The Diels-Alder reaction and frontier molecular orbitals

One of the most powerful and important reactions in organic chemistry is a ring-closing reaction, an example of which is the reaction between 1,3-butadiene and ethylene, shown in the figure below:

Figure: The Diels-Alder reaction.

This reaction, first discovered by O. Diels and K. Alder in 1928, is now known as the Diels-Alder reaction, and the ethylene is referred to as a dieneophile. The previous figure of the $\pi$ molecular orbitals for 1,3-butadiene show the HOMO and LUMO orbitals. The frontier molecular orbital approach requires that an electron pair in the HOMO of one of the molecules occupies the LUMO of the other. The tricky part here is deciding which molecule supplies the HOMO and which the LUMO. The general rule is the more $\pi$ electrons there are, the higher in energy is the HOMO. Thus, in the Diels-Alder reaction, it is the diene that supplies the HOMO to donate an electron pair to the LUMO of ethylene. Since ethylene has two electrons in a $\pi$ orbital, when an electron pair enters the LUMO, which is a $\pi^*$ orbital, the bond order in ethylene is reduced to a single $\sigma$ bond, and hence the double bond becomes a single bond. The terminal $2p$ orbitals now become involved in a rehybridization of the sp$^2$ orbitals to sp$^3$ orbitals. The $\pi$ orbital structure for both molecules is shown in the figure below:

Figure: $\pi$ orbitals in 1,3-butadiene and ethylene.

The interaction between these orbitals is shown in the figure below. The figure illustrates that they must be combined so as to lead to constructive interference between the lobes:

Figure: Interaction between the HOMO of 1,3-butadiene and LUMO of ethylene.

The mechanism of the reaction is generally considered to be concerted, meaning that the two new CC bonds form ``in concert'' rather than sequentially. The requirement of arranging the orbitals such that they add constructively or ``in phase'' in concerted reactions is part of a system of rules known as the Woodward-Hoffmann rules. These are useful guidelines for many organic reactions, however, there are cases in which they fail as in, for example, Diels-Alder type reactions of dienes on a silicon surface for the formation of molecular devices. See, for example, J. Am. Chem. Soc. 126, 13920 (2004) for a study of 1,3-butadiene on the silicon (100)-2$\times$1 surface shown below:

Figure: Idealized Si(100)-2$\times$1 surface.

In this figure, the Si-Si dimers on this surface play a role similar to that of ethylene in the traditional Diels-Alder reaction, however, the addition of 1,3-butadiene to this surface violates the Woodward-Hoffmann rules.