Anasymmetrization

A glance at the illustration of planar cyclohexene in Fig. 7.2 triggers an immediate objection The molecule is too strained to remain in plane, but would relax spontaneously from to either the cis (CJ ) or the trans (C0 conformation. As in the case of Fig. 5.6, where neither the reactant nor the product is in its stable geometry, planar cyclohexene must be understood to be a formal construct representing both accessible conformations of the molecule. The procedure was formally justified in the paper outlining the theoretical ba- 

OCAMS makes two tacit assumptions that are fundamental to all discussions of mechanism in terms of orbital symmetry  

Assume that some sym-op 5 present in G is missing in H, and that G can be generated as the direct or semidirect product of H and the two-element group E, 5. In the reaction under consideration, G is that can be generated as the direct product of either or with which comprises the identity 

Since the number of MOs in G must be the same as that in H, only one of the two linear combinations can represent ipi in the higher group. The choice of the proper sign is made on the basis of a few simple rules, that ensure continuity of the phases of the occupied orbitals along the reaction path, and guarantee that the result is independent of the particular sym-op used as anasymmetrizer  

If an atomic orbital that is included in any occupied MO is transformed in the higher group to itself, or to an AO that was symmetrically equivalent to it before anasymmetrization, it cannot be allowed to vanish in all of the anasymmetrized MOs. (A particularly useful corollary is that no MO can be allowed to disappear completely on anasymmetrization). 

---

When two AOs centered on initially non-equivalent atoms become equivalent upon anasymmetrization, any MO in which they appear, V (H), and its mirror image , S ipi(fl) are combined once with a positive and once with a negative sign. 

An AO that participates in a bonding-antibonding combination cannot be allowed to appear in one but disappear from the other on anasymmetrization. 

These rules establish the continuous one-to-one correspondence between the Born-Oppenheimer eigenstates and those of the anasymmetrized Hamiltonian which is required by theory [6, p. 12]. They are related to Goddard s [8] phase continuity rule, which, however, does not invoke symmetry explicitly. 

If the reactants and product are set up in C, all of the occupied MOs would correlate across the diagram. Alternatively, a correspondence diagram, in which the reactants are set up in C2V and the product is anasymmetrized to that symmetry point group, would show that formal desymmetrization of the pathway to C, - i.e. to the true molecular symmetry of the product -is called for. The methylene bridge of cyclopentadiene is innocuous so, as it turns out, is the bridging carbonyl group in tropone. It will become evident from subsequent examples, however, that the presence of heteroatoms and/or multiple bonds can make a substantial difference to the conclusions drawn from an orbital symmetry analysis. 

Woodward and Hoffmann [4, p. 147] characterize the reaction as an Homed [ 4a H- 2a]-cycloaddition, in which 5 /n-TC0D is formed rather than its anti isomer as a result of secondary orbital interactions. The reaction was analysed by OCAMS [5, p. 598] for a nearly-coplanar axial approach of the two CBD molecules, leading to an impossibly strained dimer with axial symmetry and a puckered central ring. The axial dimer , however, is not intended to depict a real molecule, but is a purely formal model produced by anasymmetrization to D2 of either the syn (Cs) or the anti (C2) isomer, and can thus represent them both. The correspondence diagram [5, Fig. 7] then shows that relaxation to the syn isomer involves a lower investment in distortional energy, in agreement with experiment. 

The dimer can be anasymmetrized to T>2h from the subgroup of syn-TCOD or from Cf/, that of its anti isomer. It is easily confirmed that the irreps of the four occupied MOs are just those that would be assigned to the molecule if either isomer were forced into planarity. The two lower orbitals of the reactant pair, which are the same on both sides of Fig. 7.9, correlate directly with the two upper orbitals of the product, and two non-correlating pairs of orbitals remain in both cases. Despite the apparently different nature of the bonding processes... 

Both of the pathways that allow out-of-plane relaxation of the a frame are [3s -f 3s], and are therefore allowed by the Rules. So are the other two, which are [3a + 3a], but these can be rejected as sterically unfavorable. Instead of setting up two separate orbital correlation diagrams, between the reactant and the chair and boat transition structures respectively, we make do with a single correspondence diagram in D2/1. For this purpose, the syn and anti conformers of the reactant are anasymmetrized to D2/1, in analogy to the superposition... 

The alternative choice would exchange the symmetry labels of the two anasymmetrized tt orbitals, but the electron configuration would remain the same. 

Both reaction paths are analysed in Fig. 8.11 The reaction involves 12 mobile electrons assuming that neither TS is a biradical, all we need consider are six doubly occupied orbitals four constructed from the a bonds to the migrating C atom and two tt orbitals. Both transition structures have a mirror plane passing through C7, Ci and C4, and the symmetry of their occupied MOs is characterized accordingly in the diagram on either side of the anasymmetrized reactant-product pair. 

Anasymmetrization of the occupied orbitals of norcaradiene is straightforward. The two bonds forming the cyclopropane ring involve the Pz and orbitals of C7 respectively in order for them not to vanish, the first MO must be taken as a and the second as a". The bonds to the substituent H atoms (a and b) both lie in the yz plane, so their positive and negative combinations both have to be a. C3 and C5 are reflected into each other if their Pz AOs are in-phase in the lower tt orbital they have to be taken out-of-phase in the upper. The occupied orbitals are thus anasymmetrized to [4xa - -2xa"]. As can be seen, both transition states correlate equally well with the anasymmetrized reactant-product pair, so both have to be characterized as allowed. The choice between them is evidently made by energetic factors that are not related to orbital symmetry, perhaps the better overlap of C7 s px than its Pz with the Pz AOs of C2 and Cq. 

Correspondence Between Anasymmetrized Reactant and Transition Structure... 

Figure8.14. Correspondence diagram between the anasymmetrized isomers of CBD and the square-planar TS(D2/i)...

Like the corresponding isomerization of prismane, the thermal isomerization of DB to the closed shell ground-state of benzene was found to be forbidden by the schematic correspondence diagram in Fig 5.11. It was pointed out, however, that DB crosses quite easily to the first excited triplet Ti of benzene [49, 50]. This isomerization was the first - and only - spin-non-conservative reaction analysed in the popular presentation of OCAMS [13] a more complete correspondence diagram, in which Dewar benzene was anasymmetrized and benzene desymmetrized to D2/1, followed shortly thereafter [51]. The diagram in Fig. 9.9, set up in C2V, reproduces the third one published [14, Fig. 5], differing from it in... 

First, we recognize that although formaldehyde itself is non-planar in Si, the inversion barrier is very low (1-2 kcal/mol) [28, 29], so formal imposition of planarity cannot introduce any error. The radical pathway, involving rupture of a single CH bond, does not retain C2(z) or a(zx) its highest possible symmetry is Formal symmetry can be retained for the purpose of the analysis by anasymmetrization (Section 7.1.1.1) - a particularly simple procedure in the present instance. 

---