Induced magnetic field in sp-hybridized carbon rings: analysis of double aromaticity and antiaromaticity in cyclo[2N]carbon allotropes

Abstract

The induced magnetic field of C_2N (N = 3–14) carbon rings was dissected to contributions from out-of-plane and in-plane π orbitals revealing two concurrent long range shielding or deshielding cones as a manifestation of the dual aromatic and antiaromatic character of C_4n+2 and of C_4n rings respectively. Aromaticity based on the magnetic criterion was evaluated with regard to the bonding pattern and geometrical characteristics that elucidate the influence of bond length and bond angle alteration on out-of-plane and in-plane magnetic responses. Ground state polyynic geometries of C_4n+2 rings exhibit comparable shielding cones to annulenes, decreasing the magnetic response with regard to the ring size and similar π_out and π_in diatropicity. Transition state cumulenic rings display increased aromaticity expressed by a very strong constant magnetic response and augmented π_out diatropicity with regard to π_in. The variations of the induced magnetic field are explained on the basis of frontier orbital interactions through rotational excitations, which enable further rationalization of the aromatic/antiaromatic behavior.

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Introduction

Since the discovery of buckminsterfullerene C₆₀ in 1985,¹ research on sp² carbon allotropes such as fullerenes, graphenes² and carbon nanotubes³ has sparked the interest on the design and synthesis of a plethora of novel carbon-based nanomaterials with exceptional diversity and applications in numerous fields such as optoelectronics, gas separation, energy conversion and shortage, catalysis and medicine.^4–12 Around the same time research on different carbon allotropes with mixed sp²–sp hybridization¹³ or solely sp hybridized carbon atoms¹⁴ has paved the way for the discovery of new all carbon species with promising applications such as graphdiynes^15–17 and carbon rings.¹⁸

Early in 1989 Diederich et al. reported the gas phase generation of a highly reactive C₁₈ cyclocarbon¹⁴ in which each carbon atom was bonded to two other carbons, in contrast to sp² hybridized allotropes where the bonding pattern involves three-coordinated carbon atoms. The main question that evolved to a controversy by conflicting theoretical studies was on the bonding pattern of C₁₈: is it polyynic with altering triple and single bonds, or cumulenic with equal bond lengths? Pure and hybrid DFT^19–21 as well as MP2^22,23 calculations predict cumulenic structures, whereas HF,^14,24 quantum Monte Carlo²⁵ and CCSD²⁶ calculations predict polyynic geometries. Further studies diagnosed that the DFT calculated bonding pattern depends on the amount of HF exchange of the hybrid functional,^27,28 whereas range-separated exchange nonempirical DFT schemes provide polyynic structures regardless of the type of functional that is used.²⁹ Finally in 2019 Kaiser et al.¹⁸ isolated by on-surface synthesis and structurally characterized C₁₈ using atomic force microscopy (AFM), revealing a polyynic structure with altering single–triple CC bonds putting an end to the controversy.^23,24 Moreover, the cumulenic structure has been predicted to be a transition state between two degenerate polyynic ground states with inverted single–triple bonds.^25,28,30

A unique characteristic of sp-hybridized carbon structures is the presence of two perpendicular π-electron systems³¹ extending the capabilities from sp²-hybridized carbon structures which sustain a single π-system. Diederich proposed that C₁₈ benefited from two distinctive sets of delocalized π electrons,¹⁴ one from out-of-plane (π_out) and the other from in-plane (π_in) oriented 18π orbitals (Fig. 1), both obeying Huckel's 4n + 2 rule, which is expected to exhibit interesting electron acceptor properties.²⁷ The presence of double aromaticity for C_4n+2 and antiaromaticity for C_4n cyclocarbons was confirmed via ring current analysis by Fowler et al.,¹⁹ while the large NICS values reported by Suresh²⁰ also imply the existence of double aromaticity given by the two perpendicular π-systems. The double aromaticity of small carbon rings and related boron–carbon molecules was also confirmed with the CMO-NICS analysis put forth by Schleyer et al.³² However, studies on the aromaticity of carbon rings were performed on cumulenic structures and the effect of bonding pattern on the magnetic response was not evaluated. Recently Baryshnikov et al.²⁸ employed the GIMIC method to measure the ring currents of both polyynic and cumulenic configurations of C₁₈ and reported that the polyynic structure induces a strong diatropic current which is however much weaker than the diatropic current of cumulene. Moreover, although the GIMIC method does not yet allow the dissection of ring current to contributions from different sets of orbitals, it obtained the current strengths of π_out and π_in sets by integrating the current density within distinctive contours and found that π_out orbitals induce stronger diatropic currents than π_in, both in polyynic and cumulenic geometries.

Fig. 1 Illustration of two orthogonal sets of p orbitals of C₁₈ ring displaying perpendicular to the plane (π_out) and in-plane (π_in) orientations.

Herein, we study the distinctive characteristics of double aromaticity and antiaromaticity based on the magnetic criterion³³ from cyclo[2N]carbon allotropes, given by their peculiar perpendicular π-systems. However we note that a thorough investigation of the aromaticity requires to take into consideration both electronic and energetic criteria.^34–36 We perform a detailed analysis of the magnetic response of C_2N (N = 3–14) rings which is directly related to the aromatic/antiaromatic behavior,^37–42 by means of canonical orbital contributions to the induced magnetic field (CMO-IMF) that allows the through-space visualization of shielding and deshielding cones induced by different sets of orbitals, as well as from each CMO.^43–46 Hence the magnetic field induced from each π_out and π_in set was visualized for polyynic and cumulenic geometries of C_4n+2 rings, as well as for those of C_4n rings, revealing the trends of the magnetic response of each set of MOs and the effect of ring size, bond length and bond angle. Our results show that polyynic geometries induce a cumulative magnetic field from both sets which is comparable to annulenes and declines with the evolution of ring size, whereas cumulenes induce a much stronger magnetic field independent of the ring size. Accordingly, C_4n rings display decreasing deshielding cones for both sets with regard to the ring size. These trends are associated with the inherent bonding pattern and geometrical characteristics of carbon rings, which are further explained on the basis of frontier orbital interactions through rotational excitations.

Computational methods

A number of various exchange–correlation functionals (XCs) were tested for the optimization of C₁₈ (see ESI,† Table S1). Functionals with a high percentage of HF exchange (HFX > 50%) yielded polyynic structures, whereas pure GGAs and functionals with a low percentage of HFX resulted in cumulenic structures with no bond length alteration (BLA) (Table S1, ESI†). In this work we have chosen the use of the range-separated hybrid ωB97XD⁴⁷ for the optimization of the C_2N (N = 3–14) molecules because it correctly reproduces the experimentally realized¹⁸ polyynic geometry of C₁₈ and it has been reported to have the best performance among other functionals for the description of polyynic structures.⁴⁸ Moreover, long-range corrected XC functionals like ωB97XD have been found to provide a better description of the cyclic electron delocalization in monocyclic and polycyclic aromatics with regard to conventional XC functionals.³⁶ All geometry optimizations were carried out with Gaussian09⁴⁹ using the 6-311++G(d,p) Gaussian basis set.

Nucleus independent chemical shieldings (NICS) were calculated within the GIAO formalism using several functionals, as detailed in ESI,† and the triple-ζ slater type basis set with two polarization functions (TZ2P) employing the ADF2019 software.^50,51 BHandHLYP⁵² was chosen for the quantitative analysis of dissected NICS_zz because it performs in very good agreement with ωB97XD (Table S2 and Fig. S2, ESI†) and it is available for the CMO analysis of the chemical shift in ADF. The computationally affordable PBE⁵³ was chosen for the qualitative analysis of the induced magnetic field. Although the PBE was found to overestimate the NICS, especially in polyynic structures, it accurately reproduces the qualitative features of the magnetic response with regard to the ring size and bonding pattern of C_2N rings (see ESI,† Fig. S2 and S3). Chemical shieldings were dissected to contributions from canonical MOs (CMOs) employing the NMR and EPR modules⁵⁴ of the ADF and the contributions of π_out and π_in sets of orbitals were constructed by the summation of the corresponding CMOs.

For the calculation of the induced magnetic field the molecules were placed on xy Cartesian plane and the chemical shielding was calculated in a 31³ cubic grid of points with a separation step of 0.5 Å. Extraction of CMO contributions at each point and generation of cube files per MO were performed with our custom MIMAF code.⁵⁵ Visualization of induced magnetic field isosurfaces was performed with the VMD.⁵⁶

Results and discussion

Geometrical characteristics

The ground state optimized geometries of C_4n+2 (n = 1–6) and C_4n (n = 2–6) molecules are shown in Fig. S1 (ESI†) and their corresponding geometrical parameters are given in Table 1. All molecules adopt C_(n/2)h point group symmetry representing polyynic structures with altering bond lengths and altering bond angles, except for C₆ and C₁₀ which adopt D_(n/2)h cumulenic structures with equal bond lengths and altering angles. The bond length alteration (BLA) ranges from 0.123 Å to 0.148 Å for all C_(n/2)h point group molecules, except for C₁₄ which presents a minimal BLA of 0.051 Å. On the other hand, bond angle alteration (BAA) is very large for small rings starting from 58.4° and 55.6° for C₆ and C₈, respectively, and decreases rapidly as the ring size increases, reaching a constant value of 1.7° for C_2N (N = 11–14), while the smallest BAA (0.4°) is observed in C₁₈. For C_4n molecules D_(n/2)h geometries with altering bond lengths and equal bond angles were found to be first-order saddle points, whereas higher symmetry D_nh geometries with equal bonds and angles were found to be higher-order saddle points for both C_4n+2 and C_4n molecules.

Table 1 Geometrical parameters of C_2N (N = 3–14) rings optimized at the ωB97XD/6-311++G(d,p) level

Molecule	Point Group	d 1 (Å)	d 2 (Å)	BLA^a	θ 1 (°)	θ 2 (°)	BAA^b
a Bond length alteration, d1 − d2. b Bond angle alteration, θ1 − θ2.
C6	D 3h	1.321		0.00	149.2	90.8	58.4
C8	C 4h	1.384	1.251	0.133	162.8	107.2	55.6
C10	D 5h	1.290	1.290	0.00	165.0	123.0	42.1
C12	C 6h	1.359	1.234	0.126	168.3	131.7	36.6
C14	C 7h	1.307	1.256	0.051	167.0	141.6	25.4
C14 TS	D 7h	1.281		0.00	168.1	140.5	27.6
C16	C 8h	1.364	1.216	0.148	158.9	156.1	2.8
C18	C 9h	1.346	1.223	0.123	160.2	159.8	0.4
C18 TS	D 9h	1.277		0.00	169.4	150.6	18.8
C20	C 10h	1.358	1.216	0.141	162.6	161.4	1.2
C22	C 11h	1.351	1.219	0.132	164.5	162.8	1.7
C22 TS	D 11h	1.276		0.00	171.0	156.3	14.8
C24	C 12h	1.355	1.216	0.139	165.8	164.2	1.7
C26	C 13h	1.352	1.217	0.135	167.0	165.3	1.7
C26 TS	D 13h	1.276		0.00	172.7	159.6	13.1
C28	C 14h	1.354	1.216	0.138	168.0	166.3	1.7

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For C_4n+2 (n = 3–6) molecules, QST2 and IRC calculations generated cumulenic geometries with equal bond lengths and altering bond angles, adopting D_(n/2)h point group symmetry, as transition states between two inversion related polyynic ground states²⁸ (Fig. 2). The transition states display increased BAA with regard to the respective ground states. The energy barrier for the inversion increases with the number of carbon atoms, starting from a marginal barrier of 0.12 kcal mol⁻¹ for C₁₄ and reaching 10.90, 24.42 and 37.54 kcal mol⁻¹ for C₁₈, C₂₂ and C₂₆ respectively.

Fig. 2 Relative IRC profiles of C₁₈, C₂₂ and C₂₆ depicting the interchange of single–triple bonds of polyynic ground states through a cumulenic transition state.

Total magnetic response

When an (anti)aromatic molecule is subjected to a uniform external magnetic field B⃑^ext, then closed circuits of mobile electrons induce a current density, which in turn induces a secondary magnetic field, B⃑^ind, either opposing (shielding) the external field in aromatic rings or reinforcing (deshielding) the external field in antiaromatic rings. In planar structures the molecule is oriented such that the molecular plane (xy) is perpendicular to the external field's direction (z), and hence the induced magnetic field is designated as . The induced magnetic field is typically visualized as isosurfaces of B^ind_z, forming long range shielding or deshielding cones in aromatic or antiaromatic rings respectively. The value of B^ind_z isosurface at any point is equal to the zz tensor component of nucleus independent chemical shift, NICS_zz.

NICS_zz values, presented in Table 2 and Fig. S2 (ESI†), reveal the characteristic features of carbon rings’ magnetic response. These features are as follows: (a) C_4n+2 molecules display large diatropic (negative) NICS_zz representative of aromatic character, whereas C_4n molecules display large paratropic (positive) NICS_zz representative of antiaromatic character; (b) NICS_zz of polyynic C_4n+2 (n = 3–6) and C_4n molecules decreases significantly as the ring size increases; (c) NICS_zz of the cumulenic C_4n+2 (n = 1–6) geometries increases with the ring size, showing a trend to reach a maximum value at larger rings; (d) accordingly for C_4n+2 molecules there is an increasing difference in NICS_zz between polyynic ground states and cumulenic transition states with an increment in the ring size; (e) the maximum NICS_zz among ground state C_4n+2 molecules is observed in C₁₄ which presents the minimum BLA.

Table 2 Total NICS_zz values (ppm) and contributions from π_out and π_in orbitals of C_2N carbon rings and reference molecules computed at the BHandHLYP/TZ2P level

Molecule	Total	πout	πin	Molecule	Total	πout	πin
C6	−37.7	−39.8	19.2	C8	130.9	110.5	28.2
C10	−65.2	−52.1	−3.3	C12	96.8	64.8	34.1
C14	−81.6	−49.9	−25.4	C16	57.9	27.0	31.7
C14 TS	−89.2	−56.7	−25.0	C20	33.4	15.5	18.7
C18	−44.7	−20.6	−16.4	C24	18.7	8.6	10.8
C18 TS	−101.5	−58.3	−37.2	C28	10.5	4.8	6.4
C22	−22.0	−10.7	−7.8	C8H8	107.2	90.2
C22 TS	−104.0	−58.1	−40.7
C26	−10.9	−5.4	−3.4
C26 TS	−99.3	−55.8	−39.0
C6H6	−15.9	−36.7
C18H18	−41.5	−48.0
C18H6	−8.1	−12.7	−15.7
B9H9	3.3	0.5	3.4

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The above remarks clarify controverting NICS studies of C_4n+2 rings.^19,20 The NICS values reported by Fowler et al. show an increment in diatropic NICS of C_4n+2 for n = 1–5 and then a significant decrement for n = 6, 7 due to molecular geometry obtained with B3LYP functional which predicts cumulenic structures for n = 1–5 and polyynic structures for n = 6, 7.¹⁹ On the other hand, Suresh and Remya reported increasing diatropic NICS values for all C_4n+2 (n = 1–7) trending to a maximum value due to cumulenic geometries obtained with MO6L functional.²⁰

Visualizations of B^ind_z induced by all electrons of C_2N molecules are presented in Fig. 3 using multiple clipped isosurfaces (±5 ppm and ±20 ppm) to depict the long-range effect, as well as a full isosurface of large B^ind_z values to illustrate specific characteristics of the magnetic response close to the molecular domain. C_4n+2 molecules induce strong long-range shielding cones representative of strong aromatic character. The long-range shielding cone (cyan, −5 ppm) of C_4n+2 molecules increases with the ring size but the actual trend is revealed by inspection of the strong response close to the molecular plane depicted with green isosurfaces (−50 ppm) in Fig. 3. In ground state polyynic structures the extension of the short-range response increases from C₁₀ to C₁₄ where it reaches a maximum span forming a uniform shielding cone, and then gradually decreases from C₁₈ to C₂₆, deforming to a toroidal shape inside the ring of C₂₆. On the contrary the strong shielding cone of cumulenic transition states retains its uniform long range shape. Therefore, the weakening of the magnetic response in ground states is attributable to polyynic bonding and is irrelevant to the ring size. However, the discrepancy in the magnetic response between ground and transition states for C₁₈, C₂₂ and C₂₆ is apparently evolving with the ring size. Thus, according to the magnetic criterion, the aromaticity of polyynic C_4n+2 decreases for n > 3 while it is retained for the respective cumulenic transition states.

Fig. 3 Isosurfaces of total B^ind_z of transition states D_(n/2)h C_4n+2 (n = 3–6) (bottom row), ground states C_(n/2)h C_4n+2 (n = 1–6) (middle row) and C_4n (n = 2–6) (top row).

C_4n molecules sustain strong long-range deshielding cones representative of strong antiaromatic character. However strong short-range response (+60 ppm) decreases with the evolution of ring size, starting with extended uniform deshielding cones in C₈ and C₁₂ and downgrading to a toroidal shape in C₂₄, which implies the weakening of antiaromaticity.

Magnetic responses of π_out and π_in orbitals

The total magnetic response suggests the (anti)aromatic character of C_2N carbon rings but does not give any clear information about the source of the long range (de)shielding cones and the individual role of each (para)diatropic circuit in the double aromatic character, because it is constructed by contributions from all π_out, π_in, σ and core orbitals. In order to gain insight into the origins and the duality of the (anti)aromatic character of C_2N carbon rings, we dissected the magnetic response to contributions from π_out and π_in sets of orbitals, which are exclusive sources of the strong long-range magnetic response.

In Fig. 4, the respective magnetic responses of π_out and π_in sets of orbitals of ground states C_4n+2 molecules are given. It is clear that for C₁₄ and larger rings, both π_out and π_in orbitals induce long range shielding cones, illustrating the dual source of their diatropicity. Smaller members C₆ and C₁₀ display no or a very weak in-plane magnetic response, respectively, due to a large bond angle alteration (BAA) which leads to a small in-plane orbital overlap. Specifically, the π_out orbitals of C₆ induce a shielding cone very similar to benzene and have comparable π_out–NICS_zz values (−39.8 and −36.7 ppm for C₆ and C₆H₆ respectively). The π_in orbitals of C₆ induce a weak short range diatropic response with a paratropic sphere inside the ring (+19.2 ppm at ring center), which is representative of localized CC σ bonding.⁴³ Therefore, C₆ displays equal diatropicity with benzene originating from the out-of-plane π system, and the large difference in their total NICS_zz does not arise from delocalized electrons but from localized σ bonds which have larger paratropic contributions in benzene. C₁₀ presents a strong long-range shielding cone induced from π_out orbitals with −52.1 ppm contributions at the ring center, and a considerably weaker shielding cone induced from π_in orbitals with only −3.3 ppm at the ring center, denoting a marginal in-plane π magnetic response due to a large BAA.

Fig. 4 Isosurfaces of π_out (top) and π_in (bottom) contributions to B^ind_z of polyynic ground states of C_(n/2)h C_4n+2 (n = 1–6) rings.

C₁₄ induces long range shielding cones from both π_out and π_in orbitals but the out-of-plane response is much stronger than that of the in-plane, as π_out forms an extended uniform −40 ppm isosurface, while π_in forms a condensed shielding toroid inside the ring and the π_out–NICS_zz (−49.9 ppm) is the double of π_in–NICS_zz (−25.4 ppm). This difference is attributed to the geometry of C₁₄, which presents the minimum BLA that favors the out-of-plane delocalization and significant BAA that hinders the in-plane delocalization. The corresponding transition state displays an almost identical magnetic response, both in terms of shielding cones and NICS_zz values, due to small structural differences in the two states.

For the ground states of larger rings C₁₈, C₂₂ and C₂₆, π_out and π_in display very similar shielding cones. The π_in–NICS_zz is only 2–4 ppm less diatropic than π_out–NICS_zz, denoting a marginally increased out-of-plane magnetic response. The diatropicity decreases with an increment in the ring size for both sets of orbitals. Specifically NICS_zz of π_out (π_in) decrease from −20.6 (−16.4) ppm in C₁₈ down to −10.7 (−7.8) and −5.4 (−3.4) ppm in C₂₂ and C₂₆ respectively.

In order to assess the C₁₈ diatropicity we used as reference three prototypical molecules with 18-π electrons and varying degrees of aromaticity, namely 18-annulene, 18-dehydroannulene and the BN analogue⁵⁷ B₉N₉, as presented in Fig. 5. Compared to the π response of 18-annulene (Fig. 5b), the π_out of C₁₈ induces a weaker shielding cone and presents less than half NICS_zz, denoting a considerable weaker out-of-plane magnetic response in C₁₈. If we consider both π_out and π_in orbitals of C₁₈, their cumulative shielding cone (Fig. 5a) is greater than B^ind_πz of 18-annulene, but this is due to the overestimation of PBE which predicts a total (π_out + π_in)–NICS_zz value 21 ppm to be more diatropic than NICS_πzz of C₁₈H₁₈, while with BHandHLYP it is 11.0 ppm less diatropic. Therefore, C₁₈ displays a similar or a slightly weaker diatropicity when compared to C₁₈H₁₈. Furthermore, 18-dehydroannule (Fig. 5c) is less aromatic than C₁₈ and induces a weaker π_out shielding cone (π_out–NICS_zz is −12.7 ppm), as well as a weak short-range paratropic response from 12 π_in electrons (π_in–NICS_zz is +15.7 ppm). Finally, B₉N₉ (Fig. 5d) is a characteristic example with extreme localized electrons, with both 18 π_out and 18 π_in electrons inducing weak short range diatropic cones strictly positioned on nitrogen atoms. Hence C_4n+2 (n > 3) carbon rings, although they exhibit dual source of diatropicity, show that their magnetic responses are comparable to those of classical π-aromatic annulenes.

Fig. 5 Isosurfaces of π contributions to B^ind_z of (a) C_9h C₁₈, (b) 18-annulene, (c) 18-dehydroannulene and (d) B₉N₉.

The magnetic responses of transition states, presented in Fig. 6, display constant strong shielding cones of both π_out and π_in. Specifically, π_out–NICS_zz retains a value of around −57 ppm, whereas π_in–NICS_zz amounts to ∼ −39 ppm for all transition states, implying favorable π_out delocalization. The difference in π_out and π_in can be explained by the increased BAA of cumulenic structures which hinders the in-plane overlap. Indeed, the third-order saddle point D_18h geometry of C₁₈ with zero BLA and BAA presents almost equal π_out and π_in NICS_zz values (−60.2 and −58.7 ppm respectively). However, π_in electrons in cumulenic geometries are still more diatropic than polyynic. Hence BLA is the main factor that constrains both π_out and π_in magnetic responses and BAA affects secondarily only the π_in magnetic response. Consequently, the cumulenic geometries are very diatropic from both sources of magnetic response.

Fig. 6 Isosurfaces of π_out (top) and π_in (bottom) contributions to B^ind_z of cumulenic transition states of D_(n/2)h C_4n+2 (n = 3–6) rings.

Concerning the antiaromatic C_4n (n = 2–6) rings, both π_out and π_in orbitals induce equivalent long range deshielding cones (Fig. 7), except for C₈ which exhibits only π_out deshielding cone and a weak short range paratropic response of π_in due to large BAA, which does not justify a paratropic in-plane ring current. The π_in response of C₁₄ is also weak due to large BAA but still forms a deshielding cone denoting weak in-plane paratropic current, whereas for larger rings π_out and π_in induce equivalent deshielding cones. The π_out deshielding cones of C₈ and C₁₂ are comparable to that of planar C₈H₈ but decline significantly for larger rings, as the π_in deshielding cones also do. Hence, C_4n (n = 3–6) rings exhibit a dual source of paratropicity representative of antiaromatic character which weakens significantly with an increment in the ring size.

Fig. 7 Isosurfaces of π_out (top) and π_in (bottom) contributions to B^ind_z of C_4n (n = 2–6) rings and planar C₈H₈.

CMO contributions

In order to elucidate the above remarks, we have to take into consideration the electronic structure of carbon rings, especially the frontier orbitals that principally determine the overall magnetic response. Dissection of the induced magnetic field to CMO contributions reveals that lower energy valence orbitals induce a diatropic response, whereas the highest occupied orbitals dictate the overall magnetic response.^43,44 Indeed in C_2N, the contributions of π_out and π_in to NICS_zz are linearly correlated to the contribution of the corresponding HOMOs (R² = 0.97, Fig. S4, ESI†). Hence the (para)diatropicity can be analyzed in terms of HOMO contributions, as given in Table S4 and Fig. S5, S6 (ESI†).

Generally, in antiaromatic molecules the HOMOs induce a very strong paratropic response that overwhelms the diatropicity of lower orbitals, whereas in aromatic molecules the HOMOs induce a very weak paratropic or diatropic response that adjusts the overall diatropicity and tunes the aromaticity. In turn, the paratropic response of HOMOs originates from symmetry allowed rotational excitations to unoccupied orbitals and its magnitude depends on the energy gap and the overlap of interacting orbitals.^{43,44,46,58–60}

In C_4n rings, the HOMO and LUMO of both out-of-plane and in-plane orientations are non-degenerate MOs with the same number of n nodal planes and the same symmetry. Hence the symmetry allowed HOMO_out/in → LUMO_out/in excitations, rotating the HOMOs by an angle of 2π/4n, to lead to an optimum overlap producing maximum paratropic response. For example, in C₁₆ the rotational excitations of HOMO_out and HOMO_in with 4 nodal planes rotated by 22.5° lead to a perfect overlap with LUMO_out and LUMO_in contributing with +63.3 and +64.4 ppm, respectively, to NICS_zz (Fig. 8b) and induce very strong long range deshielding cones (Fig. 8a) that dominate on the overall magnetic response. As the HOMO–LUMO gaps remain practically unchanged throughout the C_4n series (Table S4, ESI†), the decline in paratropicity originates from a weakening paratropic response of HOMOs and is attributed to the decrease in rotational overlap as the number of nodal planes increases.

Fig. 8 (a) Isosurfaces of HOMO_out and HOMO_in contributions to B^ind_z of C₁₆, ground state C_9h C₁₈ and transition state D_9h C₁₈. (b) Energy levels and contributions of HOMO_out/in → LUMO_out/in rotational excitations of C₁₆, ground state C_9h C₁₈ and transition state D_9h C₁₈.

In C_4n+2 rings, both HOMO_out and HOMO_in are doubly degenerate with n nodal planes, whereas the LUMOs are doubly degenerate with n + 1 nodal planes. In a perfect symmetry of D_nh point group a HOMO → LUMO rotational excitation would be symmetry forbidden, but in D_(n/2)h and C_(n/2)h point groups such excitations are symmetry allowed. However due to a different number of nodal planes the overlap is small, inducing a weak magnetic response that depends on the geometrical characteristics of the ring. For example in C_9h C₁₈, the frontier orbitals are distorted due to significant BLA and the HOMO → LUMO excitations display a small overlap inducing weak paratropic response (Fig. 8a) that diminishes the diatropicity of the lower energy orbitals, contributing with +27.8 and +30.6 ppm to NICS_zz of π_out and π_in, respectively (Fig. 8b). In contrast, in D_9h C₁₈ the zero BLA leads to a negligible rotational overlap of HOMO_out → LUMO_out excitation, resulting in the diatropic response of HOMO_out that adds to the overall diatropicity. On the other hand, HOMO_in → LUMO_in excitations of D_9h C₁₈ induce small paratropic contributions (+15.9 ppm) due to increased BAA that leads to a weaker diatropicity of π_in with regard to π_out. The same holds for the ground states of smaller members C₆, C₁₀ and C₁₄ with zero (or almost zero) BLA, which induce shielding cones from HOMO_out (Fig. S5, ESI†) and display increased π_out diatropicity. For C_4n+2 ground states the increment in HOMO nodal planes with the ring size results in a gradual increment in the small overlap and hence in the augmentation of HOMO paratropic contributions (Fig. S5, ESI†), causing a gradual decrement in the overall diatropicity.

Conclusions

The dissection of the induced magnetic field to contributions from out-of-plane and in-plane π orbitals revealed the dual aromatic and antiaromatic character of C_4n+2 and C_4n rings, respectively, according to the magnetic criterion. The magnetic response induced from distinctive sets of electrons is sensitive to the bonding pattern and geometrical features, where polyynic C_4n+2 ground states display similar out-of-plane and in-plane magnetic responses and comparable diatropicity to annulenes, while the cumulenic transition states exhibit a very strong diatropic character and an augmented out-of-plane magnetic response with regard to the in-plane magnetic response. Bond length alteration is the prime factor that constrains both π_out and π_in magnetic responses, whereas the bond angle alteration affects secondarily only the π_in magnetic response. A significant decline in magnetic response with an increment in the ring size is observed for both ground state aromatic and antiaromatic rings, which is attributed to decreasing paratropic contributions of HOMOs originating from rotational excitations.

Conflicts of interest

There are no conflicts to declare.

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Footnote

† Electronic supplementary information (ESI) available. See DOI: 10.1039/d0cp01252a

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