Abstract
The topology of twisted molecular rings is characterized by the linking number, which is equal to the sum of the twist—a local property of the molecular frame—and the writhe—a global parameter, which represents the bending of the molecular ring. In this work, we investigate a number of cyclic all-trans C40H40 annulenes with varying twisting numbers for a given linking number and their dications. The aromatic character is assessed by calculating ring-current strength susceptibilities using the gauge-including magnetically induced currents (GIMIC) method, which makes it possible to conduct a systematic study of the relation between the topology and aromaticity of twisted molecules. We found that the aromatic properties of the investigated Möbius twisted molecules are not only dependent on the linking number as previously suggested but also depend strongly on the partitioning of the linking number into the twist and writhe contributions.
Highlights
I n topology, a closed ribbon that is not a knot is fully characterized by its integer linking number Lk, which can intuitively be understood as the number of half twists that one end of an open ribbon has to undergo before it meets the other end to form a closed ribbon loop
The linking number in eq 1 is obtained as the sum of the twist and writhe of the closed ribbon with a given embedding.[4−6] Tw and writhing number (Wr) can take any real value, whereas the linking number is an integer
The energy levels of Möbius twisted annulenes and aromaticity rules can be qualitatively obtained by diagonalizing the Hückel molecular orbital (HMO) Hamiltonian for the molecular ring.[1,7−9] For rings with even Lk numbers, the wellknown (4n + 2) Hückel rule for aromaticity holds at the HMO level,[10,11] whereas when Lk is an odd integer, molecules with 4n π electrons are aromatic,[7,12] which is valid only for monocyclic systems
Summary
Considers only bond lengths, bond angles, and dihedrals. The structural changes due to the force field optimization are very small. Even though the differences in the HOMO−LUMO gaps are small, the studied molecules follow the general relation between ring-current strengths and HOMO−LUMO gaps.[55] Values for the HOMO−LUMO gaps calculated at the BHLYP and HF levels as well as the harmonic oscillator model of aromaticity (HOMA) index[56] are reported in the Supporting Information. Molecular rings with different topology were constructed with given Lk numbers of 0, 1, and 2 by varying Tw. The aromatic character as judged from the direction and strength of the magnetically induced ring currents was determined as a function of Tw for neutral and doubly charged molecular rings. The most twisted aromatic and antiaromatic molecular rings sustaining the strongest diatropic and paratropic ring-current strengths, respectively, were found to be the energetically lowest structures. HOMO−LUMO gaps, HOMA indices, current strengths at the B3LYP level, the total energy as a function of the twist number of the investigated molecules, and pictures with line integral convolution representations of the current densities (PDF)
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