonous increase in magnetically induced ring current strength for increasing size of the cyclo[n]carbons. The BHLYP functional yields reasonable qualitative and quantitative values for the magnetically induced ring current strength for the whole series of the studied Cn (n = 6–100) molecules, whereas the M06-2X and ωB97XD functionals yield an unexpected nonasymptotic behavior of the strength of the magnetically induced current density of C38–C50 and C44–C100, respectively. However, regardless of the employed DFT functional, the large Hückel antiaromatic (4k) and aromatic (4k + 2) cyclo[n]carbons have similar molecular structures and strengths of the magnetically induced ring current. Because the large cyclo[n]carbons sustain a very weak ring current, they are nonaromatic according to the ring current criterion.

Conclusions

A series of cyclo[n]carbons with an even number of carbon atoms in the range of n = 6–100 were studied computationally with the focus on an analysis of the magnetically induced current density, which was used for assessing their aromatic character. All studied cyclo[n]carbons (except C6, C10, and C14) have a sheer bond length alternation (BLA) corresponding to polyalkyne-type molecules. The saturation limit of the BLA, that is, when the difference in the BLA between two adjacent analogues is <0.001 Å, appears at C32–C34. The bond angle alternation (BAA) vanishes rapidly in the range of n = 6–14 and becomes negligibly small (<0.04°) for C16. Differences in the aromatic character of the (4k + 2) cyclo[n]carbons and the antiaromatic character of (4k) cyclo[4n]carbons are obtained until C34 and C32, respectively, which match exactly the saturation limit of the BLA. The BLA can thus be seen as an important structural factor affecting the electron delocalization in cyclo[n]carbons. Based on GIMIC calculations of the ring current strength, we conclude that the complete transformation of the Hückel antiaromatic (4k) and Hückel aromatic (4k + 2) systems into nonaromatic molecules occurs at n > 50, regardless of whether the molecule has (4k) or (4k + 2) π electrons in the delocalization pathway around the ring. The GIMIC calculations show that the transformation is due to a transition from a delocalized electronic structure to an electronic structure with localization in the triple bonds. The localized current density in the triple bonds suggests that the larger cyclo[n]carbons are unsaturated nonaromatic cyclic polyalkynes, which might be useful information when developing efficient protocols for synthesizing novel cyclo[n]carbons, that is, other than C18.

Acknowledgments

This work has been supported by the Olle Engkvist Byggmästare foundation (contract no. 189-0223) and the Ministry of Education and Science of Ukraine (project no. 0117U003908). The calculations were performed with computational resources provided by the High-Performance Computing Center North (HPC2N) in Umeå, Sweden, through the project “Multiphysics Modeling of Molecular Materials” SNIC 2019-2-41. The GIMIC calculations were carried out using the SKIF supercomputer at Tomsk State University. This project has also been funded by the Academy of Finland (projects 315600 and 314821). R.R.V. thanks the Tomsk Polytechnic University Competitiveness Enhancement Program (VIU-RSCABS-142/2019) for support.

Supporting Information Available

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.jpca.0c09692.

  • Structural descriptors BLA and BAA, current density plots and corresponding current strengths, and optimized geometries in Cartesian coordinates of the studied cyclo[n]carbons at different DFT levels (n = 6–100) (PDF)

The authors declare no competing financial interest.

Supplementary Material

jp0c09692_si_001.pdf (3.3MB, pdf)

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