A simple dipole isomerization model for non-equilibrium solvation dynamics in reactions in polar solvents

A simple dipole isomerization model for non-equilibrium solvation dynamics in reactions in polar solvents

Chemical Physics 90 (i984) North-Holland. Amsterdam __ 21-35 : _ ; _; : -- . .‘-1- - _ -_ 21 -. -~ I _- .__. - _ _ - _ -_- A SIMP...

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Chemical Physics 90 (i984) North-Holland. Amsterdam

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21-35

:

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; _;

:

--

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.‘-1-

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-_

21

-. -~ I _-

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A SIMPLE DIPOLE NON-EQUILIBRIUM

“_

_-

_-

--

ISOMERIZATION MODEL SOLVATION DYNAMICS

G. VAN DER ZWAN Depurrntenr of Chentrst~,

~

FOR _ . IN REAaIONS

IN POLAR _-

SO_LVENTS



’ and James T. HYNES

LT,u~ersrh of Colornrlo, Bottlrier. Colorado 80309.

UScl

Received 9 Apni 1954

A simple model _for dipole rsomenzation reactions in polar solvents is studied. A full dynamical treatment at the generalized Langevm equation level reveals several non-equdibrium solvatton regimes in which the rate constant differs considerably from the standard equdibtium solvation. transition state theory predictions. In addition. the solvent is found to be heavtly involved in the reaction coordinate, in contrast to the standard equrhbrium salvation vrew. The marked dtfferences from a Kramers Langevin equation level description are also described In each regime, the rate constant and reaction coordinate are found analytically. and typical reaction system-solvent tralectories are dIsplayed_ A related normal-mode descriptton convenient for strong reaction system-nearest-neighbor solvent dipole interactions is constructed_ In this perspective. non-equilibrium solvation effects on rates and reactton coordmates are interpreted in terms of a bias of the reactive motion towards the directton of least frictton.

1. Introduction The standard transition state theory [l] view of dipole isomerization and ionic charge transfer reactions in polar solvents asserts [2,3] that solvent dynamics play no role in the reaction rate and that the solvent is not involved in the reaction coordinate_ This view is perfectly adequate provided that the solvent molecules are equilibrated to the reacting system at all times during the transient barrier passage that constitutes the reaction [2,3]. This allows an interpretation in terms of equilibrium solvation free energies of the reactants and the transition state. But we have recently shown [2,3] that this condition can be often violated, and that the resulting non-equilibrium solvation dynamical effects can have a pronounced influence on reaction rates and coordinates. The rate constant can depend sensitively on e.g. polar solvent reorientation times, and the polar solvent can become

’ Supported in part by Grant CHE 81-13240 National Science Foundation_ a Present address: Department of Chenustry. Oregon, Eugene. Oregon 97403, USA

from

the US

University

of

heavrly involved in the reaction coordinate. In this paper we continue our study of a simple model [3] which illustrates a number of central features of the influence of polar solvent dynamics on solution reactions. The model shown in fig. 1 represents the reaction via rotation of two inner reactive dipoles (RI&) by passage over a barrier. The RDs are in turn coupled to two outer solvent dipoles (SDS) by dipolar interactions_ The interaction of the SDS with the remainder of the solvent is modelled by simple frictional damping forces on the SDS. The model bears some similarity to aspects of e.g. halogen-substituted ethane isomerizations

[31. While this model is extremely simple, it is quite rich in content and generality. The model equations of motion represent the reactive generalization of the “itinerant oscillator” equations successfully used to study aspects of molecular motion in solution [4]. More importantly; we have been able to show [5] that a more molecular level treatment of dipole isomerizations and charge transfer reactions in polar solvents [2] can be approximately reduced to the form of the model’s equations_ of motion. Thus the ideas and results we present h_ave

0301~~104/84/~03.00 Q Elsevier Science Publishers B.V. (North-Holland Physics Publishing Diviston)

G Lutz der Zwvt.

22

(a)

J.T.

H_wtes /

Non -equdtbrtwn

4

i (b)

Fig 1 Reacnon and solvent model m the (a) transItIon state and (b) reactant neighborhoods In (a) the inner reacthe dipoles interact by chemical electronic forces and dipole-dipole forces uhch gx~e a potential barrier varying as -(I/2)4.&3’. where I IS the relatne moment of inertla. ot, is the bamer frequenq and 68 is the relatne angle The outer solvent dipoies each mteract wa dipolar forces with each of the RDs. The nearest-neighbor and next-nearest-neighbor interaction spnng constants are cx and y respectwelq. The relative SD moment of mertla is I, and 619, 1s their relative displacement. In the hannomc approximation. the potential energ) is giLen m (2 I), Hhere the transkon state energy CJ’* mciudes electromc and dipolar mteracuon contnbutlons [3] The SDS are damped b) mteracuon with the remammg solvent molecules not shomn. An analogous development for the reactant regron (b) gl\es a correspondmg harmltoman H, [3]

solcation &namics

in reactmzs

tion 2. we describe the model and recast its equations of motion into a generalized Langevin equation form. In section 3. we find analytic results for the reaction rate constant k in a wide variety of non-equilibrium solvation regimes. In section 4, we carry out the corresponding analysis for the reaction coordinate. with a focus on solvent participation in that coordinate_ In section 5, we motivate and analyze an ahemate normal-mode approach to the reaction problem. Section 6 concludes the paper_

2. Model equation

description and generaked

Laugevin

We begin by describing the dynamics of the reaction-solvent model of fig. 1 in the activated transition state region. We then convert these to a generalized Lange+ description involving a time-dependent friction exerted by the solvent on the RDs. 2.I. Model description and &names

a generality that extends far beyond the simple model. In our earlier study [3], hereafter called I. we examined the four-dipole model in the limit of negligible friction on the SDS. We found the reaction rate constant and reaction coordinate. and related them to a wide variety of non-equdibrium solvation phenomena. These included (a) the effective mass regime, in which solvent inertia modifies the rate despite the rapid equilibration of the SDS to the RDs, (b) the non-adiabatic regime, where the reaction occurs despite a solvent that is nearly “frozen” out of equilibrium during the event. and (c) the polarization caging regime. in which the rate of equilibration of the slowly relaxing solvent dominates the reaction rate. In this paper, we study the Influence of the friction felt by the SDS on the reaction rate and reaction coordinate_ This SD frictional effect is quite important, as it corresponds [5] in a continuum limit to the possibility of Debye orientational relaxation of polar solvent molecules [2.5], and that relaxation can play a key role in the reaction [2]. The outline of this paper is as follows. In sec-

The various constants describing the chemical reaction barrier expenenced by the RDs and the dipolar interactions between the RDs and the SDS are catalogued in the caption of fig. 1. In the absence of friction on the SDS, the hanultonian in the transition state neighborhood is

+CY( 66 - SO,)’ + y( SB + SO,)‘] _

(2-l)

It is instructive to rewrite H + in terms of an equilibrium solvation perspective [3]. For a fixed RD configuration SO, the equilibrium average (SO,),, for the SDS is (se,),,

= (a +_u)_‘(a

- y)SB.

(2.2)

The equilibrium solvation coordinate for the solvent dipoles thus depends on the separation of the reactive dipoles: the SDS are polarized by the RDs. If the SD configuration is referenced to this equihbrium value, then H# takes the simple form

G_ van der Zwan, J T. Hines

+ +[ --ki~;qse2

+-r,o,2(sq -

/

Non,equrlrb~r~so~v~~~on

(ses),,)2] (2.3)

The picture painted by the potential energy in eq. (2.3) for H’ is the following. The RDs appear to move not onthe “bare” barrier with frequency ot,, but rather on the equilibrium solvation barrier with the equilibrium barrier frequency ub,_ (fig. 2a): 7 ubeq=

wbz

-[I(a+y)2]-*4ay.

(2.4)

The equilibrium solvation interactions make the barrier less sharp: o b eq ~-0 b. The potential-energy contribution U# -( 1/2)o&,S02 is just the potential of mean torque for the RDs which would be plotted in a standard equilibrium statistical mechanical description of the reaction. The second potential-energy term in eq. (2.3) for Hf describes the SDS oscillating with the solvent frequency o,=[Is-‘(a!+y)]“2

(2.5)

about their equilibrium configuratron in the field of the RDs at a fixed configuration 68. As we have previously stressed [3], standard discussions [l] of solution reactions implicitly focus on the analogue of the potential contributions to that we have just described_ It is imagined Hf that, at each configuration of the reacting system,

AU

ne

~I~~s~~~~+y)s6+(w-y)s8~,

[email protected]?, = (a - y)S8 -(CY + y)S6, - 1&f%J,.

) 68

/

/

I “b.eq

(01

(b)

Fig. 2. (a) Equihbrium solvation view of the transition state neighborhood Here AU represents free energy referred to the transltion state energy. (b) Non-adiabatic potential energy along the RJ3 coordinate 68, illustrating a non-adiabatic barrier (- - -) and a non-adiabatic well ( -)_ The latter results when otW IS negative.

23

(26a) (2.6b)

in which the friction on the SDS has now been included. Here T is the torque on 68 arising from solvent interaction. It is now convenient to convert to the “mass-weighted” coordmates se = I”2s8

,

se, = Iy’sos,

(2.7)

which diagonalize

the kinetic energy in Hf and of a mass point moving on the potential surface. The equations of motion become us to think

se, = gse

ASB, <*

-

= 1w;st? + T

SS = &,,se

*b.na

_

the solvent is equilibrated: This equilibrium_ solvation view effectively decouples the reacting system and the solvent.. It pictures the reaction coordinate as exclusively associated with the reaction system, and_not with the solvent. &this view is generally incorrect. it -ignore the fact _that the reacting species and the solvent are dynamically coupled. In the-RD-SD .example, a switch to the coordinates 86 and ASO, = SO, - (SO,),, _in which the RDs and SDS are decoupled in potential terms introduces a kmetic coupling in _H ? : there are cross terms in St?; and [email protected] The RD and -SD motions are thereby dynamically coupled and the reaction coordinate involves both motions. Equivalently, there are non-equilibrium solvation effects in the reaction. A correct analysis therefore requires scrutiny of the equations of motion for the RDs and SDS,

allow

AU

dy,namrcs m_reactIons

in terms

+ +ie,, - &ie, - ~58,.

(2.8a) (2.8b)

Here and henceforth we suppress the overbar notation for the mass-weighted coordinates. In addition to the solvent frequency ws already introduced, there are two frequencies in eq. (2.8) to be described. The non-adiabatic frequency ubsna is (fig. 2b) CObpa= [O~-I-‘(a+y)]1’2,

(2.9

o b,na-govems the RD motion when the solvent is “frozen” at the configuration SO, = 0 appropriate

24

G can der Znan, J T. Hyzes / Non-equlbriwn

to equilibrium solvation at an initial barrier top RD configuration 68 = 0. The “non-adiabatic” appellation contrasts to the adiabatic, equilibrium salvation situation in which the RDs rapidly respond to and solvate the moving RDs; for that case. the RD motion is governed by oi, cq. eq_ (2.4). In the non-adiabatic situation, the extreme lack of solvent equilibration makes it more difficult for the RDs to separate - the RD-SD interactions retard the RDs - and the effective barrier :s Iess sharp: at,“= < mhcrl [3]. Indeed, wt no cLtn become negative; then there is a It?eilalong the 68 Axis (fig 2b). The RD-SD coupling freqtlencr c+=

I (II,)

-l/l(a__)]Ir-

(2-10)

gauges the coupling between the RD and SD motions. For example, when the RDs separate. electrical torques develop on the SDS which tend to increase the latter-s angular separation_ As the presence of the RD-SD interaction parameters CY and y suggests. the coupling frequency measures the strength of this effect. (The presence of the difference a - y is explained in i.) -_-_ 3 7 GeneruIi=ed Lmzgeuin equation

A systematic and convenient way to describe non-equilibrium soIvation effects is to establish a generalized Langevin equation (GLE) for the RD coordinate 60. The dynamic influence of the soIvent then enters via a dynamic friction coefficient [2.3J. The first step on that route is to formally solve eq. (2 Sb) for the SD motion in terms of the RD motion. This gives (2.11) in which the solrrenr response fnnctron x ( I ) = (2&$/R

)e-‘s”‘sinh(

R = (js’ _ 4U$‘1

soiuation &namics m reamom

eq. (2.8a) and integrate by parts to give the desired GLE for the RD motion: 68(t)

= 6J2 ,$9(t)

(2.12)

governs the response of the SDS to the RDs. Here we have ignored initial-condition terms in Ms(t = 0) and @(r = 0), which are irrelevant for our purpose. The second step is to insert eq. (2.11) into

- 7).

(2-13)

Here we have ignored irrelevant initial-condition terms in S&r = 0) and Z&t = 0). If the solvent responded adiabatically to the RD motion, it would appear as if the RDs moved on the equilibrium solvation barrier with frequency this is the equilibrium solvation view. In Obcq reality. the solvent need not adjust adiabatically to the RD motion and can be driven out of equiIibrium. Such non-equiiibrium salvation effects lead to the frictionaI term in eq. (2.13), in which the time-dependent frictron coef’jciem is s(t)

= w:-l]=drX(r) = (w,j/~.$) e-f*r/r [cosh( Rt/2) + ([,/R)sinh(

Rr/2)]

_

(2.14)

In mo!ecular telrms. C(r) is the time correlation function of the fluctuating torque ST = T- (T) on the RDs due to the SDS. i.e. the equilibrium average %a!ue (I’> is subtracted out [3]: c(r)=

(2.15)

(k.T)-‘(STST(t)).

This emphasizes the key feattire that c(t) refers to the non-equrhbrium aspects of solvation; the aver.lge equil;brium salvation effect of the solvent resides in the c.$,~$? term in the GLE. Fig. 3 shows l(t) for two important limiting regimes that we will consider throughout this paper. In the underdamped solvent limit, {, -=Kw,, the solvent dipoles’ oscillatory motion is negligibly perturbed by the friction ls on them. This is reflected in an oscillatory l(t), C(f) = (~~/f$)COS

M/2),

-/hrr(+fi(r 0

fast,

1, -2x a,,

(2.16)

since the RDs are coupled to the oscillating SDS. It is important for later reference to note that the RD friction S(t) remains finite even when the friction 5; on the solvent dipoles vanishes_ In the underdamped limit, the solvent inertia is all-important and the time scale of the solvent response is set by o,- t, the inv erse solvent frequency.

G.-van der Zwan. J.T. Hynes /,Non -equdrbrrum soharlon dynizmrcstn reactions

tion. constant 5 vanishes. In contrast,leq.-(2.16) shows -that there is still dynamic friction on-the RDs in the GLE description. As we will see, the LE (2.19) has only a limited applicability to the rate problem. In addition to its time dependence, S(t) is characterized by its magnitude, or initial value.

I S(t)

=W

3. Timcdependent friction coefficient c(r) on the rcactx\e versus time in ps in theunderdamped (dtpolcs in ps-’ 1 Fig

and overdamped (- - -) solvent linuts Parameters for the calculation are. in ps- I. q =12 65. o, = 13 86: <, = Z 5. and 128 Ps -’ for the under- and o\er-damped case& rcspectixely

In the opposite. ooerdanlped solvent limit, 5; ZC=y. the SD motion is diffusive. and S(Z) approaches l(t)

25

S; B 0,.

= ( o,“/6_$)e-f’Z*,

(2.17)

-J

2 bw

-%.na

=

rycu

- y).

(2.21)

If the RDs and SDS were uncoupled, the friction would vanish; there are no non-equilibrium solvation effects. In the general case, l(t = 0) gauges this coupling. In particular, the difference of the square adiabatic equilibrium barrier and nonadiabatic frequencies reflects the coupling; to the extent that these differ, there is non-equilibrium solvation. Finally, note that eq. (2.21) holds in both the undamped and overdamped solvent limits.

3. Reaction rate co&ants

in which the diffusive soIvent relaxation time is (2.18)

7; = s;/~,’ -

In the overdamped solvent limit. the friction is quite similar to that which follows from a continuum Debye model treatment of the solvent [2,5.6]. There the analogue 5 is related to the Debye relaxation time ho. which in turn is associated with the diffusive motion of solvent dipoles. If it were true that the SDS responded quickly on the time scale of the RD motion, then the GLE would reduce to

Sli(t) = m;,_SfI(t)

- /mdr3(r)

se(t)

-0

=&.p(r)-~se(t), in which the friction constant

(2.19) is

This is an ordinary Langevin equation (LE) for the RD motion, and most of the dynamical details of the coupled RD-SD _ motion have disappeared. For example, if the SDS are undamped, the fric-

As described in section 2, the reaction coordinate in the equilibrium solvation picture is just the RD coordinate 60. The corresponding rate constant k, m thts picture is just the transition state theory (TST) result [3] kEs=(w

Req/2~)(~sRbd

Xexp[-(U+

- UR)/k,T]

_

(3-l)

Here Li * - UR is an activation energy, while ws is the soIvent frequency (2.5) in the transition state region, osR is its analogue for the reactant configuration, and mReq is the oscillation frequency of the RDs in the reactant well when the SDS are equihbrated. The precise connection of eq. (3.1) to a standard equilibrium-solvation picture is given in 1. This result is obtained [3] by averaging the one-way flux i for the RD coordinate over an equilibrium distribution at the transition state. As discussed in I, non-equilibrium solvation effects cause a breakdown in the equilibrium solvation, TST assumptions up-on which eq. (3.1) is based. In dynamic terms, there are solvent-induced recross-

G. watt der Ztvan, J. T Hones / Non - equihbrrwn so/cation dynamics in reactions

26

ings of the TST surface 66 = 0. whereas none are allowed in finding k,. Equivalently, the RD coordinate

SB is not the reaction

coordinate.

1.0

r, 8=0.5

due to

In terms of the GLE (2.13). the RD motion is governed not only by the equilibrium solvation term o’,,SB. but also by the frictlonal term involving C(r). The actual rate constant k including frictional. non-equdlbnum solvation effects is plven by the Grote-Hynes equation for the transmission coefhcient K [7]: solvent coupling.

0

=/u=d’

(3.2)

emhr’{( f).

The rate constant is self-consistently determined in terms of the Laplace transform frequency component of the time-dependent friction coefficient at -the reactive frequency X, = wb cq~_ If the fnction is negligible. non-equilibrium solvation is unimportant. K equals unity. and the eqmlibrium solvaLion rate X-, applies. Otherwise. non-equtlibrium solvation reduces the rate to a degree dependent on the dynamical friction. To analyze this. we need to investigate the transmission coefficient K. From eqs. (3.2) and (2.14). K is the solution of I

K+

p(x+Kr2)

K=

t

1 fR_Vf

22

-l

1



in it hich there are three key parameters

(3.3) /3. I- and

cs/wb,eq

Influen~c of the sohent dtpole fncrlon constmt <, on the reactiontransmlssloncoefhcwnt K for wxxk (fi = 0 5) and strong (/3 = 2) couphng. The fimte mterccpts ilt {, = 0 result from the firute RD fnctlon cocfficxnt L&IC in this hmit Here fir’ = 0 605 (see sectlon 5 of I) Fig

&A,)

3 4

response. For example, small x values correspond to very weak damping of the SDS. There is. of course, an analogy between r and s. When the SDS are weakly damped. r reflects the speed of their response; when the SDS are heavily damped, s measures that speed. All these parameters are key in what follows_ Fig. 4 shows the calculated transmisston coefficient K for two values of the coupling strength parameter fl. as a function of the friction S; on the solvent dipoles. The basic trends are best discussed in terms of whether the SDS are underdamped or overdamped. 3.1. Underdamped solvent (5, CC CO,)

s:

q.

When the SDS are underdamped. we have [, CC or

W~T;=_X/T=r(S&&&=X (3.4) The RD-SD coupling strength is measured by fl: weak and strong coupling correspond to p values much less or greater than unity. respectively_ The r parameter gauges the rate of the solvent’s inertial response. For example, for small r, the SDS vibrate rapidly and adjust quickly compared to the timescale of the RD motion. The final parameter x gauges the rate of the dissipative, frictional SD

1.

(3.5)

For r in the neighborhood of unity, this regime lies on the far left side of fig. 4. The friction {, on the SDS now plays no role, and we reduce to the case described in detail in I. For purposes of later discussion, we briefly recapitulate the behavior of K m this regime 131. There are severai important non-equilibrium solvation sub-regimes to consider_ 3.1. I. Effectwe mass regime (r -SCI, p >> I, /3r’ - 1) Here the strongly coupled solvent responds

G. can der Zwan, J. T. Hynes / Non -equihbrnan soIcarionhynamrcs m ;eaclrons

rapidly to the RD motion and [3] K = (1+

@)-r/z:

(3.6)

This reflects the kinetic inertial effect by which the effective mass‘(moment of inertia) of the RDs is increased by the inertial dragging of the SDS along in the barrier passage.

3.12. Non-adrabatic regme (r B> 1. j3 C< I, j3r’ = I) Here the sluggish SDS do not move in the barrier passage, and [3] K= (l-~)‘~=w,../GJ~eq_

(3-7)

The RDs move in the non-adiabatic field of the frozen SDS and experience the non-equilibrium barrier potential [ - (1/2)c&,68’], i.e. SS, = 0 in the hamiltonian H + (see fig. 2b).

fir’

caging

K =

regrnte (r 23 I,

/3 >> I,

large)

Here strongly coupled temporarily cage the RDs t,,,SB”] until the [( f/2) &Jz allow the reaction (see fig. [r2( /3 -

l)]

massive solvent dipoles in a non-adiabatic )veIZ SDS finally adjust to 2b) For this case [3],

-1/Z = W,/IWb ,,I.

(3-S)

and the rate is proportional to the rate os of the SD motion, since that slow motion is rate-hmiting. 3.2. Overdamped solvent (1, X- w,)

When the SDS are overdamped, then x farexceeds r and the transmisston coefficient eq. (3.3) reduces to [2] fc= [K+(l

+KX)-lPx]-l,

(3.9)

which reflects the exponentral time decay of the friction l(t), eq. (2.17). on the RDs in this limit. (Note that the inertial solvent response rate parameter r has disappeared from the problem.) Now the behavior of K is best segregated according to whether the solvent response is rapid, x = OUSTS GE 1, or whether tt is slow, x >) 1, on the scale of the equilibrium reaction barrier frequency obcq-

27

3.2.1. Raprd solvent response: adrabatic and -_ j_ . . _ -When the SD timescale-is short, x is small and KX can be neglected in eq_ (3.9) and we find .

dtffuswe limits

(~/i,,)“‘.

3.1.3. Polarizatron

_

K = [ ( px/2)2 + 11lr- - fix/2 (3.10) Since 5 is the fnction constant, this is just the result of a simple LE Kramers treatment in which the time-dependent fraction S(t) is approxrmated by the delta function &S(t) [Z]. The SDS respond quickly, but in general they exert a “drag” on the RDs. There are two important limits here. When fix = c/o b eq is small, there is negligible friction on the RDs, and we have the adiabatic, equilibrium solvation result K=

1.

(3.11)

When instead /3x is large, there is significant frictton on the RDs. and K ulttmately approaches the dlffksiue or Smoluchowski limit

(3.12)

3.2 2. Sioiv solvent response When there is sluggish solvent response x >> 1, the transmission coefficient depends critically on whether there is weak or strong coupling, i.e. I whether p -C 1 or 0 > 1 (fig. 4)_ (1) Non-adiabatic salvation (/3 -C I)_ For weak coupling, eq. (3.9) has a solution for large x given by the finite value K =

(1 -

p)l” = Wb_nn/Wb,eq.

(3.13)

The reaction proceeds despite the nearly frozen solvent by passage over the non-adiabatic barrier in fig. 2b. This is precisely the same result as eq. (3.7) for the underdamped solvent; the dynamics of the weakly coupled SDS are irrelevant when they are slow. Fig. 4shows the approach to this limit. (2) Polarization caging. @ > I). In the _strongcoupling case, the RDs are temporarily trapped in the non-adiabatic .weIl of fig. 2b. The SDS must

G. LWI der &an.

18

J.T. Hones / Non-equrhbrrwn solcation ~~nanncs

reorient to allow the reaction to proceed. and this occurs by their diffusive motion on the time scale r, a s. If we look for a solution of eq. (3.9) which vdrres as _Y- * for large X_ we find

(3.14) The sIo~_ drffusive solvent reorientation is rate limiting here. The key distinction between eq. (3.14) and the corresponding result (3-S) for the underdamped solvent case is that there the solvent rate parameter is the inertial frequency w,. whereas here tt is the diffusive relaxation rate -r,-‘_ 71 _I_-._

; - Corztrasr wttlz LE comtant-fiicttlou

predlc trorlr

If the finite response time of the solvent were srmply ignored and a LE, constant-friction treatment (2.19) were incorrectly applied to the slow solvent case. vve would recover the Kramers-type result eq. (3.10). That theory would predict that for p -C 1 - weak solvent coupling - the reaction rate would track the slowly relaxing solvent. i.e. K a T>-’ would be predicted according to the diffusive hmit. eq. (3.12). Our GLE theory instead predicts the completely different non-adiabatic result (3.13) in which the rate is mzperulous to the lengthening solvent relaxation time scale. The reaction can occur at a finite rate despite the nearly frozen solvent since there is a barrier rather than d \vell along the RD 68 axis (fig. 2b). Even in the strong-coupling case p > 1, the GLE and the LE results differ for a slov~ly relaxing solvent. The polarization caging result (3.14) combmed with eq. (3.12) shows that

P K=_KKR=_2 B-1

P p-lW; -

(3.15)

The GLE rate therefore exceeds the Kramers prediction by a coupling factor p/(p - 1) For very strong couphng j3 >) 1. this is a negligible difference_ But for only moderately strong coupling. c g. /3= 2. K can be quite a bit larger than the Kramers LE approach would have it. This can have important practical consequences in the fitting of experimental rates to reaction theory pre-

in reacttons

dictions_ Let us assume for simplicity that C a 7sa 9. i.e. that the solvent relaxation time is proportional to the shear viscosity r/_ (This behavior would follow from a Debye-Stokes-Einstein treatment [2]_) Any experimental observation that k a 17-l at high q would. on the basis of a Kramers LE theory. be interpreted as adherence to the diffusion limit (3.12). This behaviour would then suggest a way to extract the equilibrium barrier frequency w b ~ from experimental data by use of (3.12). But the GLE result (3.15) shows that the actual coefficient of c-r aq-1 a/so depends on the coupling parameter p. A Kramers picture would suggest that wb ~~ is being determined. whereas in fact [p/(p - l)]wbrg 2 wbsq is being determined. Equivalently, if abeq were known, an apparent smaller friction constant (/3 - I){/p 1~ould be determined. Fmally. the coupling value p = 1 dividing the behavror of IC between weak and strong coupling emphasizes the strikmg feature that approximate fracttorlai po\ver law dependence of IC on x. and thus on the friction constant 5 can be expected. For if we set p = 1 in eq. (3.9) we find for large s that K = _I--r/s 0: 5-‘/3 1

(3.16)

N hich is d very slow decrease of the rate. particularly vvhen compared to the Kramers Smoluchowski prediction (3.12). 4.

Non-equilibrium

salvation

and

the

reaction

coordinate

In the preceding section, we have several times alluded to the influence of the friction on the reaction coordinate. Here we describe how that coordinate can be found and discuss its meaning_ 3. I. Reactron coordinate and solvent dynamics In general, the reaction coordinate qr depends on both the RD and SD motions. For any reactive initial conditions in the transition state region. both SB and S6, will asymptotically diverge with the reaction frequency E,, of section [3]:

G. van

der

Zwan.

Fig 5. Reaction coordmatr q, and the defined in the 68. 66, coordmate space

Thus the reaction coordinate

reactive

angle

can be written as

where the reacttve angle defined by tan x+!J, = To fmd motion,

se,,/se,

$~r shown

SO,(t) =/rdrx(t

in ftg. 5 is

(r)

=

(2w:/R)

for the RD

- r)Se(r),

R = (1,’

-

e-c.‘Esinh(

RI/~)

~

4wz )I”,

(4.4)

and look for the long-time behavior then gtves the desired relation [3]

response

function.

probed

at

the

reacrive

is [3]

x(t)

(4.6)

the equilibrium time correlation of the solvent velocity and the instantaneous non-equilibrium torque on the reacting dtpoles due to the solvent dipoles. Fig. 6 shows the response function x(r) in the under- and over-damped solvent limtts. In the former case, the solvent response is oscillatory at the SD solvent frequency: x( t ) = ( O,?/os)sin us’.

condltlons

as for fig

case cannot

x(f)

3

The

eq (34). m the un- -) solvent limlts

imttal

nse of ,y in the

be seen on this scde.

z

( ~~/~5)(e-oIr/~s-

=

(u~,~JW~“~~ (t>)[,p).

e-cBr)

(4-Q

In either case, if the solvent responds rapidly on the time scale of the RD motion. eq. (4.5) reduces to

= (IJl)r’a(LY

frequency. The molecular expresston for x(t)

= (k,7-)-‘(S8,ST(t)),

Same

response functton x(r). and okerdamped (-

(4.1). This

tan Gr=lrndf e-“r’x(t)=ji(A,). (4.5) c Thus the t-eacrlon coordmate is determined by the SD

Fig 6 Solwnt derdamprd ( -)

(4.3)

_

0 x

L

owrdamprd

we can use eq. (212)

+,,

(4.2)

SO + tan qL, SO,),

29

4,

-10

q, = (1 + tan’+,)-L’Z(

.

J-11 Hynes ,I’ Non -equtlibrtwn [email protected] dynantrcs r~rreactions

(4.7)

In the latter case, there is a slow and diffusive weak response

- v)/(cy

+ y)

= tan I$? = ( StY~,/Seb,)‘q.

(4.9)

This locates the equilibrium salvation reaction coordinate, for eqs. (4.9) and (4.3) describe precise!y the relation (2.2) (when mass-weighted) which holds when the SDS are equilibrated to the RDs. Non-equilibrium solvation always reduces the reaction angle 4, from its equilibrium value. This can be seen from eqs. (4.5), (4.9), (3.2) and (2.14), which give

This means according to fig. 5 that the solvent generally lags the reacting dipoles when there is non-equilibrium solvation: for a given RD .displacement, the SD displacement is smaller in the

1

30

G. uan t&r Znan. 3-T. Hyws / Non -equdrbnwn soluarion dynamrcsm reactrons

non-equilibrium solvation reaction coordinate than if equihbnum solvation conditions obtained_ We now examine this non-equilibrium effect more explicitly.

adiabatic limit in which the solvent relaxes slowly (large x) and the coupling lies on the weak side (p -C 1). Then K = (1 - j3)lr- from eq. (3.13) and tan x$~= (tan +:>[(l

4.2.

Linzits

of rhe reaction

coordinate

In the underdamped solvent limit, eqs. (4.5) and (4.7) for 2(X,) reduce to L&A: + a,‘)-I_ This gives a solvent-dependent reaction coordinate consistent with the rate discussion of section 3.1; this was described in considerable detail in I. Here we focus on the overdanzped solvent limit. for which the appropriate rate constants are given in section 3.2. Then with eqs. (4.5) (4.8) and (4.9). the reaction coordinate IS described by tan 0, = z( A,) = (1 + hr7;)-i = (1 + K-X)-~ tan #‘;p,

- p)*/Lx]

-I_

(4.12)

The reaction coordinate hugs the 68 axis as the reactive dipoles proceed to products in the field of the nearly frozen solvent dipoles (fig. 7). This contrasts completely with the equilibrium solvatibn picture. The polarization cage limit of slow solvent response and strong coupling (!arge x, j? > 1) is both curious and instructive from the reaction coordinate viewpoint. In this limit (fig. S), tan Gc/, = [( fl-

1)/p!

tan 4:‘.

(4.13)

tan 4’;” (4 11)

so that the non-equilibrium solvatton lag is governed by the solvent relaxation time T> and the reactive frequency A,. The solvent lag is most extreme in the non-

Fig 7 Reaction coordmatc (I, (- - -_) m the non-adtabatic .ohatton hmtt [cf cq (4 l?)] and d calculated traJectory ) m the near non-adtabsttc limit TraJcctory ts tmtiatrd Cx\tth umt kelocrty I’, along SB and axes 68 M, are scaled by Lb Paramsters arc. m ps-‘. <, =125. w, =13.86. oC =12 65, = 10 Potenttat contours are shown in arbrtrq units For wbx, the potenual energy [ - (wt ,,/2)Sf?’ -r-(w,z/2)Sf?,‘- l&Isl?,]_ Note that there is a non-adtabatic barrter along 66. o’, nit> 0.

66,

A (a) I-

Fig 8 (a) An aLerage traJectory m the polarizatron cagmg regtme, illustrating the trappmg and ultimate approach to the reaction coordmate qr Trajectory is inttiatrd with umt veloctty V, along the RD axes 68, and the coordinates are reduced by this r&city. Parameters are. in ps-‘, <,= 128. Iw,,[ =lO. oJ = 13 S6 and wC= 12 65 (b) Potential-energy contours assoctatrd nith (a). in arbrtrary umts, [-(w’,_/2)S8’t(&2)SB,’ - [email protected]%0J, with the Frequenciesgiven in (a). Note that there is a non-adiabatic sell along 68 and that the barrier lies aiong a hne in the 66. St& plane.

G_ van der &an,

J-T_ Hynes /

Non :eqtahbriwG

As expected, the solvent lags the reactive dipoles. But as the coupling strength‘ /3 is increased, $; approaches the equilibrium solvation reaction coordinate angle +>! This certainly seems inappropriate for this extreme non-equilibrium solvation regime. Before proceeding, let us reassure ourselves that we have the correct picture of the dynamics. First, we note that the polarization caging rate eq. (3.14), K a Ts-‘, tells us that solve-nt motion is indeed rate limiting: the slowness of motion along the reaction coordinate 4, is certainly_due to the slowness of the solvent motion. Second, fig. 8 shows an (average) trajectory started at the transition state with initial RD velocity. The RDs are initially trapped in the non-adiabatic well and oscillate, whrle the SDS slowly move and eventually allow the reaction to proceed along qr. This is consistent with our pictorial descriptions of the polarization caging regime. How can it be then that the reaction coordinate approaches the equihbrium solvation coordinate?

soivariori dynamrcs in reactions

One of these, X,, is the positive reactive eigenvalue; the remaining two, X-,, and h,, are nonreactive’eigenvalues with negative real -parts and are related by Xii =A*, = X,. The eigenmodes u = (Nrr U”l. zl,,z) of the system are given by

u=s-x, S=

aX,(X,+A*,)

a(h,+A*,)

am:

bh,(A*,+X,)

b(A*,+X,)

bm,2 ,(4-17)

c(X,+X,)

cCi$

[ cwL

+ A,)

I

whtre a, b and c are constants whose values we do not require here. The eigenmodes have the simple exponential time dependence u,(t) t+(t)

= u,(O)e’r’,

zf,r( t) = ~~~(0) e+,

= ~~~(0) eX:‘.

(4.18)

According to eq. (4.18), if the initial values of 88, 88, and 80, are such that U,(O) = 0, then we have

4.3. Projected reaction coordmate The resolution of the difficuhy lies in the fact that in the overdamped solvent limit, the reaction coordinate q, is asymptotic, in that it is approached after a transient period (see fig. 8b). In fact, we have found that if we start the system out directly along qr, the trajectory will temporarily Ieaue q,, only to return to it at a later time. These curious features have their origin in the fact that, for the overdamped solvent, the system is described by three variables_ These are the RD coordinate 80 and velocity 86, and the SD coordinate S& governed by

0 = $38 - w,2seS- ~,s~,.

(4.14)

We now need to analyze these equations. With the definition x=

(se, s& se,),

(4.15)

the equations of motion (4.14) pose a straightforward eigenvahre problem 2 = A - X. The three eigenvalues are determined from

Fig. 9. Reaction perspecuve in the three-dtmensional space [ 66. 68, SO,) as described in text. The non-reacttve plane has mtersections n and n’ with the planes (SO,. SO) and (SO, Se) respectively_ Any motion not starting in this plane will converge to the reactive line u,. The projechon of u, in the (68, St?,) plane is the reactton coordinate qr. located by the reactive angle #,_ The reactive line and the non-reactive plane are not orthogonal, nor are ~7~and 4.. the intersectton of the non-reactive plane with the (se. se,) plane. This is a consequence of the non-orthogonahty of S in eq. (4.17). which in turn arises from the dtffering frictions on SO and SO,. _

purely non reactive motion. which ultimately decays away. This condition on U,(O) = [S X(O)], thus defines the non-reactive plane shown in fig. 9. We can now discover the conditions for purely reacrlLe motion by setting the non-reactive modes U,,(O) and u,(O) equal to zero_ Then eq. (4.18) shows that this corresponds to excitation of the pure reactive mode u,. Thus. each of the two conditions U”,(O) = 0 and u,~(O) = 0 defines a plane. and the intersection of those planes defines the reactme bne I(, shown in fig. 9. Simultaneous solution of the two equations gives l

(Ih,l’+h,Reh,)S6+(X,+Reh,)S8 [email protected],

= 0,

X$6 = se.

(4.19a) (4.19b)

which define the reactive line II,. We have now seen that in the full space of the problem. the reactive line I(, describes pure reaction without any non-reactive oscillation transients_ But what of the reaction coordinate q, that we have previously associated with the reactive motion in the first portions of this section? We now show that this IS just the prolectlon of the reactive line U, onto the plane of the variables of interest 66 and Se,_ This projectton can be found by inserting eq. (4.19b) mto eq. (4.19a) and then setting Se = 0. This gives us

=o,-‘(X~-~‘,,+~~/w,‘)se-

In the (Sf?. Se,) plane, this motion will appear as in fig. 8b as an initial series of oscillations in the polarization cage and an ultimate adherence to the reaction coordinate qr_ Note that even a trajectory initiated along q, will for similar reasons at fit depart from that axis, only to return to it after a transient period. In this way we see that while the asymptotic reaction coordinate q, is eventually reached that is identical to an equilibrium solvation reaction coordinate for large coupling p, there are severe non-equilibrium solvation polarization caging effects that occur before that stage_ If we insist on thinking only of the coordinate qr, then the polarization caging only shows up in the transmission coefficient (3.15), K = cabeq/{, as a frictional effect along that coordinate_

5. NormaLmode

perspective

As stressed in sections 2 and 3. the motions of the reactmg and solvent dipoles are coupled even in the absence of a friction 5, on the SDS. Indeed, as shown in I. the hamiltonian (2.1) can be diagonahzed to find two normal-mode (NM) coordinates which each involve 68 and St?,. here shown in fig. 10. The normal reactive mode or reaction coordinate S& carnes the system over the barrier LJ * -( 1/2)0,‘68~‘. in which w, is the reactive frequency. The bound non-reactive mode motion I%?~is osctllatory with frequency 0,. The orthogo-

(4.20)

On comparison with eqs. (4.3). (4.9) and (4.10). this tells us that the proJectron of the retrctibe Lvre II, IS JLCG rhe reactIon coordmare q,_ as indicated m fig. 9. Indeed, it is easy to verify that in the polarization caging regime, cq. (4.20) reduces correctly to eq. (4.13). We can now understand the reaction coordinate q, and its direction in the polarization caging regime. A trajectory started frcm 68 = 0 with veloctty along 68 has contrtbutions from all the modes u,. both reactive and non-reactive. The damped oscillatory non-reactive components will disappear after d transient period, and the system wiill eventually proceed along the reactive line N,.

(a)

(b)

Ftg. 10. (a) Normal-mode (NM) coordmates 80, and 50, in the absence of friction {, on the solvent dtpoles and the COITCZsponding reaction coordinate angle I$~_ Aiso shown IS the reaction coordrnate qr of section 3. m the presence of 5, and the associated angle +r (b) Potential-energy surface in NM coordinates, tllustrating the meaning of the NM non-reactive and rractivr frequennes o, and wr respectively.

f-7. can der Zman. I-T_

H’nes

/

Non-equ&ium

nal transformation to the NM coordinates from _ the RD and SD coordinates is [3]

(5-l) where & is the reaction coordinate angle displayed in fig. 10. The reaction coordinate Sf3,is a mixture of RD and SD motions even-in the absence of solvent friction 5;. Since in that case there is no recrossing of the barrier along So,, the exact rate constant is a transition state theory rate constant; its value is [3] k TSf = kl&M%,)-

(5.2)

This is lower than the equilibrium solvation rate constant k, since it includes the dynamic coupling between the RDs and the undamped SDS [3]. In this section, we analyze the rate constant problem in the presence of solvent friction {, from a NM viewpoint. The rate constant k will then differ from eq. (5.2), and will depend on l._. While all our results will be equivalent to those of sections 3 and 4, the NM viewpoint is an important one to pursue. The motivation here is as follows_ Many solution reactions should involve strong coupling to the nearest solvent molecules - e.g. those in the first solvent shell. These latter molecules in turn interact with subsequent solvent shells. Clearly cases can arise where one would wish (a) to treat the reacting system and first solvent shell at a detailed level and (b) to approximately represent the first solvent shell - remaining solvent interactions at a less detailed, e.g. frictional, level. For the present model, the first step corresponds to diagonalizing the undamped RD-SD system, and the second step corresponds to accounting for {,_ We now find the rate constant and reaction coordinate in the NM perspective. 5.1. Equations of motion

The (66, Se,) equations of motion (2.8) can be converted to equations of motion for the NM variables (Se,, 84) with the transformation (5.1). We find ‘(5.3a)

soluatton dynamics in reactions

sg) = -&it?,

-

{*nsfi; - ~,,SB,, _

33

_

(5.3b)

in which the various friction coefficients depend both on the SD friction_ S; and the NM reactive angle &r: S, = S; sin’+,,

5,” = S; cc&~,

Jk = l,, = S, cos Cp,sin 6 -

(5 -4)

For example, the direct NM reactive mode friction constant 5, depends on the amount of SD coordinate character in Se,. The reactive and non-reactive coordinates are evidently frictionally coupled [S-10]. Inspection of fig. 10 shows the origin of this coupling_ In the NM S& motion, both the RDs and the SDS move, but only the latter feel a direct friction, proportional to S,_ The SDS thus decelerate compared to the RDs, and this excites a Se,, component of the motion_ This coupling will be more pronounced, the larger is the SD friction constant <,_ We will see that due to this coupling, the actual reaction coordinate differs from the NM reactive mode Se,. A GLE for the NM reactive (r) mode SS, can be established by formally solving eq. (5.3b) for the NM non-reactive (n) mode Sf?,, and inserting that result into eq. (5.3a). This gives se,(r)=ofse,(r)-_ldd7~~(f-7)s8,(7).

(5.5)

in which the r-mode friction coefficient is

X [cosh( Rt/2) R = (l,‘,

- (<,,/R)sinh(

- 4w,z)?

Rt/2)]

, (5-6)

The instantaneous delta friction contribution is a direct r-mode friction, while the second delayed contnbution is due to r-n mode coupling. It is instructive to note that when the GLE is approximated by an LE, Se,(t)=o:_Se,(l)-(~md~r;,(~))Se,(t),

(5.7)

the entire mode coupling effect disappears, since one has s;=i”dtS;(f)=<,,

(5-g)

34

G. oan der Zwan. 3-T. H’ner / Non -equdrbnurn soharron d_~namrcsin reacrtons

due to the oscillatory behavior of the mode coupling portion of the friction (5.6). As we will see. the reaction rate constant often depends in an important way on the mode coupling. and the LE approximation is a poor one. 5.1 Rate cottstan~ The SD friction S; will cause the actual reaction frequency rl r to differ from the NM reaction frequency c+ In oiher words. the actual rate differs from its TST value k,, eq. (5.2). due to the frictionally induced recrossing of the NM reaction coordinate barrier top 80, = 0. In particular. the Grote-Hynes analysis applied to the GLE (5.5) gives the transmission coefficient gauging this effect as

(5.9) With this and eqs. (5.2) and (3.2). we transmission coefficient K, in the NM is related to the equilibrium solvation transmission coefficient K = k/k,, of

see that the perspective perspective section 3 by (5.10)

K, = (%.&JJK-

If the SD friction {, is negligible, then c, + 0 and we recover X-= li,, or r1, = w,. As shown in 1. this leads to the rate constant results sum-

0

L

I 4

I 8 S&w

Fig. 11. Dynamical trammlsslon coefficientL, verbusthescaled reacwe mode direct fnctton CO~GUII <, for t\\o values of the couplmg parameterfl. weak coupling. p = 0 5: strong coupling. p = 7. Parameters are ldenttcal to those of fig 4 Note that s’, is zero when rS = 0. (Sate the contrast to fig -I )

marized in section 3.1. But for finite I,, the friction on 86, will lead to recrossing of the barrier along 86,. and the transmission coefficient K, = A,/w, will be less than unity. Fig. 11 shows the results for a representative calculation illustrating this solvent effect. We will focus in what follows on the large SD friction limit <, --, 30. Here the solvent relaxes very slowly, and the frictionally induced coupling between the Se, and Se,, is very pronounced_ We already know from the perspective of section 3.2 that for large 1, we can expect two distinct regimes: non-adiabatic solvation and polarization cage trapping. We now see how this appears in NM language. 5.Z. I. Poiarizatron cage trapping

in this strong-coupling regime, the friction {,( t ) should grow with c,, and we expect to find K, = AJ_w, to vary as S;-‘_ If we accordmgly set 11, = {[ ‘A f in eq. (5.9 ) and use (5.6), we fmd for large <, that (5.11) This contrasts with the corresponding result extracted if the simplified LE (5.7) - which ignores the mode coupling - were used instead. That result is Kf” = Wr/Srr,

(5.12)

i.e. d transmission coefficient of the standard diffusive, Smoluchowski form [compare eq. (3.12)]. The GLE result (5.11) shows that the actual rate is grearer than the LE would have it. This is a consequence of the frictlonal coupling between Se, and Sf?a_As will be made more explicit below, the actual reactive motion is biased by the coupling towards the dtrecttott of least friction [8,10]. The reaction involves less motion of the highly damped SDS than is described by the undamped NM reaction coordinate 66, in fig. 10. This brings in the So,., motion in an important way, and is reflected in eq. (5.11) by the diffusive reiaxation time l*,/wz of the St?, coordinate_ Of course, eq. (5.11) must be equivalent to the polarization caging result (3.14), which is refer-

G_ con der Zwon. J-T_-Hynes

/

Non-equilrbnum

enced to the -equilibriums solvation rate constant k,. That this is indeed the case is readily established from eqs. (5.11), (5.10) and (3X4). 5.2.2. Non-adiabatic salvation Section 3.2 tells us that in this limit the rate constant k wiU remain finite as 5; increases. To extract the rate, we look for a finite solution of eq.

(5.9) as 5, --) co; we find

which is independent of c,_ Once again, the rate 1s greater than if mode coupling were neglected; indeed, K,(&.,, = 0) = O! In the absence of coupling, the reaction would have to proceed directly along Se,; along that coordinate, the direct friction constant 5;,( a S,)increases inexorably as {, increases. If the reaction proceeded in this way, the rate would ultimately vanish as {, climbed. But the coupling allows the reaction to seek out a lower friction route. As we know from section 4, this is motion along the 68 axis over the non-adiabatic barrier [ -(I/2)& &X9*]_ Along that direction, there is no direct friction - since that acts only on the SD coordinate St?,- and we get a rate independent of S,. Finally, one can easily verify that eq. (5.13) with eq. (5.10) is equivalent to the nonadiabatic result (3.13).

The frictional mode coupling induced bias of the actual reaction coordinate toward the least friction axis 66 discussed above can be quantified. Here we only briefly sketch the analysis. The actual reaction coordinate qr in the presence of mode coupling can be located in the NM system (Sq, Se,,) by the relation q, = (1 + tan’y,) -1’2(se,

+ tan y, se,),

(5.14)

where the angle yr is determined by the non-reactrve coordinate response friction x,(t): /0

oOdte-“rrxr(

sB,(t)=~Tx,(t-T)se~(T).

m ~eoctmns

relate y, to both & iocatmg. the NM reactive coordinate and 9,. iocating the actual reaction coordinate. In this way, one-can show_that: (a) the actual reactive coordinate is as close as or closer IO the RD axis ihan is the NM reactive coordinate Se,, due to the bias to the lower friction direction. (b) This bias is most extreme in the non-adiabatic limit, for which q, is along 68, as in section 4. (c) The reactive coordinate in the polarization caging regime is located by tan #, = (w,?/u:)(/~ - 1)/p, just as in section 4.

6. Concluding

remarks

In this paper, we have analyzed the reaction rate constant and reaction coordinate for a simple model of dipolar isomerization in polar solvents_ We have described in detail several non-equilibrium solvation regimes - non-adiabatic solvation and polarization caging - in which the rate differs considerably from a standard equilibrium solvation description and the solvent is heavily involved in the reaction coordinate. As we show elsewhere [5], these results can be immediately taken over to describe more molecularly detailed treatments of dipole isomerization and ionic charge transfer reactions.

References

5.3. Reactton coordinate

tan yr =

solcotioiz &nom&

t),

(5.15)

The response x,(t) is found by solving eqs. (5.3) and (5.15). Simple geometry can then be used to

111E.M.

Kosower, Physical organic chemistry (Wdey, New York, 1968). T-H. Lowry and KS. Richardson, Mechanism and theory in organic chemistry, 2nd Ed. (Harper and Row. New York, 1981). 121G. van der Zwan and J.T. Hynes, J. Chem. Phys. 76 (1982) 2993; Chem. Phys. Letters 101 (1983) 367 c31G van der Zwan and J-T. Hynes, J. Chem. Phys. 78 (1983) 4174 141E Mauricio. S. Yelasco, C. Girardet and L. Galatry. J. Chem. Phys. 76 (1982) 1624. 151G. van der Zwan and J-T. Hynes, to be submitted for publication. VI H. Friihiich Theory of die&tics (Clarenddn Press, Oxford, 1958) 171RF. Grote and J.T. Hynes, J. Chem. Phys. 73 (1980) 2715. (81 G. van der Zwan and J-T. Hynes, J. Chem. Phys. 77 (1982) 1295. [9] RF. Grote and J-T. Hynes, J. Chem. Phys. 74 (1981) 4465; 75 (1981) 2191. [lo] S.H. Northrup and JA. McCammon, J. Chem. Phys. 78 (1983) 987.