4.3. The Woodward-Hoffmann Rules

In order to understand the Woodward-Hoffmann rules for determining the stereochemistry of several different types of concerted reactions, let us first consider the ground and excited states of a "normal" reaction. The ground state energy rises continuously to the transition state and then falls to the product. Often, this is the only state shown in such reaction profile diagrams. The excited state is not generally involved in the reaction, but often has a minimum above the transition state, as shown in Figure 4.4:

Figure 4.4 Energy profiles for the ground and excited states of a "normal" reaction.

The reaction occurs in this case when the system gains enough energy to be able to cross the barrier to the product side. Let us, however, now consider a second reaction where the ground state energy profile does not lead directly to the product ground state, as shown in Figure 4.5:

Figure 4.5 Energy profiles for a reaction in which the ground state of the starting material does not lead to that of the product.

In this case, if the system stays in the same electronic state, it will gain energy continuously along the reaction path and lead to the excited states of the product. Similarly, if we were to start with the excited state of the reactant, the system could lose energy continuously to reach the ground state of the product. Strictly speaking, the two states can only cross if they have different symmetries, otherwise they will mix with each other near the crossing point to give a transition state for the ground state slightly below the state crossing. However, this activation energy is usually much higher in energy than those typically found for processes like that shown in Figure 4.4. These reactions are known as allowed reactions and those as shown in Figure 4.5 as forbidden reactions. The names are far stricter than the real situation; forbidden reactions can sometimes occur quite easily. However, the vast majority of reactions follow the allowed path if they have two alternatives.

In order to translate the above picture, which uses plots of the energies of different electronic states, into one that uses orbitals, we should consider a model case in which two electrons and only two orbitals are involved on either side of the reaction and in which we only consider the ground state and the doubly excited state in which both electrons occupy the higher orbital. This situation is shown in Figure 4.6 for a forbidden reaction:

Figure 4.6 Orbital occupancies for a forbidden reaction as shown in Figure 4.5.

Because the red and the green orbitals change their energy ordering, retaining their occupations along the reaction paths leads directly to the excited states. This can be expressed for the ground state in terms of orbital correlations, which are shown in Figure 4.7 for allowed and forbidden reactions:

Figure 4.7 Orbital correlations for allowed and forbidden reactions.

Thus, the criterion for an allowed reaction in terms of state correlations diagrams like Figures 4.4 and 4.5 is that the ground states of reactant and product correlate with each other. For orbital correlation diagrams like those shown in Figure 4.7 the corresponding criterion is that no orbital correlation line crosses the border (the dashed line) between occupied and virtual orbitals. We will now consider three different sorts of reaction that can be treated in this way.

4.3.1. Electrocyclic Reactions

Electrocyclic reactions are those in which a p-system with N p-bonds and M rings is converted into a new system with N-1 p-bonds and M+1 rings (or the reverse). Some common examples are shown in Figure 4.8 below:

Figure 4.8 Selected examples of electrocyclic reactions.

The net result of such electrocylclic reactions is the interconversion of a p- and a s-bond. The reactions usually proceed towards formation of the s-bond because s-orbitals are lower in energy than p-, but formation of a strained ring may reverse this energetic effect. Let us now consider a prototype electrocyclic reaction, the ring-closure of s-cis butadiene to cyclobutene (Figure 4.9):

Figure 4.9 Ring-closure of s-cis butadiene to cyclobutene.

This reaction can proceed in one of two ways. The two terminal CH₂-groups can either rotate in the same direction to give a conrotatory process (Figure 4.10):

Figure 4.10 Conrotatory ring-closure.

or they can rotate in opposite directions to give the disrotatory ring closure (Figure 4.11):

Figure 4.11 Disrotatory ring-closure.

These two processes give the same product (cyclobutene) for the prototype reaction, but different stereoisomers if the butadiene is asymmetrically substituted with methyl groups. The purpose of the Woodward-Hoffmann rules for such reactions is to rationalise the stereochemistry of the products for both thermal and photochemical reactions.

In order to understand the orbital process involved in the ring-closure, let us consider the orbitals that change during the reaction. These are the four p-MOs of butadiene, which are transformed into two p- and two s-orbitals in cyclobutene, as shown in Figure 4.12.

Figure 4.12 The MOs of butadiene and cyclobutene that are interconverted during the ring-closure. The dashed lines indicate the border between occupied and unoccupied MOs.

This figure gives schematic diagrams of the MOs in which all contributions are treated as being from pure p-orbitals. They can be compared with the AM1-calculated MOs for s-cis-butadiene and cyclobutene. We must now decide how best to convert the relevant MOs of butadiene into those of cyclobutene in a continuous process. This is not difficult as we know that the two terminal CH₂-groups of the butadiene must rotate to make cyclobutene, but that the two CH-groups remain planar throughout the reaction. Therefore we must retain the orbital contributions for these two carbon atoms with the same relative phases as they have in butadiene in cyclobutene (because they cannot change their relative phases by rotating). This analysis implies that we must make doubly occupied orbitals from doubly occupied and unoccupied from unoccupied, as suggested by the orbital correlation diagram shown in Figure 4.7. Therefore, the p-HOMO of cyclobutene (Y_2C in Figure 4.12) must be derived from the lowest p-MO of butadiene (Y_1B in Figure 4.12). Similarly, the p-LUMO of cyclobutene (Y_3C) must be derived from the highest p-MO of butadiene (Y_4B). Our analysis of the geometrical process required to complete the ring-closure is therefore reduced to converting Y_2B into Y_1C and Y_3B into Y_4C. If our analysis is correct, we will be able to use the same movement (either conrotatory or disrotatory) to perform both transformations.

In order to convert Y_2B into Y_1C, we can rotate the two CH2-groups in the same direction (i.e. perform a conrotatory ring-closure), as shown in Figure 4.13:

Figure 4.13 The conrotatory process that converts Y_2B into Y_1C.

Exactly the same movement also converts Y_3B into Y_4C, as shown in Figure 4.14:

Figure 4.14 The conrotatory process that converts Y_3B into Y_4C.

These orbital correlations give the following diagram for the thermal, conrotatory ring-closure reaction (Figure 4.15):

Figure 4.15 Orbital correlation diagram for the conrotatory ring-closure.

This diagram corresponds to an allowed reaction, as shown in Figure 4.7, but what would happen if we try to perform the disrotatory process? In this case, we must convert Y_1B into Y_1C and Y_4B into Y_4C (check these processes yourself), so that the p-MOs now correlate differently, as shown in Figure 4.16:

Figure 4.16 Orbital correlation diagram for the disrotatory ring-closure.

This diagram is representative of a forbidden reaction, as shown in Figure 4.7. We can thus conclude that the thermal ring-closure of butadiene to cyclobutene is an allowed conrotatory process, but a forbidden disrotatory one. It should thus occur in a conrotatory fashion, as is observed experimentally.

The orbitals processes described above are quite independent of the electronic state of the reaction system, so can be used without change for the photochemical process. In this case, one electron is promoted from the HOMO to the LUMO of both butadiene and cyclobutene, so that, although the orbitals correlations are not changed, we now have three different types of MOs; doubly occupied, singly occupied and unoccupied. This means that, instead of the one border between doubly occupied and unocuppied MOs found for the thermal reaction, we now have two that may not be crossed by an orbital correlation line. This leads to the two diagrams shown in Figure 4.17:

Figure 4.17 Orbital correlation diagrams for the photochemical ring-closure reaction: conrotatory (forbidden, left) and disrotatory (allowed, right).

Now the disrotatory process is allowed and the conrotatory one forbidden - exactly the reverse of the thermal process. This result is quite general; photochemical electrocyclic reactions always occur in the opposite sense to their thermal equivalents.

We can generalise the results obtained above. Remember the rules outlined for determining the p-MOs of linear systems in Sect. 3.3.1. Remember also that N p-MOs are converted into N-1 in an electrocyclic ring-closure. This means that, of all the p-MOs, only the "extra" one in the acyclic system, the HOMO, cannot be retained in the smaller, ring-closed p-system. This is therefore the MO that we should use to form the new s-MO. Thus a generic orbital correlation diagram (Figure 4.18) for such a process is as follows:

Figure 4.18 Generic orbital correlation diagram for electrocyclic ring-closure reactions.

Although the orbitals crossing shown above are not real, this diagram gives us a convenient way to treat electrocyclic reactions. We can determine the nature of the process (conrotatory or disrotatory) by assuming that the p-HOMO must be converted to the new s-MO. This was the hypothesis that was originally published by Woodward and Hoffmann. It means that, because the symmetry of the HOMO alternates as one more double bond (or two electrons) is added, the conrotatory or disrotatory nature of electrocyclic reactions also alternates. This can then be formulated as a set of rules, as shown in Table 4.1:

Table 4.1 The Woodward-Hoffmann rules for electrocyclic reactions.

The same principles can now be used to describe other types of concerted reactions.

4.3.2. Cycloadditions

Cycloadditions are reactions in which two p-systems are added to each other so that two new s-bonds are formed between their termini to give a new ring (Figure 4.19):

Figure 4.19 Schematic representation of cycloaddtion reaction.

The best known example, which we will treat here, is the Diels-Alder reaction, the prototype of which is the addition of ethylene to butadiene (Figure 4.20):

Figure 4.20 Diels-Alder cycloaddition between ethylene and butadiene.

There are two ways in which a chain can be added to a planar p-system. If the two new bonds are formed from the same side of the plane of the p-system, the addition is said to be suprafacial; an addition from opposite sides is antarafacial (Figure 4.21):

Figure 4.21 Suprafacial (left) and antarafacial (right) addition to p-systems.

As cycloadditions occur between two p-systems, each can undergo either suprafacial (s) or antarafacial (a) addition to give four different possibilities, s+s, a+a, s+a and a+s. The Woodward-Hoffmann rules for cycloadditions distinguish between these four modes according to the numbers of electrons involved.

For similar reasons to those outlined above for electrocyclic reactions, we only need to consider the frontier orbitals of the two systems. The HOMOs and LUMOs of butadiene and ethylene form the two new s- and two new s*-MOs. These s-MOs are grouped in bonding and antibonding combinations, as for cyclobutene above. The p-HOMO and p*-LUMO of cyclohexene are derived directly from Y_1B and Y_4B of butadiene (Figure 4.12). The two CH-groups of butadiene remain in the new p-system, as shown in Figure 4.22:

Figure 4.22 The interconversion of the p-MOs in the Diels-Alder reaction.

The frontier orbitals of butadiene and ethylene and the s-MOs of cyclohexene are shown schematically in Figure 4.23. Once again, these schematic orbitals should be compared with those calculated with AM1 for s-cis-butadiene, ethylene and cyclohexene.

Figure 4.23 The orbitals that form the new s-bonds in the Diels-Alder reaction.

The two sets of orbitals can be divided into two classes according to whether they are symmetric or antisymmetric (in the sense that these terms are used in Sect. 3.3.1). This classification is indicated by the designations A and S in Figure 4.23. There are two occupied orbitals on each side of the equation, one symmetric and one antisymmetric, and similarly, A and S unoccupied MOs. Furthermore, the HOMO of butadiene is antisymmetric, as is the LUMO of ethylene, whereas the LUMO of butadiene and the HOMO of ethylene are symmetric. Thus, we can conclude that the two p-systems interact via two HOMO-LUMO two-electron attractive suprafacial interactions, as shown in Figure 4.24. These two interactions need not be equally important. In most real Diels-Alder reactions, the olefin, or dienophile is substituted with electron-accepting groups. This sinks the energy level of its LUMO, making it a good electron acceptor. The interaction between the low-lying olefin LUMO and the diene HOMO then becomes the dominant stabilizing factor early in the reaction. However, it is possible to reverse this trend completely by using a very electron-rich olefin and a very electron-poor diene to give a so-called inverse Diels-Alder reaction. The orbitals obtained from the HOMO-LUMO interactions can be compared with the relevant s-MOs of cyclohexene.

Figure 4.24a The interaction between the butadiene HOMO and the ethylene LUMO.

Figure 4.24b The interaction between the butadiene LUMO and the ethylene HOMO.

Thus, the Diels-Alder reaction is a thermally allowed reaction in which the two components both add suprafacially and in which one has four p-electrons and one has two (a [4s + 2s] reaction). The Woodward-Hoffmann rules for cycloadditions with I p-electrons in one component and J in the other are summarised in Table 4.2:

Table 4.2 The Woodward-Hoffmann rules for cycloaddition reactions.

Now, look at the molecular orbitals for the prototype Diels-Alder transition state in order to identify the six reacting orbitals outlined above.

4.3.3. Sigmatropic Rearrangements

Sigmatropic rearrangements are reactions in which a substituent that is hyperconjugating with a p-system migrates from one end to the other. In a sense, the bridged ethyl cation considered in Sect. 3.4.1 could be considered as the mid-point of a sigmatropic 1,2-hydrogen shift, but we will consider the model 1,3-hydrogen shift in propene. As for cycloadditions, sigmatropic rearrangements can either occur in a suprafacial manner, in which the shifting group remains on the same side of the p-system, or antarafacially, where the shifting group changes faces of the p-system (Figure 4.25):

Figure 4.25 Suprafacial (left) and antarafacial (right) addition to p-systems.

Sigmatropic reactions are best understood by considering their transition states, which are approximated as consisting of the two radicals obtained by breaking the bond to the migrating group. Thus, for the propene 1,3-hydrogen shift, a hydrogen atom must migrate across an allyl radical. The singly occupied MO (SOMO) of the allyl radical is shown schematically in Figure 4.26 below:

Figure 4.26 The SOMO of the allyl radical.

The hydrogen atom, which has a singly occupied s-orbital, cannot shift suprafacially because at the transition state it would lie in the nodal plane of the allyl-SOMO and thus lose all bonding interaction (Figure 4.27):

Figure 4.27 Schematic transition state for the suprafacial 1,3-hydrogen shift in propene.

For the antarafacial shift, on the other hand, some overlap can be retained, even if the geometry of the transition state is very strained (Figure 4.28):

Figure 4.28 Schematic transition state for the antarafacial 1,3-hydrogen shift in propene.

Thus, the suprafacial shift is forbidden and the antarafacial one allowed. In fact, neither takes place without dissociation in propene itself because of the strain in the antarafacial transition state, but the fact that no suprafacial shift occurs is rationalised by the above arguments.

The Cope-rearrangement consists of a 1,3-shift of an allyl radical across the face of another (Figure 4.29):

Figure 4.29 Schematic representation of the Cope-rearrangement.

The two radical SOMOs can interact strongly in the suprafacial + suprafacial transition state, as shown in Figure 4.30. (there are now two p-systems and so the same four stereochemical possibilities exist as for cycloadditions).

Figure 4.30 The interaction between two allyl SOMOs in the Cope-rearrangement transition state.

This reaction is an example of a 1,3-shift that is suprafacial for both components and involves two p-systems, each with 3 electrons. The MOs of 1,5-hexadiene and the Cope-rearrangement transition state show the reacting orbitals. Table 4.3 gives the Woodward-Hoffmann rules for sigmatropic rearrangements between p-systems with I and J electrons:

Table 4.3 The Woodward-Hoffmann rules for sigmatropic rearrangements.

Lewis Acid/Lewis Base Interactions
Selectivity; Frontier MO Theory
The Woodward-Hoffmann Rules