The Journal of
Physical Chemistry
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0 Copyright, 1990, b y the American Chemical Society
VOLUME 94, NUMBER 13 JUNE 28, 1990
LETTERS Dynamic Solvent Effects in Adiabatic Electron-Transfer Reactions: Role of Translational Modes B. Bagchi,* A. Chandra, Solid State and Structural Chemistry Unit, Indian Institute of Science, Bangalore 560 01 2, India
and G . R. Fleming Department of Chemistry and James Franck Institute, University of Chicago, Chicago, Illinois 60637 (Received: December 4, 1989; In Final Form: March 13, 1990) A molecular theory of the dynamic solvent effects on an outer-sphere adiabatic electron transfer reaction is presented. The theory properly includes the role of the microscopic correlations present in a dense dipolar solvent in the dynamics of an electron-transferreaction. The theory predicts that the translational modes of the solvent, hitherto neglected in most discussions on electron-transfer reactions, can significantly enhance the rate of a n adiabatic electron-transfer reaction.
Introduction The possible role of polar solvent dynamics in influencing the rate of an electron-transfer reaction has been a subject of several theoretical studies in recent years.14 While the majority of the theoretical predictions are yet to be confirmed, recent experimental results seem to indicate a definite role of polar solvent dynamics in determining the rate of an outer-sphere electron-transfer reactiona5 Several computer simulation studies6*' have also ad( I ) (a) Zusman, L. D. Chem. Phys. 1980,49,295. (b) Alexandrov, I. V. Cfiem. Phys. 1980, 51.449. (2) Calef, D. F.; Wolynes, P.
G.J . Phys. Chem. 1983, 87, 3387. (3) Hynes, J . T. J . Pfiys. Cfiem. 1986, 90, 3701. (4) (a) Sumi, H.; Marcus, R. A. J . Cfiem. Phys. 1986, 84, 4272. (b) Nadler, W.; Marcus, R. A . J . Chem. Phys. 1987,86, 3096. (c) Sparpaglione, M.; Mukamel, S. J . Chem. Phys. 1988.88, 3263. (d) Rips, 1.; Jortner, J . J . Chem. Phys. 1987,87,2090. (e) Zusman, L. D. Chem. Phys. 1988,51, 119. ( 5 ) (a) Nielson, R . M.: McManis, G.E.; Weaver, M. J. J . Phys. Chem. 1989. 93. 4703. (b) McManis, G. E.; Weaver, M. J. J . Chem. Phys. 1989, 90,912. (c) Simon, J. D.; Su,S.G.J . Chem. Phys. 1987,87, 7016. (d) Su, S. G.;Simon, J . D. J . Cfiem. Pfiys. 1988, 89, 908. (e) Kosower, E. M.; Hupper, D. Annu. Rev. Phys. Cfiem. 1986, 37, 127. (f) Kahlow, M. A.; Jarzeba. W.; Kang, T. J.; Barbara, P. F. J . Cfiem. Phys. 1989, 90, 151. (6) Bader, J . S.; Chandler, D. Chem. Phys. Letr. 1989, 157, 501. 0022-3654/90/2094-5197$02.50/0
dressed the problem of solvent involvement in the dynamics of electron-transfer reactions in dipolar solvents. In this Letter we present a molecular theory of dynamic solvent effects on an outer-sphere adiabatic electron-transfer reaction. This theory extends the earlier theoretical studies'-4 in several directions. First, microscopic descriptions of the structure and the dynamics of the solvent are included self-consistently. Most of the earlier studies were based on a continuum model description of the solvent. For dynamic solvent effects, such continuum model description can be seriously flawed because the continuum model includes only the long wavelength relaxation processes whereas in electron transfer, the intermediate wavelengths are quite important. Second, we include, for the first time, the effects ofthe translational modes of the solvent in the dynamics of electron transfer. We show that these translational modes can have a significant effect on the dynamics of electron transfer; especially, they can significantly enhance the rate over that given by the rotational modes alone. (7) Zichi, D. A,; Ciccotti, G.; Hynes, J. T.; Ferrario, M. J . Phys. Chem. 1989, 93, 6261.
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Models We shall consider two different models of electron-transfer reactions in this work. In the first model, the reaction involves an isolated redox couple Ox e Red (1) This model has been studied recently by McManis and Weaver,8 who used the dynamic mean-spherical approximation (DMSA) of Wolynes9 to find the dynamic solvent effects on the adiabatic barrier-crossing frequency. As pointed out by McManis and Weaver,s this reaction can be thought of as an electrochemical exchange process, with the reactant located sufficiently far from the metal surface. The net free energy driving force for this reaction is zero, so that the free energy of activation, AG#, is the “intrinsic” outer shell (Le., solvent) part of the barrier energy. In this case, the reaction coordinate is the solvation energy of the reaction system. In the second model, we consider the model electron-transfer reaction.
+
A-l/2B+1/2 A+I/ZB-1/2 (2) for a solute pair AB, with A = B. This model has been studied recently by Zichi et al.,’ who performed a molecular dynamics simulation to study departures of electron-transfer rates from the Marcus theoryiO*Iipredictions. Significant deviation was found in the adiabatic limit because of dynamic solvent effects. The reaction coordinate for this reaction is assumed to be given by
A E ( t ) = I d r [EoP(r)- EoR(r)].6P(r,t)
(3)
where b R ( r ) and b P ( r ) are the bare electric fields of the reactant and the product, respectively, and 6P(r,t) is the polarization fluctuation at time t . AE(t) is the difference, for a given polarization, of the Coulomb potential energy of interaction between the solvent and the reactant and the product solute pairs. As noted by Zichi et al.,7 the reaction transition state is located at AE = 0. For both the models, we shall consider only the adiabatic limit (both weak and strong) and follow the standard prescription for adiabatic dynamics.*v7
Theory A simple and elegant theoretical formulation to include dynamic solvent effects on an adiabatic electron-transfer reaction has been presented recently by H y n e ~ .The ~ merit of Hynes’ formulation is that the two different limiting situations, namely, the weakly and the strongly adiabatic reactions, can be treated within the same framework. In this theory, the rate of a symmetric adiabatic electron-transfer reaction is given by keF1 = kb-‘ 2kw-‘ (4) where kb is the rate of crossing the activation barrier and kw is the rate of solvent polarization relaxation within the potential wells. Both k b and kw can be expressed in terms of the reaction coordinate time correlation function A,(t) defined by 40) = ( W O ) W t )) i / ( W O ) A W ) ) i (5) where the average is over the solvent degrees of freedom at equilibrium in the presence of the reaction system in the state i. The rate constant kb is given by the Grate-Hynesi2 formula for a parabolic barrier top, and kw is given by the expression recently provided by H y n e ~ .For ~ numerical evaluation of the rate constant we need to calculate the frequency-dependent friction, CR(w),and also the well and the barrier frequencies, wR and Wb, respectively. The frequency-dependent friction can be calculated from a generalized Langevin equation for the motion of the reaction coordinate in the reactant potential weL3 The reactant well frequency,
+
(8) McManis, G. E.; Weaver, M . J . J . Chem. Phys. 1989,90, 1720. (9)Wolynes, P. G . J . Chem. Phys. 1987,86, 5133. (10) Marcus, R. A. J . Chem. Phys. 1956,966, 979. (II) Marcus, R. A . Annu. Rev. Phys. Chem. 1964,IS, 155. (12)Grote, R. F.; Hynes, J. T. J . Chem. Phys. 1980,73, 2715. For recent developments, see: Straub, J . E.; Borkovec, M.; Berne, B. J . Chem. Phys. 1986,84,1788. Pollak, E.;Grabert, H.;Hanggi, P. J . Chem. Phys. 1989,91. 4073.
Letters wR, can also be calculated from the reaction coordinate time
correlation function, AR(t), as discussed by Zichi et ai.’ and by H y n e ~ .Thus, ~ the problem of calculating the rate constant now reduces to the calculation of the reaction coordinate time correlation function, AR(t). We next present microscopic calculations of A R ( t ) for both the models considered here. In the first model, the reaction coordinate is the solvation energy of an ion in a dipolar liquid, and in the second model it is that of a dipole. The reaction coordinate time correlation function is given by
where &(k) is the Fourier transform of the bare electric field of an ion (for model I) or of a dipole (for model 11). The longitudinal polarization 6PL(k,t) is defined by the expression 6PL(k,t) = l i d r e”.’Jdw
[h?(w)]
6p(r,w,t)
(7)
where & ( w ) is a unit vector with orientation w and @ is the magnitude of the dipole moment of a solvent molecule. It has been shown e l ~ e w h e r e ’that, ~ * ~ in ~ a linear theory, the decay of the polarization time correlation function is exponential: (6P,(-k) 6PL(k,t))R = (6PL(-k) 6PL(k))Re-f/TL(k) (8) where the wave-vector-dependent relaxation time is given byi3 7L(k)
=
(2&)-’[(1 p’(kq)2)(1 - ( P o / ~ ) ( C A (+~ )2CD(k)))l-’ (9) where DR is the rotational diffusion coefficient of the dipolar solvent molecules and p’ = DT/2DRc2,DT and c being the translational diffusion coefficient and the diameter of solvent molecules, respectively. p’is a measure of the relative importance of the translational modes in solvent polarization relaxation. CA(k) and CD(k)are Fourier transforms of the anisotropic partsi5of the two-particle direct correlation function of the dipolar solvent. The mean square longitudinal polarization fluctuation is given by13e
where N is the total number of solvent molecules and h,(k) and hD(k) are the usual anisotropic parts of the two-particle pair correlation function. h , ( k ) and hD(k) are related to CD(k) and C,(k) by the Ornstein-Zernike re1ati0n.I~ Since the bare field of an ion or of a dipole is known, it is now straightforward to write analytic expressions of the reaction coordinate time correlation function, AR(t), for both the models. The explicit expression of & ( t ) for model I1 is given by AR(t) = x ( r ) / x ( o ) (1 la) with X(t) =
r L
where j l ( z ) is the spherical Bessel function of order unity and (13) Chandra, A.; Bagchi, B. (a) Chem. Phys. Lett. 1988,151,47;(b) Proc. Indian Acad. Sci. 1989,101,83; (c) J . Phys. Chem. 1989,93, 6996; (d) J . Chem. Phys. 1989,91,2594;(e) Chem. Phys. Lett., in press. (0 Bagchi, 9.; Chandra, A. J . Chem. Phys. 1989,90,7338;(8) Chem. Phys. Lett. 1989, 155, 533. (14)Bagchi, B. Annu. Reu. Phys. Chem. 1989,40, 115. ( 1 5 ) Gray, C. G.; Gubbins, K. E. Theory of Molecular Fluids; Clarendon: Oxford, 1984;Vol. 1.
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Letters
30t
(11 R = 1 0
0.0 I
0.0
0.2
I
I
0.4 P‘
-
0.6
1
I
1
1
I
I
I
02
04
06
08
I
0.8
Figure 1. Dependence of the barrier-crossing frequency, Y, of a weakly adiabatic electron-transfer reaction (for model I given by eq 1) on the translational diffusion of the solvent molecules. The calculated values of Y are plotted against p’for two different values of the solute-solvent size ratio, R. The barrier frequency, wb. is equal to 4wR, where wR is the well frequency (given by eq 12). The dashed line indicates the rate predicted by the continuum model theories, as discussed in the text. The values of the dielectric constant (to) and the reduced density (pOu3!of the solvent are 18.0 and 0.8, respectively. The values of the longitudinal relaxation time ( T ~ and ) the Debye relaxation time ( T ~ are ) O.076DR-’ and I .368DR-I,respectively, where DR is the rotational diffusion coefficient of the solvent. @e2/u= 140.0.
argument z ; q = ka; and R is the solute-solvent size ratio, R = 2a/a, where a is the radius of the artificial sphere that surrounds the reaction system. In deriving eq 11 the field of the dipolar reacting system has been replaced by that of a point dipole located at the center of the sphere that encompasses the reacting system.
Numerical Results We now present numerical results of our calculations of the effective barrier-crossing frequency, u, defined by the relation k,, = u exp[-PAG#]
0 01 00
P’
(12)
The explicit expression of Y can be obtained from eqs 4-1 I . In our formalism, the free energy of activation, AG#, for model I is given by the expression
where 9 = ka. The above expression correctly reduces to the corresponding dielectric continuum formula for the intrinsic barrier16 when c(k) is approximated by toand when R is very large. We have discussed elsewhere’3cthat the replacement of c(k) by to can be a poor approximation when the reactant-solvent size ratio is small ( R < 1). For model 11, the expression of the barrier height is given by the expression for the solvation energy of a dipole.’3b In the numerical calculations, we have considered both the weakly adiabatic and the strongly adiabatic cases. In the former case, we have taken cdb = 4cdR, and for the strongly adiabatic case, we have taken a b = wR. We have calculated the frequency-dependent friction, CR(s), by using the generalized Langevin equation for the motion of the reaction coordinate in the overdamped limit, thus ignoring the acceleration term. In all our numerical calculations we have used the mean-spherical approximation (MSA) for the two-particle direct correlation f ~ n c t i o n . ’ This ~ is fairly accurate at low dielectric constant. Figures 1 and 2 show the results of our calculations of u for model I . I n Figure I , the dependence of u on the translational parameter, p’, in the weakly adiabatic limit (wb = 4 0 R ) is shown. It is seen that the rate of electron transfer increases with an increase in the value of p’. The increase in the electron-transfer (16) The dielectric continuum formula for the intrinsic barrier involving an isolated redox couple is AG# = (e2/8a)(l - eo-’) in a liquid of nonpolarizable molecules.
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Figure 3. Dependence of the barrier-crossing frequency, Y, of a weakly adiabatic electron-transfer reaction (for model 11, given by eq 2) on the translational diffusion of the solvent molecules. The calculated values of v are plotted against p’for two different values of the solute-solvent size ratio, R. The value of the barrier frequency wb is equal to 4wR. The values of the dielectric constant and the reduced density of the solvent are the same as in Figure 1. The values of the longitudinal relaxation time and the Debye relaxation time are the same as in Figure 1.
rate is higher for smaller solute size. This is because when the size of the reactant is small, the reaction probes more of the intermediate wavevector processes than when it is large. The relaxation of the solvent polarization at intermediate wavevectors becomes increasingly faster as the value of the translational parameter, p’, is increased. This large effect of the translational modes on the intermediate-wavelength processes comes from two sources. First, small spatial displacements, which have a pronounced effect at these length scales, are quite fast (because rate varies as DTk2). Second, because of strong intermolecular correlations between nearest neighbors, orientational relaxation is rather In Figure 1, we have also plotted the prediction of the continuum model (which includes only the k = 0 mode contribution toward the solvent polarization relaxation) for the rate, given by u = (2mL)-I, by a dashed line. The calculated rate is less than the continuum model prediction for p ’ = 0 but becomes greater as p ’ is increased. Figure 2 shows the dependence of u on translation but in the strongly adiabatic limit (wb = wR). In this limit the electron transfer occurs at a lower rate than that in the weakly adiabatic case and also the dependence on p’ is somewhat weaker. The reason is that, in the weakly adiabatic limit, the reaction probes the high-frequency components of the solvent response. At large p’, translational modes populate these high-frequency components, so the dependence on p’is stronger for a weakly adiabatic reaction
J . Phys. Chem. 1990, 94. 5200-5202
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Figure 4. Dependence of the barrier-crossing frequency, Y , of an weakly adiabatic electron-transfer reaction (for model 11) on the solute-solvent size ratio, R. The calculated values of Y are plotted against R for two different values of the translational parameter, p’. The values of the barrier frequency, the dielectricconstant, and the reduced density of the solvent are the same as in Figure 3. The values of the longitudinal relaxation time and the Debye relaxation time are the same as in Figure I.
than that for a strongly adiabatic reaction. Also, the reactive friction is less for the former than for the latter. Figures 3 and 4 show the results of our calculations for model 11. In Figure 3, the dependence of v on p’is shown for a weakly adiabatic reaction. The results of this figure are rather similar to those of Figure 1 except that here the dependence on p’is much stronger. This is because the reacting system is modeled as a point dipole, so the reaction coordinate does not contain any contribution from the zero wavevector pr~cesses.’~ The intermediate wavevector processes are more important in this model and hence the stronger dependence on p’than observed for model I. The dependence of Y on the solutesolvent size ratio, R , is shown in Figure 4. It shows that, for zero translational contribution (p’ = 0), the rate of electron transfer increases with increases in the value of R. However, for a sizable translational contribution @’>0.5), the
rate of electron transfer decreases with increasing R. The reason for the above behavior is as follows. As we increase the value of R, the reaction coordinate receives a greater contribution from the small wavevector processes. These small wavevector processes occur at a faster rate than the large wavevector processes when the translational contribution is absent. Thus, in the absence of translation, the rate of electron transfer increases with increase in R . However, in the presence of a sizable translational contribution, the intermediate wavevector processes occur at a faster rate than the small wavevector processes. So in the presence of finite translation, the rate of electron transfer decreases as the value of the reactant-solvent size ratio, R , is increased. We have also calculated the rates predicted by the transitionstate theory for both models. For the weakly adiabatic case, the Grote-Hynes formula12gives a rate that is typically 0.8-0.9times the transition-state-theory value. For a strongly adiabatic reaction, the friction effects are more important and we get kcH = (0.50.6)kTST.Note that these numbers are in good agreement with the computer simulation results of Zichi et al.’ It is interesting to note that, in this one-dimensional model, kTSThas a dependence on p’that enters through wR. This explains the weak p’dependence of the ratio kCH/kTST. The dramatic effect of the translational diffusion on the rate of an electron-transfer reaction has a simple physical origin. At small length scales, the collective orientational relaxation of a dense dipolar liquid is slow in the absence of translational contribution.’3a However, the translational modes are especially effective at small length scales. These two effects combine to give rise to the marked dependence of the rate constant on p’ In conclusion, we have considered here two somewhat different models of outer-sphere adiabatic electron-transfer reactions and have shown that, for both models, the translational modes of the dipolar solvent can significantly enhance the rate of the reaction which is demonstrated here for the first time. Moreover, these dynamic effects are found to be significantly dependent on the adiabaticity of the reaction. Acknowledgment. This work was supported in part by NSF, USA (G.R.F.), CSIR, INDIA (A.C.), and INSA, INDIA (B.B.). B.B. is a Homi Bhabha Fellow for 1989-1991.
Electron Impact Mass Spectrometry of Cholesterol in Supersonic Molecular Beams Aviv Amirav Sackler Faculty of Exact Sciences, School of Chemistry, Tel- Aviu University, Ramat Auiv 69978, Israel (Received: March 1, 1990)
The effect of vibrational supercooling on the electron impact mass spectrometry of cholesterol was studied in supersonic molecular beams. An extensive mass spectral simplification was observed at 20 eV electron energy. The parent undissociated molecular ion becomes the dominant peak even at 70 eV electron energy but the conventional complex fragmentation pattern is retained in spite of the large vibrational cooling. Some of the unique aspects of electron impact mass spectrometry in supersonic molecular beams are mentioned.
Since the pioneering demonstration of Smalley et al.,’ supersonic molecular beams (SMB) revolutionized the research of optical spectroscopy and intramolecular dynamics. The extreme molecular vibrational-rotational cooling considerably simplifies laser-induced fluorescence spectra of large polyatomic molecules, ( 1 ) Smalley, R. E.; Ramakrishna, B. L.; Levy, D. H.; Wharton, L. J . Chem. Phys. 1974, 61. 4363.
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eliminates sequence congestion, and provides clear information on the initial and final ro-vibronic quantum states2 On the other hand, vibrational temperature effects on electron impact induced ion fragmentation are well-known and documented in electron (2) (a) Amirav, A.; Even, U.; Jortner, J. Chem. Phys. 1980,51, 31. (b) Amirav, A.; Even, U.; Jortner, J. J . Chem. Phys. 1981, 75, 3770. (e) Levy, D.H.;Wharton. L.; Smalley, R. E. Arc. Chem. Res. 1977. 10, 134.
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