Nonequilibrium solvation dynamics in solution reactions

Nonequilibrium solvation dynamics in solution reactions G. van der Zwan and James T. Hynes Citation: The Journal of Chemical Physics 78, 4174 (1983); doi: 10.1063/1.445094 View online: http://dx.doi.org/10.1063/1.445094 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/78/6?ver=pdfcov Published by the AIP Publishing

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Nonequilibrium solvation dynamics in solution reactionsa) G. van der Zwan and James T. Hynes Department of Chemistry, University of Colorado, Boulder, Colorado 80309

(Received 10 August 1982; accepted 29 September 1982) We

.c~nst~uct.

a theoretical framework for the description of nonequilibrium solvation and solvent tn. the reaction coordinate for solution reactions. The framework is illustrated by a model of reactl~e dl~le Isomenmt~on. ~e sh~w that a multidimensional reaction coordinate picture is equivalent to a one d~mens\Onal descnptlon. tn whIch a generalized friction characterizes and quantifies nonequilibrium ~olvat\On effects ?~ the ~eact\On rate. ~h~ adiabatic regime where equilibrium solvation and mean potential ~deas are correct IS IdentIfied. Several dlsttnct regImes of non equilibrium solvation are identified and described tn molecular terms. In the effective mass regime, equilibrium solvation ideas give the reaction barrier curvature .7orrect~:, bu.t solvent inertia modifies the barrier passage rate. In the nonadiabatic regime, the sol~ent IS . frozen dun~g the b~mer passage and cannot provide equilibrium solvation. In the polarization cagl~g r.eglme, t~e reacttng specIes adjust to the moving solvent, rather than vice versa, and the solvent is heavIly tnvolved In the reaction coordinate. The rate constant in each of these regimes is related to reactive and solvent dynamICs. partl~lpat~on

I. INTRODUCTION

The degree of solvation relevant to a reaction transition state and more generally to a reaction path in solution is an old and vexed question. Indeed, this problem has been called the most "delicate" in transition state theory. 1 At one extreme, standard transition state theory assumes that the transition state has an equilibrium solvation: The solvent is equilibrated to the high energy activated complex. This provides the "thermodynamic" basis for the treatment of solvent effects on rates presented in many texts. 2 In this view, the reaction coordinate is determined solely by the chemically reacting species and not by the solvent. The other extreme is illustrated by electron transfer reactions in polar solvents. Here the reaction coordinate is usually assumed to be determined solely by the solvent, and a nonequilibrium solvent polarization applies at the transition state. 3 One expects, of course, that there is a large intermediate regime in which the solvent partiCipates to some extent in the reaction coordinate, and there is some degree of nonequilibrium solvation in the transition state and its neighborhood. Indeed, this qualitative view has been frequently suggested for reactions involving ionic and dipolar reacting species in polar solvents; examples include proton transfers, 4(a) SN2 reactions, 4(b) ion pair interconversions, 4(cl and isomerizations. 4(d) Unfortunately, no general theoretical framework to describe and quantify solvent effects in this intermediate regime in terms of the underlying reactive and solvent dynamics has yet emerged. Recently we began the investigation of this regime for simple models of charge transfer and reactive dipole rotation in a continuum polar solvent. 5 An important limitation of that study was that the continuum level treatment of the solvent did not explicitly introduce a coordinate gauging the solvent motion. Thus, an ex-

a)

Supported in part by NSF Grant CHE81-13240.

plicit solvent-dpendent reaction coordinate was not exposed, and nonequilibrium solvation effects had to be inferred in a slightly indirect fashion. In this paper, we consider a model for the chemical isomerization of two dipoles strongly coupled to their polar solvent neighbors, which are in turn coupled to the remaining solvent. The model is a generalization, for reaction, of the "itinerant oscillator" model used successfully to study nonreactive molecular motion in solution. 6 This particular model contains features that are likely to be of importance for, e. g., isomerizations of haloalkanes in polar solvents. 5,7 The more fundamental relevance of the model is that it allows, for the first time, an explicit and detailed analysis of the solvent partiCipation in the reaction coordinate, the associated degree of nonequilibrium solvation, and the resulting influence on the reaction rate. The formalism we introduce goes well beyond the confines of the special model used to illustrate it. We show that the multidimensional (here two dimensional) reaction coordinate picture is equivalent to a generalized Langevin description, in which a frictional term accounts for and measures the nonequilibrium solvation. We also examine the applicability of the popular statistical mechanical concept of the "mean potential" often employed in reaction problems. 8,9 We connect this concept to equilibrium solvation; we find, however, that there is an important regime where the equilibrium picture painted by the mean potential is incorrect, even though the solvent rapidly adjusts to the reacting system's motion. This is just one example of the several nonequilibrium solvation regimes that we identify and characterize. The outline of this paper is as follows. In Sec. n, we describe the reaction model and equilibrium solvation therein. In Sec. Ill, we find the standard transition state theory rate constant and contrast it with the actual rate constant. We construct the generalized Langevin description in Sec. IV and apply it to the interpretation of the rate and reaction coordinate in Sec. V. We conclude with some remarks in Sec. VI.

© 1983 American Institute of Physics J. Chem. Phys. 78(6), Part II, 15 March 1983 0021-9606/83/064174-12$2.1 0 4174 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 130.37.164.140 On: Thu, 06 Mar 2014 11:40:12

G. van der Zwan and J. T. Hynes: Solvation dynamics in solution

4175

Hf. = Ut. + i [106 2 + I s 09 ~ -Iw~ 08 2

\~

+ a(08 - 08.)2 + ,,(08 + 08.)2] ,

I

(b)

(0 )

FIG. 1. Reaction and solvent model in the (a) transition state and (b) reactant neighborhoods. The inner reactive dipoles /.Ll and /.L2 are separated by a distance l, while the outer solvent dipoles /.La (leftmost) and IJr, (rightmost) are separated from ILt and j./oz, respectively, by ls' The remaining solvent molecules are not shown. Interaction between /.Ll and j./oz is divided into chemical (e. g., electronic) and dipolar terms.

(2.4)

where we have momentarily ignored the friction on IJ.a and IJ.&. Here the relative solvent coordinate 08. = 8a - 8& measures the relative displacement of the SD and Is =Ia /2 is their reduced moment of inertia. The coefficients a and" are, respectively, the nearest and next-nearest neighbor spring constants for the RD-SD interaction: 2a = 1.-3p.p..;

2y =

U. + l)-3p.p..

(2.5)

•

The transition state energy Ut. includes both the chemical term and all electrostatic interactions at the tranSition state

U:.

Uf.= U~+ 1-3p.2_4(a+,,)

(2.6)

j

the solvent stabilizes the tranSition state. II. REACTION MODEL AND EQUILIBRIUM SOLVATION

The equations of motion for the RD and SD coordinates corresponding to Hf. and including outer solvent dipole damping are

A. Model description In our baSic reaction model (Fig. 1), the inner reactive dipoles (RD) IJ.1 and IJ.2 are governed by a "chemical" potential barrier U~b -1/2 ~2082 in the relative angle 08 = 8 1 - 8 2 , The reduced moment of inertia is 1= It/2. These dipoles repel each other via the dipoledipole potential Ut~P(08) = 1-3IJ.1· IJ.2"1-3p.2(1_~082) ,

(2.1)

here expanded for small displacements. This interaction modifies both the frequency wt and the interaction energy to give the RD potential energy Uao (08) = U: .. + (p.2/13) -

0/2) W~082;

w~ = w~ 2 + 2p.2/13 , (2.2)

I08=Iw~08-(a+,,)08+(a-,,)08.

j

Is 08 s = (a -,),)08 - (a + ,,)08. -1.,.08••

(2.7a) (2.7b)

Here ,. is the friction constant incorporating the dissipative effects of the remaining solvent on the two SD. These equations have the structure of the "itinerant oscillator" equations,6 generalized to the reactive case. In this paper, we focus on the limit where ,. is small and plays no key role; Eq. [2. 7(b)] then becomes I.08 s = (a -,),)08 - (a + ,,)08. ,

(2.7c)

with which we work. Finite ,. is treated in a subsequent paper. 10

valid in the neighborhood of the chemical barrier top 08 = O. We call w& the barrier frequency. The reaction then involves the relative motion of the RD over the barrier defined by Eq. (2.2).

Equation [2. 7(a)] shows that, in addition to the barrier torque IW~08 tending to separate the RD, there is a torque T on 08 arising from the interaction with the solvent:

We now turn to our solvent model. One expects that the most important interactions of the reaction system are with its nearby solvent neighbors, represented in Fig. 1 by a pair of outer solvent dipoles (SD) IJ.a and p.&. These, in turn, interact with the remainder of the solvent. We will crudely model this latter influence below by a frictional damping on ~a ...old p.&. The SD interact with the RD via the dipolar potential

The reSisting torque - «(I + ,,)08 arises from the attraction to the SD of the RD as the latter separate off the barrier top. This torque is present even if the SD are held in their equilibrium (for 08 = 0) position 08. = O. The second torque, (a -,,)08., dynamically couples the RD coordinate 08 and the SD coordinate 08.. Focus, e. g. , on solvent dipole IJ. a as the RD IJ.I and IJ.2 separate: IJ. a experiences a torque driving it along with IJ.I> but at the same time feels a lesser torque driving it in the direction of IJ.2. Similarly, solvent dipole p.& is subjected to torques from the RD tending to drive it in the direction of P.2. The competition just described is measured by (0: -')'), the difference in spring constants for, e. g., the a-1 and a-2 interactions.

UsdlP = 1.-3( P.a • IJ.1 + P. 2' IJ.&) + (i. + l)-3(P.4 • IJ.2 + IJ.1 • IJ.&). (2.3)

We ignore the a-b interaction for simplicity. We assume that U~IP is sufficiently strong (» kB T) that the SD execute small oscillations about their equilibrium positions anti parallel to the nearest reactive dipole, e. g. , 8~

= 81 +

7T,

8:q = 8 2 + 7T

•

At these relative equilibrium pOSitions, the total attractive RD-SD dipole interaction energy is - 2p.p.s [1.- 3 + (ls + l)-3].

T= -(a+')')08+(a-,,)08••

An analogous analysis can be carried out when the RD are in the neighborhood of the reactant (R) configuration 8 1 - 82 = 7T (Fig. 1). This gives the Hamiltonian H1/.

With the assumptions above, the system Hamiltonian in the vicinity of the barrier top, or transition state, is

(2.8)

= U1/. + i

[10°1 +I.OO~ + Iwi08i + a(081/. -158'1/.)2

-')'(081/. + 08 8 1/.)2] ; (2.9)

J. Chern. Phys., Vol. 78, Part II, No.6, 15 March 1983


4176


Here 08R = (6 1 - 82 -1T) measures the RD separation from the equilibrium reactant configuration, and 06.R = (6. - 66 -1T) measures the SD separation in this configuration. The sign of')' in HR is reversed compared to H~: the individual dipole pairs (a, 2) and (1, b) repel in the reactant configuration, whereas they attract in the transition state configuration. The constant energy term inHR'

~u

(2. 10)

includes the chemical interaction of the antiparallel dipoles 1 and 2 and all electrostatic terms in the reactant configuration. The reactant frequency wR for the relative motion of 1 and 2 is modified by the dipolar attraction from the value w~ due to the chemical interaction between them. B. Equilibrium solvation

The traditional notion of "solvation" involves the concept of equilibrium in an inextricable fashion. Equilibrium solvation in the transition state for our model is described by demanding at a given value of the RD coordinate 06, that the SD are equilibrated to this 06 value. The relevant distribution of 06, is then the spacial equilibrium form, conditional on 08, Peq(08.108)=

{f

d06.ex P[-,8U.(06.108)]}-1 (2. 11)

xexp[-,8U.(08.106)] ,

where ,8 = (kB T)-I and where the potential energy of 06. for given oe is [cf. Eq. (2.4)] U.(08. 106) = H(a + ')')08! - 2(a -')')06.06] .

(2.12)

When the RD are separated by 06, the SD are "polarized," with an equilibrium separation given by (08'>09 =

J

d08. Peq (06.1 06}06. = (a + ,),)-I(a -,),)06 .

(2. 13)

We note, for later reference, that this does not involve the moments of inertia; only potential energy considerations are involved. In fact, we can rewrite the total potential energy in the barrier region Hamiltonian H~ in terms of this equilibrium polarization as (Fig. 2)

U~(08 ' 06} = U~ + .!.{1 rw 2 -~] 06 2 s 2 [b l(a + ')'} w~

FIG. 2. The equilibrium solvation view of the transition state and its neighborhood. Here t.08. =08.- (68.)fi9' Also, here and in the following figures t. u == u - u~. The corresponding view for the reactant configuration has a well of frequency W R.eq along {j()R and a well of frequency W.R along t.08 sR =08 sR - (08 sR )fi9R'

This mean frequency is reduced from the ''bare'' barrier frequency Wb by the equilibrium solvation of the RD by the SD; for when the SD are equilibrated, the total average torque on 06 is [cf. Eq. (2.8)] (T tot >09 =1w~06 - (a + ')'}06 + (a-')') (08'>09 =1{w~-[I(a+')'}]-14a')'}08=lw~.eq06.

(2.16)

The equilibrated SD provide an attractive potential well [4ay/2(a +')'}]06 2 for the RD, which "softens" w~ to w~.eq. This well is proportional to both the inner and outer interaction spring constants a and ')'; if either were zero, the RD #J.I and #J.2 would be independently solvated, and no modification of the barrier frequency would result. In rather standard statistical mechanical parlance, 11 the potential of mean torque W~(oe} for our problem gives the same picture as Eq. (2. 14}. W"(09} is the potential energy for the RD coordinate 06 when the SD are equilibrated to the former's fixed configuration. More precisely, one has

exp[-,8W~(09)]a:

f

d09.exp[-i3U~(09, 09.}]. (2.17)

A short calculation again produces the mean barrier frequency:

= (a + ,),}/1•. (2.14)

This describes, in potential energy terms, an oscillation of the SD with frequency w. about their equilibrium position in the field of the RD fixed at separation 08. If 06. were exactly its average (06'>09' i. e., if the solvent were equilibrated, then Eq. (2. 14) shows that the potential for the inner reactive coordinate 08 would be governed by the equilibrium solvation barrier frequency (Fig. 2) (2. 15)

W~(08) = const - (I/2)w~.eq 06 2

;

- aW~(08)/a09 = (Ttot)oe = 1w~.eq 09 .

(2.18)

Thus, Fig. 2 is exactly the picture provided by standard mean potential ideas. Similar considerations apply in the neighborhood of the reactant configuration. The potential of mean torque WR(09 R ) and the equilibrium solvation or mean RD frequency wR • eq are found to be WR(06 R ) = const + (1/2)wi.eq 06i ; w~ •• q = wi - [1(a -y) ]-1 4ay .

(2.19)

J. Chern. Phys., Vol. 78. Part II, No.6, 15 March 1983


G. van der Zwan and J. T. Hynes: Solvation dynamics in solution The equilibrium solvation of the RO by the SO softens the bare reactant well 1/2 wi158~, due to the repulsion between the individual dipole pairs (a, 2) and (1, b). The total potential energy is

4177

(0)

1-

4 (0+)'1

U1/(081P 158.1/) = U1/ +Hlwi,eq158~ + l.w~1/ [08'1/ -(158 s1/>081/]2} ;

u~

(08'1/>681/ = (a -y)-I(a + y)1581/ , (2.20) in which w'1/ is the oscillation frequency of the SO about the equilibrium polarized average value.

______- L_____ _

w;1/ = (a -y)/l.;

-4(0-1'11

(b)

The description of the reaction model and equilibrium solvation therein is now complete. We next turn to the evaluation of the rate constant. III. REACTION RATE CONSTANT

A. Equilibrium solvation view Figure 2 suggests that the relative RO angle 58 is the reaction coordinate. Although we will see presently that this view is often incorrect, it is quite instructive to pursue this standard line of thought. If the true reaction coordinate in the barrier region were indeed 158, then 08 would be an unstable reactive normal mode: Any angular motion in the forward direction over the barrier would lead ineluctably to reaction. We can recognize this as just the condition for transition state theory (TST) to apply to the reactive coordinate 08. 12 The rate constant for this equilibrium solvation (ES) perspective is then the equilibrium average of the oneway flux over the barrier at 58 = 012 ,13: (3. 1)

Here 15 and h are the delta and step functions, respectively. The average is over the equilibrium distribution [ f dr exp(-~H1/) ]-1 exp( - tl~) of the RO and SO in the transition state region, normalized by the reactant equilibrium distribution. A short calculation gives k ES = (w1/,.J27f)(wsl/ 1 w.) exp[ - /3(U# - U1/)] .

(3.2)

This standard TST form is just what we would expect on the basis of the equilibrium solvation Fig. 2. The prefactor (w1/, eq /27f) of the obvious activation energy factor is proportional to the mean oscillation frequency of the RO in the reactant configuration. In qualitative terms, this is an "attempt" frequency for the reaction. The ratio wsRlw. is an entropic contribution arising from the differing frequencies of SD oscillation orthogonal to 08 in the reactant and transition state configurations. Equation (3.2) for the rate constant easily lends itself to the sort of discussion of equilibrium solvation common in the literature. 2,7 To see this, let us introduce the corresponding rate constant without solvent effects as

(3.3)

i. e., the limit of kE8 as the solvent spring constants vanish: a, y - O. The ratio of kES to ko then defines a transfer free energy 15(~"), 2(c>,14

FIG. 3. Definition diagrams for the thermodynamic interpretation of the rate in the equilibrium solvation view. (a) Energies of solvent stabilization. (b) Solvent influence on reactant vibration. (c) Orthogonal solvent vibrations.

kEsiko = (W1/, eq 1 W1/)( w.1/I w.)

xexp{-/3[(U" - U~) - (U1/ - U~)]}

=exp[ -/315(AG")] = exp{-/3[15(AU") - T15(~")]} , (3.4)

whose ingredients are shown in Fig. 3. The more polar transition state is stabilized more than the less polar reactants by the solvent interaction energy. The more subtle entropy effects are twofold. First, w1/,.q 1 w1/ < 1 favors the reactants: The solvent softens the reactant dipole oscillation and increases the available phase space. This effect seems not to have been noticed in the literature. The second entropy effect is one commonly invoked in qualitative terms in solvent effect discussions. 2 The orthogonal solvent vibration frequency ratio wS1/1 w.

= [(a -

y)/(a + y) ]112

is less than unity and favors the reactants: The solvent motion in the more polar transition states is more restricted than in the less polar reactant configuration. Equation (3.2) has resulted from an equilibrium solvation, one dimensional view in which 58 is the presumed reaction coordinate. It does not, however, give the true rate constant for the model (given the model assumptions of Sec. 11). In reality, the RD coordinate 58 is dynamically coupled to the solvent SO coordinate 08.; this crucial feature is totally missed in the equilibrium perspective. Indeed, Eq. (2.7) tells us that as the RO separate (58 >0) in a barrier crOSSing attempt, opposing torques develop. These solvent torques can reverse the RD relative motion and, thus, induce a barrier recrossing. This vitiates the central TST assumption upon which kE8 is based. Alternately stated, 58 is not the reaction coordinate, due to dy-

J. Chem. Phys., Vol. 78, Part II, No.6, 15 March 1983 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 130.37.164.140 On: Thu, 06 Mar 2014 11:40:12

4178

G. van der Zwan and J. T. Hynes: Solvation dynamics in solution AU

VIs 88 s

(3.8)

88 r

which works out to be 88 n

(3.9)

The key question now is: how is k related to kES [Eq. (3.2)] which assumes equilibrium solvation? This relationship can be found by a Redlich-Teller l5 type analysis described elsewhere. 16 The result is

88r

(0)

(3.10)

(b)

FIG. 4. (a) Reactive and nonreactive modes in the massweighted coordinate system. (b) Potential diagram illustrating the reactive and nonreactive mode frequencies. Here kinetic and potential energies are both diagonal, as opposed to Fig. 2; there the velocities are coupled, so that Fig. 2 is misleading concerning the dynamics.

namical solvent coupling or "nonequilibrium solvation." We now find the true rate constant that includes this solvent effect.

B. True reaction coordinate and rate constant Motion along the actual reaction coordinate 09 r in the tranSition state region (Fig. 4) is a completely unstable oscillation, i. e., a translation driven by a parabolic barrier -I/2w~09~ defining the reactive frequency wr • Motion orthogonal to this direction defines the nonreactive coordinate 09" in which motion is oscillatory with the nonreactive frequency wn • Both De r and 09. can be found by a straightforward normal mode analysis of the equations of motion (2. 7a) and (2. 7c): I/2 159]' 09 r,n =Nr,n [II/209+(wZ±w2 s r,n, )-1(11S )-1/2(a_')I)I8 S, 2 2 2 2 2W2r." = ±[Wb2 _ CW21 51 + [(Wb _ CWs )2 + 4wb Ws - (II.)-116ay ]1/2 •

The ratio of k to kES is just the ratio of the reactive mode frequency wr and the mean barrier frequency wb • eq • If it were true, as the equilibrium solvation piCture suggests, that the reactive frequency is equal to the mean frequency, then k would just equal kES ' As Table I shows, however, the true rate constant k can be less than kES ' The dynamic solvent torques on the inner reactive dipole coordinate that induce barrier recrOSSing cause this reduction; the dynamical transmission coefficient k/k ES < 1. In equivalent language, there are nonequilibrium solvation effects, which invalidate the picture painted by the standard thermodynamic rate interpretation. We now begin their complete dynamical description. IV. FRICTION AND NONEQUILIBRIUM SOLVATION

The rate reduced by the solvent suggests that some sort of "friction" acts on the RD coordinate 09. To show that this is indeed the case and to relate the friction to nonequilibrium solvation and the rate are the burdens of this section.

A. Generalized Langevin equation (3.5)

Here C = I -1(J + I.) and Nr and Nn are normalization constants. The angle CPr defining the normal mode orientations in Fig. 4 is given by tan CPr = (J./I)1/2 809. /809 = {(JI.)-1/2(w~ + w;)}-I(a -')I)

This can be verified directly by insertion of the relevant frequencies for the present model.

To begin the derivation of a frictional equation for 09, we first formally solve Eq. (2.7c) for 09.(t) in terms of 09 at earlier times, and then insert the result

into Eq. (2.7a). This yields the equation of motion 59(t) =

[W~- (Cl;r)

]09(t)+(Cl;r

)fof dTX(T)09(t-T) ,

(3.6)

The reaction coordinate 09 r clearly involves the motions of both the RD and the SD. As the former separate, the latter are driven by the ensuing torques, and the solvent participates in the reaction coordinate. This cannot happen in the equilibrium solvation picture if we think along the lines of Fig. 2. Equations (2.13) and (2.7c) show that then the torque on the solvent, T. = -(a+')I)[09.-(09.)G8]' vanishes identically.

(4.1)

where we have ignored irrelevant initial value terms in 09.(0) and 08.(0). The last term here is a dynamical "reaction field" contributionI6 • 17 R(t) arising from the dynamical coupling to the solvent:

fo

R(t)=r 1(a-')I)

X(t)

f

dTX(T)09(t-T);

= (J.w.t 1( Cl -')I) sin w.t

(4.2)

•

A similar development in the reactant region gives the normal mode frequencies of oscillation there: TABLE I. Rate constant ratio 'Yla = 1/8. a

2 _ 2 2 [( 2 2 )2 2 WR 1.R2 - WR + CW'R ± WR + CW'R

klkEs

vs

r=",g,eq/w; for 1.11

= 10 and

(3.7) We can now find the true rate constant k. Since 09 r is the true reaction coordinate, the TST assumption of Sec. InA applied to it is exact, 12.13 and so we have

r

0.1

0.5

1

5

10

20

0.38

0.39

0.4

0.54

0.70

0.85

"Calculated from Eq. (3.10).

J. Chem. Phys., Vol. 78, Part II, No.6, 15 March 1983



Here X is a solvent response function to which we will later return. According to our previous discussion, equilibrium solvation effects are described by the mean barrier frequency Wb,eq [Eq. (2.15)]. To introduce this into Eq. (4. 1), we first note that

1W!,eq=lw~-(a+y)+(a-y)

1"o

dtX(t) ,

(4.3)

so that we can write 6iHt)

[f.t

dTX(T)69(t-T) _

(4.10)

~,.

dTX(T)69(t)]

(4.4) An integration by parts, with 69(0) = 0, brings this to the desired form 68(t) = w~

,

eq

69(t) -

f

t 0

•

dTl;(T) 69(t - T) ,

(4.5)

in which the time dependent friction coefficient ,(t) is ,(t) =rl(a -y)

~,.

dTX(T)

= (II.w~)-l(O! _y)2 cos w.t

.

(4.6) The friction depends on the dynamic RD-SD coupling coefficient (a -y) identified in Sec. IIA. Equation (4. 5) is a generalized Langevin equation (GLE) for the RD coordinate motion. 5,13 The solvent's dynamical influence resides in the friction coefficient. To be more precise, we show in the Appendix that

= 03/1) (6T6T(t»

,

(4.9)

for positive initial velocities 68, and Eq. (4.8) reduces to kES' Eq. (3.1). As discussed in Sec. III, it then appears as if equilibrium solvation holds. Our GLE, however, shows that in general the solvent friction leads to a breakdown of that assumption, and thus to a deviation of k from k EB • The methods of Grote and Hynes allow this deviation to be related to the friction 13 ; evaluation of Eq. (4.8) for the present model gives

= W~,eq69(t) +rl(O! -y) x

,(t)

j(t) = 686[69(tI69 = 0)] = 6B6(68t) = 6(t) ,

4179

(4.7)

i. e., the friction is the time correlation function of the fluctuating torque 6T:::. T - (T) on the RD coordinate 69 due to the SD. Note that the average or mean torque (T) :::. (T)GB' which arises when the solvent dipoles are equilibrated to 1ilJ, is subtracted in the fluctuating torque; that average solvation effect resides in the mean frequency term. The RD coordinate GLE thus describes equilibrium solvation effects through w~,.q o59(t) , and the dynamical effects of nonequilibrium solvation by the generalized friction. We now express the rate constant in these terms.

B. Rate constant and friction

Hynes and co-workers5 ,13 have shown that the true reaction rate constant k, which we found for our special model by normal mode analysis at Eq. (3.8), is related to the underlying molecular dynamics by the time correlation formula (4.8) This involves the initial flux j = 696[059] at the barrier top 059 = 0 and its subsequent value j(t) as determined by the dynamics. If 69 were the true reaction coordinate, then as described in Sec. III, passage of 69 over the barrier at 69 :::. 0 in the forward direction would always lead directly to reaction. In that case, the flux j(t) in Eq. (4.8) would be the "free flight" flux

The reactive frequency w, is thus determined by the frequency component, at frequency w" of the time dependent friction ,(w,) = [

o

dtexp(- w,t) ,(t)

= (II.W~)-l(O! _y)2(w; + w!)-l w, •

(4. 11)

The reaction probes the solvent response at the reactive frequency. It is easily verified that Eqs. (4.10) and (4.11) reproduce the normal mode analysis results Eq. (3.5) and (3.10). For a reaction, the power of these equations is that they show (a) that a multidimensional reaction system can be understood in terms of a friction acting in one dimension and (b) that the friction is a measure of nonequilibrium solvation. As to point (b), note that if , is negligible, Eq. (4.10) gives the equilibrium solvation result w, = Wb,eq' We will pursue the full implications of Eq. (4.10) in Sec. V. First we pause to emphasize point (a).

C. Reaction coordinate and solvent dynamics

The solvent participation in the reaction coordinate is, by Fig. 4, gauged by the angle cP, [Eq. (3.5)]. The reaction coordinate is "two dimensional". We now connect this to the solvent response to and the friction on the RD, which are one dimensional concepts. For any reactive initial conditions, the motion of the mass weighted coordinates I 1/2 69 and 1F2 69 • must approachthereactioncoordinateforlongtimes. If we denote such asymptotic directions by [ 112 059,. and 1!/2 60._, respectively, then Fig. 4 shows that (4.12) We should therefore examine the response of the SD coordinate [ s1l2 1i9 s (t) to motion in the RD coordinate I 11 2059(t). The combination of Eqs. (2.7c) and (4.2) gives this relation as Is1/260s(t)

= (J./I)1/2

fo

t

dTX(T)I1/269(t-T) ,

(4.13)

in which xU) is the solvent response function defined previously at Eq. (4.2). Now for asymptotic times, both 69. and 69 diverge with a rate given by the reactive frequency o59(t) - 059,. exp(wrt);

60.(t) -059 .. exp(w.f).

(4.14)

When we insert these solutions into Eq. (4.13) and concurrently let t - 00, we find

J. Chern. Phys., Vol. 78, Part II, No.6, 15 March 1983


4180


tany= (1./1)1/2

f