There has been a recent revival of interest in Möbius topologies in aromatic conjugated molecules. Several suggestions for Möbius-like systems in small and medium sized rings have been made,

A clear maximum in the adjacent CCCC dihedrals is observed, indicating a significant degree of localisation of the twist. Two maxima are observed for the HCCH measurements, corresponding to two circuits of the ring back to the starting point, as is of course appropriate for a Möbius ring. We note particularly that the two maxima in this case are not equal in value. Inspection of the 3D geometry reveals that although there is only a single periphery, localisation of the twist means it has both an inside and an outside edge, each with different properties. This observation in turn reflects on another well known property of formal Möbius strips, their non orientable properties, i.e. the distinction between left- and right-handedness cannot be preserved consistently over the whole surface. In a time-independent analysis, the localisation of the twist means the topology is indeed orientable. We also note that whilst the maximum value of dihedral angle between two adjacent benzo rings is approximately 35 degrees, the minimum value is actually of opposite sign. The difference between the maximum and minimum values increases as the ring size decreases. It remains to be established whether this localisation phenomenon will vanish for an infinitely large cyclacene.

To evaluate whether the twist localisation was a feature of
the Hamiltonian used, we also re-optimised for n=8 at the *ab
initio* level using an STO-3G or 3-21G bases, obtaining
essentially identical results compared with AM1 (Figure 2),
implying the localisation may be insensitive to the precise
nature of the potential energy function. Further analysis of
the relationship between the potential function and propensity
towards twist localisation will be reported in a subsequent
article.

Localisation of the twist also impacts upon other computed
properties of the cyclacene surface. The highest occupied
molecular orbital (HOMO) and HOMO-1 (Figure 2) differ in energy
by 0.2 eV, and are both delocalised over approximately 12 of
the 15 benzo rings, but the HOMO (and LUMO) has little density
in the region of twist, whilst the lower energy HOMO-1 shows
complementary behaviour, having larger coefficients in the
region of the twist. The conventional expectation is that
reducing orbital overlap by geometrical distortion will raise
rather than lower the energy of a molecular orbital. This
orbital behaviour are also seen for the *ab initio* 3G and
3-21G wavefunction. We note here a
recent report^{5} relating to the dynamics of creation
and annihilation of soliton pairs upon photo excitation of
polyene chains. The analogy with twist and charge localisation in
the Möbius cyclacene system is currently being investigated.
Finally in this section, we note that the computed molecular
electrostatic isopotential (Figure 3) indicates the negative
region of the potential occurs on the inside concave surface of
the aromatic p system.

To probe the factors that might affect the characteristics of the twist in these systems, we also looked at the properties of the charged systems, arguing that adding or removing electrons would remove electrons from bonding p-p orbitals, or add them to antibonding orbitals, hence making the backbone more flexible and potentially influencing the ability of the ring to twist. The results indicate the single twist localisation is little changed by changing the electron occupancy by twelve electrons (Figure 7). The charged double twist systems (Figure 8) show that the effect of the charge is to increase the difference between the minimum and maximum twist values, indicating that making the benzo-ring more flexible does have a small effect on the localisation. If greater flexibility increases the localising effect, then more rigidity would be required to remove it. It is difficult to see how this could be achieved chemically. Finally in this series we note the charged triple twist systems (Figure 9) show the localisation phenomenon to be almost eliminated, this being more true of the negatively charged system.

n | Charge | t | Energy, kcal mol^{-1} |
---|---|---|---|

8 | 0 | 1 |
528.0
(-1205.5326)^{a}/[1213.6331]^{
b} |

15 | 0 | 1 | 551.0 |

15 | +6 | 1 | 2168.4 |

15 | -6 | 1 | 914.0 |

15 | 0 | 2 | 594.4 |

15 | +6 | 2 | 2230.0 |

15 | -6 | 2 | 975.2 |

15 | 0 | 3 | 701.2 |

15 | 0 | 3 | 728.1 |

15 | +6 | 3 | 2293.7 |

15 | -6 | 3 | 1040.9 |

Figure 2. HCCH and CCCC Dihedral angles for Cycloacenes, n=15, t=1 and n=8, t=1.

Figure 3. (a) AM1 HOMO (-6.73ev) and (b) HOMO-1 (-6.93eV) for Cyclacene, n=15, t=1.

Figure 4. AM1 Computed Molecular Electrostatic Potential for Cyclacene, n=15, t=1.

Figure 5. CCCC Dihedral angles for Cyclacene, n=15, t=1-3.

Figure 6. Peripheral C-C bond lengths Cyclacene, n=15, t=3 for (a) low energy form (b) high energy form.

Figure 7. HCCH Dihedral angles for Cyclacene, n=15, t=1, with charge 0, +6 and -6.

Figure 8. CCCC Dihedral angles for Cyclacene, n=15, t=2, with charge 0, +6 and -6.

Figure 9. CCCC Dihedral angles for Cyclacene, n=15, t=3, with charge 0, +6 and -6.

Figure 10. Calculated Mulliken charge distributions on the H atoms for Cyclacene, n=15, t=1, with charge 0, +6 and -6.

Figure 11. Calculated Mulliken charge distributions on the combined HC atoms for Cyclacene, n=15, t=1, with charge 0, +6 and -6.

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