Solvent Barriers in Unimolecular Ionizations. 2. Electron Transfer

J. Phys. Chem. 1994, 98, 5460-5470

5460

Solvent Barriers in Unimolecular Ionizations. 2. Electron Transfer Perspective for Alkyl Iodide Ionizations Jeffery R. Mathis and James T. Hynes' Department of Chemistry and Biochemistry, University of Colorado, Boulder, Colorado 80309-021 5 Received: January 28, 1994"

-

+

The analysis of the alkyl iodide ionizations R I R + I- begun in I (preceding paper in this issue) is continued by focusing on a nuclear (r) coordinate-dependent electron transfer (ET) perspective. Considerations presented within provide an understanding of the factors that determine, e.g., whether or not charge transfer plus bond breakage occurs in a stepwise or concerted fashion. Two different E T rate constants are investigated: a conventional r-dependent E T rate kFT-generalized to these sN1 ionizations-and a new ET rate kfT. kfT accounts for the r-dependence of the solvent barrier location and thus incorporates into the rate analysis dividing surface curvature or, equivalently, the strong transition state electronic structure variation along the nuclear coordinate. This added component-which is not present in the standard E T rate-is essential in providing a reasonable estimate of the ionization rate while still attempting to retain an ET perspective. k7Tcan be more than an order of magnitude smaller than kfT and does not provide a reasonable estimate of the ionization rate. In the absence of the electronic structure variation effects, kfT reduces to kFT. Comparison of kfT with the variationally optimized result k, from I shows that they are comparable, but with kfT being slightly larger. Trajectory calculations, carried out to test the no-recrossing assumption for the kfT dividing surface, show that there is some recrossing. This recrossing correction brings kfT into accord with k,, so that k, is the best estimate of the actual rate. The results are applicable to other reaction classes, including the SRNIreaction mechanism and inner sphere electron transfer reactions.

1. Introduction

In the preceding paper in this issue, hereafter I, the reaction rates of the solution phase alkyl iodide unimolecular ionizations RI

-

R+

+ I-;

R = (CH,),C, (CH,),HC

(1.1)

were investigated by employing methods focusing primarily on the alkyl group-iodine nuclear separation as the reaction coordinate. However, as was shown in I, these alkyl iodides are unique in that, in addition to exhibiting activation barriers along their nuclear coordinate, they also show activation barriers along the solvent coordinate. This suggests that a nuclear coordinatedependent electron transfer (ET) a p p r ~ a c h l -might ~ be appropriate to calculate the reaction rate. In this paper, we carry out the reaction rate analysis from this nuclear coordinate-dependent ET perspective, where the reaction coordinate is viewed as being primarily along the solvent ~oordinate.*-~ A very brief review of some relevant information presented in I will be given in order to facilitate the present analysis. The evolving solute electronicstructure in solution, discussed in section 2 of I, is described by the two valence bond state wave function

in terms of orthonormal covalent state $c(r) [RI] and ionic state $I(r) [R+I-] basis wave functions,& the vacuum energies for which were also discussed in section 2 of I. The two state coefficients ( c c , ~depend ) not only on the-nuclear separation r but alsopn the orientational polarization Po,solvent coordinate s, since Po,fluctuations can alter the solute charge distribution.6 The general free energy function, in terms of the two coordinates r and s, is given by49536a G(r,s) = v(r,s) - '/,AV(r,s)x - P(r)y

+

1/4AG:(r)yZf+ AGPs2 - k,T 1n[r/rol2 (1.3) @

Abstract published in Advance ACS Absrracts, May 1, 1994.

where all quantities have their same meaning as discussed in section 2 of 1. Figure 1 shows the free energy surface contours of eq 1.3 for t-BuI and i-PrI in CH3CN solvent, respectively. Also displayed are the locations of the equilibrium solvation states (cf. eq 2.12 in I), collectively termed the equilibrium solvation path (ESP). The solid lines corresopnd to the equilibrated reactant (s = 0) and product (s i= 1-1.5) well minima, while the dashed lines in Figure 1 correspond to the solvent barrier top locations. Figure 2 shows the free energy profiles along s for the two alkyl iodides in CH3CN (cf. Figure 5 in I) at three representative r values: r C r*, r = r * , and r > r*, where r* is the saddle point location in r. From these figures, one can see that both the solvent barrier location and height for the alkyl iodides are r-dependent. In fact, analytic expressions for these quantities as a function of r can be easily obtained. For the solvent barrier location st(r) in s as a function of r, we have

(x) ( ) AG,(r) * I 2 1 -x:(r)

st(r) =

2

obtained by the second-order expansion around s = 1 1 2 of eq 2.1 8 in I. The approximate free energy Gt(r) of the solvent barrier location can also be obtained by using the perturbative result of eq 1.4 combined with eq 1.3. However, it is found that an even simpler expression agrees with the exact result (eq 1.3) to within *0.2 kcal/mol, given by Ct(r) = G[r,st(r)] = Gz(r)

1 AG,(r) = -[AGW(r) 4AGAr)

+ AG,(r) - P(r) + AC,(r)I2

solely in terms of the equilibrated covalent free energy G?(r) (cf. eq 2.14 in I), the r-dependent generalization of the Marcus3

0022-3654/94/2098-5460$04.50/0 0 1994 American Chemical Society

The Journal of Physical Chemistry, Vol. 98, No. 21, 1994 5461

Solvent Barriers in Unimolecular Ionizations. 2

-31'

S

3.5

3.25

3

S

2.15

Figure 2. Solvent coordinate free energy profiles for (a) r-BuI and (b) i-PrI in acetonitrile solvent at three representative r values in the free energy surface saddle point vicinity (cf. Figure 1). The solvent barrier height and location in s are r-dependent.

2.5

2.25

0

0.25

0.5

0.75

1

1.25

1.5

S

Figure 1. Free energy contour surface for (a) r-BuI and (b) i-PrI in acetonitrile. The solid lines are the reactant and product solvent well minima,while the dotted linedenote-sthe solvent barrier location. Contour lines are 2.0 kcal/mol apart and begin at -52.0 kcal/mol near the reactant state. equation for the diabatic barrier height in the solvent coordinate AGM(r) and the electronic coupling @(r)(cf. eq 2.9 in I). The sum of AGM(r) and the electronic coupling term is approximately the solvent barrier height (the second-order perturbative expression contains two additional terms proportional to the square of @ and AGq = Gf4 - GZ, respectivelf). Clearly, this state of affairs is quite reminiscent of r-dependent electron transfer (ET) reactions.1-3 All the usual ingredients indicative of adiabatic electron transfer reactions are present in the alkyl iodide systems, even though the reaction is not typical in the sense that here a bond is being broken. The coupled charge transfer plus solute nuclear reorganization process present with the ionizations eq 1.1 are, however, characteristic of inner sphere electron transfer rea~tions,~J where there is usually a ligand exchange between two metal centers, accompanied by a change in oxidation state. In this respect, the sN1 ionizations of eq 1.1 can be regarded as examples of inner sphere processes. The natural question to ask then for systems such as t-BuI and i-PrI in eq 1.1 is whether their ionizations can indeed be viewed as an r-dependent E T reaction, with the reaction coordinate being primarily the solvent coordinate, or if the reaction coordinate is

still primarily r, as is the case with other sN1 systems investigated.4-5 We should emphasize that since the electronic coupling magnitude @ is relatively large (-2-4 kcal/mol) compared to typical outer sphere electron transfer reactions,'-' the ionizations of eq 1.1 are completely electronically adiabatic, and we need not beconcerned with nonadiabaticcorrection factors for the reaction rate,9J0 which have been the subject of numerous investigations in the literature for both standard ET and nuclear coordinate dependent E T reaction r a t e ~ . ~ - ~ s ~ J ~ Two ET rate constant expressions will be evaluated. The first, k$, corresponds to a standard method of evaluating the rate expression for bar,rier crossing in the solvent coordinate a t fixed r a n d then averaging over r to obtain the total rate expression.lJ The second, kf?,uses a coordinate transformation in its evaluation so that the solvent barrier locations depicted in Figure 2 all have the same location with respect to one of the new coordinates, which then serves as the dividing surface for reactants and products. The perpendicular flux through this surface is then derived and subsequently used to evaluate a transition state theory (TST) rate constant.11 This approach effectively incorporates dividing surface curvature into the reaction rate constant, as opposed to reaction path curvature which was investigated in section 4C of I.l*J3 Equivalently stated, the changing solute electronic structure is explicitly incorported into the second E T rate expression, whereas we will see that this feature is not incorporated in the conventional ET expression. It will be demonstrated that, as a result, the two E T rate constants are not equivalent and in some cases differ by more than an order of magnitude.

5462

Mathis and Hynes

The Journal of Physical Chemistry, Vol. 98, No. 21, 1994

TABLE 1: Ri Values for Various Rate Comparisons in Various Solvents R , = kmT/kET .I .-,

C6HsCI

CHZC12

(CH3)zCO

CH3CN

80.5 6.7

16.3 5.5

10.6 4.9

9.9 4.9

61.8 7.2

23.0 7.2

17.1 6.9

16.2 6.8

Use of the Hamiltonian

..a

t-BuI i-PrI Rz = k:T/ k7T t-Bd i-PrI RJ = kVmS'/kFT t-BuI i- Pr I

1.30 0.93

0.71 0.77

0.62 0.7 1

0.61 0.7 1

After considerable analysis of the results here and in I, it will be shown that the variational rate, kv ( c j Table 3 of I), is the best estimate to the actual rate in all solvents, despite the ET characteristics displayed in Figure 2. In solvents of very low polarity, both ET rate constants fail to accurately assess the reaction rate. However, in higher polarity solvents, the new ET rate kFT requires only a small recrossing correction to match the more accurate kv and is thus a quite reasonable estimate (kfT again fails). The reason for the approximate agreement between these two approaches is that, due to the explicit incorporation of the transition state electronic structure variation, the physical picture of the ionizations of eq 1.1 for the two approaches turns out to be nearly the same when viewed in the proper coordinate system, despite the seemingly "orthogonal" approaches used to evaluate each one. The format of this paper is as follows. In section 2, we derive the conventional expression for nuclear coordinate-dependent E T reaction rates, as generalized to the present alkyl iodide S N ~ ionizations. The new ET rate expression is derived in section 3, which incorporates the strong electronic structure variation coupling present in the reaction system and is subsequently compared to the conventional result, as well as to those in I. Concluding remarks and applications to other reaction classes are given in section 4.

2. Standard r-Dependent Electron Transfer The conventiona11-3E T reaction mechanism-as here extended to the S N ionization ~ examples-would be viewed as first a bond extension to reach a given r, followed by barrier crossing in the solvent coordinate:

H(r,s,p,,p,) = Pr2 + Ps2 + G(r,s) 2r, 2PLs

in the evaluation of eq 2.3, which was derived and discussed in the Appendix of I, and subsequent rearrangement gives for the integrand ingredients in eq 2.2

A&r)

= Gt(r) - G&(r)

['

"-1

2

= 27r 27rkBT

112

[rdr

exp[-(Gt(r) - GR)/kBT]

(2.7) where eqs 2.5 and 2.6 have been combined to yield the simplified integrand, and the integration is only over the region where the solvent barrier is present, as denoted by the initial (rin) and final (rfin) r values. The rotational entropy term is included in the expression for G in eq 1.3. This term cancels in AGJ(r), so only K,(r) contains it via AG&(r). The rotational term can be extracted from K,(r) so that kFT is equivalently (2.8)

where p,(r) is the analog of eq 2.5 but without the rotational entropy term. Equation 2.8 now has the form usually found in the literature in connection with conventional r-dependent ET reactions (modulo the spatial restrictions).l-3 It would be desirable to have a simpler harmonic approximation for this rate expression to evaluate and discuss. However, it is found that Gt(r)is generally too anharmonic (especially for i-PrI) and that the integration limits in eq 2.7 cannot be extended to i m without serious error. Appendix A discusses these aspects in more detail. The full numerical result for eq 2.7 will therefore be used for all subsequent rate comparisons. Comparison of kyT with kv (cf. Table 3 and section 4C of I)

R , = k,/kfT

where the integration is to.be carried out over the r range where the solvent barrier is present. Expressions for K,(r) and k,l(r) can be obtained by evaluating the one-way flux across the solvent barrier location st(r) (cf, eq 1.4 and Figure 2) and then averaging over r:

(2.6)

where AGl(r) is the r-dependent solvent barrier height, given in terms of the free energy Gf(r) of the solvent barrier location for which an approximate expression is given by eq 1.5, and GR is the alkyl iodide reactant state free energy. Since the electronic coupling is comparatively large, the ionization is completely electronically adiabatic, and k,l(r) does not contain nonadiabatic correction f a c t o r ~ . ~Both J ~ K,(r) and k,l(r) have the forms anticipated above for them. The total rate constant is then

kFT = Jdr (r/rq)*G(r) kel(4

Here RI* denotes the alkyl iodide with sufficient equilibrium free energy GA(r), i.e. sufficient bond extension, to reach the vicinity of the solvent barrier ( c j Figures 1 and 2). K,(r) is an equilibrium probability per unit length of reaching that free energy, or equivalently, a differential equilibrium constantlf with units of reciprocal length. k,l(r) governs the electron transfer step and should be in a simple approximation the probability for attaining the solvent barrier multiplied by a solvent reactant well frequency factor. The total rate constant kyT is then the integral'-3

(2.4)

(2.9)

compiled in Table 1, gives values significantly greater than unity in all solvents for both alkyl iodides. This result is intriguing, in that it is ambiguous at this point which rate constant is the best estimate for the actual rate: a naive TST approach could suggest that the smaller of the two rate constants-here kfTwould be the best estimate. We will however now reach the opposite conclusion.


Solvent Barriers in Unimolecular Ionizations. 2 TABLE 2 Calculated Saddle Point Frequencies. for Alkyl Iodide Ionization in Several Solvents CsHsC1

CH2C12

10.2 51.0 2.3 224.1 229.6 2.2

10.5 61.2 6.2 103.5 119.8 5.4

12.9 26.1 2.2 152.5 154.4 2.2

12.1 56.5 11.2 61.2 82.6 8.3

(CH&CO

CH3CN

10.6 67.2 7.8 91.2 112.8 6.3

10.7 68.7 8.2 89.3 112.2 6.5

12.1 71.1 14.1 61.3 93.1 9.3

12.1 74.7 14.7 61.6 95.9 9.4

t-BuI *I WII

(_~,2)’/~ ut +p2)L/z *t

i-PrI *I

/I

(_w,2)’/2 *t (;WA*)’/~ *t

3. Transition State Theory Electron Transfer Rate Expression

and wI were determined by the All quantities in units of ps-l. methods discussed in section 4A of I. In order to decide which rate constant is better, we reason as follows. kf’ is nor a TST rate constant because the flux through its dividing surface-the plane perpendicular to the plane completely in s that contains the points = st(r)-is evaluated at fixed r and then averaged over r; there is thus no single dividing surface used in the evaluation of k’: but rather an entire sequence of them each defined a t a given r. In effect, kf’ represents a “one-dimensional” rate approximation to a process that is inherently two-dimensional, in that there is both solvent reorganization and solute nuclear motion. On the other hand, the dividing surface used to evaluate k, in I is continuous over the entire free energy surface, incorporating the energetic and coupling effects of both r and s coordinates and as such is a well-defined TST rate expression. Thus, the TST argument1I-l3 that the lowest rate is the best estimate does not apply here, since the dividing surfaces for the two rate constants are not related to each other in any way. It appears that the larger rate kv should be chosen, since given a choice between two reaction “paths” (although there is no strict path for kf’), the reaction system will take the one that corresponds to the larger rate. We now determine why k$ is smaller than k,. As discussed above, there is no single reaction path on the free energy surface corresponding to k p . Instead, there are a series of reactant product transitions across the solvent barrier, each occurring at a different constant value of r. Since the solvent barrier height is r-dependent, it is useful to designate the effective transverse degree of freedom for kf’ to be r with the free energy Ct(r). Why k:’ is smaller than k, can then be qualitatively understood by comparing their transverse well frequencies, which effectively describe the reaction “windows” in the TS vicinity for each mechanism. Accordingly, we define for the E T mechanism the frequency -+

Wt =

( M y a2Gt0,t a3 = (&-wb,”.’)

this, providing a further argument in favor of the k, choice. As solvent polarity increases, the solvent barrier for the present systems can exist for a greater range in r, so the integration limits in eq 2.7 are larger, and thus kFT becomes larger relatively. The R1 values in Table 1 directly reflect this: kFT is relatively much smaller than k, in weak polarity solvents because the solvent barrier and the associated ET reaction window is disappearing, and k$ is a progressively more extreme underestimate of the rate constant.15 In view of all this, we must reject k p as a useful description of the reaction rate constant. We now attempt to improve the situation while still retaining an E T perspective.

(2.10)

As discussed above, kfT is not a TST rate expression since there is not a singledividing surface, but rather an entire sequence of them, each defined at a given rand perpendicular to the plane in s that contains the point s = sf(r). In this section, the r-dependence of the solvent barrier location in s, and thus the strong s-r potential coupling, is explicitly accounted for in order to derive a TST rate constant. The procedure is to first convert the sequence of distinct dividing surfaces used in kFT into a single continuous dividing surface, which effectively accounts for dividing surface curvature. Next, the correct perpendicular velocity through this surface is found, and then the TST rate constant is evaluated. A. Derivation. To begin the derivation of this second E T rate expression, k;’, we introduce the shifted relative solvent coordinate A(r,s) = s - st(r)

where st(r) is the location in s of the r-dependent solvent barrier top given approximately by eq 1.4. The free energy surface G(r,s) is then transformed to G(r,A), with the solvent barriers located now at constant A = 0. As an example, Figure 3 displays the G(r,A) surface contours in the saddle point vicinity for i-PrI in CH3CN. The A = 0 dividing surface, which contains the saddle point {r*,s*), can be readily identified. It is important to recognize that A contains one or more singularities corresponding to the point of points where the solvent barrier disappears. Thus, A is not defined for either iodide in the small r region near the reactant state where there is no solvent barrier and again is undefined for t-BuI at larger separations where its solvent barrier disappears. These aspects will not however preclude a well-defined rate calculation. Next, the velocity normal to the A = 0 surface is needed. This amounts to asking for the velocity normal to the st(r) curve, as illustrated in Figure 4, and is just the time derivative of A

A = s - y(r)k

(3.2)

where y(r)

where the second equality is an exact result and shows the relation of u t to the quantities introduced in section 4A of I. Comparing the saddle point wI values with those of ut,Table 2 shows that ut > wL in all cases, so the effective ET reaction window is more narrow-and often much more narrow-than that associated with the SRP, and kf’ is consequently smaller than kv. Stated differently, the SRP is entropically favored over the kf’ mechanism. Further, R l decreases with increasing solvent polarity. In the limit of ”zero” solvent polarity (Le., Po, = 0), there is no solvent barrier, and kyT must be zero. Yet reaction should still be possible, even if slow, and the finite k, value appropriately reflects

(3.1)

dst(r)/dr

(3.3)

is a coupling factor between r a n d s motions and represents the local direction of the A = 0 dividing surface in (r,s) coordinates. It is important to stress that y(r) is also the gradient of the solvent barrier ionic character (cf. eqs 2.12 and 2.13 of I) and is only present due to the changing solute electronic structure induced by the solvent.16 In fact, y(r) can be formally related to the solute-solvent coupling factor g and solvent frequency w, introduced in section 4A of I according to (3.4)

5464

Mathis and Hynes


TABLE 3: Maw Factors Evaluated at the Saddle Points in Various Solvents CsHSCl

CHzCl2

(CH3)zCO

CH3CN

29.6 0.14 8.2 X lo3

31.3 0.17 7.3 X lo3

t-BuI

3.05

16.0 0.002 3.6 X loS

PI'

kA(r')" pr(r*)b

23.2 0.06 1.5 X 104 i-PrI

20.6 29.1 36.5 38.5 0.004 0.53 0.83 0.91 pr(r*)b 1.6 x 105 1.8 x 103 1.4 x 103 1.4 x 103 a Units are 1 X 1W9g cm2. For comparison, the moment of inertia of an acetonitrile molecule is -8.4 X 10-39 g cm2 [Maroncelli, M. J . Phys. Chem. 1991, 94, 20841. Units are g/mol. Solute reduced mass pr is 39.39 and 32.17 g/mol for t-BuI and i-PrI, respectively. PIo

3

@A(r*)a

OS !+

2.95

2.9

2.85

Delta Figure 3. G(r,A)free energy surfacecontoursin the saddle point vicinity for i-PrI in acetonitrile. The dividing surface used in the evaluation of k;* is located at A(r) = 0.

The evaluation of eq 3.5 requires that the Hamiltonian of eq 2.4 be transformed to the new (r,A) coordinate system; in addition, the momentum associated with the normal velocity A needs to be found, since the integrations in eq 3.5 are conveniently carried out over momentum variables. These manipulations are carried out in Appendix B, and the result for eq 3.5 is

'I

-

I A=S

- dst(r) dr

F.

-

1

r

l

4

8.7

3.5

2.9

3.1

3.3

(4

r Figure 4. Schematicdiagram for determiningthe velocity perpendicular to the st(r) dividing surface. 6 is the angle of the tangent vector off the r axis.

This formal relation is valid as long as the r-dependent condition aG/as = 0 is satisfied and therefore applies all along the A = 0 dividing surface. The r-dependent notation for the solute-solvent coupling factor g(r) and the square solvent frequency u?(r) (cf. eqs 4.7 and 4.6 of I, respectively) has been written explicitly to show this. To eliminate any possible confusion, absolute magnitudes of g(r) and u?(r) have been displayed, since both of these quantities are negative. y(r) is negative; the solvent barrier first appears near s 1.25-1.5 (the product) and, as r increases, moves toward s = 0 (the reactant), as can be clearly seen in Figure 1. Its magnitude however is largest in the r region where the solvent barrier first appears since there the solvent barrier top is almost flat and uS(r)is very small. Note also that A is roughly parallel t o r in the vicinity where the solvent barrier first appears, while only at larger separations-where the solvent barrier location does not change much with r-is A along s. We then define the associated TST rate constant to be the equilibrium average one-way flux across the A = 0 surface, normalized with respect to the reactant state:"

B. Comparison with In comparing eq 3.6 with eq 2.7, one can see that the only difference is the new mass factor, ir(r) = pr + r2(r)ps = pr[l

+ ?(r)(~s/~r)I

(3.7)

associated with the transverse coordinate r, involvingthe electronic structure factor y ( r ) . Accounting for dividing surface curvature effectively decouples r and s in potential energy terms; the coupling is shifted to the kineticenergy, eq B.7, and nonlinearly mixes their respective masses, with the result that the new mass associated with the transverse coordinate is 2-3 orders of magnitude larger (cf. Table 3) than the old mass pr. Since, for example, near the TS the ratio of p, to pr is not that large (-5-7 A*, cf. Table 3), this large disparity is due to the electronic structure factor y2(r). In Figure 5,pr(r) for the two alkyl iodides in CH3CN is displayed as log[ir(r)/pr] in order to show its more-than-exponential r-dependence. The masses for both iodide cases have maximum values when their respective solvent barriers first appear, because the electronic structure factor y2(r) is largest there (cf. eq 3.4 and Figure 2). As r increases, pr(r) initially decreases for both alkyl iodides, corresponding to a decrease (in magnitude) in y(r), and then increases again for r-BuI because its solvent barrier disappears, while tending toward ccr for i-PrI because its solvent barrier location in s becomes constant (cf. Figure 1). As might be anticipated, the reactive coordinate mass MA(r) (cf. eq B.17), whose saddle point values are also compiled in Table 3, PA(^) = /+ps/ir(r) = ~ s [ 1+ T * ( ~ ) ( P ~ / P ~ ) I - '(3.8) is now 2-3 orders of magnitude smaller than p,. pA(r) of course does not appear in the final answer for kFT, since the flux is defined only in terms of the velocity A. We have stressed in section 2 that the E T rate constant kyT seriously underestimates the rate. We now determine if this major deficiency is repaired in the TST description afforded by kFT. Table 1 shows the ratios


7.5

-

I

t-BUl

5.0

2.5

-

-

0 2.8

Figure 5. Mass factors &r) as a function of r for t-BuI (-) and i-PrI (- -), A logarithmic scale is used because of the magnitudes, and to demonstrate its more-than-exponentialbehavior. t-BuI displays a "cup-

2.9

A

3

3.1

r Figure 6. r-dependentrate constants kaT(r),ky(r)for i-PrI acetonitrile. k:T(r) is much larger than kaT(r)due to the new mass factor &r), eq 3.1.

like" behavior because the solvent barrier for it disappears, causing the

Electronic structure factor y ( r ) in eq 3.4 to become large at two points. generally larger than that of i-PrI because its solvent barrier is characterized by a smaller wg (cf. Table 2), which by eq 3.4 means that y(r) is larger.

pr(r) for r-BuI is

ET

ET

R2 = k2 l k l

(3.9)

using the numerical results of eqs 2.7 and 3.6. R2 is relatively constant for i-PrIbecause the difference in its r integration limits (rfin-rin) does not change appreciably as solvent polarity is varied. For both iodides, however, kFTis significantly larger than kBTin all solvents, an effect solely attributable to pr(r), eq 3.7, which itself arises from the potential decoupling-and thus the kinetic coupling-of r and s motions in the new r, A coordinate system. At this point, it is important to discuss in greater detail the reasons why R2 is greater than unity and its implications. For these S Nionizations, ~ two processes take place: (a) motion of the solvent coordinate from covalent reactant to ionic product configurations and (b) motion of the solute coordinate corresponding to a bond being broken and the accompanying charge migration. kfT accounts for the former, as can be seen from eq 2.7 involving the r-dependent solvent barrier height AGi(r), but disregards the latter. On the other hand, k:T incorporates both a and b via the condition that the dividing surface A = 0 is crossed in a perpendicular fashion. In other words, ky fully accounts for the bidimensionality of the reaction, while kfT incorporates basically only the solvent coordinate. However, note what has really happened by first adopting the r,A coordinate system and then evaluating the perpendicular flux through the A = 0 dividing surface. As shown in Figure 4, perpendicular crossing of the A = 0 surface in the r,A coordinate system is the same as crossing the sf(r) line in the r,s system perpendicularly and as such involves both rands motions. Indeed, at smaller r where the solvent barrier first appears (cf. Figures 1 and 4), crossing st(r) there involves r motion almost exclusively. It is only at large r, where the solvent barrier location in s does not changeappreciably, that s is the dominate motion. Therefore, although a series of profiles obtained from Figure 3 along A at fixed r would show double solvent wells separated by a barrier similar to those in Figure 2 (except that the solvent barrier location now does not change) and as such would give the idea of r-dependent adiabatic electron transfer profiles, these same profiles projected onto the r,s system (cf. Figure 1) and viewed as a function of their arc length would in fact show only a single barrier, similar to the SRP free energy projections displayed in Figure 9 in I. The r-dependent ET perspective seen clearly in the r,A coordinate system is effectively lost when viewed in the r,s system. The formalism "fools" us into believing that the rate

is being evaluated in an r-dependent E T fashion, but in reality, k:T is quite similar to the SRP rate constants discussed in I in that both r and s motions are being incorporated in the rate, in complete contrast with the unidimensional kFT. Continuing now with the analysis of kFT, we reexpress the integrand in eq 3.6 as the product of a differential equilibrium constant K,(r) and a solvent barrier rate constant k,l(r) as in eqs 2.2, 2.5, and 2.6, but with pr(r) appearing with kei(r):

Here w:, the frequency of the solvent reactant well in the (r,A) coordinate system (cf. Figure 3), is defined as

Figure 6 shows the two integrands of eqs 2.7 and 3.6, Le., k z ( r ) , for i-PrIin CH&N as an example and allows for a very clear comparison between k y and kFT. The effect of ir(r) is quite pronounced. In entropic terms, the very large value of the transverse mass translates into a much wider window, as expressed by the transverse frequency

-

(which should be compared to eq 2.10) through which reactant product crossings ace made, which then accelerates the rate. TS values for LOA and ut are also given in Table 2. Finally, note that ky reduces to k y when the transition state electronic structure does not change; Le., y(r) = 0. Why the transition state electronic structure change is so large for the alkyl iodides is traceable to the pronounced r-dependence of the solvent barrier location ins, which itself is the result of the covalent and equilibrated ionic diabatic potential curves having opposite slopes, as shown in Figures l a and 6a of I. As r increases, the covalent curve moves up in energy, while the equilibrated ionic curve moves lower in energy, which then produces an r-dependence for the diabatic crossing point in the solvent coordinate (the electronic coupling does not shift the solvent barrier location

5466

Mathis and Hynes


appreciably from the diabatic crossing point). As the solvent barrier top moves from the initial product solvent well vicinity to the final reactant solvent well vicinity, its electronic structure also changes-in this case rather abruptly-from a product-like to a reactant-like charge distribution. This effect can be seen clearly in Figure 2. By contrast, for typical outer sphere electron transfer reactions, e.g., M/; Mk Mfl Mzk+' where M I , M ~ are transition metal complexes and j,k their total charges, the electronic structure variation of the solvent barrier top should not be as large, since the potentials for the reactant and product complexes along the nuclear separation coordinate will have the same slope. In other words, the reactant chargesj,k usually have the same sign as the product charges j - l,k 1; there is no significant reaction asymmetry in the solvent coordinate as the nuclear separation is varied, and consequently no significant solvent barrier top electronic structure change. Under these conditions, kFT should then provide an accurate estimate of the reaction rate. kfT can therefore be regarded as a generalization of the conventionall-3 kyT to account for reaction asymmetry and the resulting electronic structure variation of the solvent barrier top location in the solvent coordinate. C. Comparison with k,. The analysis above indicates that kfT is a much improved estimate of the rate compared to k p . We now ask how k$ fares compared to k, of I and reopen the issue as to whether k, is in fact the best rate estimate or if instead kT ! is. Comparison in Table 1 of kfT with k,,

+

-+

+

+

R , = k,/kfT

(3.13)

shows that kET is larger than k, except for a single example: t-BuI in C6HsCI. In this particular case, the solvent barrier is not onlyvery small-less than 0.1 kcal/mol (cf.Table 1 in 1)-but the range in r where it exists spans only -0.5 A. The solvent 0), so k$ goes to zero. Since barrier is disappearing (Le. ?or the actual rate should remain finite, even if slow, we can dismiss kT ! as a useful description for t-BuI in solvents of very low polarity. But, this simple argument cannot be applied for the remaining examples; k y is larger than k, in all cases except that mentioned above, so again we are faced with an ambiguity as to which rate represents the best estimate. However, in contrast to the scenario in section 2, the ET rate is now the larger. In order to resolve the issue, Figure 7 displays a close-up view of the d ( r ) dividing surface used for kfT for i-PrI in CH3CN. One can observe that, in this local region, it appears that the perpendicular potential gradient along the kZET dividing surface is either positive or negative, depending on the position relative to the saddle point. That is to say, to the left of the saddle point r > r*,a trajectory crossing the dashed dividing surface in Figure 7 in the positive direction will encounter a repulsive force and will recross. To the right of the saddle point r < r*, the same effect will occur for a trajectory crossing the dividing surface with a negative velocity. This indicates that low kinetic energy kFT trajectories probably would recross, with reactant-reactant recrossings predominant to the left of the saddle point and product-product recrossings predominant to the right." On the basis of Figure 7, one would then expect kZETto indeed be an overestimate of the reaction rate. A transmission coefficient from a trajectory calculation

-

K

= k,/ kFT

(3.14)

should then be equal to the ratio R3. We have tested this reasoning by calculating K directly from a classical trajectory simulation for i-Prl in CHsCN, using the Hamiltonian of eqs 2.4 and B.16. The employed procedure is as follows. The average total free energy (kinetic plus potential) ( H )= J d r H exp[-H/kBT)/Jdr exp[-H/kBU, along the kFT

S

Figure 7. Close-up view of the free energy surface for i-PrI in CH$N (cf Figure lb). The kFTdividing surface is denoted by (- -). One can see that the potential gradient perpendicular to the dividing surface is positive for trajectoriescrossing to the left of the saddlepoint and negative ~ for trajectoriescrossingto the right of the saddle point. The k z dividing surface from I, denoted by (-),shows a positive potential gradient only for trajectories crossing to the right of the saddle point and as such should be a better estimate to the rate than kfT. However, as there would be some recrossing of the km dividing surface, the best estimate to the rate is provided by k,. Contour lines are 0.5 kcal/mol apart and are not

mass-weighted. surface is found to be (H)= 1 . 4 7 k ~ Tabove the saddle point, which shows that the potential free energy along this surface is not quite harmonic in the relevant region, in agreement with the results in Appendix A. Trajectories were then initialized from a Boltzmann distribution subject to the above average total free energy. The initial potential free energy was first specified by choosing the coordinate r from the distribution 2.845 Ir I3.1, corresponding to the energetically accessible region (cf. Figure 6). Since the G(r,A=O) = Gt(r) well is not harmonic, a simple random distribution for the initial value of r was employed. The surplus free energy was then given to PN (always positive) and Pr (cf. eq B.16) and subsequently related back to pr and ps via eqs B. 11-B. 13. After this initialization procedure, the trajectories along the kFTdividing surface were then propagated forward and backward in time according to previously developed methods.l8J9 A fourth-order Runge-Kutta integration routine was used with a step size of 0.00 1 ps to carry out the calculation,*Oand a 0.05-ps time scale proved sufficient for determining trajectory fates in either direction. Further, a 0.5% energy leakage criterion was imposed on all trajectories, with no failures, and the forward and backward propagations were terminated when the potential energy fell to 1.5kBTbelow the initial free energy (usually before the 0.05-ps duration expired). K was then computed for an ensemble of trajectories byl8J9

-

(3.15) wherewi = exp[-E,,./kBU (Eto,,= total energy) is theBoltzmann weighting factor, and qi is either 0 for a recrossed trajectory or 1 for a reactant product single cross. A group of 10 ensembles of 100 trajectories each gave an average of K = 0.76 f 0.05, thus showing that there is recrossing of the k$ surface. Further, the type of recrossings observed completely agree with what is expected from Figure 7: only reactant-reactant recrosses were observed to the left of the saddle point ( r > r*),and only product-product recrosses were observed to the right ( r < r * ) . We have not performed a trajectory calculation for t-BuI because a close-up

-



such as an excited state cyclohexane molecule, the reaction proceeds in a stepwise fashion: first the conformational change followed by the charge transfer, or vice versa. Our results seem in contrast with this point of view, with the concerted charge transfer plus bond breakage ( k f T ) being preferred over the stepwise bond extension followed by charge transfer (k$). However, there is the important difference in the present systems that there are no stable intermediates; the reaction can only go from reactants to products, and does not become trapped (or "gatednga)in a free energy surface local minimum corresponding to some stable intermediate (cf. Figure 1). For the alkyl iodide ionizations occurring in low polarity solvents, kFT underestimates the reaction rate constant and indeed vanishes in the limit of zero solvent polarity, due to the fact that the solvent barrier disappears; the possibility of an ET mechanism no longer exists. This same limiting behavior is also exhibited by k$. Therefore, we can dismiss both k!T and kFT as useful descriptions of the ionization rate constant in low polarity solvents, since the ionization rate should still be finite, even if small. The variationally optimized result k, from I is unambiguously the best estimate to the reaction rate in very low polarity solvents. However, in higher polarity solvents, comparison of kFT with k, from I shows that they are comparable, but with k$ slightly larger: k f T / k , -1.1-1.6 depending on the alkyl iodide (kyT is still considerably smaller than k,). A trajectory calculation carried out to test the no-recrossing assumption for k$ showed that some trajectories did recross the kfT dividing surface, so that despite the r-dependent electron transfer characteristics for these alkyl iodide sN1 ionizations, k, again provides the most accurate rate constant estimate (although from a practical standpoint, k y and k, are nearly equivalent). Why kfT is larger than k, is basically because the dominant aspect of the reaction is breaking the carbon-iodine bond and not the electron transfer. In fact, the k y formalism itself shows that in one coordinate system, the reaction resembles an r-dependent electron transfer reaction, but when viewed in the r,scoordinate systemused in thedetermination of k,, the r-dependent E T perspective is lost (cf. section 3C). Because of this commonality, k;T and k, did not differ substantially. The results of the alkyl iodide analysis presented in this paper and its predecessor have ramifications that extend well beyond the sN1 ionization reaction class. For example, it is shown in section 3 that kFT is a generalization of the conventional result kFT accounting for electronic structure variation of the transition state location in the solvent coordinate with nuclear separation r and reduces to kFT in the absence of these effects. The pronounced transition state electronic structure variation itself in the present cases is the result of large reaction asymmetry in 4. Concluding Remarks the solvent coordinate along r. The covalent and equilibrated ionic diabatic potentials for the alkyl iodides have opposite slopes, In this paper we have continued the analysis of the alkyl iodide and as r increases, the covalent potential moves up in energy (a ionizations R I R+ + I-, begun in I, by focussing on a nuclear bond is being broken), while the equilibrated ionic potential moves coordinate dependent electron transfer (ET) perspective. Two lower in energy (increasing solvent stabilization). The crossing different ET rate constants were investigated: a c~nventional~-~ point in the solvent coordinate then changes its location ins rather r-dependent E T rate kET,generalized to these sN1 ionizations, dramatically as r increases. In the presenceof thesoluteelectronic and a new rate k;T. kb, T accounts for the r-dependence of the coupling, the change of location in the solvent coordinate of the solvent barrier location and thus incorporates dividing surface diabatic crossing then results in a very abrupt change in the curvature, or equivalently the solvent barrier top electronic electronic structure of the solvent barrier top, going from a structure variation, into the rate analysis. This added component product-like to a reactant-like charge distribution in a relatively of kFT-which is not present in the standard kyT rate-is narrow range in r. The new k f formalism fully accounts for this essential in providing a reasonable estimate of the ionization rate effect. For typical outer sphere electron transfer rea~tions,l-~ while still attempting to retain an ET perspective. The conventional result kfT is shown to be smaller than kFT by at least a (4.1) factor of 7 and up to a factor of 60 and as such is an unreliable estimate of the ionization rate. where MI, Mz are, e.g., transition metal complexes and j , k are It is generally believedg that when an ET system has both the initial charges for them; the potential energy along the nuclear charge transfer and conformational changes associated with it,

of its kfT dividing surface exhibits the same characteristics as that for i-PrI in Figure 7; to the left of the saddle point, there is a barrier in front of the kFT dividing surface, and to the right there is a barrier behind it. One can then safely infer that a trajectory calculation for it would give K = R3. Reference 21 contains additional information regarding this calculation. The agreement with the R3 entry for i-PrI in CH3CN in Table 1 is strong evidence supporting our reasoning detailed in the preceeding paragraphs and thus that k, is the best estimate to the actual rate. However, the result that kZET requires only a small recrossing correction means that the approach used to compute it is quite reasonable. Indeed, by the arguments in section 3B, incorporation of the strong electronic structure changes effectively means that the r-dependent ET characteristics are lost; the main difference between kfT and k, is that their respective dividing surfaces are slightly offset from one another. As an additional note, we are now in a position to discuss graphically why kZH is also an overestimate, and in particular why the ordering k$ > kZH > k, is observed (cf. Table 3 of I and Table 1). Figure 7 also displays the dividing surface used to compute kZH. By examination of the local potential gradients along this dividing surface, one can see that it too will produce an overestimate of the rate due to some recrossing, although not to the extent that it would be with the kZET dividing surface. To the left of the saddle point (r > r*), the potential gradient is nearly zero along the solid line, so that there should be little or no recrossing in this region. This represents an improvement compared to kFT,for which it was argued that low kinetic energy trajectories would recross for it in this region. However, to the right of the saddle point ( r < r*), the potential gradient along the solid line is positive, so that low kinetic energy trajectories would feel a barrier in front of them and thus would probably recross. Therefore, one would expect that kZH is also an overestimate of the rate, although not as much as is kFT. The data in Table 1 of this paper and Table 3 of I show that k;T > kZH > k, (in high polarity solvents), in agreement with this reasoning. In summary, by attempting to retain an r-dependent electron transfer perspective for the alkyl iodide S Nionizations ~ and yet still evaluate correctly a transition state theory rate constant by defining a single continuous dividing surface (A = 0, Figure 3) in the new r,A coordinate system, we have in effect lost an ET perspective when viewed in the original r,s coordinate system. We must conclude that when a chemical reaction has both charge transfer and bond cleavage or formation taking place in a concerted fashion, the coupling of the solvent and solute motions precludes any description in terms of standard r-dependent electron transfer ideas,l-3 even though its free energy surface may suggest the contrary.

-

5468

Mathis and Hynes

The Journal of Physical Chemistry, Vol. 98, No. 21, I994

separation coordinate for the reactant and product configurations, and in particular their corresponding slopes, will be quite similar since both are usually composed of Coulombic contributions combined with nuclear repulsive contributions. The reaction asymmetry is thus expected to be small, and consequently the transition state electronic structure variation will also be small. For these types of reactions, we do not expect that the kyT rate estimate would be seriously in error. However, when there is significant charge character change in going from the reactant to products, such as would be the case for two neutral complexes reacting to give two charged complexes, e.g., Fe(CN)6 reacting with a neutral organic molecule,8 then the conventional kFT approach would give a serious underestimate of the reaction rate, or equivalently an overestimate of its activation free energy. The formalism presented within accounts for this type of charge variation resulting from reaction asymmetry, and therefore would give an accurate rate estimate. In addition to outer sphere electron transfer processes outlined above, there is also another general organic reaction mechanistic category for which the analysis presented here would be applicable. This is the S R Nmechanism ~ (substitution, radical, nucleophilic, unimolecular). As was first elucidated in 1966 by Kornblum and Russell,22 molecules that tend to undergo reaction by this mechanism are usually very unreactive to sN1 or sN2 substitution.*-23 The S R Nmechanism ~ is characterized by an initial electron transfer from, e.g., an external source or a radical ion generated in situ, to the reactant, such as R - X,to form (R-X)-. This radical ion complex then dissociates to form R’ and X-, which in the presence of an external nucleophile N forms the substituted R - N product. Specific examples of systems that have been shown to react in this fashion usually have phenyl rings or nitro groups present within the molecule, which act to stabilize the radical ion intermediate by delocalizing the electron.24 What are not known in any detail concerning S R Nreactions ~ are what factors determine whether or not the reaction proceeds in a stepwise fashion, with electron transfer being followed by ratedetermining dissociation, or in a concerted fashion, with both electron transfer and dissociation being rate determining. Our analysis here shows that in order for electron transfer to be rate determining for the S R Nreaction ~ mechanism, there must be (a) a sufficiently small electronic coupling so that a solvent barrier can exist and an electron transfer thus possible and (b) sufficiently small reaction asymmetry in the solvent coordinate as the nuclear separation changes, which produces a small transition state electronic structure change as a function of the nuclear coordinate. If either of these criteria are not met, we have demonstrated in section 3 that electron transfer characteristics are no longer present, and the reaction is a concerted electron transfer-bond breaking process. More recent work on the S R Nreaction ~ mechanism, both experimentally and theoretically, has been provided by S a v b n t and co-workers25 in studies of a series of dissociative electron transfer reactions. These share in common all aspects of the S R Nmechanism, ~ but the emphasis is placed on the electron transfer step rather than the entire process leading to a substituted product (which is more of a synthetic interest23). However, their analysis has been couched in terms of standard Marcus theory for outer sphere electron transfer processes.3 As a result, the effects exposed here of the strong electronic structure variation-whose incorporation is central to providing an accurate estimate of the reaction rate from an E T perspective-are not included in the approach of these authors. It would be of interest to further pursue the ET analysis presented here for these types of reactions. Additional reaction classes for which our analysis may be applicable are inner sphere electron transfer reactions,7-8 which involve, e.g., both electron transfer and ligand exchange, and possibly twisted intramolecular charge transfer (TICT) com-

plexes, which involve, e.g., (dimethylamino)benzonitrile.26 Both of these reaction classes may exhibit significant charge variation of the transition state in the solvent coordinate as a function of nuclear separation, or the N - C rotation angle in the case of TICT, as there are energy requiring bond breakages combined with energy stabilizing effects from the solvent due to the presence of electric charge. Acknowledgment. This work was supported in part by N S F Grants CHE-8807852 and CHE-9312267. We also thank Dr. Hyung J. Kim and Dr. Arnulf Staib for useful discussions. Appendix A. Additional Analysis for

kf“

In this Appendix, we attempt to simplify eq 2.7. First, the r integration limits in eq 2.7 are extended to f m and Gt(r) is expanded harmonically as

Gt(r) * Gt

+ 1/2p,o+2(r* - r)’

(A. 1)

where ut2 = (pr)-l [a2Gt(r)/ar2] * is the frequency of the @(r) well (cf. eq 2.10) evaluated a t the saddle point. The result is 0

kFTz--w~ 0, exp[-AG*/k,T] 2?r O+ It is found that this approximate expression is an overestimate of the numerical answer to eq 2.7 by 540% for i-PrI and 20-75% for t-BuI, with the better agreement for both iodides coming in CH3CN solvent. Thus, the two approximations, harmonic expression of Ct(r) and extension of integration limits, are not justified. To determine which approximation, if any, is the “worst”, the harmonicexpansion of @(r) is retained, but the finite integration limits are accounted for by the use of error functions. Thus,

where erf(zfin,in)is the error function with arguments

It is found that the error function result is a better estimate than the initial one above for theexact eq 2.7 for t-BuI (-3%difference in all solvents), but worse for i-PrI (5-65%), and in all cases is an underestimate of eq 2.7. Appendix B. Transition State Theory Electron Transfer Rate Evaluation Here the mathematical manipulations required to evaluate eq 3.5 are performed. It is convenient to use mass-weighted coordinates (? = prl/2r, 3 = pa112s)since this avoids the unit disparity present in the rectilinear coordinate system. Therefore, unless explicitly stated otherwise, all following equations and their ingredients refer to the mass-weighted coordinate system. To evaluate the transition state theory rate constant, one usually determines first the dividing surface and then the correct perpendicular velocity through it. The dividing surface in this case has been given by eq 3.1, so we will now proceed to determine the correct perpendicular velocity (and momentum). The Lagrangian in mass-weighted coordinates is

L = 1/2[p + S2]

- G(r,s)

(B.1)

which serves as !he starting point. The velocity vectors normal N a n d tangent T to the st(r) curve are (cf. Figure 4)


N

= -y(r)i.

+ S;

T = i.


+ y(r)S

(B.2)

where y ( r ) ,thelocaldirectionof thest(r) curve, is now understood to be the mass-weighted analog of eq 3.3 and as such is unitless. Squaring each member of eq B.2 and subsequent substitution into eq B.1 gives the kinetic energy in terms of the new velocities N a n d T,

In terms of PI and PA,PNand PTare given by (cf. eqs B.5 and B.ll)

One can then attempt to make an F2(r,A,p~,&) canonical t r a n ~ f o r m a t i o nto~ ~find the new coordinates according to

aF,/ar = P, = (1 The momenta PN,PTassociated with the velocities N,Tare then defined by

and can be expressed in terms of the old momenta pr and p s as (cf. eq 2.4)

Now we ask what the Hamiltonian looks like in the r,A coordinate system, and, in particular, whether or not the perpendicular momentum PNas derived above actually appears. To answer this, the Lagrangian approach used above is again employed. The time derivative of the new coordinate A(r,s) = s - st(r) (cf. eq 3.1) is

A = s - y(r)i.

(B.6)

which when combined with the Lagrangian eq B.l yields L = 1/2[(1

+ r2(r))PT; dFJdA = PA = P N

+ y(r)PT

(B.13)

However, the conditions specified for F2 in eq B.13 are incompatible with a solution, due to the r-dependence of y(r). Examination of the mixed second derivatives of F2, which should not depend on the order in which they are taken, shows this point:

More directly, the Poisson bracket using eqs B.11 and B.12 does not vanish. This means that no global coordinates conjugate to the specified momenta P, and PAcan be found. We are then reduced to introducing PNinto the Hamiltonian of eq B.10 via the first part of eq B.12 simply as a change of integration variable foregoing any dynamical interpretation per se. This gives

The subscript eff means that this is an effective Hamiltonian; A and PN are not canonically conjugate. Further, it can be easily demonstrated that the momentum variable transformation (P,,PA) (P,,PN) has a unitary Jacobian determinant (cf. eq B.12). At this point, it is instructive to convert eq B.15 back to the rectilinear coordinate system before actually evaluating eq 3.5. The result is

-

+ r2(r))b2 + A2 - Zy(r)Ai.] - G(r,A)

The momenta for the system are then defined as

P, = a q a A = [A

where the mass factors

+ y(r)i.];

P,= aqai. = [(I + r2(r))i. + y(r)A] are associated with r and A motions, respectively, and where PN/PA(r) = A is the velocity perpendicular to the st(r) curve. Equation 3.5 then becomes

Solution for the time derivatives in terms of P, and P A

=

IP,

- Y(r)PA]; A

= [( 1

+ r2(r))PA - y(r)Pr]

allows finally for the new Hamiltonian to be given by 0 2

H= 2

D 2

rA + -(1 + r 2 ( r ) ) - y ( r ) P P A + G(r,A) 2

(B.lO)

Use of eq B. 16 and classical evaluation of the reactant partition function gives the final result eq 3.6.

where the new momenta P,,PA are in terms of the old:

References and Notes

This transformation shows that shifting the solvent coordinate in an r-dependent fashion changes the momentum conjugate to r and does not affect the old momentum ps. Consequently, the required momentum PNperpendicular to the st(r) curve (cf. eq B.5) does not appear in the transformed Hamiltonian of eq B.lO. An additional transformation to introduce PNinto eq B.10 is necessary.

(1) (a) Levich, V. G .Adu. Electrochem. Electrochem. Eng. 1966,4,249. (b) Tembe, B. L.; Friedman, H. L.; Newton, M. D. J . Chem. Phys. 1982,76, 1490. (c) Newton, M. D. In Mechanistic Aspects of Inorganic Reactions; Rorabacher, D. B., Endicott, J. F., Eds.; American Chemical Society: Washington, D.C., 1982. (d) Marcus, R. A.; Siders, P. J . Phys. Chem. 1982, 86,622. (e) Marcus, R. A. Int. J . Chem. Kinet. 1981,13,865. ( f ) Newton, M. D. Int. J. Quantum Chem. Symp. 1980,14, 363. (2) (a) Sutin, N . In Electron Transfer in Inorganic,OrganicandBiological Systems; Bolton, J . R., Mataga, N . , McLendon, G., Eds.; Advances in Chem. Series 228; American Chemical Society: Washington, D.C., 1991. (b) Kakitani, T.; Yoshimori, A.; Mataga, N. In Elecrron Transfer in Inorganic, Organic and Biological Systems; Bolton, J . R., Mataga, N., McLendon, G., Eds.;Advances in Chem. Series 228; American Chemical Society: Washington,

Mathis and Hynes

5470 The Journal of Physical Chemistry, Vol. 98, No. 21, 1 994 D.C., 1991. See also: Gertner, B. J.; Ando, K.; Bianco, R.; Hynes, J. T. Chem. Phys., in press. (3) (a) Marcus, R. A. J. Chem. Phys. 1956,24,966. (b) Marcus, R. A. Ibid. 1956, 24,979. (c) Marcus, R. A. Faraday Discuss. Chem. Soc. 1960, 29.21. (d) Marcus, R. A. Ibid. 1960, 29, 29. (e) Marcus, R. A. J. Chem. Phys. 1963,38,1858. (f) Marcus, R. A. Annu. Rev. Phys. Chem. 1964.15, 155. (g) Albery, W. J. Annu. Rev. Phys. Chem. 1980, 31, 227. (h) Sutin, N. Prop. Inorp. Chem. 1983. 30. 441. (i) Newton. M. D.: Sutin. N. Annu. Rev. Piys. Ciem. 1984, 35; 437. Marcus, R: A.; Sutin, N: Biochim. Biophys. Acta 1985, 811, 265. (4) Mathis, J. R.; Kim, H. J.; Hynes, J. T. J. Am. Chem. SOC.1993,115, 8248. (5) (a) Kim, H. J.; Hynes, J. T. J . Am. Chem. SOC.1992,114, 10508. (b) Kim, H. J.; Hynes, J. T. Ibid. 1992,114, 10528. (c) Hynes, J. T.; Kim, H. J.; Mathis, J. R.; Timoneda, J. J. i. J . Mol. Liq. 1993, 57, 53. (d) Kim, H. J.; Bianco, R.; Gertner, B. J.; Hynes, J. T. J. Phys. Chem. 1993,97,1723. (6) (a) Kim, H. J.; Hynes, J. T. J . Chem. Phys. 1992,96,5088. In the context of the self-consistent field approximation, see: (b) Kim, H. J.; Hynes, J. T. J . Chem. Phys. 1990,93,5194. (c) Kim, H. J.; Hynes, J. T. Ibid. 1990, 93, 5211. (7) (a) Kochi, J. K. Ace. Chem. Res. 1992,25,39. (b) Creutz, C. Prog. Inorg. Chem. 1983, 20, 1. (8) Eberson,L. Electron Transfer Reactions in Organic Chemistry;Springer-Verlag: Berlin, 1987. (9) (a) Ratner, M. A. In Perspectives in Photosynthesis. The Jerusalem Symposia on Quantum Chemistry and Biochemistry; Jortner, J., Pullman, B., Eds.; Kluwer Academic Publishers: Dordrecht, The Netherlands, 1990; Vol. 22. (b) Hoffman, B. M.; Ratner, M. A. J . Am. Chem SOC.1987,109, 6237. (c) Brunschwig, B.; Sutin, N. J. J. Am. Chem. Soc. 1989,111,7454. (IO) See, e.g.: (a) Brunschweig, B. S.;Logan, J.; Newton, M. D.;Sutin, N. J. Am. Chem. SOC.1980,102,5798. (b) Siders, P.; Marcus, R. A. J. Am. Chem. Soc. 1981, 103, 741. (c) Isied, S.S.;Vassilian, A.; Wishart, J. F.; Creutz, C.; Schwartz, H. A.; Sutin, N. J . Am. Chem. SOC.1988, 110, 635. (d) Isied, S. S.;Vassilian, A,; Magnuson, R. H.; Schwartz, H. A. J . Am. Chem. SOC.1985, 107,7432. (11) See, e.g.: (a) Eyring, H. J . Chem. Phys. 1935, 3, 107. (b) Evans, M. G.; Polanyi, M. Trans. FaradaySoc. 1935,31,875. (c) Wigner, E. Trans. Faraday Soc. 1938, 34, 29. (d) Glasstone, S.;Laidler, K. J.; Eyring, H. Theory of Rate Processes; McGraw Hill: New York, 1941. (e) Chandler, D. J. Chem. Phys. 1978,68,2959. (f) Johnston, H. S.Gas Phase Reaction Rate Theory; Ronald Press Co.: New York, 1966. (12) (a) Lee, S.;Hynes, J. T. J. Chem. Phys. 1988,88,6853. (b) Lee, S.;Hynes, J. T. Ibid. 1988, 88, 6863. (13) For more recent accounts of solution phase variational treatments, see: (a) Schenter, G. K.; McRae, R. P.; Garrett, B. C. J . Chem. Phys. 1992, 97,9116. (b) Truhlar, D. G.; Schenter, G. K.; Garrett, B. C. Ibid. 1993,98, 5756. (c) Schenter, G. K.; Messina, M.; Garrett, B. C. Ibid. 1993,99, 1674. However, the developments above do not take full nonlinear account of the most important solvent variable u, which is connected to the rapidly evolving electronic charge distribution in the reaction system." (14) (a) Kim, H. J.; Lee, S.-Y.; Mathis, J. R.; Gertner, B. J.; Hynes, J. T. Manuscript in preparation. (b) Kim, H. J. Manuscript in preparation. (1 5) It should be noted that Rl strongly depends on the choice of u : , as uL is rather close to its value. Use of a larger up will tend to decrease Ri.

u)

(16) The complete expression for y ( r ) is

where the prime denotes an r-derivative and all quantities are evaluated along the path in r corresponding to the solvent barrier location. At the saddle point, the first term within the square brackets is much larger than the second 2-60 A-1 compared with -0.5 A-1, while the prefactor involving the scaled reorganization free energy is equal to unity, so that y ( r ) really is the gradient of the solvent barrier ionic character and not just dependent upon it. The second term in brackets results from the coordinate shift introduced by eq 2.3 of I. It should be emphasized that the relation for y ( r ) , eq 3.4, in terms of the solute-solvent coupling g and square solvent frequency u,2, respectively, is exact. (17) A similar situation occurs for the CI- + CH$l S Nreaction ~ in water. (18) (a) Bergsma, J. P.; Gertner, B. J.; Wilson, K. R.; Hynes, J. T. J . Chem. Phys. 1987.86, 1356. (b) Gertner, B. J.; Bergsma, J. P.; Wilson, K. R.; Lee, S.;Hynes, J. T. Ibid. 1987,86, 1377. (c) Gertner, B. J.; Wilson, K. R.; Hynes, J. T. Ibid. 1989,90,3537. (d) Huston, S. E.; Rossky, P. J.; Zichi, D. A. Ibid. 1989, I l l , 5680. (e) Keirstead, W. P.; Wilson, K. R.; Hynes, J. T. J. Chem. Phys. 1991, 95, 5256. (19) (a) Anderson, J. B. J. Chem. Phys. 1973, 58, 4684. (b) Keck, J. Discuss. Faraday SOC.1962,33,173. (c) Keck, J. C. J . Chem. Phys. 1960, 32, 1035. (20) Press, W. H.; Vetterling, W. T.; Teukolsky, S.A,; Flannery, B. P. Numerical Recipes in Fortran, 2nd ed.; Cambridge University Press: Cambridge, U.K., 1992. (21) For the 10 100-trajectoryensembles, averagequantitiesareasfollows (energies referenced to the saddle point free energy and kBT): (H) = 1.51 f0.14; (KE) = 1.08f0.12; (PE) =0.44+0.04; ( r ) =2.92+0.01;average number of recrosses = 34 4; product-product recrossea = 12 f 3; reactantreactant recrosses = 22 4. ( r ) above is the same as that computed from ( r ) = J d r r exp[-H/kaTI/Jdr exp[-H/kaTl = 2.92 A. (22) (a) Kornblum, N.; Michl, R. E.; Kerber, R. C. J . Am. Chem. Soc. 1966,88,5662. (b) Russell, G. A.; Danem, W. C. J. Am. Chem. SOC.1966, 88, 5663. (23) See, e.&: (a) Lowry, T. H.; Richardson, K. S. Mechanism and Theory in Organic Chemistry, 3rded.; Harper and Row: New York, 1987. (b) Cary, F. A.; Sundberg, R. J. Advanced Organic Chemistry, Part A, 3rd ed.; Plenum: New York, 1990. (24) For experimental accounts of the S m l mechanism, see, e.g.: (a) Symons, M. C. R. J. Chem. Res. (S) 1978,360. (b) Russell, G. A,; Mudryk, B.; Ros, F.; Jawdosiuk, M. Tetrahedron 1982,38, 1059. (c) Russell, G. A,; Mudryk, B.; Jawdowsiuk, M. J . Am. Chem. Soc. 1981,103,4610. (d) Rossi, R. A.; Santiago, A. N.; Palacious, S.M. J. Org. Chem. 1984, 49, 3387. (25) (a) B e r t h , J.; Gallardo, I.; Moreno, M.; Savhnt, J.-M. J . Am. Chem. Soc. 1992,114,9576. (b) Andrieux, C. P.; Le Gorande, A.; Savhnt, J.-M. J. Am. Chem. Soc. 1992, 114, 6892. (c) Kuhn, A.; Illenberger, E. J . Phys. Chem. 1989,93,7060. (d) Andrieux, C. P.; Gelis, L.; Mtdebielle, M.; Pinson, J.; Savhnt, J.-M. J . Am. Chem. Soc. 1990, 112, 3509. (e) SavCant, J.-M. Ado. Phys. Org. Chem. 1990, 26, 1. (26) Fonseca, T.; Kim, H. J.; Hynes, J. T. J. Mol. Liq., in press. (27) See. e.&: Goldstein, H. Classical Mechanics. 2nd ed.: AddisonWesley: Reading, MA, 1980; Chapter 9.

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Solvent Barriers in Unimolecular Ionizations. 2. Electron Transfer

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