The recognition of a leak in the lowest excited singlet potential energy surface halfway along the path of ground-state-forbidden pericyclic reac- tions is due to Zimmerman (1966). He argued at the Huckel level of the- ory, at which cont'igurational wave functions do not interact. The correla- tion diagram drawn Ibr the high-symmetry reaction path then exhibits a touching of S,,, S,, and Sz surfi~ces (Figure 6.1 la), since (I+,, [I), .,, and
@)=I: are all degenerate at the pericyclic geometry.
Van der Lugt and Oosterhoff (1969) took into account electron repulsion at the PPP level for the specific example of the conversion of butadiene to cyclobutene and noted that along the high-symmetry path the surface touch- ing becomes avoided (Figure 6.1 I b), with a minimum in the S , surface rather than the S, surface as drawn in the older Longuet-Higgins-Abrahamson (1965) and Woodward-Hoffmann (1969) correlation diagrams. This "peri- cyclic minimum" results from an avoided crossing of the covalent G and D
configurations (a, and @I=;:). Van der Lugt and Oosterhoff pointed out that it will provide an efficient point of return to S,, and a driving force for the reaction.
Subsequently, from an ab initio study of a simple four-electron-four- orbital model problem (H,), Gerhartz et al. (1977) concluded that the peri- cyclic "minimum" thitt results if symmetry is retained in the correlation diagram is not a minimum along other directions, specifically the symmetry- lowering direction that increases the diagonal interactions in the perimeter, stabilizing the S, statc and destabilizing the S,, state (Section 4.4.1). They identified an actual S,-SO surface touching (conical intersection) in H,, in a sense returning to the original concept of Zimmerman (1966), but at a less symmetrical geometry. They proposed that analogous diagonal bonding in the pericyclic perimeter is a general feature in organic analogues of the sim- ple Hz + Hz reaction they studied, in which the simple minimum actually was not a minimum hut a saddle point (transition structure) between two conical intersections (Figure 6.1 Ic). Assuming, however, that in the low- symmetry organic analogues the surface touching probably would be weakly avoided, they conlinucd to retkr to the global region of the pericyclic funnel, including the diagonally distorted structure, as "pericyclic minimum." They proposed that becausc of the diagonal distortion at the point of return to the S,, surface, the mechanism now accounted not only for the ordinary 12, +
2,] pericyclic reactions, such as the butadiene-to-cyclobutene conversion, but also for x[2, + 2,] reactions, such as the butadiene-to-bicyclobutane con- version. Although the exact structure at the bottom point of the pericyclic funnel remained unknown, it was clear from the experiments that it is a funnel in the sense of being extremely efficient in returning the excited mol- ecules to the ground state.
Finally, in a remarkable series of recent papers, Bernardi, Olivucci, Robb and their collaborators (IWO-1994) demonstrated that the S,-S, touching actually is not avoided even in the low-symmetry case of real organic mol- ecules, and they confirmed t he earlier conjectures by computing the actual geometries of the funnels (conical intersections) in the S , surface at a rea- sonable level of ab initio theory. They also pointed out that still additional reactions can proceed through the same pericyclic funnel, such as the cis- trans isomerization of butadiene.
Sections 6.2. I. I and 6.2.1.2 describe the pericyclic funnel in more detail for two particularly important and illuminating examples: the cycloaddition of two ethylene molecules and the isomerization of butndiene. We rely heav- ily on the results of recent ad initio calculations (Bernardi et id., IC)!90u; Oli- vucci et al., 1993).
Figure 6.11. Schematic correlation diagrams for ground-state-forbidden pericyclic reactions; a) HMO model of Zimmerman (1%6), b) PPP model of van der Lugt and Oosterhoff (1%9). and c) real conical intersection resulting from dii~gonal interac- tions. The two planes shown correspond to the homosymmetric (y) and heterosym- metric (6) case. Cf. Figure 4.20.
6.2.1.1 The Cycloaddition of Two Ethylene Molecules
First, we consider the face-to-face addition of two ethylene molecules. The orbital correl;~tion c l i ; ~ ! r;1111 li)r thc I~igh-hvn~rnctl-y path ;~n:~logous to tho rec-
6.2 PEKICYCLIC REACTIONS tangular H z + H z path discussed in Section 4.4.1 wils given in Figure 4.15.
The resulting configuration and state correlation diagriims are essentially the same as those for the electrocyclic ring closure of hutadiene discussed in Section 4.2.3. At the halfway point (rectangular geometry, four equal reso- nance integrals along the perimeter), the situation corresponds to a perfect biradical, analogous to the n system of cyclobutadiene. (Cf. Section 4.3.2.) The order of the states is T below G (S,,), followed hy D (S,) and S (Sz) at higher energies. As discussed in Section 4.4.1 for the model case of H,, one can use either of the simplified approaches, 3 x 3 CI or 2 x 2 VB, to un- derstand the nature of the distortion that leads from the rectangular geome- try to the conical intersection of S , with S,,. The use of the former permits a discussion of substituent effects, and we shall consider it first.
From Figure 4.20b it is apparent that a perturbation (3 produces a hetero- symmetric biradicaloid and promises to lead to thc critically heterosym- metric situation (i, = c'i,,). whcre S,, and S, ;we d e g c n c ~ i t e and u real conical inlersection results. f;rom 1;igur-e 6.12 i t is seen that ;I diagonal interiiction differentiates the otherwise degenerate energies of thc localized nonbonding orbitals of the cyclobutadiene-like perfect biradical and thus produces a 6 perturbation. This suggests that the conical intersection may be reached by a rhomboidal distortion of the pericyclic geometry. which decreases the length of one diagonal and increases that of the other and therefore corre- sponds to either an enhanced 1,3 or an enhanced 2.4 interaction. For unsub- stituted ethylenes, these two cases are equivalent by symmetry, and the peri- cyclic funnel consists of two conical intersections separated by a transition state iit the rectangular geometry as shown in Figure 6.13 (Hernardi et al.,
I99oa.h; Bentzien iind Klessinger, 1994).
In the case of the [2 + 21 photocycloaddition ol' substituted ethylenes, the two situations remain equivalent for head-to-head addition (1,2 arrange- ment of the substituents) but are no longer equivalent for head-to-tail addi- tion (1.3 arrangement of the substituents), since the latter affects the energy difference between the two localized forms of the nonbonding orbitals. Ac-
2.L interaction 1.3 ~nteraction
Figure 6.12. The effect of di;igonal interactions on the energy of the degenerate loc;~liir.ed nonhonding orbitals of ;I cyclohut;idicnc-like pcrlkct hiradical.
Figure 6.13. Pericyclic funnel region of ethylene dimerization, showing two equiv- alent conical intersections corresponding to 1.3 and 2,4 diagonal interactions and the transition state region at rectangular geometry (a = 0). The curves shown for a = 0 correspond to the van der Lugt-Oosterhoff model (by permission from Klessinger,
19951.
cess to onc of the otherwise equivalent conical intersections will thus be favored. Electron-donating substituents will reinforce the effect of the diag- onal interaction in the case of 1,3 substitution and will counteract it in the case of 2.4 substitution. The opposite will be true of electron-withdrawing substituents (BonaCiC-Koutecky et al., 1987). The overall effect of substi- tuents on the kinetics and product distribution in [2 + 21 photoaddition of olefins is however complicated by the possible intermediacy of an excimer or an exciplex, which will typically be more stable for the head-to-head ar- rangement of substituents. This is discussed in more detail in Section 7.4.2.
Depending on the extent of diagonal interaction, the rhomboidal distortion may lead to preferential diagonal bonding in contrast to pericyclic bonding after return to the ground state, resulting in an x[2, + 2,] addition product rather than a [2, + 2,l product. Examples of x[2 + 21 cycloaddition reactions also are discussed in Section 7.4.2.
In terms of the 2 x 2 V B model, the argument for rhomboidal geometries as favored candidates for a conical intersection (Bernardi, 1990a,b) goes as follows. (Cf. Figure 6.14.) The exchange integral KR (cf. Example 4.13), which corresponds to the spin coupling in the reactants, will decrease rap- idly with decreasing 1.2 iind 3.4 overlap as the intrafi-agment CC distance
PHOTOCHEMICAL REACTION MODELS
Figure 6.14. Schematic representation of possible geometries at which the conical intersection conditions may be satisfied. Dark double-headed arrows represent K , , light arrows K,,, and broken arrows K , contributions (adapted from Bernardi et a]., 1990a).
increases. The exchange integral K p , which corresponds to the product spin coupling, will increase rapidly with increasing 1,3 and 2,4 overlap, as the interfragment distance is decreased. K x will be quite small for rectangular geometries. While one might satisfy Ku = K, at these geometries, Ku = K x can never be satisfied. Figure 6.14 gives a schematic representation of pos- sible geometries on the (F - 2)-dimensional hyperline, where the conditions imposed by the equations of Example 4.13 may be satisfied and El = E,, =
Q. The Coulomb energy Q will be repulsive in the region of structures b through d in Figure 6.14 because of interference of the methylene hydrogen atoms. Thus, in agreement with the argument based on the two-electron- two-orbital model, the favored region of conical intersection for the [2 + 21
cycloaddition is found at the rhomboidal geometries.
6.2.1.2 The Isomerization of Butadiene
Monomolecular photochemistry of butadiene is rather complex. Direct ir- radiation in dilute solution causes double-bond cis-trans isomerization and rearrangements to cyclobutene, bicyclo[l. l .O]butane, and I-methylcyclo- propene (Srinivasan, 1968; Squillacote and Semple, 1990). Matrix-isolation studies have established another efficient pathway, s-cis-s-trans isomeriza- tion (Squillacote et al., 1979; Arnold et al., 1990, 1991 ).
Scheme 2 illustrates the phototransformations of s-cis-butadiene. Until recently, it was not clear whether the reaction paths leading from the two conformers to the various observed photoproducts are different already on the S, surface or whether they do not diverge until al'ter return to S,,. The latter alternative is now believed to be correct. As suggested by the results obtained for the H, model system (20 x 20 C1 model, see Section 4.4). a simultaneous twist about all three CC bonds provides the combination of perimeter and diagonal interactions (Figure 6.15) needed to reach a funnel ( S , S , degeneracy). Return to S,, through this geometry accounts for the for- mation of both cyclobutene and of bicyclo[l.l.O]butane (Gerhartz et al., 1977). The twist around the central CC bond was first invoked by Bigwood and BouC (1974). Both the 3 x 3 CI and the 2 x 2 VB models account for the presence of this pericyclic funnel in S,. Recent extensive calculations
Scheme 2
(Olivucci et al., 1993, 1994b; Celani et al., 1995) put these conjectures on firm ground and suggest very strongly that all the different photochemical transformations of butudienes listed in Scheme 2, including cis-trans iso- merization, cyclization. bicyclization, and s-cis-s-trans-isomerization, in- volve passage through a common funnel located in a region of geometries characterized by a combination of peripheral and diagonal bonding (Figure 6.1%-f). Moreover, contrary to previous expectations, these calculations demonstrate that in the funnel region the S,-S,, gap not only is very small but vanishes altogether in a large region of geometries. The funnel therefore
Figure 6.15. n-Orbital interactions in butadiene; a) planar geometry, p AOs, b) high- symmetry disrotatory pericyclic geometry, peripheral interactions along the perim- eter, and ckf) geometries of two equivalent funnels; diagonal interactions are shown in c) and d); peripheral inreractions in e) and f).
338 I'li( )'l'OCHI:bI ICAL. KEAC'I'ION MODELS
6.2 PERlCYCLlC REACTIONS 339
corresponds to a true S,-S,, touching. that is, u conical intersection of the kind expected for a critically heterosymmetric biradicaloid geometry.
For unsubstituted butadienes, the 1.3- and 2,4-diagonal interactions are
equivalent (Figure 6 . 1 5 ~ and d), as are the two possible movements of the 8 terminal methylene groups above or below the molecular plane. Thus, the
pericyclic funnel consists of a total of four equivalent conical intersection regions, communicating pairwise through the two high-symmetry disrota- tory "pericyclic minima" investigated by van der Lugt and Oosterhoff (1969). The funnel can be reached from either of the vertically excited ge- ometries, s-cis- and s-!runs-butadiene, after efficient crossing from the ini- tially excited optically allowed B symmetry (S) state to the much more co- valent A symmetry (D) state. Three points of minimal energy within the funnel, dubbed s-trans, s-cis, and central conical intersection, have been identified within the conical intersection region, as shown in Figure 6.16.
However, the lowest-energy geometries within the funnel region are prob- ably never reached in the S, state, since rapid return to S,, most likely ensues
5 - cisoid central s - transoid
Figure 6.16. The funnel region for butadiene isomerization: a) cross section of the excited-state potential energy surface for P = a:, where a, and a2 correspond to rotations about the two C==€ bonds of butadiene and /3 to rotation about the single bond and b) optimized geometries of the three points of minimal energy within the funnel (by permission from Olivucci et a]., 1993).
once the edge of the funnel region has been reached. There are two stereo- chemically different pathways connecting the spectroscopic minima with the central funnel region. As expected from the classical Woodward-Hoffmann correlation diagrams (Section 4.2. I ) , the disrotatory pathway is energetically mare favorable than the conrotatory one. (Cf. Sections 7.1.3 and 7.5.1.)
In the absence of molecular dynamics calculations, it is impossible to make quantitative statements about the probability with which the various product geometries will be reached on the S,, surface once the molecule has passed through the funnel. Bonding is possible both by increasing further the strength of the diagonal interaction to yield bicyclobutane (6 perturba- tion, x , in the branching space) and by increasing the strength of either pair of opposite-side pericyclic interactions to yield either cyclobutene or buta- diene (y perturbation, x, in the branching space). In the latter case, an s-cis or an s-trans product may result, depending on the direction of twist along the central CC bond, and either a cis or a trans geometry may result at the double bonds in suitably labeled butadienes. This ambiguity accounts for the lack of stereospecificity in the ring opening of labeled cyclobutenes, which is believed to occur through the same funnel (Bernardi et al., 1992~).