Möbius aromaticity

In organic chemistry, Möbius aromaticity is a special type of aromaticity believed to exist in a number of organic molecules. In terms of molecular orbital theory these compounds have in common a monocyclic array of molecular orbitals in which there is an odd number of out-of-phase overlaps, the opposite pattern compared to the aromatic character to Hückel systems. The spatial configuration of the orbitals is reminiscent of a Möbius strip, hence the name. The smallest member of this class of compounds would be trans-benzene. Möbius molecular systems were considered in 1964 by Edgar Heilbronner by application of the Hückel method, but the first such compound was not synthesized until 2003 by the group of Rainer Herges.

Ansatz Wavefunction & Hückel-Möbius Energy

For the Mobius geometry, the boundary conditions differ from the standard particle in a ring problem. Supposing to have a strip of length $L_x$ and $L_z$, we can see that general Mobius boundary conditions for the $\psi$ wavefunction are:

• $\psi(x,0)=\psi(x,L_z)$
• $\psi(0,z)=\psi(L_x,-z)$

or using the spherical azimuthal angle $\phi$:

$\psi(\phi)=-\psi(\phi+2\pi)$

For an $N$-carbons, the proposed ansatz linear combination of atomic orbitals (LCAO) is:

$|{\psi_\lambda}\rangle=\sum_{j=0}^{N-1}{{c^\lambda_j}|{\varphi_j}}\rangle=\sum_{j=0}^{N-1}{e^{i\lambda\phi_j}|{\varphi_j}}\rangle=\sum_{j=0}^{N-1}{e^{{i\lambda2\pi j/N}}|{\varphi_j}}\rangle$

where $\phi_j=2\pi j/N$ is the angle at each $j$-th carbon atom and $\varphi_j$ is the $j$-th AO. Thus, for circular carbon rings, the general Mobius boundary condition can be rewritten as:

$c_{j+N}^\lambda=-c_j^\lambda$

Using this equation and the Euler rule we can find the right $\lambda$ value satisfying previous boundary conditions:

$e^{i\lambda 2\pi(j+N)/N}=-e^{i\lambda 2\pi j/N}$

$e^{i\lambda 2\pi}=-1$

$\lambda_k=\frac{2k+1}{2}\;\;\; k=0,1,2,\ldots,(N-1)$

From the last equation we see that to fulfil the general boundary conditions, $\lambda$ must be a half-integer number. The coefficients of the ansatz become:

$c_j^{(k)}=e^{i\pi(2k+1)j/N}$

From figure above, it can also be seen that the overlap between two consecutive $p_z$ AOs is at a constant angle $\omega=\pi/N$, and for this reason resonance integral $\beta^\prime$ it's considered as a constant into the Huckel matrix we will write later. It could be simply written as:

$\beta^\prime=\beta\cos(\pi/N)$

where $\beta$ is the standard Huckel’s resonance integral value (the one with $\omega=0$). Nevertheless, the presence of a $C_2$ axis as the only symmetry element brings to a full phase change at the end of the ring, e.i. between the first and the $N$-th carbon atoms. For this reason, in the Huckel matrix the resonance integral between carbon $1$ and $N$ is $-\beta^\prime$.
For the generic $N$ carbons Mobius system, the Huckel matrix $\mathbf{H}$ is:

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

Eigenvalues equation can now be solved. Since $\mathbf{H}$ is a $N\times N$ matrix, we will have $N$ eigenvalues $E_k$ and $N$ MOs. Defining the variable

$x_k=\frac{\alpha-E_k}{\beta}$

we have:

$\begin{pmatrix} x_k& 1 & 0 &\cdots& -1 \\ 1 & x_k& 1 & \cdots & 0 \\ 0 & 1 & x_k & \cdots & 0 \\ \vdots &\vdots &\vdots &\ddots &\vdots \\ -1 &0& 0 & \cdots & x_k \end{pmatrix} \cdot \begin{pmatrix} c_1^{(k)} \\ c_2^{(k)} \\ c_3^{(k)} \\ \vdots\\ c_N^{(k)} \\ \end{pmatrix}=0$

Hence we obtain a system of $N$ equations, in which the first one ($k=0$) and the last one ($k=N-1$) have a $-1$ coefficient:

$\begin{cases} x_0c_1^{(0)}+c_2^{(0)}-c_N^{(0)}=0\\ \vdots\\ c_{j-1}^{(k)}+x_kc_j^{(k)}+c_{j+1}^{(k)}=0\\ \vdots\\ c_{N-1}^{(N-1)}+x_{N-1}c_N^{(N-1)}-c_1^{(N-1)}=0 \end{cases}$

All these equations can be easily solved using Euler's rule, leading to

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

hence

$E_k=\alpha+2\beta^\prime\cos{\frac{(2k+1)\pi}{N}}=\alpha+2\beta^\prime\cos{\frac{2\pi\lambda_k}{N}}$

Hückel-Möbius aromaticity

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

The phenylene rings in this molecule are free to rotate forming a set of conformers: one with a Möbius half-twist and another with a Hückel double-twist (a figure-eight configuration) of roughly equal energy.

Transition states

Möbius systems are also found in transition states. The determination of a transition state as Möbius or Hückel is involved in deciding if a reaction with 4N or 4N+2 electrons is allowed or forbidden. This uses the Möbius-Hückel concept.

Compound 6

The compound (6 in the image below) was synthesized in several photochemical cycloaddition reactions from tetradehydrodianthracene 1 and the ladderane syn-tricyclooctadiene 2 as a substitute for cyclooctatetraene.

Intermediate 5 was a mixture of 2 isomers and the final product 6 a mixture of 5 isomers with different cis and trans configurations. One of them was found to have a C₂ molecular symmetry corresponding to a Möbius aromatic and another Hückel isomer was found with C_s symmetry. Despite having 16 electrons in its pi system (making it a 4n antiaromatic compound) the Heilbronner prediction was borne out because according to Herges the Möbius compound was found to have aromatic properties. With bond lengths deduced from X-ray crystallography a HOMA value was obtained of 0.50 (for the polyene part alone) and 0.35 for the whole compound which qualifies it as a moderate aromat.

Conversion

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.

Difference demonstration

The difference was demonstrated in a hypothetical pericyclic ring opening reaction to cyclododecahexaene. The Hückel TS (left) involves 6 electrons (arrow pushing in red) with C_s molecular symmetry conserved throughout the reaction. The ring opening is disrotatory and suprafacial and both bond length alternation and NICS values indicate that the 6 membered ring is aromatic. The Möbius TS with 8 electrons on the other hand has lower computed activation energy and is characterized by C₂ symmetry, a conrotatory and antarafacial ring opening and 8-membered ring aromaticity.

Cyclononatetraenyl

Another interesting system is the cyclononatetraenyl cation explored for over 30 years by Paul v. R. Schleyer et al. This reactive intermediate is implied in the solvolysis of the bicyclic chloride 9-deutero-9'-chlorobicyclo[6.1.0]-nonatriene 1 to the indene dihydroindenol 4. The starting chloride is deuterated in only one position but in the final product deuterium is distributed at every available position. This observation is explained by invoking a twisted 8-electron cyclononatetraenyl cation 2 for which a NICS value of -13.4 (outsmarting benzene) is calculated.

Bond length alternation

In 2005 the same P. v. R. Schleyer questioned the 2003 Herges claim: he analyzed the same crystallographic data and concluded that there was indeed a large degree of bond length alternation resulting in a HOMA value of -0.02, a computed NICS value of -3.4 ppm also did not point towards aromaticity and (also inferred from a computer model) steric strain would prevent effective pi-orbital overlap.

Metallapentalenes

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. 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.

See also

• Barrelene
• Baird's rule