Figure 6.3 shows a schematic representation of two surfaces of the ground state (S,) and of an excited state (S, or T I ) and of various processes following initial excitation. The thermal equilibration with the surrounding dense me- dium requires a sojourn in some local minimum on the surface for approxi- mately I ps or longer. This may happen for the first time while the reacting species is still in an electronically excited state (Figure 6.3, path c), or it may occur only after return to the ground state (Figure 6.3, path a). In the former case, the reaction mechanism can be said to be "complex," in the latter,
"direct." Direct reactions such as direct photodissociations cannot be quenched. The first ground-state minimum in which equilibrium is reached need not correspond to the species that is actually isolated. Instead, it may correspond to a pair of radicals or to some extremely reactive biradical, etc.
(Figure 6.3, path k). Ncvertheless, this seems to be a reasonable point at which the photochemical reaction proper can be considered to have been
0.1 A QUALI'IXI'IVE PHYSICAL MODEL 315
Figure 6.3. Schematic representation of potential energy surfaces of the ground state (S,,) and an excited state (S, or TI) and of various processes following initial excitation (by permission from Michl. 1974a).
completed, even though the subsequent thermal reactions may be ofdecisive practical importance.
If return t o So from the minimum in S , o r T , originally reached by the molecule is slow enough for vibrational equilibration in the minimum to oc- cur first, the reaction can be said to have an excited-sttrte intermediate. The sum of the quantum yields of all processes that proceed from such a mini- mum, that is, from an intermediate, cannot exceed one.
Those minima in the lowest excited-state surface that permit return t o the ground-state surface so rapidly that there is not enough time for thermal equilibration are termed "funnels." (Usually. they correspond to conical in- tersections, cf. Section 4.1.2.) By definition, direct photochemical reactions without an intermediate proceed through a funnel. Sometimes, a funnel may be located on a sloping surface and not actually correspond t o a minimum.
Since different valleys in S,, may be reached through the same funnel de- pending on which direction the molecule first came from, the sum of quan- tum yields of all processes proceeding from the same funnel can differ from unity. The reason for the difference between an intermediate and a funnel is that a molecule in a funnel is not sufficiently characterized by giving only the positions of the nuclei-the directions and velocities of their motion are needed a s well (Michl, 1972).
Example 6.2:
I'assage of a reacting system to the S,, surface through a funnel in the S, surface is common in organic photochemical reactions, but the quantitative description of such an event is not easy. Until recently, it was believed that the funnels mostly correspond to surface touchings that are weakly avoided, but recent work of Bernardi, Olivucci, Robb, and co-workers (1990-1994) has shown that the touchings are actually mostly unavoided and correspond to true conical intersections. This discovery does not have much effect on the description of the dynamics of the nuclear motion, since the unavoided touching is merely a limiting case of a weakly avoided touching. Only when the degree of avoidance becomes large, comparable to vibrational level spacing, does the efficiency of the return to So suffer much. The expression "funnel" is ordinarily reserved for regions of the potential energy surface in which the likelihood of a jump to the lower surface is so high that vibrational relaxation does not compete well.
The simplest cases to describe are those with only one degree of freedom in the nuclear configuration space. In such a one-dimensional case, the probabil- ity P for the nuclear motion to follow the nonadiabatic potential energy surface
Figure 6.4. Schematic representation of potential energy surfaces So and S, as well as So+S, excitation (solid arrows) and nuclear motion under the influ- ence of the potential energy surfaces (broken arrows) a) in the case of an avoided crossing, and b) in the case of an allowed crossing (adapted from Michl, 1974a).
316 PHUrOCHEMICAL KEACTION MODELS (i.e., to perform a jump from the upper to the lower adiabatic surface) is given approximately by the relation of Landau (1932) and Zener (1932):
P = exp [ ( - n2/h) (AEZIvAS) ]
Here, AE is the energy gap between the two potential energy surfaces at the geometry of closest approach, AS is the difference of surface slopes in the region of the avoided crossing, and v is the velocity of the nuclear motion along the reaction coordinate. Thus, the probability of a "jump" from one adiabatic Born-Oppenheimer surface to another increases, on the one hand, with an in- creasing difference in the surface slopes and increasing velocity of the nuclear motion, and on the other hand, with decreasing energy gap AE. The less avoided the crossing, the larger the jump probability; when the crossing is not avoided at all, AE = 0 and therefore P = 1. This is indicated schematically in Figure 6.4.
In the many-dimensional case, the situation is more complicated (Figure 6.5). The arrival of a nuclear wave packet into a region of an unavoided or weakly avoided conical intersection (Section 4.1.2) still means that the jump to the lower surface will occur with high probability upon first passage. How- ever, the probability will not be quite 100%. This can be easily understood qualitatively, since the entire wave packet cannot squeeze into the tip of the cone when viewed in the two-dimensional branching space, and some of it is forced to experience a path along a weakly avoided rather than an unavoided crossing, even in the case of a true conical intersection.
An actual calculation of the S,+S, jump probability requires quantum me- chanical calculations of the time evolution of the wave packet representing the initial vibrational wave function as it passes through the funnel (Manthe and
Figure 6.5. Conical intersection of two potential energy surfaces S, and So;
the coordinates x, and x, define the branching space, while the touching point corresponds to an (F - 2)-dimensional "hyperline." Excitation of reactant R yields R*, and passage through the funnel yields products P, and P, (by per- mission from Klessinger, 1995).
Koppel, 1990), or a more approximate semiclassical trajectory calculation (Herman, 1984). In systems of interest to the organic photochemist, ~ i r ~ ~ t i l t a - neous loss of vibrational energy to the solvent would also have to be included, and reliable calculations of quantum yields are not yet possible. I t is perhaps useful to provide a simplified description in terms of classical trajectories for the simplest case in which the molecule goes through the bottom of the funnel, that is, the lowest energy point in the conical intersection space.
The trajectories passing exactly through the "tip of the cone" (Figure 6.5) proceed undisturbed. They follow the typically quite steep slope of the cone wall, thus converting electronic energy into the energy of nuclear motion. This acceleration will often be in a direction close to the x, vector, that is, the di- rection of maximum gradient difference of the two adiabatic surfaces. (Cf. Ex- ample 4.2.) Trajectories that miss the cone tip and go near it only in the two- dimensional branching space have some probability of staying on the upper surface and continuing to be guided by its curvature, and some probability of performing a jump onto the lower surface and being afterward guided by it.
However, in this latter case, an amount of energy equal to the "height" of the jump is converted into a component of motion in a direction given by the vector
X, (direction of the maximum mixing of the adiabatic wave functions, cf. Ex- ample 4.2), which is generally not collinear with x, and is often approximately perpendicular to it (Dehareng et al., 1983; Blais et al., 1988). After the passage through the funnel, motion in the branching space, defined by the x,, xz plane, is most probable. Of course, momentum in other directions that the nuclei may have had before entry into the funnel will be superimposed on that generated by the passage through the funnel.
Figure 6.6. Schematic representation a) of the transition state of a thermal reaction and b) of the conical intersection as a transition point between the excited state and the ground state in a photochemical reaction. Ground- and excited-state reaction paths are indicated by dark and light arrows, respec- tively (adapted from Olivucci et al., I994b).
ti. 1 A QUALII'A'rIVE PHYSICAL MODEL 319 A bifurcated reaction path will probably result, and several products are
often possible as a result of return through a single funnel. (Cf. Figure 6.5.) In summary, the knowledge of the arrival direction, the ~nolecular structure as- sociated with the conical intersection point, and the resulting type of molecular motion in the x,, x, plane centered on it provide the information for rational- izing the nature of the decay process, the nature of the initial motion on the ground state, and ultimately. after loss of excess vibrational energy, the distri- bution of product formation probabilities. Depending on the detailed reaction dynamics, the sum of the quantum yields of photochemical processes that pro- ceed via the same funnel could be as low as zero or as high as the number of starting points.
To appreciate the special role of a conical intersection as a transition point between the excited and the ground state in a photochemical reaction, it is useful to draw an analogy with a transition state associated with the barrier in a potential energy surface in a thermally activated reaction (Figure 6.6). In the latter, one characterizes the transition state with a single vector that corre- sponds to the reaction path through the saddle point. 'The transition structure is a minimum in all coordinates except the one that corresponds to the reaction path. In contrast, a conical intersection provides two poss~ble linearly inde- pendent reaction path directions.
Often. the minimum in S, or TI that is originally reached occurs near the ground-state equilibrium geometry of the starting molecule (Figure 6.3. path c); the intermediate corresponds to a vibrationally relaxed approximately vertically excited state of the starting species, which can in general be iden- tified by its emission (fluorescence or phosphorescence, Figure 6.3, path d).
Quenching experiments can help decide whether the emitting species lies on the reaction path or represents a trap that is never reached by those mole- cules that yield products. However, often the minimum that is first reached is shallow and thermal energy will allow the excited species to escape into other areas on the S, or TI surface before it returns to So (Figure 6.3, path e). This is particularly true for the TI state due to its longer lifetime. In the case of intermolecular reactions the rate also depends on the frequency with which diffusion brings in the reaction partner. The presence of a reaction partner may provide ways leading to minima that were previously not ac- cessible, for example, by exciplex formation. Other possibilities available to a molecule for escaping from an originally reached minimum are classical energy transfer to another molecule, absorption of another photon (cf. Ex- ample 6.1), triplet-triplet annihilation (cf. Example 6.4), and similar pro- cesses.
Example 6.3:
For reactions proceeding from the S, state. intersystelli crossing is frequently a dead end. Thus. irradiation of truru-2-methylhexadiene (4) in acetone (3) yields the oxetanes 5 and 6 through stereospecific addition of the ketone in its
'(n,n*) state (S,) to the CC double bond, whereas intersystem crossing into the '(n,n*) state (T,) prevents oxetane formation. The '(n,n*) state is quenched by triplet energy transfer to the diene, which then undergoes a sensitized trans- cis isomerization to 7 (Hautala et al., 1972).
Example 6.4:
A well-studied example of a photoreaction involving excimers is anthracene dimerization (Charlton et al., 1983). Figure 6.7 shows part of the potential en- ergy surfaces of the supermolecule consisting of two anthracene molecules.
Singlet excited anthracene ('A* + ' A ) can either fluoresce (monomer fluores-
Figure 6.7. Schematic potential energy curves for the photodimerization of anthracene (adapted from Michl. 1977).
320 PHOTOCHEMICAL REACTION MODELS cence hv,), intersystem cross to the triplet state ('A* + 'A), or undergo a bimolecular reaction to form an excimer I(AA)*, which can be identified by its fluorescence (hv,). Starting from the excimer minimum and crossing over the barrier, the molecule can reach the pericyclic funnel and proceed to the ground-state surface leading to the dimer (24%) or to two monomers (76%). In this case intersystem crossing to the triplet state does not necessarily represent a dead end, because two triplet-excited anthracene molecules may undergo a bimolecular reaction to form an encounter pair whose nine spin states are reached with equal probability. (Cf. Section 5.4.5.5.) One of these is a singlet state '(AA)** that can reach the pericyclic funnel without forming the excimer intermediate first. The experimentally observed probability for the dimer for- mation via triplet-triplet annihilation agrees very well with the spin-statistical factor (Saltiel et a)., 1981).
From Example 6.4 it can be seen that the molecule may also end up in a minimum o r funnel in S , o r T, that is further away from the geometry of the starting species. This then corresponds t o a "nonspectroscopic" minimum o r funnel (Figure 6.3, minimum f) such a s the pericyclic funnel of the an- thracene dimerization in Figure 6.7, o r even to a spectroscopic minimum of another molecule o r another conformer of the same molecule (Figure 6.3, minimum i). Reactions of the latter kind can sometimes be detected by prod- uct emission (Figure 6.3, path j). (Cf. Example 6.5.)
In many photochemical reactions return t o the ground-state surface S, occurs from a nonspectroscopic minimum (Figure 6.3, path g) that may be reached either directly o r via one o r more other minima (Figure 6.3, se- quence c,e).
ecules can move over potential energy barriers that would be prohibitive at thermal equilibrium. Since the vibrational energy may be concentrated in specific modes of motion, a molecule can move over barriers in these fa- vored directions and not move over lower ones in less-favored directions.
Chemical reactions (i.e., motions over barriers) that occur during these pe- riods are called "hot. "
Example 6.5:
Reactions of hot molecules are known for the ground state as well as for ex- cited states. Thus, after electronic excitation of 1,Cdewarnaphthalene (8) in a glass at 77 K emission of 8 as well as of naphthalene (9) produced in a photo- chemical electrocyclic reaction is observed. The intensity of fluorescence from 9 relative to that from 8 increases if the excitation wavelength is chosen such that higher vibrational levels of the S, state of 8 are reached. This suggests that there is a barrier in S,. (Cf. Example 6.1.) If some of the initial vibrational energy is utilized by the molecule to overcome the reaction barrier before it is lost to the environment, 9 is formed and can be detected by its fluorescence.
If the energy is not sufficient or if thermal equilibrium is reached so fast that the barrier can no longer be overcome, fluorescence of 8 is observed. The situation is even more complicated because at very low temperatures a quan- tum mechanical tunneling through the small barrier in S, of 8 occurs (Wallace and Michl, 1983).
6.1.3 "Hot" Reactions
Even in efficient heat baths, there are usually several short periods of time during a photochemical reaction when the reactant vibrational energy is much higher than would be appropriate for thermal equilibrium. One of these occurs at the time of initial excitation, unless the excitation leads into the lowest vibrational level of the corresponding electronic state. The amount of extra energy available for nuclear motion is then a function of the energy transferred t o the molecule in the excitation process, that is, a func- tion of the exciting wavelength. Further, such periods occur during internal conversion (IC) o r intersystem crossing (ISC), when electronic energy is converted into kinetic energy of nuclear motion, o r after emission that can lead t o higher vibrational levels of the ground state.
The length of time during which the molecule remains "hot" for each of these periods clearly depends on the surrounding thermal bath. In ordinary liquids at room temperature it appears to be of the order of picoseconds. (Cf.
Section 5.2.1 .) Although short, this time period permits nuclear motion dur- ing lo2-1W vibrational periods. During this time, a large fraction of the mol-
Thermal o r quusi-equilibrium reactions, a s opposed to hot ones, can be usefully discussed in terms of ordinary equilibrium theory a s a motion from o n e minimum t o another, using concepts such a s activation energy, activa- tion entropy, and transition state, a s long a s the motion remains confined t o a single surface. "Leakage" between surfaces can be assigned a tempera- ture-independent rate constant. In this way jumps from one surface to an- other through a funnel may be included in the kinetic scheme in a straight- forward manner.
Example 6.6:
For the photoisomerization of 1.4-dewarnaphthalene (8) to naphthalene al- ready discussed in Example 6.5 the following efficiencies could be measured in an N, matrix at 10 K (Wallace and Michl, 1983): q1.(8) = 0.13, qsT(8) = 0.29, qR. = 0.14, qRd + qrcl = 0.44, qJ9) = 0.047, and qrs(9) = 0.95. Here, the indices Ra and Rd refer to the adiabatic and diabatic reaction from the singlet state, ret refers to the nonvertical return to the ground state of 8, and TS refers to the intersystem crossing to the ground state of 9. M;~king the plausible as- sumption that triplet 8 is converted to triplet 9 with lo()% efficiency. these data
1 I /\ QLII~LI 1AI IVE PHYSICAL MODEL 323
Figure 6.8. Schematic representation of the potential energy surfaces relevant for the photochemical conversion of 1.4-dewarnaphthalene to naphthalene.
Radiative and nonradiative processes postulated are shown and probabilities with which each path is followed are given (by permission from Wallace and Michl, 1983).
may be combined to produce the graphical representation shown in Figure 6.8.
Here the probabilities with which each path is followed after initial excitation of a molecule of 8 into the lowest vibrational level of its IL, (S,) state are shown. Using the values r,.(8) = 14.3 ? I ns and T , (9) = 177 ? 10 ns for the fluorescence lifetimes, rate constants can be derived f~.om Equation (5.8) for the various processes. Thus, one obtains k,. = qlltl = 9.1 x 10" s ' for the fluorescence of 8 and k, = (q,, + qWd + vrr,)ltk = 4.1 x 10' s I for the rate of passing the barrier.