Talk:Möbius aromaticity

Suggested references moved from "see also" in article
I am moving the below material directly from the "See also" section of the article. - Lady  of  Shalott  04:09, 9 January 2010 (UTC)

''Missing References; and Note: Moebius Systems occur more widely than in traditional poly pi rings. For example, in pericyclic transition states one can find Moebius or Hueckel electronics and determine which'' is of lower energy.

47. "On Molecular Orbital Correlation Diagrams, the Occurrence of Möbius Systems in Cyclization Reactions, and Factors Controlling Ground and Excited State Reactions. I," Zimmerman, H. E. J. Am. Chem. Soc., 1966, 88, 1564-1565.

48. "On Molecular Orbital Correlation Diagrams, Möbius Systems, and Factors Controlling Ground and Excited State Reactions. II," Zimmerman, H. E. J. Am. Chem. Soc., 1966, 88, 1566-1567.

106. "The Möbius-Hückel Concept in Organic Chemistry. Application to Organic Molecules and Reactions," Zimmerman, H. E. Acc. Chem. Res., 1971, 4, 272-280.

Howard E. Zimmerman 19:09, 8 January 2010 (UTC)

The Article as written has revised history
This article has attributed to Edgar Heilbronner what he did not propose. Heilbronner stated that Möbius Transition States could never be lower in energy than Hückel ones in complete contrast to the statements in this article. It was only in the 1966 Missing References (and later also) references that the 4N versus 4N+2 M-H criteria were proposed.

Howard E. Zimmerman 15:28, 13 June 2010 (UTC) —Preceding unsigned comment added by Hezimmerman (talk • contribs)

Mathematics
I'm a bit confused about the derivation given for the wavefunctions and energies. The ansatz as given is not normalized. That's a minor point. But related to this, I think coefficients are missing here.

It seems obvious, though not explicitly stated, that the e^(i\lambda\phi_j) (essentially a rotation) is to account for the twisting of the orbitals, so that when \lambda\phi_j reaches \pi, then we have a complete phase inversion. However, the \varphi_je^(i\lambda\phi_j) are the basis functions, are they not?

The Mobius aromatic orbitals should have differing lobe sizes (modulus of coefficients) and nodal patterns, like their Huckel aromatic counterparts, so shouldn't each of these kets have a coefficient c_j in front of them in the ansatz, so that the wavefunction is \sum_j c_j|\varphi_j\rangle\exp(i\lambda\phi_j)? Please help me understand this. The derivation is not particularly clear, so I wanted to revise it, but as soon as I started, I found that I could not quite follow it.

Best, Alsosaid1987 (talk) 03:29, 8 December 2017 (UTC)

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:
 * $$|{\psi_\lambda}\rangle=\sum_{j=0}^{N-1}{{c^\lambda_j}|{\varphi_j}}\rangle=\sum_{j=0}^{N-1}{e^{i\lambda\phi_j}|{\varphi_j}}\rangle=\sum_{j=0}^{N-1}{e^|{\varphi_j}}\rangle$$.

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 fulfill 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} $$.