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Introduction
In the previous article, we took a look at the Woodward-Hoffmann rules by the frontier orbital method. In this article, we will be looking at these rules from another point of view, using a different method. Of course, for those who have read the previous article, we are talking about the Mobius-Huckel method. Take note that I would consider this simply as enrichment because it does not really link to topics we have learnt before. In order for us to understand this method, we also discuss Mobius aromaticity and Huckel aromaticity.
The frontier orbital method explores how cycloaddition reactions are disallowed if opposite lobes interact with each other, and because of this, only certain types of cycloaddition are allowed. Strikingly, we also noted the idea that a thermal cycloaddition could be disallowed while a photochemical cycloaddition could be allowed, because photochemical excitation changes the lobes themselves.
This article, being an enrichment, and not containing any other topics, should be quite a short one. At the time of writing this, I am aiming for at least a thousand words, although this could change by the time I conclude. The article will also be tagged under ‘addition reactions’, and I hope everyone appreciates the indirect link between this article and addition reactions.
Before we delve into what the Mobius-Huckel rule is exactly, we have to understand how this builds, albeit just a little, on previous knowledge we have. Some may find the name ‘Huckel’ familiar; indeed, it is related to Huckel’s rule of aromaticity (Fig. 1), which I have previously explained in this article, an introduction to aromaticity.
Fig. 1: Huckel’s rule of aromaticity.
Simply, Huckel’s rule suggests that aromaticity is achieved for a close to planar molecule if it possesses (4n + 2) π electrons. Furthermore, antiaromaticity is achieved if the molecule possesses 4n π electrons. From here, we refer to this concept as Huckel aromaticity, in order to provide a better comparison to Mobius aromaticity.
Mobius Aromaticity
Let us slowly get to the topic of the Mobius-Huckel method. Before that, we have to understand Mobius aromaticity, and before we do that, we need to look at the orbitals of benzene in detail. Benzene has 6 p orbitals, the structure of which is depicted below (Fig. 2). Notice how between the two lobes, there is an area where electrons cannot be present.
Fig. 2: Structure of a p orbital.
In the lobes, there is a set probability of electrons being present. However, electrons may not be present in the nodal plane. Electrons pass between the lobes through quantum tunneling (which is outside the scope of our discussion). In benzene, which is aromatic, there are 6 p orbitals, giving rise to a nodal plane in the plane of the benzene molecule (Fig. 3). Interactions between orbitals give rise to Fig. 4.
Fig. 3: Orbital diagram of benzene.
Fig. 4: Orbital diagram after accounting for interactions between orbitals.
Now, we can finally learn about Mobius aromaticity. In Huckel aromaticity, as we have explored in Fig. 4, the lobes are shaped like a cylinder (or more accurately, a torus), if we take into account that the lobe is three-dimensional. In Mobius aromaticity, the p orbitals are unusually twisted such that they bear similarity to a Mobius strip (Fig. 5). The molecule may still be monocyclic, but the orbitals are ‘twisted’ (Fig. 6).
Fig. 5: Illustration of a Mobius strip.
Fig. 6: Orbital diagram in Huckel vs Mobius aromaticity (1).
Now, interestingly, because electrons are still delocalized throughout the structure, such a molecule would also be aromatic. Due to the nature of Mobius aromaticity, a structure with 4n π electrons is aromatic, while a structure with (4n + 2) π electrons is antiaromatic, exactly the opposite of Huckel’s rule. The reason for this is explained by the Mobius-Huckel circle mnemonic (Fig. 7).
Fig. 7: Mobius-Huckel circle mnemonic applied to benzene.
To fill at least one energy level of orbitals, at least two electrons are required for the Huckel system while at least four electrons are required for the Mobius system, because each orbital (represented by a line) holds two electrons and some are degenerate (i.e. equal in energy level). Keep in mind that for each system to be stable, an energy level’s orbitals have to be fully-filled.
In the case of the Huckel system of benzene, we need at least two electrons (4n + 2, where n = 0), otherwise the second energy level has to be filled (again, 4n + 2, where n = 1). For the Mobius system, we need at least four electrons (4n, where n = 1).
Finally, we will look at an example of a mobius-shaped molecule. An example of a Mobius-aromatic compound is trans-C9H9 (Fig. 8), as is explained by its structure.
Fig. 8: Structure of trans-C9H92.
Mobius-Huckel Method
At last, we will be looking at the method itself. First, some basic terminology. A set of p orbitals which combine to form a molecular orbital is known as a basis set. When the p orbitals interact in the transition state, it is possible that no sign inversion occurs (constructive interference), or sign inversion occurs (destructive interference). In the entire transition state of a pericyclic mechanism, there will be several p orbitals interacting.
As long as there are an even number of sign inversions (including zero), the system is a Huckel system. If there are an odd number, then, the system is a Mobius system. We have illustrated the transition state of the Diels-Alder reaction below; notice how there are zero sign inversions because similar lobes interact with one another (Fig. 9). As such, it is a Huckel system.
Fig. 9: Orbital transition state of the Diels-Alder reaction (3).
How does this help us understand whether a pericyclic reaction is feasible? If a pericyclic reaction involves a Huckel system, the total number of electrons moving in the reaction must be 4n + 2. Likewise, if a Mobius system is involved instead, the total number of electrons has to be 4n. Note that this is only for thermal pericyclic reactions; for photochemical ones, the requirements are reversed.
Finally, we can then apply the Mobius-Huckel method to see whether the formation of cyclobutane from the dimerization of ethene is feasible. It is a Huckel system, because no sign inversions are present. Thus it must have 4n + 2 electrons for the reaction to be feasible. However, it only has four electrons, thus it is not feasible. However the photochemical reaction is feasible because the requirements are reversed.
Attribution
1: “Mobius vs Huckel”, by V8rik at en.wikipedia, licensed under CC BY-SA 3.0.
2: “sketch of trans-C9H9+”, by Alsosaid1987, licensed under CC BY-SA 4.0.
3: “DA reaction in total synthesis”, by Chrito23, licensed under CC BY-SA 3.0.