14-9 Qualitative Molecular Orbital Theory of Reactions
Chapter 14 Qualitative Molecular Orbital Theory well as in π MOs. Indeed, if one performs EHMO calculations on these molecules, one finds that σ -type MO energies change much more than does the π HOMO energy.
This comes about because of the fairly close approach by some of the hydrogens in these molecules. Much of this σ energy change cancels out among the several σ MOs.
In propene, the cancellation is so complete that the π HOMO energy change is almost the same as the total EHMO energy change. In butadiene, however, the π HOMO accounts for only about one-third of the total energy change. Even though the frontier orbital method has an astonishing range of qualitative usefulness (we shall see more applications shortly), it is clear that caution is needed.
EXAMPLE 14-5 How should the bond lengths and angles of propene change if the methyl group rotates from the stable conformation (Fig. 14-16a) to the unstable one (Fig. 14-16b)?
SOLUTION The HOMO for form b has a bit more negative overlap, which requires a slightly larger normalizing coefficient. This increases the influence of the HOMO, so we expect a slight shortening of the two out-of-plane C–H bonds and the C=C bond, a lengthening of the C–C single bond, and a slight opening of the H–C–H angle for the out-of-plane C–H bonds. No changes are predicted for the in-plane C–H bond lengths or angles as a result of HOMO renormalization.
(However, other MOs are also being renormalized, so other changes will result. However, the HOMO-induced changes should dominate.)
14-9 Qualitative Molecular Orbital Theory of Reactions
It has been found possible to extend and amplify QMOT procedures so that they apply to chemical reactions. One of the most striking examples of this was application to unimolecular cyclization of an open conjugated molecule (e.g., cis-1,3-butadiene, closing to cyclobutene). This type of reaction is called an electrocyclic reaction. The details of the electrocyclic closure of cis-1,3-butadiene are indicated in Fig. 14-17.
If we imagine that we can keep track of the terminal hydrogens in butadiene (perhaps by deuterium substitution as indicated in the figure) then we can distinguish between two products. One of them is produced if the two terminal methylene groups have rotated in the same sense, either both clockwise or both counterclockwise, to put the two inside atoms of the reactant (here D atoms) on opposite sides of the plane of the four carbon atoms in the product. This is called a conrotatory (c˘on · r ¯o · t¯a · tory) closure. The other mode rotates the methylenes in opposite directions (disrotatory) to give a product wherein the inside atoms appear on the same side of the C4 plane.
A priori, we do not know whether the reaction follows either of these two paths.
Figure 14-17 depicts processes where both methylene groups rotate by equal amounts Figure 14-17 Two idealized modes of electrocyclic closure of cis-1,3-butadiene.
Section 14-9 Qualitative Molecular Orbital Theory of Reactions as the reaction proceeds. This is an extreme case of what is known as a concerted
process. The two processes occur together, or in concert. The opposite extreme is a nonconcerted, or stepwise process, wherein one methylene group would rotate all the way (90◦) and only after this was completed would the other group begin to rotate.
This process would lead to an intermediate having a plane of symmetry (ignoring the difference between D and H), which means that the second methylene group would be equally likely to rotate either way, giving a 50–50 mixture of the two products pictured in Fig. 14-17.
One can make a case for the reaction having some substantial degree of concertedness (by which we mean that the second methylene should be partly rotated before the first one is finished rotating). The reaction involves destruction of a four-center conjugated π system and formation of an isolated π bond and a new C–C σ bond. Energy is lost in the dissolution of the old bonds, and gained in formation of the new ones. Therefore, we expect the lowest-energy path between reactants and products to correspond to a reaction coordinate wherein the new bonds start to form before the old ones are completely broken. But the new σ bond cannot form to any significant extent until both methylene groups have undergone some rotation. Thus, concertedness in breaking old bonds and forming new ones is aided by some concertedness in methylene group rotations.
(Note that concertedness does not necessarily imply absence of an intermediate. If the reaction surface had a local minimum at a point at which both methylenes were rotated by 45◦, it would not affect the argument at all.)
Because they knew that many electrocyclic reactions are observed to be stereospecific (i.e., give ∼100% of one product or the other in a reaction like that in Fig. 14-17), Woodward and Hoffmann [12] sought an explanation of a qualitative MO nature. They used frontier orbitals and argued how their energies would change with a con- or disrotatory motion, due to changes in overlap. For butadiene in its ground state, the HOMO is the familiar π MO shown in the center of Fig. 14-18. The figure indicates that the interaction between p–π AOs on terminal carbons is favorable for bonding in the region of the incipient σ bond only in the conrotatory case. Therefore, the prediction is that, for concerted electrocyclic closure, butadiene in the ground state should prefer to go by a conrotatory path. When the reaction is carried out by heating butadiene (thermal reaction), which means that the reactant is virtually all in the ground electronic state, the product is indeed purely that expected from conrotatory closure.
One can also carry out electrocyclic reactions photochemically. The excited buta diene now has an electron in a π MO that was empty in the ground state. This MO was the lowest unoccupied MO (LUMO) of ground-state butadiene, pictured in Fig. 14-19.
One can see that the step to the next-higher MO of butadiene has just introduced one Figure 14-18 The HOMO of ground state cis-1,3-butadiene as it undergoes concerted closure by either mode.
Chapter 14 Qualitative Molecular Orbital TheoryFigure 14-19 The HOMO of the first excited state of cis-1,3-butadiene as it undergoes closure by either mode.
more node, reversing the phase relation between terminal π AOs, and reversing the predicted path from con-to disrotatory. Experimentally., the photochemical reaction is observed to give purely the product corresponding to disrotatory closure. (It is not always obvious which empty MO becomes occupied in a given photochemical experiment. One assumes that the LUMO of the ground state is the one to use, but there is some risk here.)
EXAMPLE 14-6 Is the LUMO ← HOMO transition dipole-allowed for cis-1,3 butadiene?
SOLUTION The HOMO is antisymmetric for reflection through the symmetry plane that bisects the molecule, and the LUMO is symmetric for this reflection. This is the only symmetry reflection plane where the MOs have opposite symmetry, so the transition is dipole-allowed (and is polarized from one side of the molecule towards the other). The group theory approach for this C2ν molecule is that the HOMO has a2 symmetry, the LUMO has b1 symmetry, their product has b1 symmetry, and, since x also has b1 symmetry, the transition is allowed and is x-polarized (where x is colinear with the central C–C bond).
One might worry about the fact that we are looking at only a part of one MO, thereby ignoring a great deal of change in other MOs and other parts of the molecule. However, much of this other change, while large, is expected to be about the same for either of the two paths being compared. The large overlap changes between p–π AOs on terminal and inner carbon atoms, for instance, are about the same for either mode of rotation.
This approach, then, is focused first on the frontier orbitals, which are guessed as being most likely to dominate the energy change, and second on those changes in the frontier orbitals that will differ in the two paths.
This method is trivially extendable to longer systems. Hexatriene closes to cyclo hexadiene in just the manner predicted by the frontier orbitals. The only significant change in going from butadiene to hexatriene is that we go from four to six π electrons.
This means that the HOMO for hexatriene has one more node than that for butadiene (or, the HOMO for a 2n π -electron system is like the LUMO for a 2n−2 π-electron system insofar as end-to-end phase relations are concerned). The net effect is that the predictions for hexatriene are just the reverse of those for butadiene. That is, hexatriene closes thermally by the disrotatory mode and photochemically by the conrotatory mode.
The general rule, called a Woodward–Hoffmann rule, is this: the thermal electrocyclic reactions of a k π -electron system will be disrotatory for k = 4q + 2, conrotatory for k = 4q (q = 0, 1, 2, . . . ); in the first excited state these relationships are reversed.7 It is possible to treat electrocyclic reactions in another way, namely, via a two-sided correlation diagram approach. This was first worked out by Longuet-Higgins and Abrahamson [14]. Only orbitals (occupied and unoccupied) that are involved in bonds being made or broken during the course of the reaction are included in the diagram. For butadiene, these are the four π MOs already familiar from simple H¨uckel theory. For cyclobutene, they are the two π MOs associated with the isolated 2-center π bond and the two σ MOs associated with the new C–C σ bond. These orbitals and their energies are shown in Fig. 14-20.
In cyclobutene, the σ and σ ∗ MOs are assumed to be more widely split than the π and π ∗ because the pσ AOs overlap more strongly. Also, the σ MO is assumed lower than π1 of butadiene. However, these details are not essential. All we have to be certain of is that we have correctly divided the occupied from the unoccupied MOs on the two sides. The dashed line in Fig. 14-20 separates these sets.
Next we must decide which symmetry elements are preserved throughout the ideal ized reactions we wish to treat. Let us consider first the reactants and products. These have C2v symmetry, that is, a two-fold rotational axis, C2, and two reflection planes σ1
Figure 14-20 MOs associated with bonds being broken or formed in the electrocyclic closure of (a) cis-1,3-butadiene to (b) cyclobutene.
7See Woodward and Hoffmann [13, p. 45].
Chapter 14 Qualitative Molecular Orbital Theory
Figure 14-21
Sketches illustrating that the conrotatory mode (b) preserves the C2 axis while the disrotatory mode (a) preserves the reflection plane σ1.
and σ2 containing the C2 axis (see Fig. 14-21). A conrotatory twist preserves C2, but, during the intermediate stages between reactant and product, σ1 and σ2 are lost as symmetry operations. A disrotatory twist preserves σ1 but destroys C2 and σ2. Therefore, when we connect energy levels together for the disrotatory mode, we must connect levels of the same symmetry for σ1, but for the conrotatory mode, they must agree in symmetry for C2. The σ2 plane applies to neither mode and is therefore ignored.
The symmetries for each MO are easily determined from examination of the sketches in Fig. 14-20, and are given in Table 14-3. These assignments lead to two different correlation diagrams, one for each mode. It is conventional to arrange these as shown in Fig. 14-22.
TABLE 14-3 Symmetries for C2v MOs
MO
σ1
C2
MO
σ1
C2
Butadiene
Cyclobutene
π1
Sa
A
σ
S
S
π2
A
S
π
S
A
π3
S
A π ∗
A
S
π4
A
S
σ ∗
A
A a S is symmetric; A antisymmetric.
Section 14-9 Qualitative Molecular Orbital Theory of ReactionsFigure 14-22 A pair of two-sided correlation diagrams (one for each mode) for the electrocyclic reactions of cis-1,3-butadiene: (a) cyclobutene; (b) butadiene; (c) cyclobutene.
There is curve crossing in these diagrams, but it is always lines of different symmetry that cross, and so no violation of the noncrossing rule occurs.
If we are considering a thermal reaction, the lowest two π MOs of butadiene are occupied. These correlate with the lowest two MOs of cyclobutene if the conrotatory mode is followed, and the thermal conversion of cis-butadiene to cyclobutene by a conrotatory closure is said to be symmetry allowed. The other mode correlates π2 with an empty cyclobutene MO (π ∗). Taking this route moves the reactant toward doubly excited cyclobutene. (Even though we might anticipate de-excitation somewhere along the way, the energy required in early stages would still be much higher than would be needed for the symmetry-allowed mode.) This is said to be a symmetry-forbidden
reaction.
If we now imagine photo-excitation of cis-butadiene to have generated a state asso ciated with the configuration π 2π
1
2π3, and trace the fate of this species for the two Chapter 14 Qualitative Molecular Orbital Theory modes of reaction, we note that the disrotatory route leads to cyclobutene in the configuration σ 2π π ∗ while the conrotatory mode gives σ π 2σ ∗. Both of these are excited, but the former corresponds to the lowest excited configuration (π → π∗) while the second corresponds to a very high-energy excitation (σ → σ ∗). Therefore, the former is “allowed” (since it goes from lowest excited reactant to lowest excited product) and the latter is “forbidden.”
The two-sided correlation diagrams of Fig. 14-22 thus lead to the same predictions as the frontier orbital maximization of overlap approach. The difference between these approaches is as follows: The frontier-orbital approach requires sketching the HOMO and then judging overlap changes upon nuclear motion using QMOT reasoning.
The two-sided correlation diagram approach requires sketching all the MOs (occupied and unoccupied) of both reactant and product involved in bonds breaking or forming, ordering the corresponding energy levels, and finding symmetry elements preserved throughout the reaction. Once all this is done, the levels are connected by correlation lines without reliance on QMOT reasoning. Some qualitative reasoning enters in the ordering of energy levels (levels with more nodes have higher energy), but the two-sided correlation diagram technique is the more rigorous method of the two and tends to be preferred whenever the problem has enough symmetry to make it feasible. Reliance on frontier orbitals is more common for processes of lower symmetry.
Concern is sometimes expressed about the apparent restrictions resulting from use of symmetry in correlation diagram arguments. One can imagine the butadiene cyclization occurring with less-than-perfect concertedness, the two methylene groups rotating by different amounts as the reaction proceeds. But that would destroy all symmetry elements. Will our symmetry-based arguments still pertain to such an imperfectly concerted reaction coordinate? Again, if we label certain sites by substituting deuteriums for hydrogens as shown in Fig. 14-17, the symmetry will be destroyed. Do our predictions still apply? One can answer these questions affirmatively by reasoning in the following way. If we had a collection of nuclei and electrons, and we could move the nuclei about in arbitrary ways and study the ground-state energy changes, experience tells us that the energy would be found to change in a smooth and continuous way. We can think of the energy as a hypersurface, with hyperdimensional “hills,” “valleys,” and “passes.” Now, in a few very special nuclear configurations, identical nuclei would be interrelated by symmetry operations, and we would be able to make deductions on group-theoretical grounds. Such deductions would only strictly apply to those symmetric configurations, but they would serve as indicators of what the energy is like in nearby regions of configuration space. Thus, the correlation diagram indicates that a perfectly concerted thermal electrocyclic reaction of butadiene will require much less energy to go conrotatory as opposed to disrotatory. The inference that a less-perfectly concerted reaction will have a similar preference is merely an assumption that it is easier to pass through the mountains in the vicinity of a low pass than a high one. Experience also leads us to expect that substituting for H a D (or even a CH3) will have little effect on the MOs, even though, strictly speaking, symmetry is lost. In essence, we work with an ideal model and use chemical sense to extend the results to less ideal situations, just as we do when, in applying the ideal gas equation of state to real gases, we avoid the high-pressure, low-temperature conditions under which we know the oversimplifications in the ideal gas model will lead to significant
error.
Section 14-9 Qualitative Molecular Orbital Theory of Reactions
It is possible to combine information on orbital symmetries and energies to arrive at state symmetries and energies. Then one can construct a correlation diagram for states.8 We now demonstrate this for the dis- and conrotatory reactions just considered.
Each orbital occupation scheme is associated with a net symmetry for any given sym metry operation. Character tables could be used to assign these symmetries, but this is not necessary. All we need to use is the fact that, in multiplying functions together, symmetries follow the rules: S × S = S, A × A = S, S × A = A. Thus, any doubly occupied MO in a configuration will contribute symmetrically to the final result. To ascertain the net symmetry, then, we focus on the partly filled MOs. The symmetries for C2 and σ1 of ground and some excited configurations of butadiene and cyclobutene are listed in Table 14-4. Included are the cyclobutene configurations that result from intended correlations of various butadiene configurations. (For instance, π 2π 2 butadiene has an
1
2
TABLE 14-4 Symmetries and Intended Correlations of Some Configurations of Butadiene and Cyclobutene
Cyclobutene
Symmetry for “intended” configuration
Configuration
σ1
C2
Con
Dis
Butadiene π 2
2
1 π2
Sa
S π 2σ 2 σ 2π ∗2 π 2 1 π2π3
A
A π 2σ σ ∗ σ 2π ∗π π 2 1 π2π4
S
S π 2σ π ∗ σ 2π ∗σ ∗
π
2
1π2 π3
S
S π σ 2σ ∗ σ π ∗2π
π
2
1π2 π4
A
A π σ 2π ∗ σ π ∗σ ∗2 π 2 1 π 2
S
S σ 2σ ∗2 σ 2π 2
3
...
Cyclobutene σ 2π 2
S
S σ 2π π ∗
A
A σ π 2π ∗
A
S σ π 2π ∗
A
S σ π 2σ ∗
A
A σ 2π ∗2
S
S σ 2π ∗σ ∗
S
A
...
a S is symmetric; A antisymmetric.
8Actually, we shall be looking at simple products of MOs, or configurations. Each configuration is associated with one or more states and gives the proper symmetry for these states as well as an approximate average energy of all the associated states. Hence, the treatment described here gives a sort of average state correlation diagram.
It might be more accurately called a configuration correlation diagram.
Chapter 14 Qualitative Molecular Orbital Theory intended correlation with σ 2π ∗2 cyclobutene if the disrotatory mode is followed. This is inferred from the orbital correlation diagram, Fig. 14-22.)
Assuming that the energies of states associated with these configurations fall into groups roughly given by sums of orbital energies, we obtain the two-sided diagram shown in Fig. 14-23. Only a few of the configurations are interconnected, to keep the diagram simple. Note that the ground-state configuration of butadiene correlates directly with the ground state of cyclobutene for conrotatory closure, but has an intended correlation with a doubly excited configuration in the disrotatory mode. This intended correlation would violate the noncrossing rule by crossing another line of S symmetry, so that the actual curve turns around and joins onto the ground state level for cyclobutene.
The effect of the intended correlation with a high-energy state is to produce a significant barrier to reaction. The figure shows that, for the first excited configuration, the highenergy barrier occurs for the opposite mode of reaction. State correlation diagrams Figure 14-23 A state or configuration correlation diagram for the electrocyclic closure of cis1,3-butadiene.
Section 14-9 Qualitative Molecular Orbital Theory of Reactions thus convert a “symmetry-forbidden” orbital correlation diagram into a high-activationenergy barrier: the conclusions are the same using either diagram.
Another kind of reaction that is formally closely related to the electrocyclic reaction is the cycloaddition reaction, exemplified by the Diels–Alder reaction between ethylene and butadiene to give cyclohexene (I). Such reactions are classified in terms of the number of centers between the points of connection. Thus, the Diels–Alder reaction is a [4 + 2] cycloaddition reaction. One can conceive of several distinct geometrical possibilities for a concerted mechanism for such a reaction. The two new σ bonds can be envisioned as being formed on the same face (suprafacial) (II) or opposite faces (antarafacial) (III) of each of the two reactants. The various possibilities are illustrated in Fig. 14-24. Qualitative MO theory is used to judge which process is energetically most favorable. One has a choice between the two-sided correlation diagram and the frontier orbital approach. We demonstrate the latter9 since it is simpler. Both methods lead to the same conclusion. In the course of this reaction, electrons become shared between the π systems of butadiene and ethylene. This is accomplished, to a rough approximation, by interaction between the HOMO of butadiene and the LUMO of ethylene and also between the LUMO of butadiene and the HOMO of ethylene. Let us consider the former interaction. The MOs are shown in Fig. 14-25 and the overlapping regions are indicated for the four geometric possibilities. Inspection of the sketches indicates that the two MOs have positive overlap in the regions of both incipient σ bonds only for the [4s + 2s] and [4a + 2a] modes. Therefore, the prediction is that these modes proceed with less activation energy and are favored. Now let us turn to the other pair of MOs, namely the LUMO of butadiene (IV) and the HOMO of ethylene (V).
9See Hoffmann and Woodward [15].
Chapter 14 Qualitative Molecular Orbital Theory
Figure 14-24
Four suprafacial–antarafacial combinations possible for the Diels–Alder 2 + 4 cycloaddition reaction.
Figure 14-25 Overlaps between HOMO of butadiene and LUMO of ethylene resulting from four interactive modes pictured in Fig. 14-24. The geometries for the four modes would all differ. These drawings are highly stylized.
Note that, for each MO, the end-to-end phase relationship is reversed from what it was before. Two symmetry reversals leave us with no net change in the intermolecular phase relations. It is easy to see, therefore, that these MOs also favor the [s, s] and [a, a] modes.
In cycloaddition reactions of this sort, one need analyze only one HOMO-LUMO pair in order to arrive at a prediction. Extension to longer molecules or to photochemical cycloadditions proceeds by the same kinds of arguments presented for the electrocyclic reactions.
Another type of reaction to which qualitative MO theory has been applied is the sigmatropic shift reaction, where a hydrogen migrates from one carbon to another and simultaneously a shift in the double bond system occurs. An example is given in Fig. 14-26.
The usual treatment of this reaction10 involves examining the HOMO for the system at some intermediate stage in the reaction where the hydrogen has lost much of its bonding to its original site and is trying to bond onto its new site. At this stage, the HOMO of the molecule becomes like that of the nonbonding MO of an odd alternant hydrocarbon (Fig. 14-27) with a slightly bound hydrogen on one end. The suprafacial mode is favored in this particular case because the hydrogen can maintain positive overlap simultaneously with its old and new sites—the new bond can form as the old Figure 14-26 The two possible distinct products resulting from a shift of a hydrogen from position 1 to position 5 in a substituted 1,3-pentadiene. Groups A, B, C, D are deuterium atoms, methyl groups, etc., enabling us to distinguish the products.
Figure 14-27 Phase relations in the HOMO for (a) suprafacial and (b) antarafacial [1, 5] sigmatropic shifts.
10See Woodward and Hoffmann [16].
Chapter 14 Qualitative Molecular Orbital Theory bond breaks. This is not possible for the antarafacial [1, 5] shift. However, the [1, 7] shift prefers the antarafacial mode.
Many other types of chemical reaction have been rationalized using qualitative MO theory. The association of SN 2 reactions with Walden inversion (i.e., the adding group attacks the opposite side of an atom from the leaving group) is rationalized by arguing that an approaching nucleophile will donate electrons into the LUMO of the substrate.
The LUMO for CH3Cl is shown in Fig. 14-28. A successful encounter between CH3Cl and a base results in a bond between the base and the carbon atom, so the HOMO of the base needs to overlap the p AO of carbon in the LUMO of Fig. 14-28a. Attack at the position marked “1” in the figure is unfavorable because any base MO would be near a nodal surface, yielding poor overlap with the LUMO. Therefore, attack at site 2 is favored. As the previously empty LUMO of CH3Cl becomes partially occupied, we expect a loss of bonding between C and Cl. Also, negative overlap between the forming C-base bond and the three “backside” hydrogens should encourage the latter to migrate away from the attacked side, as indicated in Fig. 14-28b.
The tendency of a high-energy, occupied MO of the base to couple strongly with the LUMO of a molecule like CH3Cl depends partly on the energy agreement between these MOs. If they are nearly isoenergetic, they mix much more easily and give a bonded combination of much lower energy. Molecules where the HOMO is high tend to be polarizable bases. A high-energy HOMO means that the electrons are not very well bound and will easily shift about to take advantage of perturbations. Such bases react readily with molecules having a low-energy LUMO (Fig. 14-29a). This corresponds to a “soft-base-soft-acid” interaction in the approach of Pearson [17]. When the HOMO and LUMO are in substantial energy disagreement, orbital overlap becomes less important as a controlling mechanism, and simple electrostatic interactions may dominate. This Figure 14-28 (a) The LUMO of CH3Cl. (b) Positive overlap between HOMO of base B and LUMO of CH3Cl increases antibonding between C and Cl and also repels H atoms from their original positions.
Section 14-9 Qualitative Molecular Orbital Theory of ReactionsFigure 14-29 LUMO–HOMO interactions and splitting for (a) nearly degenerate levels, (b) well separated levels.
is a “hard-acid-hard-base” situation. Thus, we expect QMOT rules to apply to soft-soft, rather than hard-hard interactions.
The reader may, by this time, begin to appreciate the very wide scope of QMOT and the large number of variations of a common theme that have been used. In this chapter we have given only a few representative examples. We have not described all the variations or all types of application. For fuller treatment, the reader should consult specialized books on this subject, some of which have been referred to in this chapter [18].
14-9.A Problems14-1. Calculate to first order the electronic energy of a hydrogen atom in its 1s state and in the presence of an additional proton at a distance of 2 a.u. What is the total energy to first order? Repeat for distances of 1 and 3 a.u. (See Appendix 3.)
14-2. Evaluate and graph the effects of dividing Haa ± Hab by 1 ± S for each of the following cases: Haa = 0, −5, −10, +10, −20. In each case, let Hab = −5, S = 0.5.
Does the QMOT rule that antibonding interactions are more 14-3. Table P14-3 is a list of electron affinities (in electron volts) of certain molecules and atoms. Can you rationalize the molecular values relative to the atomic values using QMOT ideas?
TABLE P14-3 Atomic and Molecular Electron Affinities (in electron volts)
H (0.75)
Cl (3.61)
C2 (3.4)
O2 (0.45)
Cl2 (2.38)
C (1.26)
Br (3.36) CN (3.82)
F2 (3.08)
Br2 (2.6) N (0.0 ± 0.2) I (3.06)
N2 (−16)
S2 (1.67)
I2 (2.55)
O (1.46)
S (2.08) CO (