Möbius aromaticity

In

Intermediate 5 was a mixture of 2

It was pointed out by Henry Rzepa that the conversion of intermediate 5 to 6 can proceed by either a Hückel or a Möbius transition state.^[6]

The difference was demonstrated in a hypothetical

Another interesting system is the cyclononatetraenyl cation explored for over 30 years by

In 2005 the same P. v. R. Schleyer

A Hückel-Möbius aromaticity switch (2007) has been described based on a 28 pi-electron porphyrin system:^[12][13]

The

In 2014, Zhu and Xia (with the help of Schleyer) synthesized a planar Möbius system that consisted of two pentene rings connected with an osmium atom.[14] They formed derivatives where osmium had 16 and 18 electrons and determined that Craig–Möbius aromaticity is more important for the stabilization of the molecule than the metal's electron count.

Transition states

In contrast to the rarity of Möbius aromatic ground state molecular systems, there are many examples of

From the figure above, it can also be seen that the interaction between two consecutive ${\displaystyle p_{z}}$ AOs is attenuated by the incremental twisting between orbitals by ${\displaystyle \cos \omega }$, where ${\displaystyle \omega =\pi /N}$ is the angle of twisting between consecutive orbitals, compared to the usual Hückel system. For this reason resonance integral ${\displaystyle \beta ^{\prime }}$ is given by

${\displaystyle \beta ^{\prime }=\beta \cos(\pi /N)}$,

where ${\displaystyle \beta }$ is the standard Hückel resonance integral value (with completely parallel orbitals). Nevertheless, after going all the way around, the Nth and 1st orbitals are almost completely out of phase. (If the twisting were to continue after the ${\displaystyle N}$th orbital, the ${\displaystyle (N+1)}$st orbital would be exactly phase-inverted compared to the 1st orbital). For this reason, in the Hückel matrix the resonance integral between carbon ${\displaystyle 1}$ and ${\displaystyle N}$ is ${\displaystyle -\beta ^{\prime }}$.
For the generic ${\displaystyle N}$ carbon Möbius system, the Hamiltonian matrix ${\displaystyle \mathbf {H} }$ is:

${\displaystyle \mathbf {H} ={\begin{pmatrix}\alpha &\beta '&0&\cdots &-\beta '\\\beta '&\alpha &\beta '&\cdots &0\\0&\beta '&\alpha &\cdots &0\\\vdots &\vdots &\vdots &\ddots &\vdots \\-\beta '&0&0&\cdots &\alpha \end{pmatrix}}}$.

Eigenvalues for this matrix can now be found, which correspond to the energy levels of the Möbius system. Since ${\displaystyle \mathbf {H} }$ is a ${\displaystyle N\times N}$ matrix, we will have ${\displaystyle N}$ eigenvalues ${\displaystyle E_{k}}$ and ${\displaystyle N}$ MOs. Defining the variable

${\displaystyle x_{k}={\frac {\alpha -E_{k}}{\beta '}}}$,

we have:

${\displaystyle {\begin{pmatrix}x_{k}&1&0&\cdots &-1\\1&x_{k}&1&\cdots &0\\0&1&x_{k}&\cdots &0\\\vdots &\vdots &\vdots &\ddots &\vdots \\-1&0&0&\cdots &x_{k}\end{pmatrix}}\cdot {\begin{pmatrix}c_{1}^{(k)}\\c_{2}^{(k)}\\c_{3}^{(k)}\\\vdots \\c_{N}^{(k)}\\\end{pmatrix}}=0}$.

To find nontrivial solutions to this equation, we set the determinant of this matrix to zero to obtain

${\displaystyle x_{k}=-2\cos {\frac {(2k+1)\pi }{N}}}$.

Hence, we find the energy levels for a cyclic system with Möbius topology,

${\displaystyle E_{k}=\alpha +2\beta ^{\prime }\cos {\frac {(2k+1)\pi }{N}}\quad (k=0,1,\ldots ,\lceil N/2\rceil -1)}$.

In contrast, recall the energy levels for a cyclic system with Hückel topology,

${\displaystyle E_{k}=\alpha +2\beta \cos {\frac {2k\pi }{N}}\quad (k=0,1,\ldots ,\lfloor N/2\rfloor )}$.

See also

• Barrelene
• Baird's rule
• Bicycloaromaticity

References