9660
J. Phys. Chem. 1995,99, 9660-9667
ARTICLES Dynamic Effects in Spin-Correlated Radical Pair Theory: J Modulation and a New Look at the Phenomenon of Alternating Line Widths in the EPR Spectra of Flexible Biradicals Nikolai I. Avdievich and Malcolm D. E. Forbes* Venable and Kenan Laboratories, Department of Chemistry, CLM 3290, University of North Carolina, Chapel Hill, North Carolina 27599 Received: January 20, 1995; In Final Form: March 3, 1995@
Expressions for contributions to TI and T2 from modulation of the exchange interaction (J)in spin-correlated radical pairs (SCRPs) are derived and incorporated into SCRP theory for simulation of direct detection EPR spectra of alkane chain biradicals. The T2 term is consistent with that previously derived by Luckhurst to explain the temperature-dependent alternating line widths observed in EPR spectra of stable nitroxide biradicals. Alternating line-width patterns are observed for symmetric bis(alky1) biradicals at high temperatures, where the rapid modulation of J is caused by conformational jumping. However, in unsymmetrical systems such as acyl-alkyl biradicals, J modulation manifests itself at low temperatures as an increased broadening of the spectral lines with increasing distance of the transition from the center of the spectrum, with no alternation. Several experimental examples are presented, simulated, and discussed. Contrary to a suggestion by Maeda et al., J modulation is shown to have only a minimal effect on TI in SCRP theory for the case of biradicals undergoing rapid conformational interconversion. A connection is proposed between the magnitude of the relaxation matrix elements and the biradical conformational distribution and chain dynamics.
Introduction More than 30 years ago, steady-state EPR spectroscopy of nitroxide biradicals allowed some of the first direct estimates of scalar electron spin exchange interactions.’ An interesting feature often observed in flexible nitroxide biradical EPR spectra was an alternation of the line widths in the five-line pattern indicative of strong exchange, Le., when J was significantly larger than the hyperfine coupling term.2 The observed effect was extremely dependent on temperature, leading to the hypothesis that modulation of J by conformationaljumping was re~ponsible.~ Several researchers published theoretical treatments in the 1960s which accounted well for the appearance of this effect in most of the experimental spectra published at that time.4 An excellent review by Hudson and Luckhurst summarizes both theoretical and experimental efforts on this problem5 In their review it was suggested that the magnitude of the line broadening could be quantitatively related to the conformational dynamics of the system if some additional parameters could be determined. These included the average J value and the correlation time re for a given fluctuation of the J value. Because the effect was observed in the large J limit, it was not possible to extract a precise value for J from the steady-state EPR spectrum, and dynamic conformational models that would have led to a well-defined value for re were not available at that time. The spin-polarized acyl-alkyl and bis(alky1) biradicals obtained upon Norrish I cleavage of cyclic ketones, under investigation in our laboratory for the past several years, are now routinely used for the study of the effect of molecular structure on exchange interactions6 To date, dynamic effects such as J modulation have not been observed in TREPR spectra
* To whom correspondence @
should be addressed. Abstract published in Advance ACS Abstuacrs, June 1, 1995.
0022-365419512099-9660$09.00/0
of unstable biradicals. If such phenomena become observable, spin-polarized biradicals have several immediate advantages over their nitroxide analogues:’ First, the phase of the polarization (absorption (A) or emission (E)) allows the sign of the J coupling to be determined. Second, the J coupling can be quantitatively estimated from intensity patterns even in cases when it is much larger than the hyperfine interactions or g-factor differences. Because of this, the range of accurately measurable couplings is about 5 orders of magnitude (10-5-1 cm-I). Third, they are structurally simple compared to nitroxides and should exhibit conformational behavior that is very similar to the corresponding alkane chain. With the tremendous advances in understanding of both static and dynamic properties of alkane chains over the past 2 decades,s it should be possible to establish a more quantitative relation between the magnitude of the modulation of J in flexible biradicals, the correlation time re, and the observed EPR effects. Biradicals are excellent models for the study of electronic couplings in general, and dynamic effects on such couplings are of great interest. For example, Tang recently reported theoretical results on fluctuating electron transfer matrix elements and found marked differences in rate between static and dynamic system^.^ Of direct relevance to the work reported here are the papers of Bittl and Schulten,Io who published a detailed analysis of the effect of fluctuating J couplings on the magnetic field effects observed by Weller et al.lI in charge recombination reactions involving flexible triplet precursors. Their approach, using the stochastic Liouville equation and a “static ensemble” of J couplings, could easily be modified to calculate the TREPR spectrum with J modulation for simple cases involving small numbers of hyperfine interactions and small J couplings (Le., no S-T- mixing). The relationship between the conformationally averaged J coupling measured by EPR and magnetic field effects in charged donor-acceptor 0 1995 American Chemical Society
Spin-Conelated Radical Pair Theory
SCHEME 1
12,
/T’
systems or in chemically induced nuclear polarization studies of neutral biradicals is complex and has been discussed by Closs et aLI2 In that work it was concluded that EPR was the most accurate technique for the measurement of a motionally averaged J in flexible systems. The existence of spin-correlated radical pairs (SCRPs) and biradicals and their easy observation by time-resolved electron paramagnetic resonance (TREPR) spectroscopy, have led to an intense effort to measure exchange interactions (J)in flexible alkane chain biradical~,’~ radical pairs (RPs) confined to micelle^,'^ and other systems of restricted dimen~ionality.’~ The ability to measure the average J coupling in flexible biradicals over a wide range of values is a major development and is a direct result of the presence of non-Boltzmann spin-state populations in these species. Rudimentary SCRP theory, put forward independently in 1987 by Closs et a1.I6 and Buckley and co-worker~,’~ has accounted fairly well for most spectra published to date. However, the spectral simulations are rarely perfect fits and several recent theoretical reports have recently suggested reasons for this. These include population relaxation (TI processes),I8 dynamic effects (T2 proce~ses),’~ and the presence of a dipolar coupling in addition to the exchange coupling between the unpaired electrons.20 An additional important explanation arose from an elegant theoretical development by Norris and co-workers,2’ who clearly showed that the SCRP polarization and “ordinary” CIDEP (doublet state) polarization should be calculated simultaneously to best account for the TREPR spectrum when J is changing with time. Because of the major effort in our laboratory aimed at the precise measurement of average exchange interactions in biradicals, improvement of the fitting procedure is of paramount importance. While we and are actively considering other processes affecting the shape and intensity of SCRP spectra, in this paper we focus on dynamic effects. In particular, we wish to present a theoretical treatment for the effect of relaxation (TI and T2) in spin-polarized biradical EPR spectra when there exists a modulation of the exchange interaction via conformational jumping. After deriving expressions for the relaxation parameters and inclusion of the new populations and line shapes into the well-established SCRP theory, we will present examples which clearly show that it is sometimes impossible to fit biradical TREPR spectra without consideration of this process. We will also discuss the connection of our results to the earlier work on nitroxides and to the chain dynamics of the biradicals.
Theory We begin with a basic description of the biradical spin dynamics in order to define terms used in the relaxation calculation. It is appropriate to use the spin Hamiltonian (eq l), used previously in the development of SCRP the0ry.’~3’~ The Hamiltonian consists of the Zeeman and hyperfine terms (ui,u,) for each side of the biradical, plus the exchange
interaction J. The J term is written to give a singlet ground state which is known to be the case for all the biradicals used for this study. The constants and operator symbols in eq 1 have their conventional meanings. Use of the high-field approximation (a ai,uj (the “singlettriplet” transitions). It should be noted that the populations of the eigenstates in eq 2 are non-Boltzmann, and the resulting spectra show all of the observable transitions to be in emission ( E ) or enhanced absorption (A). Population relaxation (TI processes) by unconelated mechanisms (hyperfine or g-factor anisotropy) and correlated mechanisms (dipolar interaction) have been considered in detail by de Kanter and co-workersZ2and will not be considered further here. For a complete description of the inclusion of the de Kanter relaxation results into SCRP spectral simulation routines for biradicals, the reader is referred to two of our earlier paper^.^^.'^ The theory to be developed here is based on the fact that motion due to rapid conformational jumping of an alkane chain biradical causes modulation of J and that the effect of this modulation on the intensity and line shapes of the SCRP spectra can be treated as a relaxation process. To describe this relaxation, we rewrite the exchange interaction portion of the Hamiltonian into time-independent and time-dependent terms as shown in eq 4:
Avdievich and Forbes
9662 J. Phys. Chem., Vol. 99, No. 24, 1995 H,(t) = (7
+ V(t))(’/, + 2S,.S2) = (J + V(t))F
(4)
The term 7 is the time-averaged value of the exchange interaction obtained from the SCRP simulation routine. It is a parameter manipulated to best fit the experimental spectrum. For simplicity we will henceforth write it as J , but it is to be understood that this is an average value over all possible conformations of the biradicalSz3 Equating the time and conformational averages implicitly assumes the condition of conformational equilibrium. This is necessary to ensure the validity of the theory which follows. We have rewritten the spin operator ( l / 2 2SlaS2) in order to simplify expressions involving the matrix elements of F, presented below. Additional assumptions implicit in eq 4 are (1) that any anisotropy of J is averaged out due to fast rotation of the end groups of the chain and (2) that the time average of V(t) is equal to zero. Following Redfield theory,24T I and T2 can be obtained from the elements of the relaxation matrix R,, and R,, respectively, as shown in eq 5:
+
T2-’(l-2,
2-4,
1-3,
+
[( 1 f (J/Q))2k(0) (q2/Q2)k(20)](7b)
--
-
+
T2-’(1-2,
This equation shows that the time dependence of V can be carried by the term re, which, as mentioned above, is the correlation time for a particular value of J to exist in the biradical. Clearly re is dependent on conformational motion of our flexible system, and this will be discussed in more detail below. The static perturbation (V(0)2)will be written henceforth as (p)for simplicity, where the symbols ( ) represent an ensemble-averaged value. Calculating all the matrix elements of the spin operator F in the basis of wave functions ll), 12), 13), and 14) and then substituting into eq 5, we get
To write these relaxation times in terms of spin Hamiltonian parameters q, J , and B we make use of the trigonometric relations 2 sin2 8 = (1 - AB) and 2 cos2 8 = (1 JlB). Then we obtain the following expressions for T1-I and T2-I, which are specific for the particular transitions indicated in parentheses:
+
-
A negative sign in eq 7b corresponds to transitions 1 3 and 3 4 as indicated in Scheme 1, and a positive sign to transitions 1 2 and 2 4. For our specific case of negative J values, the former two transitions are forbidden (“triplet-singlet”), while the latter two are allowed (“triplet-triplet”). These equations can be reduced further by considering the following: From eq 6 it obvious that k(0) = (p)ze, and that k(2B) = (p)ze/ (1 4Q2te2). Since the term !2 is exactly determined by spin Hamiltonian parameters J and q (see eq 3c), we know that in most cases it will be less than a few hundred gauss. Below we will give the estimation of temore detailed consideration, but for present purposes we state that it should be on a time scale similar to an alkane chain conformational jump, say a few tens of picoseconds. This means that for the biradicals under consideration here, 4Q2z,2 > rl and eq 9b reduces to the simple expression ( V )= J12(t1/zz). From these estimates of te and ( V ) we can make the
J(ri) = J,, exp(-A(r, - ro))
following comments, keeping in mind that this is an oversimplified model: First, re should have only a minor dependence on biradical chain length and can be approximated for long-chain biradicals as being equal to T I , Le., about 2 x lo-” s (re will be dominated by the shorter value of TI and z2 whenever they differ by more than about a factor of 5). We note here how close this value is to that of an alkane chain conformational jump time, as predicted above when re was first introduced. Second, the use of Redfield theory seems to be validated for the case of these alkane chain biradicals because re should for these structures always be much smaller than the value of either q-’ or (d(V*))-I. This is true at least for the longer chain length alkyl biradicals (eight or more carbon atoms in length) where J is always less than a few hundred gauss. If, as stated above for the two-site model, ( V )= JI~(TI/Z~), then we can consider J I to be the “contact” value of J , say about lo4 G. This leads to a calculated value for (d(V))-’of about 8 x lo-” s, in support of the previous statement regarding the validity of Redfield theory. We now consider ways to improve our estimate of ( V )using a model that is more realistic for alkane chains. The ( V )term represents the ensemble averaged value of the square of the fluctuation of the exchange interaction due to conformational motion. We define ( V )as ((J(r) where J(r) is the value of the exchange interaction at any particular conformation and J is the time average value, which was equated above with the conformational average value. Since our biradicals have many millions of conformations available to them, we must consider the equilibrium distribution of these conformations and how J changes as a function of them. It is important at this point to ask whether or not we are working with such a distribution experimentally, because this dictates whether or not the time and conformational averages are the same. Two pieces of evidence strongly suggest that our chains are at equilibrium: (1) molecular dynamics calculations of alkane chains almost always put the equilibration time scale for chains of less than 20 carbon atoms at about 10 Since we are observing the TREPR spectrum at least 100 ns after the creation of the biradicals and our observation time is on the order of 100 ns, it is clear that even our 25 carbon chain should be equilibrated by this time. Also, in the absence of J modulation we see sharp lines in the EPR spectrum. If our chains were not equilibrated, presumably we would begin to observe a superposition of spectra attributable to more than one J coupling, which has never been unambiguously observed to be the case for these structures. To model ( ( J ( r ) - a*)we begin with the average J value. For the purposes of demonstration, we will assume that the J value is attributable only to mechanisms which depend on the end-to-end distance of the biradical such as through-solvent coupling, Le., through-bond coupling will be ignored. We know from previous work that this is probably not correct, but for most of the chain lengths considered here through-bond coupling is probably the minor component. A complete discussion of all the possible coupling mechanisms and their distance and temperature dependencies is outside the scope of this paper, but we believe the correct trend in the data will emerge from our analysis, and so we will leave the more complicated problem of mechanism and dynamic effects for future consideration. A suitable end-to-end distance dependence for J is expressed conveniently in parametric form following de Kanter et aL2* and written in eq 11 for a specific distance ri:
a2),
Here JOis the value of J at ro, the contact distance defined above (typically 3.5 A), and A is an exponential falloff parameter generally agreed to be about 1 A-1 (we will use 1.1 A-1 in our calculations). We have calculated end-to-end distance distributions by Monte Carlo methods developed originally by Nairn and Braun3I but modified by us to include excluded volume effects.32 Figure 3 shows three end-to-end distance distributions, with n = 10 (Figure 3A), n = 13 (Figure 3B), and n = 25 (Figure 3C). It is both intuitive and clear from Figure 3 that the distribution function becomes much broader with increasing n. Values for J can be computed from these distributions by averaging the function in eq 11 over the distribution using distances ri and probablities P(rJ from the Monte Carlo calculation^.^^ The average J relevant to our spectral simulations is then given by eq 12:
a2),
To obtain the value of ( V )= ((J(r) we divide up an end-to-end distance distribution into small slices i (0.25 A thick), compute the value of J in each slice (J(ri)) from eq 11, and then subtract the average J computed from eq 12. It is then possible to go back through the distribution using eq 13 to calculate an average value of (V).
Table 2 lists J and d(V2) calculated from the distributions in Figure 3, along with the value of (V)rein gauss for each chain length and temperature, which we can now compare with the value of (V)zeobtained from the simulation program used to fit the spectra in Figures 1 and 2. The experimentally observed trends are reproduced quite adequately for the Cl3 and C25 biradicals, but for the shorter Clo chain this model accounts for neither the magnitude of the effect nor the temperature dependence. The most likely reason for this is that the throughbond coupling, which becomes more important in the shorter chains, has been ignored. Below room temperature, this may be quite important at this chain length, as it has been shown that the through-bond and through-solvent coupling mechanisms have opposite temperature coefficients.26 More extended conformations, favored at lower temperatures, have stronger through-bond coupling. These chains also have longer end-toend distances and therefore the through-solvent component is weaker. This means that modulation of J from its average value will be different for chains moving into the long distance part of the distributions in Figure 3 than those moving into the shorter distance region. For the C13 biradical the temperature dependence expected from a “through-solvent only” model is observed, and with some refinements to the calculations plus data from additional chain lengths (especially longer ones) very good numbers for V and ze could be obtained. Such refinements might be provided by dynamic rather than equilibrium Monte Carlo calculations, in addition to the inclusion of the throughbond mechanism. A final comment on the parameter (V)zecomes from the observation that the line broadening expected from J modulation was observed at both high and low temperatures for our alkane chain biradicals. The correlation time re should be shorter at higher temperatures, leading to the expectation that the magnitude of the modulation would be smaller. However, the mean change in end-to-end distance (and therefore also V) per conformational jump should become larger with increasing
Avdievich and Forbes
9666 J. Phys. Chem., Vol. 99, No. 24, 1995
biradical is due to a lengthening of the correlation time. Unfortunately, spectra of the 1,25-bisalkylbiradical at more than one temperature were not obtainable because of sample depletion (eight scans were required to obtain the spectrum in Figure 1A). Future work on this problem includes a thorough temperature dependence of the J modulation effect in this and other longchain biradicals, which should lead to a deeper understanding of the connection between the static (V) and dynamic (z,) properties of these important molecules.
2500 1051
1.875
id
i
i;
J
I !
I +
2 1.250 Id I
6.242
I
104 -
Summary 1 " " 1 " " 1 " " 1 " " 1 " " 1
0 2.500
105
1.875
ld
5
10
15
20
25
30
35
20
25
30
35
r(A)
1
-i
0
5
10
15
Acknowledgment. We thank G. R. Schulz and S. R. Ruberu for providing the TREPR spectra shown in Figures 1 and 2. This work was supported by National Science Foundation (Divisions of Chemistry and Materials Research) through Grant No. CHE-9200917 and through a Young Investigator Award to MDEF (Grant No. CHE-9357108). N.I.A. thanks Dr. N. N. Lukzen for helpful discussions.
2.553 10
1.875 10
6.242 10'
4 0
References and Notes
5
10
20
15
25
30
35
r (A)
Figure 3. Normalized equilibrium plots of probability vs end-to-end distance for n-alkane chains from the Monte Carlo program of ref 29a, but modified to include excluded volume effects by removing such forbidden conformations from consideration. A total of l o 6 samples were run for each of the three chain lengths: (A) 10 carbon atoms (solid line is for -7 "C and the dashed line is for -43 "C, (B) 13 carbon atoms (solid line is for -6 OC and dashed line is for -44 "C), (C) 25 carbon atoms at 40 "C.
TABLE 2: Results of Monte Carlo Calculations biradical
temp ("C)
(J)" (G)
d(V)(G)
(V)teb (G)
ClO
-43
-58 -84 -14 -22 -75
150 323 173 22 1 1000
8 37 11 17 352
ClO
-1
cI3
-44 -6 +40
cI3 c25
Modulation of J has been shown to be an important dynamic effect in SCRP theory. Our experimental examples and simulations clearly show that this phenomenon can have a strong influence on the shape and temperature dependence of biradical TREPR spectra. Accurate simulation of some SCRP spectra is impossible without the inclusion of this relaxation mechanism, and for other spectra it has become clear that the use of a single line width for each transition will not provide the best possible fit. Perhaps most revealing is that the old problem of "altemating" line widths from the early work on nitroxide biradicals is just a special case of this phenomenon, Le., that where the nuclear spins are identical on each side of the biradical. Finally, T I relaxation by this mechanism has been shown to have no discernible effect on the TREPR spectra of biradicals undergoing rapid conformational interconversion.
Computed using JO = 3.4 x lo4 G for the acyl-alkyl biradicals, x lo4 G for the bis(alky1) biradicals, A = 1.1 A-', and ro = 3.5 A. Computed using te= 2 x lo-" s. a
JO = 4.8
temperature. Since the distance dependence is exponential but V is taken to the square in determining 7 ' - ' , it appears that (v)is the dominant term above room temperature and at longer chain lengths. On the other hand, at low temperatures, V should decrease, leading us to conclude that the appearance of the line broadening at lower temperatures for the Clo acyl-alkyl
(1) (a) Reitz, D. C.; Weissman, S . I. J. Chem. Phys. 1960, 33, 700. (b) Buchachenko, A. L.; Golubev, V. A,; Meiman, M. B.; Rosantsev, E. G. Dokl. Akad. Nauk. SSSR 1965, 163, 1416. (2) (a) BriBre, R.; Dupeyre, R. M.; Lemaire, H.; Morat, C.; Rassat, A. Bull. Chim. Soc. Fr. 1965, 3290. (b) Dupeyre, R. M.; Lemaire, H.; Rassat, A. J. Am. Chem. Soc. 1965, 87, 3771. (3) (a) Glarum, S . H.; Marshall, J. H. J. Chem. Phys. 1967, 47, 1374. (b) Hudson, A.; Luckhurst, G. R. Mol. Phys. 1967, 13, 409. (4) (a) Luckhurst, G. R. Mol. Phys. 1966, 10, 543. (b) Buchachenko, A. L.; Golubev, V. A.; Medzhidov, A. A,; Rosantsev, E. G. Teor. Eksp. Khim. 1965, 1, 249. (c) Johnson, C. S . , Jr. Mol. Phys. 1967, 12, 25. (d) Parmon, V. N.; Zhidomirov, G. M. Mol. Phys. 1974, 27, 367. (5) Hudson, A,; Luckhurst, G. R. Chem. Rev. 1968, 69, 191. (6) (a) Forbes, M. D. E.; Bhagat, K . J. Am. Chem. Soc. 1993, 115, 3382. (b) Forbes, M. D. E.; Ruberu, S . R. J. Phys. Chem. 1993,97, 13223. ( c ) Forbes, M. D. E. J. Am. Chem. Soc. 1993, 115, 1613. (7) (a) Closs, G. L.; Forbes, M. D. E. J. Am. Chem. Soc. 1987, 109, 6185. (b) Closs, G. L.; Forbes, M. D. E. J. Phys. Chem. 1991, 95, 1924. (8) (a) Cassol, R.; Ferrarini, A.; Nordio, P. L. J. Phys.: Condens. Marrer 1994, 6 , A279. (b) Guamieri, F.; Still, W. C. J. Compur. Chem. 1994, 15, 1302. (9) Tang, J. J. Chem. Phys. 1993, 98, 6263. (10) (a) Bittl, R.; Schulten, K . J. Chem. Phys. 1986, 84, 9. (b) Bittl, R.; Schulten, K. Chem. Phys. Lerr. 1988, 146, 58. (11) Busmann, H.-G.; Staerk, H.; Weller, A. J. Chem. Phys. 1989, 91, 4098 and references therein. (12) Closs, G. L.; Forbes, M. D. E.; Piotrowiak, P. J. Am. Chem. SOC. 1992, 114, 3285. (13) (a) Forbes, M. D. E. J. Phys. Chem. 1993, 97, 3390. (b) Forbes, M. D. E. J. Phys. Chem. 1993, 97, 3396. (14) Jenks, W. S.; Turro, N. J. Res. Chem. Inrermed. 1990, 13, 237. (b) Turro, N. J. Pure Appl. Chem., in press. ( c ) Bagranskaya, E. G.; Sagdeev, R. Z. Prog. React. Kinet. 1993, 18, 63. (15) (a) Forbes, M. D. E.; Ruberu, S . R.; Dukes, K. E. J. Am. Chem. SOC.1994, 116, 7299. (b) Forbes, M. D. E.; Dukes, K. E.; Myers, T. L.; Maynard, H. D.; Breivogel, C. S.; Jaspan, H. B. J. Phys. Chem. 1991, 95, 10547. (c) Forbes, M. D. E.; Myers, T. L.; Dukes, K. E.; Maynard, H. D.,
Spin-Correlated Radical Pair Theory J. Am. Chem. SOC.1992, 114, 353. (d) Kroll, G.; Pliishau, M.; Dinse, K.P.; van Willigen, H. J. Chem. Phys. 1990, 93, 8709. (16) Closi, G. L.; Forbes, M:D. E.; Noms, J. R., Jr. J. Phys. Chem. 1987, 91, 3592. (17) Buckley, C . D.; Hunter, D. A,; Hore, P. J.; McLauchlan, K. A. Chem. Phys. Lett. 1987, 135, 307. (18) Maeda, K.; Terazima, M.; Azumi, T.; Tanimoto, Y .J. Phys. Chem. 1991, 95, 197. (19) Hore, P. J.; Hunter, D. A. Mol. Phys. 1992, 75, 1401. (20) Shushin, A. I. Chem. Phys. Lett. 1991, 183, 321. (21) (a) Noms, J. R., Jr.; Moms, A. L.; Thumauer, M. C.; Tang, J. J. Chem. Phys. 1990, 92, 4239. (b) Wang, Z.; Tang, J.; Nonis, J. R., Jr. J. Magn. Reson. 1992, 97, 322. (22) de Kanter, F. J. J.; den Hollander, J. A.; Huizer, A. H.; Kaptein, R. Mol. Phys. 1977, 34, 857. (23) It should be noted here that equating the time average with the conformational average raises many questions, especially since we have yet to specify a mechanism for J , i t . , through-bond, through-solvent, etc. In the discussion we present a more thorough justification for this, but in order to proceed with the presentation of the theory we state that under certain conditions, which are usually met in the systems studied here, the
J. Phys. Chem., Vol. 99, No. 24, 1995 9667 time average and the conformational average are one and the same. (24) Redfield, A. G. IBM J. Res. Dev. 1957, 1, 19. (25) Shushin, A. I. Chem. Phys. Lett. 1991, 181, 274. (26) Forbes, M. D. E.; Closs, G. L.; Calle, P.; Gautam, P. J. Phys. Chem. 1993, 97, 3384. (27) Freed, J. H.; Fraenkel, G. K. J. Chem. Phys. 1963, 39, 326. (28) Salikhov, K. M.; Molin, Yu. N.; Sagdeev, R. Z.; Buchachenko, A. L. Magnetic Spin Effects in Chemical Reactions; Elsevier, Amsterdam, 1984, p 48. (29) Forbes, M. D. E.; Schulz, G. R. J. Am. Chem. SOC. 1994, 116, 10174. (30) (a) Daivis, P. J.; Evans, D. J. J. Chem. Phys. 1994, 100, 541. (b) Klatte, S. J.; Beck, T. L. J. Phys. Chem. 1993, 97, 5727. (31) Naim, J. A,; Braun, C . L. J. Chem. Phys. 1981, 74, 2441. (32) Forbes, M. D. E.; Le, T. T., unpublished results. (33) Closs, G. L.; Forbes, M. D. E., Kinetics and Spectroscopy of Carbenes and Biradicals; Platz, M. S., Ed.; Plenum: New York, 1990, p 51. JP950210B
