Curly arrow pushing is one of the essential tools of a mechanistic chemist. Many a published article will speculate about the arrow pushing in a mechanism, although it is becoming increasingly common for these speculations to be backed up by quantitative quantum mechanical and dynamical calculations. These have the potential of exposing the underlying choreography of the electronic dance (the order in which the steps take place). The basic grammar of describing that choreography tends to be the full-headed curly arrow for closed shell systems and its half-barbed equivalent for open shell systems. An effectively unstated and hence implicit rule for closed shell systems is that only one curly arrow is used per breaking or forming bond, i.e. electrons move around bonds in pairs. So consider the following reaction (inspired by a posting on Steve Bachrach’s blog)
This is very much a hypothetical mechanism, or a thought-experiment if you will. Three nitrosonium cations decide to get together to swap their partners. Each diatomic molecule swaps e.g. one oxygen for another during this exchange reaction (it could easily be studied experimentally of course using isotopic substitution). Three sets of three curly arrows have been used, shown in different colours above. One set of these arrows at least has plenty of analogy in the real world; representing a π2s+π2s+π2s cycloaddition reaction. The other two sets represents rotation of the in-plane π-set and the in-plane σ-set. What about the choreography? Can all three sets move at the same time? If so, they would provide an exception to the rule above; three bonds would concurrently change their order from 3 to 0; the other three the reverse of 0 to 3.
What does quantum mechanics say about this? Well, a well defined, synchronous concerted transition state can indeed be found (B3LYP/6-31G(d), DOI: 10042/to-2905) It has one imaginary frequency (click on the above diagram to view the animation) which does indeed perform the bond transposition function required! It has the form of the so-called Kekule mode (deriving from a mode found in benzene which involves shortening of the lengths of three bonds, and lengthening of the other three, much in the manner of the resonance named after Kekule; see e.g. DOI: 10.1039/B911817A for more details). Of course, describing it as a change in the bond orders 3 → 0/0 → 3 is simplistic; the bond order in the nitrosonium cation itself is almost certainly somewhat less than three. But clearly, the implicit rule that mechanistic arrow pushing should not involve more than one arrow departing from or arriving at any one bond can be broken. I will leave it to the reader of this blog to see what happens when you try to rearrange the choreography of the above reaction. Try pushing first one set of three arrows, then another and a final third. What do you get? (the why of the dance is almost certainly due to electrostatic repulsions between the three nitrosonium cations).