In addition to the solvent-induced electron spin relaxation by independent spin flips at two well-separated radical centers (spin-lattice relaxation), which is quite slow in the absence of paramagnetic impurities (ordinarily, on the order of 10'-106 s-I in monoradicals), there are two important mecha-
220 4 POTENTIAL ENERGY SURFACES: BARRIERS, MINIMA, AND FUNNELS 4.3 BlRADlCALS AND BIRADICALOIDS 22 1 nisms of spin-flipping in triplet biradicals and biradicaloids. The first of these
is due to the difference of the g values of the unpaired electron spins in the presence of a magnetic field, and to their hyperfine interaction with the spin of magnetic nuclei located in the vicinity (Closs et al., 1992; for earlier views see Doubleday et al., 1989). It is well known from the theory of CIDNP (chemically induced dynamic nuclear polarization, Salikhov et al., 1984).
The second important mechanism for intersystem crossing is spin-orbit cou- pling. (Cf. Section 1.3.2.)
The "g-value difference and hyperfine coupling mechanism" is important in biradicals in which the two radical centers are relatively far apart (1,6- biradicals and longer), and the spin-orbit coupling is weak at almost all bir- adical geometries, and the T, and So levels are almost exactly degenerate (at most a few cm- ' apart).
Example 4.9:
in long bifadicals the singlet-triplet separation falls off roughly exponentially with distance (De Kanter et al., 1977; for an updated view see Forbes and Schulz, 1994) and with the number of intervening bonds (Closs et al., 1992). In a strong magnetic field, e.g., in EPR experiments, only the To component of the triplet state is usually sufficiently close to So for fast crossing.
The coupling of each unpaired electron to the magnetic nuclei in its vicinity is fundamentally a local effect, and is nearly independent of the distance be- tween the two radical centers. Each nuclear spin substate in each of the three low-energy triplet levels of the biradical behaves essentially independently, ex- cept that ordinary spin relaxation processes may cause transitions among them and more slowly, into the singlet level, as mentioned earlier. Since the singlet- triplet splitting is so small at nearly all conformational geometries of the long- chain biradical, even the very small g-value and hyperfine terms (typically 0.01 cm-I) cause spin function mixing, and each sublevel becomes an "impure"
triplet, with a small weight of singlet character. The weight oscillates very rap- idly in time, but only its average value is important. When the conformational motion of the chain that proceeds over small barriers on the triplet surface brings the two radical centers into close vicinity such that the two singly oc- cupied orbitals interact, the covalent perturbation y suddenly increases greatly.
This causes the singlet-triplet splitting to increase in magnitude well above the mixing terms, whose effect thus becomes negligible, and the new spin eigen- functions of the system once again become an essentially pure triplet or an essentially pure singlet, with probabilities given by the respective weights of these two spin functions in the "impure" triplet at the time immediately before the encounter of the radical ends. Most of the time, the decision falls in favor of the triplet, and the nuclei then feel forces dictated by the T potential energy surface. This is typically repulsive and causes the radical centers to separate.
No reaction occurs; the "impure" triplet biradical is reformed, and continues to live to try to recombine another time. After a few hundred or thousand excursions into geometries characterized by a large value of y. the decision
falls in favor of the singlet; that is, intersystem crossing occurs. In this event, the spin-orbit coupling mechanism of intersystem crossing, which tends to be important at just these geometries, as discussed later, provides additional sin- glet-triplet mixing and may play a significant role, but it does not discriminate between the various nuclear substates. From now on, the nuclear motion is governed by the shape of the So surface, which typically slopes steeply down- hill to a ring-closure or a disproportionation product without a barrier. Singlet product formation from all molecules that have undergone intersystem cross- ing is thus virtually guaranteed. This is not to say that the So surface does not have small conformational barriers at various geometries, just like the triplet- only that these are not likely to lie between the geometry at which intersystem crossing was forced by a large increase in the So-T, splitting, and the geometry of a singlet product.
Note that the region of geometries where the covalent interaction just sets in is particularly favorable for a jump between surfaces: it contains the turning point for vibrations on the T surface, which just begins to rise, as well as on the So surface, which just begins to drop, and the Franck-Condon overlap of the vibrational wave functions is sitable. At the same time, the amount of energy that needs to be transferred from electronic to vibrational motion in the non-Born-Oppenheimer jump from one to the other surface is still relatively small. A motion along a direction that increases the magnitude of the covalent perturbation is therefore well suited for promoting intersystem crossing.
The rate of intersystem crossing is then proportional to the frequency of end-end encounters and to the weight of the singlet spin function in the "im- pure" triplet, dictated by a combination of spin-orbit and hyperfine coupling terms. Because of the latter, it may be different for each nuclear spin sublevel.
At times, this leads to spectroscopically detectable nuclear spin polarization in the products, and to large isotope effects.
The second mechanism of intersystem crossing in biradicals and biradi- caloids, spin-orbit coupling, requires a significant degree of covalent inter- action between the two radical centers, through space or through bond. It needs to be considered in biradicals of any length at geometries in which the two radical centers are close, becomes important at all geometries in radicals of intermediate length, and is dominant in short biradicals, which occur as intermediates in numerous photochemical reactions (Chapter 7). In 1,2-, 1-3-, and 1,Cbiradicals the So-T splitting tends to be relatively large at most geometries, from a few to several thousand cm-' (Doubleday et al., 1982, 1985; Caldwell et al., 1988) and the g-value and hyperfine effects are then negligible.
In the absence of magnetic field, the three triplet levels are separated in energy by the zero field splitting (roughly 10-2-10-1 cm-' in short biradi- cals) and are quantized with respect to the molecular axes dictated by the principal directions of the electron spin-spin dipolar coupling tensor primar- ily responsible for the zero-field splitting. (Spin-orbit coupling affects the
222 4 POTENTIAL ENERGY SURFACES: BARRIERS, MINIMA, AND FUNNELS 4.3 BlRADlCALS AND BIRADICALOIDS 223 zero-field splitting, too.) Their spin functions are labeled Ox, 0,. and 0, and
are given by
while the singlet spin function I: is given by
The three sublevels of the triplet couple to the So singlet to different degrees, dictated by the singlet-triplet separation and by the matrix elements eO,
l f o , and e o , respectively. The nuclear spin states are immaterial. Often, it is assumed that the relaxation between the three levels is rapid, or that all three are initially populated more or less equally, and the "total spin-orbit coupling strength" of the S,, state is then defined as (eo2 + H$02 +
eoZ)'". This can be thought of as the length of the "spin-orbit coupling vector" Po with components e, u = x, y, z. In short organic biradicals, values above about 0.1 cm-' are considered large, and typical magnitudes do not exceed a few cm- I .
The original qualitative conclusions of Salem and Rowland (1972) con- cerning spin-orbit coupling in biradicals were based on the two-electron- two-orbital model of biradicals. They have been supported more recently to a remarkable degree by the ab initio calculations mentioned earlier, as long as one permits the radical centers to delocalize somewhat into the nearby bonds and does not insist on the original simplest interpretation in which they were strictly viewed as carbon 2p orbitals, thus precluding through- bond contributions. In the two-electron-two-orbital model, the description of the presumably dominant one-electron part of spin-orbit coupling is rela- tively simple (Salem and Rowland, 1972; Michl and BonaCiC-Kouteckq, 1990; Michl, 1991). If the singlet state functions Si and the triplet state func- tions TI, (u = x, y, z) are written as
with the spin functions Z and 8, given by Equations (4.7) and (4.8), and if the spin-orbit coupling operator of Equation (1.42) is used, one obtains, after performing the spin integration (Example 1.8),
where the sum is over all atoms p. Because of the presence of the factor Ir~l 3 , contributions from pairs of atomic orbitals that are both located on
atoms other than p can be ignored when evaluating the p-th term in the sum.
The magnitude of the coefficient Ci,+ does not depend on the choice of the nonbonding orbitals q and (p' as localized, delocalized, or even intermediate, as long as they are real and orthogonal. In the following, we adopt the most convenient choice, the most localized orbitals X, and x,. For these, the hole- pair functions can be identified with the zwitterionic functions if the biradi- cal carries no formal charges at the radical centers. The form of Equation (4.11) most useful for our purposes (Michl, 1991) is finally arrived at by the standard procedure of correcting for the effects of the electrons of the fixed core, including inner-shell electrons, and for the effect of the neglected two- electron part of the spin-orbit operator, by introducing an atomic spin-orbit coupling parameter, t,, whose magnitude increases roughly with the fourth power of the atomic number 2,. For the spin-orbit coupling vector Po we obtain
The sum runs over all atoms p. Each term includes only contributions from those pairs of atomic orbitals in which at least one of the partners is located at the atom in question. The x, y, and z components of the vector operator 911, which differs from the angular momentum operator ip only by the factor hli, are slat, alaq, and a/at, respectively, where 5, q, and 5 are the angles of rotation around the x , y, and z axes passing through the p-th nucleus. The action of this operator on atomic orbitals located on atom p is as follows:
The s functions are annihilated, and for p functions,
(Cf. Example 1.8.) This result shows that in the 3 x 3 model, the spin-orbit coupling vector depends on three factors: the coefficient Ci.+ of the in-phase
I(& + d) character of the singlet state, the spin-orbit coupling parameter 5,
(heavy atom effect), and the spatial disposition of the orbitals X, and x,. The actual intersystem crossing rate will also depend on the Franck-Condon- weighted density of states. (Cf. the Fermi golden rule, Section 5.2.3.)
First of all, we consider the significance of the presence of Cia+, the coef- ficient of the in-phase combination of the two hole-pair functions in the So wave function, in Equation 4.12. In the simple model for a perfect biradical, this in-phase combination is exactly equal to the wave function of the S, state, and it does not enter into those of the So and S, states at all. Thus, in this approximation, So does not spin-orbit couple to the triplet. The same is true in weakly heterosymmetric biradicaloids (0 < 6 < 6,). in which the in- phase hole-pair character is shared by S, and s,, but not So, and the former two spin-orbit couple to T, but So does not. In strongly heterosymmetric
224 . , 4 POTENTIAL ENERGY SURFACES: BARRIERS, MINIMA, AND FUNNELS biradicaloids (6 > a,), the in-phase hole-pair character is shared by S, and S,. (Cf. Figure 4.20b.) Now, So spin-orbit couples to T, but in such donor- acceptor pairs the two states are ordinarily far removed from each other in energy and the So-T intersystem crossing is relatively slow. The S, state of such species is nearly degenerate with T, but does not spin-orbit couple with it in this approximation.
The situation changes dramatically in the presence of a covalent pertur- bation, y # 0, which causes the in-phase hole-pair function to mix into the So state. In the resulting homosymmetric or nonsymmetric biradicaloid, T may spin-orbit couple to So, and the total spin-orbit coupling strength will be proportional to the coefficient Cis + of the in-phase hole-pair spin function in the wave function of So. This is sometimes expressed in a simplified way (Salem and Rowland, 1972) such as "T-So spin-orbit coupling in a biradical requires ionic character in the So state," but it may be more accurate to state that it requires covalent interaction (nonzero resonance integral) between the orthogonal localized orbitals at the radical termini, since it is only an admixture of the in-phase and not the out-of-phase combination of the two hole-pair ("ionic") configurations that matters. The simultaneous presence of a polarizing perturbation in nonsymmetrical biradicaloids does not facil- itate spin-orbit coupling, but might have a positive effect on the intersystem crossing rate by affecting the S,T gap.
Note that the geometries needed for the required covalent perturbation are not likely to be located at energy minima in the triplet surface, since after all, the triplet state is antibonding with respect to the two radical cen- ters. At least a small thermal activation barrier to reaching the optimal spin- orbit coupling geometries thus is to be expected. This could well be different in different conformers, possibly leading to temperature-dependent stereo- selectivity and isotopic selectivity in product formation even in those cases in which the conformers as such had identical free energies.
The tendency of the spin-orbit coupling matrix element to be largest just at the covalently perturbed biradicaloid geometries, which lie on the path to singlet products, is undoubtedly at least partly responsible for the perception that intersystem crossing in triplet biradicals ordinarily gives closed-shell singlet products rather than floppy singlet biradicals. (Cf. Example 4.11 .)
Next, we consider the significance of the sum over atoms on the right- hand side of Equation 4.12. Each atom contributes a vector <XuIv"l~b>, weighted by its spin-orbit coupling parameter I , and this clearly reflects the
"heavy atom effect." However, it indicates equally clearly that a contribu- tion from an atom does not need to be large just because the atom is heavy:
the vector contribution from that atom might have a small length or an un- fortunate direction that cancels contributions from other atoms. (Note that
"inverse" heavy atom effects are therefore possible; cf. lhrro et al., 1972b.) This brings us to the last factor to consider-the size and direction of the atomic vector contributions <xUl@lXb>. The size can only be significant if the coefficients of the 2p orbitals located on atom p in at least one of the most localized orbitals X, and xb are large (and indeed, in the first approxi-
4.3 BlRADlCALS AND BlWlCALOlDS 225
mation only the two atoms carrying the radical centers were considered, Salem and Rowland, 1972), but this condition is not sufficient. Specifically, in view of Equation (4.13), the z component of the vector contribution from atom p consists of two parts that are added algebraically. The first will be large when the coefficient on the p,, orbital in 2, is large, and when one or more of the atomic orbitals that have a large overlap with p, (including p, itself) have large coefficients in xu. The second will be large when the coef- ficient on the p, orbital in xb is large, and when one or more of the atomic orbitals (including p,,) that have a large overlap with p,, have large coeffi- cients in xu. The signs of the two contributions are dictated by the signs of
the orbital coefficients, by the signs in Equation (4.13), and by the signs of the overlaps. Similar results hold for the x and y contributions to the atomic vector provided by atom p.
Example 4.10:
The atomic vectors <X,l@"'Xb> contain through-space and through-bond con- tributions. Their origin is most easily visualized for conformations in which the axes of the singly occupied orbitals at the two radical centers are mutually perpendicular. It is somewhat unfortunate that these are often just the confor- mations at which the covalent perturbation y of a perfect biradical is zero, making C,., vanish, and causing the So-T, spin-orbit mixing to be negligible even if the resultant of the atomic vectors is large (it is then the experimentally uninteresting S2-T, coupling that is large). This is indeed the case in both of the following simple examples.
Figure 4.25. Sum-over-atoms factor in the spin-orbit coupling vector HSo a) in orthogonally twisted ethylene and b) in (0.90") twisted trimethylene biradi- cal, using Equation (4.12) and (4.13); most localized orbitals x,,. xh and non- vanishing atomic vectorial contributions from x,, (white: through-space, black:
through-bond).
226 4 I'WEN'I'IAL ENERGY SUKI.'t\CCS: UAKKIL..IIS, M I N I M , AND FUNNELS 4.3 BlRADlCALS AND BIRADICALOIDS 227 In orthogonally twisted ethylene, the localized orbitals X, and X, are related
by symmetry. Each is primarily located on a free p orbital of one CH, group and is hyperconjugatively delocalized into the CH bonds of the other CH, group (Fig. 4.25a). The principal axes of the zero-field-splitting tensor are dic- tated by symmetry. Only the z component of the atomic vectors <X.191'lX,>, directed along the CC axis, is different from zero, and spin-orbit coupling will mix S , with only one of the three triplet components. TI:. We can therefore locate the x and y axes arbitrarily as long as all three axes are mutually or- thogonal, and we choose them to lie in the planes of the two CH, groups.
The nonbonding orbitals then have the form
where c, * c,, c, > 0.
The resulting atomic vectors for the hydrogen atoms are negligible; those for the two carbon atoms (p = 1,2) are large and equal. We illustrate the evaluation
<xnI9!Ixb>:
<x'Je:lxh> = c ~ < ~ l r 1 9 ~ I ~ ? v > - cl('2<~l.!19!l~Iy>
- c,c,<p,,l~jls, - s,> + stnull icrtns
where the small terms either contain a product of two small coeffi- cients, clc3. andlor contain no atomic orbitals located on C(I) (e.g., c l c , < P , l ~ l I ~ , > ) .
The first term on the right-hand side is of the through-space type. It would be present even if there were no hyperconjugation, that is, even if the non- bonding orbitals were strictly localized on C(1) and C(2), respectively (c, = 1, cz = c, = 0). The second and third terms would vanish in this limit and are of the through-bond type.
The action of the operator 9: on a n-symmetry orbital on its right is to rotate it by 90" around the z axis in the clockwise sense when viewed against the direction of z (cf. Equation 4.13); for instance,
Thus, the through-space term makes a positive contribution to the z compo- nent of the atomic vector <Xe1911~b>. and the two through-bond terms make negative contributions. I t is impossible to evaluate the sign of the resultant without at least a rough numerical evaluation of the opposing contributions:
the through-space term contains the overlap integral S,, as a relatively small multiplier while the through-bond terms contain the small coefficient c, or c, as a multiplier, but there are more of them. In a minimum basis set approxi- mation, a computation shows that the through-bond terms dominate. Inclusion of the two-electron part of the spin-orbit operator in the calculation reduces the magnitude of the result by a factor of about two. hut does not change its sign.
Note that the opposed signs of the through-space and through-bond contri- butions are a result of the nodal properties of the orbitals x,, and xb. As is seen in Figure 4.25a, they have a node separating the main part of the orbital from the minor conjugatively delocalized part. However. the presence of the node
is sensitive to the details of the structure, and so is the relative sign of the through-space and through-bond contributions. For instance, in orthogonally twisted H,N-BH,, the empty orbital located on boron does not have such a node, and the dominant through-bond contribution, due to the one-center term on the N atom, now has the same sign as the through-space contribution.
The through-bond terms will not always dominate, but are likely to be quite generally important. In our second example, a singly twisted 1,3-trimethylene biradical (Figure 4.25b), the leading term ( p = 3) arises from the hyperconju- gation of the singly occupied orbital of the CH, group on C(l), whose hydro- gens are twisted out of the CCC plane, with the C(2)C(3) bond. In the coordi- nate system of Figure 4.25b. chosen so as to diagonalize the zero-field splitting tensor, X, then contains small contributions from the p3, and p, orbitals on C(3). an atom on which X, has its dominant component, p,,. The action of either the 9; or the operator rotates p,: into overlap with x,. As a result, both the x and y components of the atomic vector <X,1931X,,> are nonzero, and the through-bond mechanism of spin-orbit coupling will be the primary cause of the mixing of both TI, and TI, with So.
Example 4.11:
The magnitude of the total spin-orbit coupling strength can change dramatically as a function of biradical conformation (Carlacci et al., 1987). The through- space part can be roughly approximated by
where w is the acute angle between the axes of the p orbitals containing the unpaired electrons and S is their overlap integral. The overlap S is related to the resonance integral between these two orbitals and thus to the coefficient of the in-phase combination of the two hole-pair functions in the So wave func- tion, Co.+. The factor sin w originates from the operator 9~ in the matrix ele- ment <X,,lefillxb>. which rotates the orbital xh according to Equation (4.13) be- fore its overlap with x,, is taken.
No such simple generally valid approximation is currently available for the through-bond part of the spin-orbit coupling strength, which results from the delocalization of even the most localized form of the nonbonding orbitals into the a skeleton, onto nearby carbon atoms p located between the radical cen- ters. This produces nonzero coefficients on the p orbitals on these atoms and permits them to contribute to the sum in Equation (4.12). A similar mechanism operates when atoms carrying lone pairs, such as oxygen. are located between the radical centers. The low-lying electronic states of biradicals of this type are more numerous and the spin-orbit coupling is less likely to be properly described by the simple model that led to this equation. The availability of additional states involving promotion from the lone pairs appears to make spin- orbit coupling particularly effective. Note that the placement of lone-pair car- rying or heavy atoms into positions that do not lie between the two radical centers cannot be expected to have much effect, since then only one of the most localized nonbonding orbitals has significant coefficients on their orbit- als.