A qualitative anticipation of the location of minima and funnels on excited- state potential energy surfaces, in particular the S, and TI surfaces, is in general more difficult than the estimation of the minima on the ground-state surface So that correspond to stable ground-state species. There are three basic types of geometries where one would intuitively expect minima andlor funnels in S, and T,-namely, near ground-state equilibrium geometries, at exciplex and excimer (complex) geometries, and at biradicaloid geometries.
Minima on excited-state surfaces in the region of ground-state equilibrium geometries may be referred to as spectroscopic minima. Spectroscopic tran- sitions from the ground state to such minima or vice versa are in general easy to observe, even if the probability may be somewhat constrained by the Franck-Condon principle. Excitation from the ground state to spectro- scopic minima is approximately vertical. Generally, upon return from these minima to the ground state by fluorescence, phosphorescence, or radiation- less processes, an excited molecule will end up in the initial minimum of So, and the excitation-relaxation sequence corresponds to a photoptlysical pro- cess without chemical conversion. In particular in larger molecules such as naphthalene, the promotion of an electron from a bonding into an antibond- ing MO causes such a small perturbation of the total bonding situation that its effect on the equilibrium geometry of the excited state is relatively small.
It can, however, also be more significant. Thus, formaldehyde is pyramidal in the I(n,n*) as well as in the '(n.n*) state, and not planar as in the ground state. (Cf. Section 1.4.1 .)
Exciplex minima can be viewed as a particular case of spectroscopic min- ima. Their presence in the excited surfaces reflects the fact that molecules are more polarizable. more prone to charge-transfer processes, and gener- ally "stickier" in the excited state. These minima occur at geometries that correspond to a fairly close approach of two molecules at their usual ground- state geometries, for example, the face-to-face approach of two n systems generally associated with excimers and exciplexes. The ground-state surface normally does not have a significant minimum at the complex geometry when the very shallow van der Waals minimum is disregarded, except in the so-called charge-transfer complexes and similar cases. Radiative or nonra- diative return from this type of minimum in an excited state to the ground state leads to the reformation of the two starting molecules in an overall photophysical process.
If a pair of nearly degenerate approximately nonbonding orbitals is oc- cupied with a total of only two electrons in the simple MO picture of the ground state, the molecule is called a biradical or a biradicaloid. Various types of biradicals and biradicaloids will be discussed in some detail in Sec- tion 4.3. A usual prerequisite for the presence of degenerate orbitals is a specific nuclear arrangement that is referred to as biradicaloid geometry.
Minima and funnels at such biradicaloid geometries, that is, biradicaloid minima or funnels, normally are reactive minima or funnels. They are typi- cally characterized by a small or zero (in the case of critically heterosym- metric biradicaloids, see Section 4.3.3) energy gap between the potential energy surfaces; a usually fast radiationless transition from the excited state to the ground state; and normally a shallow minimum in the ground-state surface, if any. A molecule in a biradicaloid minimum generally is a very short-lived species. After return to So, deeper minima at geometries other than the initial one will usually be reached, so the net process corresponds to a photochemical conversion. The relevance of biradicaloid minima for photochemical reactivity was first pointed out by Zimmerman (1%6, 1969) and van der Lugt and Oosterhoff (1969). They represent funnels or leaks in the excited-state potential energy surface (Michl, 1972, 1974a).
Biradicaloid geometries are in general highly unfavorable in the ground state, since the two electrons in nonbonding orbitals contribute nothing to the number of bonds. As a result, the total bonding is less than that ordinar- ily possible for the number of electrons available. When the geometry is distorted to one in which the two orbitals are forced to interact, a stabiliza- tion will result. This is evident from Figure 4.4, which shows that the two orbitals combine into one bonding and one antibonding orbital, and both
Figure 4.4. Orbital energy scheme for two orbitals, cp, and q2, which are occupied with a total of two electrons: a) for biradicaloid geometries cp, and cp, are degenerate;
from the three possible singlet configurations only one is shown; b) if the degeneracy is removed the ground state will be stabilized; c) excited configurations will usually be destabilized. The doubly excited configuration and the triplet configuration will be destabilized correspondingly.
188 4 POTENTIAL ENERGY SURFACES: BARRIERS, MINIMA, AND FUNNELS 4.1 POTENTIAL ENERGY SURFACES 189
electrons can occupy the bonding orbital. In the excited state, however, any distortion from a biradicaloid geometry that causes the two orbitals to inter- act, and split into a bonding and an antibonding combination, is likely to lead to destabilization, since only one of the two electrons can be kept in the bonding MO, while the other one is kept in the antibonding MO, and the destabilizing effect of the latter predominates.
Here, we consider the two simplest and most fundamental cases of bi- radicaloid geometries, reached from ordinary geometries by stretching a sin- gle bond or by twisting a double bond. These two cases together underlie much of organic photochemistry.
The stretching of an H, molecule provides an illustration of the dissocia- tion of a prototypical single bond (see Figure 4.5): The minimum in So occurs at small internuclear distances (74 pm) and is definitely not a biradicaloid minimum since the a, MO is clearly bonding and the a,*. MO clearly anti- bonding at this geometry. In the dissociation limit, the So state corresponds to a pair of H atoms coupled into a singlet and has a high energy, the same as the TI state. In the latter, the coupling of the electron spins is different, but this has no effect on the energy since the H atoms are far apart. The T I
Figure 4.5. Energies a) of the bonding MO a, and the antibonding MO a:, as well as b) o f the electronic states of the H, molecule as a function of bond length (sche- matic). On the left, the states are labeled by the MO configuration that is dominant at the equilibrium distance; on the right, they are labeled by the VB structure that is dominant in the dissociation limit. Rydberg states are ignored (by permission from Michl and BonaCiC-Kouteckv, 1990).
state has its minimum at the infinite separation of the two H atoms where both MOs are nonbonding. The fully dissociated geometry thus clearly cor- responds to a biradicaloid minimum. The S, state has a minimum at 130 pm, that is, at intermediate nuclear separations. For this geometry the MOs a, and c$ are much less bonding and antibonding, respectively, than for the ground-state geometry, so the minimum can also be referred to as biradical- oid. The difference in the minimum geometry of T I and S, is easily under- stood when it is realized that S, and S, are of zwitterionic nature and may be described by the VB structures HGH@ and H@H@. (See Example 4.3.) The location of the minimum in S, represents a compromise between the tendency to minimize the energy difference between a, and a:, favoring a large internuclear distance, and to minimize the electrostatic energy corre- sponding to the charge separation in the zwitterionic state, favoring a small internuclear distance.
Even though H, may not appear to be of much interest to the organic chemist, the importance of Figure 4.5 cannot be overemphasized, and we shall refer to it frequently in the remainder of the text. This is because it represents the dissociation of a single bond in its various states of excitation, and the breaking of a single bond, usually aided by the bond's environment, underlies most chemical reactions. In a sense, Figure 4.5 represents the sim- plest orbital and state correlation diagrams, to which Section 4.2 is dedi- cated. Introduction of perturbations by the environment of the bond con- verts its parts a and b, respectively, into correlation diagrams for the orbitals (e.g., Figures 4.10 and 4.1 I ) or the states (e.g., Figure 4.13) of systems of actual interest for the organic photochemist. In Section 4.3 we shall examine the wave functions of an electron pair in its various states of excitation in more detail, and shall describe the relation between the MO-based descrip- tion (left-hand side of Figure 4.5b) and the VB-based description (right-hand side of Figure 4.5b). A recent illustration of the same principle in the case of a C--C bond, more directly relevant to organic photochemistry, is pro- vided by the contrast between the Franck-Condon envelopes of the So-S, and the So-T, transitions in [I. I . l]propellane (1) (Schafer et al., 1992).
These make it clear that the length of the central bond is nearly the same in the ground state and the '(a,#) excited state, which can be thought of as a contact ion pair. In contrast, in spite of the constraint imposed by the tri- cyclic cage, the central bond is substantially longer in the '(a@) state, which can be thought of as a repulsive triplet radical pair.
However, it is also important to note the essential limitations of the dia- grams presented in Figure 4.5: they apply to the dissociation of single bonds
190 4 POTENTIAL ENERGY SURFACES: BAKHIEKS, MINIMA, AND FUNNELS 4.1 POTENTIAL ENERGY SURFACES 191
that ( I ) are nonpolar and (2) connect atoms neither of which carries lone- pair AOs or empty AOs nor is involved in multiple bonds to other atoms.
Thus, as we shall see later (Section 6.3.3), they apply to what is known as nonpolar bitopic bond dissociation. For example, Figure 4.5 applies to the dissociation of the Si-Si bond in a saturated oligosilane (see Section 7.2.3), but not to the dissociation of the C-SO bond in an alkylsulfonium salt, to the dissociation of the C--0 bond in an alcohol (see Section 6.3.3), or to the dissociation of the C--C bond adjacent to a carbonyl group. (See Section 6.3.1 .) Introduction of an electronegativity difference between the two bond termini causes changes in the course of the correlation lines in Figure 4.5 and is discussed in Section 4.3. Introduction of lone pairs, empty orbitals, or adjacent multiple bonds introduces new low-energy states and is dis- cussed in Section 6.3.3.
The twisting of the ethylene molecule provides an illustration of a proto- typical n bond dissociation (Figure 4.6): The minimum in So occurs at a planar geometry (twist angle of 0") and is definitely not a biradicaloid mini- mum since the nu MO is clearly bonding and the $ MO clearly antibonding at this geometry. In contrast, the TI, S,, and S, states have a minimum at 90"
twist, where both MOs are nonbonding. This obviously is a biradicaloid min-
Figure 4.6. Energies a) of the bonding MO n and the antibonding MO n* and b) of the n-electronic states of ethylene as a function of the twist angle 8. On both sides the states are labeled by the MO configuration dominant at planar geometries; in the middle, they are labeled by the VB structure that is dominant at the orthogonally twisted geometry (by permission from Michl and BonaCiC-Kouteckg, 1990).
imum, and the wave functions of the molecular states are analogous to those discussed above for the case of H,. The So and TI states are represented by VB structures containing a single electron in each of the carbon 2p orbitals of the original double bond, prevented from interacting by symmetry, and not by distance as in the Hz case. The S, and S, states are of zwitterionic nature and are represented by VB structures containing both electrons in one of these orbitals and none in the other.
Unlike the biradicaloid minima located along the dissociation path of a a bond, one of which occurred at a loose (TI) and one at a tight (S,) geometry, those located along the dissociation path of a n bond are at nearly the same geometry. The reason for this difference can be readily understood in terms of the simple model of biradicaloid states outlined in Section 4.3.
Figure 4.6 again represents a very simple correlation diagram, and is of similar fundamental importance for reactions involving the dissociation of a n bond as Figure 4.5 is for reactions in which a a bond dissociates. It un- derlies correlation diagrams such as that for the cis-trans isomerization of stilbene (Figure 7.3). Its use is subject to similar limitations as the use of Figure 4.5: ( I ) The course of the potential energy curves changes when the electronegativity of the two bond termini differs (e.g., Figures 4.22 and 7.6), and such modifications are essential for the understanding of the relation of the potential energy diagrams of protonated Schiff bases and TICT (twisted internal charge transfer) molecules to those of simple olefins. (2) The num- ber of low-energy potential energy surfaces increases when one or both bond termini carry lone pairs or empty orbitals, or if they engage in additional multiple bond formation. Examples of such situations are the cis-trans iso- merization of Schiff bases (Figure 7.8) and azo compounds (Figures 7.10 and 7.11).
A more complete discussion of the results embodied in Figures 4.5 and 4.6 than can be given here is available elsewhere (Michl and BonaCiC-Kou- teckl, 1990).
Geometries with large distances between the radical centers such as the TI state of Hz are called loose biradicaloid geometries; those with small dis- tances such as the Sl state of Hz are called tight biradicaloid geometries.
Since biradicaloid minima in TI occur preferentially for loose geometries, whereas those in Sl occur for tight geometries, return from the S, and T, to the ground state So will frequently take place at different geometries. This presumably quite general behavior may be one of the primary reasons for the difference in the photochemistry of molecules in the singlet and triplet states (Michl, 1972; Zimmerman et al., 1981).
Example 4.3:
The simplest model to describe a a a s well as a n bond is the two-electron two- orbital model, which will be used frequently in the remainder of the text. The MO and the VB treatment of the H, molecule may serve as an example. In the
192 4 POTENTIAL ENERGY SURFACES: BARRIERS, MINIMA, AND FUNNELS