Cycloadditions: 2 + 2 cycloadditions with ketene

Ketenes form an exception to the Woodward-Hoffmann rule, as they exhibit concerted thermal reactions with ethylene-type compounds, to form cyclobutanones.
In fact, exception is not the right word, as the behaviour of ketenes can also be explained by orbital considerations.

The point is that ketenes have two, perpendicular pi systems, and both of them participate in the reaction. E.g. a computation on ketene itself results in two unoccupied MO's very close in energy. One of them (left picture) can be described as the LUMO (pi*) of the C=C system, the other one (right picture) as the LUMO (pi*) of the carbonyl group.

ketene LUMO and LUMO+1
(click for vrml)

For a closer look we take the reaction between methylketene and propene. From a MOPAC calculation we derive a crude transition state for this reaction, in which we notice that the two C=C bonds are not parallel, but make an angle with eachother of about 40°.

(Try to orientate the structure on the left as in the pictures below.)

If we look at the orbitals for this transition state we see why this is.
in the HOMO-1 we see overlap between the ethylene pi orbital with the pi* of the C=C bond in the ketene. The two red lobes form a new sigma bond.
(Clicking on these pictures will load the VRML files.)

The HOMO of the system shows the formation of the other sigma bond, from the ethylene pi orbital and a combination of the two red lobes of the pi* orbitals shown above.
It would be impossible to reach the upper red lobe of the C=C pi* alone, but the combination is distributed over a larger part of space.

In these pictures one sigma bond is formed at the bottom side of the ketene molecule, the other sigma bond at the top side. With the assistance of the carbonyl pi* we have accomplished what is called an antarafacial reaction.
In the propene part the bonds are formed at one side of the pi system, which is called suprafacial.

The Diels-Alder reaction that we have shown before is clearly a supra/supra reaction, in both reactants the new bonds are formed on the same side of the pi system.

Herewith we arrive at another extension to the Woodward-Hoffmann rules:
While the 'normal' (i.e. supra/supra) 2 + 2 (or 4 + 4) reaction is forbidden, the antara/supra combination is allowed.

For geometric reasons the 2 + 2 combination cannot reach the antarafacial orbitals, unless, as in ketene, other orbitals assist.

Normally an antarafacial reaction has consequences for the stereochemistry of the product. In ketene however the carbonyl carbon doesn't carry enough substituents to make this visible in the resulting cyclobutane.

For a closer look we take the reaction between methylketene and propene. From a MOPAC calculation we derive a crude transition state for this reaction, in which we notice that the two C=C bonds are not parallel, but make an angle with eachother of about 40°. (Try to orientate the structure on the left as in the pictures below.)
		If we look at the orbitals for this transition state we see why this is. in the HOMO-1 we see overlap between the ethylene pi orbital with the pi* of the C=C bond in the ketene. The two red lobes form a new sigma bond. (Clicking on these pictures will load the VRML files.)
The HOMO of the system shows the formation of the other sigma bond, from the ethylene pi orbital *and a combination of the two red lobes of the pi orbitals** shown above. It would be impossible to reach the upper red lobe of the C=C pi* alone, but the combination is distributed over a larger part of space.