Condition for fractional-power viscosity dependence of the average rate constant of solution reactions influenced by slow solvent fluctuations

Chemical Physics ELSEVIER

Chemical Physics 212 (1996) 9- 27

Condition for fractional-power viscosity dependence of the average rate constant of solution reactions influenced by slow solvent fluctuations Hitoshi Sumi Institute of Materials Science, Unioersi~ of Tsukuba, Tsukuba. lbaraki 305, Japan Received 12 March 1996

Abstract It has rigorously been shown that the average rate constant, derived from the mean lifetime, of solution reactions including first- and second-order ones can be expressed as l / ( k x s T- l + kf- l) under the condition of initial thermalization of reactants [H. Sumi; J. Phys. Chem. 95 (1991) 3334 and 100 (1996) 4831]. In this formula, kxs T represents the rate constant expected from the transition state theory (TST). Since TST assumes that thermalization of reactants is always maintained in the course of reaction due to fast solvent fluctuations, kTs.r does not depend on how fast the thermalization is. On the other hand, kf (> 0) depends on the time ~" for the thermalization, decreasing in many cases as a fractional (less-than-unity) power of ~-- J as ~" increases. Usually, ~- is proportional to the solvent viscosity r/. TST is invalidated in the large ~- region of kf << kTs x where 1/(kTs x- 1 .~_kf- l) approaches kf. In this region, reaction becomes controlled by slow speeds of solvent fluctuations, and the fractional-power dependence becomes observed as that of the average rate constant. In fact, kf represents the rate constant with which solute-solvent rearrangements most favorable for reaction are realized as a result of solvent fluctuations. The fractional-power dependence of kf on "r- 1 occurs when the rearrangements, described by a coordinate X, are not unique but distributed. To be more exact, when an intrinsic reactivity at each X is written as ki(X), it occurs when ( [ d k i ( X ) - l / d X ] 2 ) e / ( k i ( X ) - 1 ) 2 diverges for first-order reactions, where ( . . . ) e represents the thermal average in X in the reactant state, with a similar condition for second-order reactions.

1. Introduction The traditional theory for chemical-reaction rates is the transition state theory (TST). It has been applied with great success to chemical reactions in various fields extending from chemistry to biology [l]. Recently, however, much attention has been aroused on solution reactions whose rates show a solvent-viscosity dependence unexpected from TST [2-4]. It is tacitly assumed in TST that fluctuations in the reactant state are so fast that thermal equilibration has always been attained among substates in the reactant state when reaction occurs. In this situation,

reactant populations in the transition state are in thermal equilibrium with those in the reactant state, and the rate constant is determined only by an efficiency of transmission from the transition state to the product state. Then, the rate constant becomes independent of the speed with which thermal equilibration is attained in the reactant state. The inverse of the speed is called the thermalization time in the reactant state. In solution reactions, the speed decreases as the speed of solvent fluctuations decreases, that is, as the viscosity of solvents increases with the thermalization time usually becoming proportional to the solvent viscosity. In many solution

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H. Sumi / Chemical Physics 212 (1996) 9-27

reactions it has been observed that the rate constant decreases as the solvent viscosity is increased. This fact has been interpreted as showing that TST is invalidated in these reactions since the rate constant depends on the thermalization time in the reactant state through the solvent viscosity [2-4]. These solution reactions can be found in first-order reactions including biological enzymatic reactions [5], electron or proton transfer reactions [6], atom-group transfer reactions [7] and isomerization reactions [8] and also in second-order reactions [9,10]. They cover a variety of solution reactions, while solution reactions have been regarded as most typical in chemical reactions. This shows that a general theory for chemical-reaction rates has not yet been established. This is a reason why much attention has been aroused on this subject. Slow fluctuations of molecular arrangements in the solvated structure of a solute molecule seem to play a decisive role in these reactions mentioned above in the case of first-order reactions. In this case, the reactant state is composed of substates with various arrangements of solvent molecules different little by little in the solvated structure. The arrangements are transformed among themselves during solvent fluctuations. Thermal equilibration among them in the reactant state has been established as a result of solvent fluctuations when reaction occurs, so long as reaction is sufficiently slow. Since solvent fluctuations give rise to fluctuations of the solvated structure of a solute molecule, the speed of the solvatedstructure fluctuations decreases as the solvent viscosity increases. The reason why TST is invalidated in many solution reactions mentioned earlier is, therefore, that the solvated-structure fluctuations are not so fast in solvents with large viscosities that thermal equilibration in the reactant state cannot be maintained during reaction. In this situation, the rate constant is controlled by a slow speed of the solvated-structure fluctuations, decreasing as their speed decrease, that is, as the solvent viscosity increases. This is the behavior that has been observed, as mentioned earlier. Beside the solvated-structure fluctuations around a solute molecule, another kind of fluctuations in the solute-solvent system is also important. They are fluctuations of atomic arrangements, represented by intramolecular vibrations, in a solute molecule. It has

been observed that when a sudden change in electron cloud in a solute molecule in solvents is induced by photo-excitation, it gives rise to reorganization due to these two kinds of fluctuations, as derived from temporal variation in energy and width of optical spectra [11,12]. A solution reaction is always associated with an electronic change in a solute molecule. The reaction is, therefore, accompanied by reorganization along coordinates of these two kinds of fluctuations in the solute-solvent system. This means that both the solvated-structure fluctuations and the intrasolute vibrational fluctuations are effective in activating the solute-solvent system from the reactant state to the transition state for reaction. In solution reactions, it is important also that the former is much slower than the latter. We thus see that solution reactions are induced by cooperation of these fast and slow fluctuations in the reactant state. Although we describe solution reactions mainly in terms of first-order reactions hereafter, the view mentioned above can be applied also to second-order reactions in solution as explained later in Section 2. On the basis of this view, it was shown by the present author [13] ~ that the rate constant of solution reactions can be expressed as k = l / ( k T r l T + k f ~),

with k f > 0 ,

(1.1)

where kTsT represents the rate constant expected from TST while kf represents a term causing deviation of the rate constant k from krs r. In (1.1), kTsr does not depend on the thermalization time in the reactant state, as noted earlier, while kf does. To be more exact, kf depends on the thermalization time ~of the solvated-structure fluctuations, as kf ~ 7"- a ~ ,q-a,

with 0 < a ~< 1,

(1.2)

where it was taken into account that z is usually proportional to the solvent viscosity r/. When kTsv << k t in low-viscosity solvents, (1.1) reduces to k = kvs v, recovering TST. When kf << kTsv in high-viscosity solvents, on the other hand, (1.1) reduces to k = kf, invalidating TST. In this non-TST regime, it has in fact been shown theoretically [13] that thermal equilibration in the reactant state has not been maintained in the course of reaction because of slow solvent fluctuations, and reac-

i See also Refs. [9] and [10] for second-orderreactions.

H. Sumi / Chemical Physics 212 (1996) 9-27

tion has become controlled by their slow speed. In this regime, the rate constant k approximated by kf decreases in proportion to a fractional (less-thanunity) power of r / - l , following (1.2), with the increase of the solvent viscosity r/. This fractional r/ dependence of the rate constant is in fact what has been observed as the most typical feature in the TST-invalidated solution reactions mentioned earlier [5-8,10]. Since the solvent viscosity decelerates the speed of the solvated-stmcture fluctuations of a solute molecule and the rate constant approaches k~ in this regime, kf can be interpreted as a measure of the speed with which solvated structures most favorable for reaction are attained as a result of solvent fluctuations. Since (1.1) with (1.2) covers both the TST regime realized in low-viscosity solvents and the non-TST regime in high-viscosity solvents, as noted above, (1.1) can be regarded as a general expression of the rate constant of solution reactions which has been sought for. It was only in the non-TST regime, however, that (1.1) with (1.2) was checked with the observation of only k = kf ct r/-~ with 0 < a ~< 1 in many solution reactions mentioned earlier. A complete experimental confirmation of (1.1) with (1.2) has recently been obtained by thermal Z / E isomerization reactions of substituted azobenzenes and Nbenzylidene-anilines [14-16] in several solvents. In these experiments, the solvent viscosity r/ was increased over 10 8 times under pressure which was varied up to several hundreds MPa (nearly equal to several thousands atm). The observed rate constant k showed the non-TST regime in the high-pressure, high-viscosity region, since k decreased with the increase of r/ in proportion to a fractional (lessthan-unity) power of r/-1 over its variation of 10 3 times there. In the low-pressure, low-viscosity region ranging over 77 variation also of 10 3 times, k showed the TST regime, since the pressure dependence of k can reasonably be described by the concept of activation volume in TST there. In the transition region between the two regimes, k can nicely be fitted by (1.1) with (1.2). Thus, the general expression for the rate constant of solution reactions was experimentally confirmed over viscosity variation of 10 8 times in these reactions. It was the Kramers theory [17] presented as early as 1940 that took into account, for the first time,

11

deviation of the reactant distribution from the thermally equilibrated one in the reactant state, not assuming a priori that the thermalization time there is short enough. He considered that surmount over the transition-state barrier from the reactant-state potential well as well as thermalization there is accomplished only by diffusive (that is, Brownian) motions of reactants along a one-dimensional reaction coordinate. In this model, however, the rate constant decreases with the increase of the thermalization time ~in proportion to z- 1. Since z is usually proportional to the solvent viscosity r/, the Kramers theory cannot describe the observation that the rate constant of many solution reactions decreases with increasing r/ in proportion to a fractional (less-than-unity) power of r/-~. To understand the observed 77 dependence slower than 77- ~, approximately two streams of theories have been developed. One was initiated by Grote and Hynes [18], while another by Sumi and Marcus [19]. The latter considered two-dimensional reaction coordinates, taking into account not only the solvated-structure fluctuations which were modeled by diffusive motions as in the Kramers model, but also much faster intrasolute vibrational motions as inducing solution reactions. The theory by the present author [13], giving (1.1) with (1.2) as the rate constant, is based on this model, and clarifies the expression of the rate constant given by this Sumi and Marcus's two-mode model. Grote and Hynes, on the other hand, rely on a one-dimensional reaction-coordinate model as in the Kramers one, but consider it as important that friction felt by reactants decreases when they surmount the transition-state barrier very fast. If this effect is important, friction effective for reaction is smaller than that expected from the hydrodynamic viscosity r/ appropriate for sufficiently slow motions. Then the rate constant for a given r/ should not be as small as given by the Kramers theory. Since the speed of reactants surmounting the transition-state barrier can be estimated by the (imaginary) frequency at the barrier top, the Grote and Hynes model has been called the model of frequency-dependent friction. It has been investigated in Ref. [20] whether the Grote-Hynes theory can describe the experimental data on the thermal Z / E isomerization reactions, although their rate constant has been known to have a viscosity dependence expected from the Sumi-

12

H. Sumi / Chemical Physics 212 (1996) 9-27

Marcus model, as mentioned earlier. It was shown therein that if this is the case, we can derive from the experimental data, without adjustable parameters, a concrete value of the correlation time among microscopic random motions of solvent molecules, which cause friction on the solvated-structure fluctuations. The correlation time thus derived was too long to be realized in liquids. Thus, (1.1) with (1.2) given by the Sumi-Marcus model remained as a candidate for the general expression of the rate constant of solution reactions. As shown in the next section, the relaxation of the solvated-structure fluctuations is at least of the order of one ps or longer except an exceptional case of electric fluctuation in water with an electric dipole moment. The solvated-structure fluctuations are, therefore, much slower than microscopic solvent motions whose correlation time is at most of the order of 0.1 ps. In this situation, friction exerted on the solvated-structure fluctuations by microscopic solvent motions can be regarded as lying in the limit describable by hydrodynamics. This means that the idea of frequency-dependent friction in the GroteHynes theory does not seem necessary in describing solution reactions in usual cases [3,4]. In (1.1) giving a general expression for the rate constant of solution reactions, kf can be called the solvent-fluctuation-controlled rate constant, since the rate constant k approaches kf in the non-TST regime for kf << kTsT where reaction is controlled by a slow speed of solvent fluctuations. To be more exact, kf can be interpreted as a measure of the speed with which solvated structures most favorable for reaction are attained as a result of solvent fluctuations, as mentioned earlier. The most characteristic feature of kf is given by the situation of a < 1 in (1.2), in contrast to the Kramers theory which gives k at zat r/-~ in the limit of the non-TST regime. The present work is devoted to clarifying a condition for the situation. It is essential in this condition that the solvated structures most favorable for reactions are not unique but distributed, as will be shown later.

2. Diffusion-reaction equation

the

In Sumi and Marcus' two-mode model, not only solvated-structure fluctuations of a solute

molecule but also intrasolute vibrational motions contribute to reaction. The characteristic time of intrasolute vibrational motions can be estimated by their period, and is at most of the order of 0.1 ps. The solvated-structure fluctuations are diffusive motions in solvents, in contrast to vibrational (that is, ballistic) motions of the former. The characteristic time of the solvated-structure fluctuations can have various values, depending on the situation of a solute molecule. It is of the order of one ps or longer for a small solute molecule, as estimated by time resolved absorption and fluorescence spectra for dimethyl-stetrazine and MnO 4 in solvents [11,12]. Small moieties protruding from the surface of a large molecule such as proteins perform rotational diffusive fluctuations in solvents. The relaxation time of these motions is of the order of 10 ps to 1 ns, measured by Doppler broadening of the Rayleigh scattering of MSssbauer radiation [21]. Comparatively-rigid large domains around a cleft in the protein structure perform hinge-bending fluctuations in tune with the solvated-structure fluctuations. The relaxation time of these motions is of the order of 10 to 100 ns, measured by the same method as above [21]. In association with electron-transfer reactions in polar solvents, orientational reorganization of polar solvent molecules takes place around solute molecules. In this case, the solvated-structure fluctuations are excited a n d / o r damped by microscopic solvent motions not only through viscosity, but also through electric fields generated by orientational fluctuations of polar solvent molecules [22,23]. The latter gives rise to a spherical Coulombic field fluctuating around a charged solute molecule, contributing to energy fluctuations of the solute-solvent system. The relaxation time of the field is given by r DS o / 8 s, where ~'D denotes the Debye relaxation time when polar solvents are represented by a dielectric continuum with the optical and the static dielectric constants 80 and e s respectively [24]. This longitudinal dielectric relaxation time has a magnitude of the order of 1-100 ps with an exceptional case of the order of 0.1 ps for water. We thus see that the solvated-structure fluctuations of a solute molecule are much slower than intrasolute vibrational fluctuations except an exceptional case. Since reorganization along coordinates of the both kinds of fluctuations takes place after reac-

H. Sumi / Chemical Physics 212 (1996) 9-27

tion, both of them contribute to promote reaction. Intrasolute vibrational motions are fast enough and also cannot be decelerated by the increase of solvent viscosities. It is considered, then, that when reaction is induced by intrasolute vibrational motions, thermal equilibration has always been attained between the transition and the reactant states at each coordinate value of the much slower solvated-structure fluctuations. In this situation, we can envisage an intrinsic reactivity, given by a rate constant of reactions induced by intrasolute vibrational motions, at each coordinate value of the solvated-structure fluctuations. The intrinsic reactivity should be determined by TST in the situation mentioned above. The solvated-structure fluctuations, on the other hand, are slow and can be decelerated by the increase of solvent viscosities. Then, thermal equilibration cannot always be maintained, during reaction, among reactant populations at various coordinate values of the solvated-structure fluctuations. Taking the simplest one-dimensional model, we can introduce a single coordinate value X for describing the solvated-structure fluctuations. Reactant populations at each coordinate value X at time t are written as P(X; t). The intrinsic reactivity at each coordinate value X is described by a rate constant ki(X) with different values for different X. Then P(X; t) should satisfy the following diffusion-reaction equation

OP(X;t) ~) ( 0 1 dV(X)) -D---+-- P(X; t) Ot OX ~X kBT dX - ki( X ) P ( X; t),

(2.1)

where the first term on the right-hand side describes an increase of reactant populations at X by diffusion, with diffusion constant D, in a potential V(X), while the second term describes a decrease of reactant populations at X by reaction. Since reaction is taken into account by the second term, V(X) in the first term can be a single-well potential with '

-~bX , around the bottom of V ( X ) , (2.2) with a curvature b ( > 0). Motions on the potential

V(X) describe the solvated-structure fluctuations, and their relaxation time is given by

~"= kBT/ba.

(2.3)

13

Eq. (2.1) gives the fundamental starting point from which the rate constant formula (1.1) with (1.2) can be derived [13]. The fundamental equation describing second-order reactions in solution is essentially the same as (2.1) with slight modifications explained below. Then, we can expect that the rateconstant formula for second-order reactions has the same form as (1.1) with (1.2), as in fact verified in Refs. [9,10]. In second-order reactions, a species of molecules, say R, is subject to reaction with another species, say S, during diffusion in solvents in the field of a mutual interaction potential U(r) which depends on the mutual distance r between R and S. It is usually assumed that reaction takes place at each mutual distance r with an intrinsic reactivity ki(r) which depends also on r. Then, the distribution function Q(r; t) for the R at a position with distance r from an S at time t should satisfy a three-dimensional diffusion-reaction equation

OQ(r;t) ( ) = D V " VQ+---~-Q v u - k i ( r ) Q , Ot kBT (2.4) where V represents the three-dimensional gradient operator and the diffusion constant D is given by a sum of diffusion constants of R and S. Since ki(r) depends on space variables only through the mutual distance r, so does Q(r; t), and (2.4) is essentially a one-dimensional diffusion-reaction equation such as (2.1). Only differences are that r must be larger than the encounter distance a given in a situation of closest contact between the two molecules, and also that a multiplication factor of 47rr 2 must be taken into account in the r integration from a to ~. In second-order reactions, the inverse of the mean lifetime of R is linear in the concentration of S when it is sufficiently small. The coefficient of iinearity gives the average rate constant in second-order reactions, and it can be shown [9,10] to have the same general form (1.1) with (1.2) as in first-order reactions. The distribution Q(r; t) thermalizes in the field of the potential U(r) in second-order reactions so long as reaction of R with S is sufficiently slow. The thermalization is attained as a result of solvent fluctuations also here, since they enable these solute molecules to diffuse. The thermalization time is in-

14

H. Sumi / Chemical Physics 212 (1996) 9-27

versely proportional to the diffusion constant D, which is inversely proportional to the solvent viscosity r/ from the hydrodynamic Stokes-Einstein-Debye relation. In the following, first-order reactions are treated unless otherwise is explicitly noted.

3. Average rate constant as the inverse of the mean lifetime

As the initial condition for (2.1), let us assume that distribution of X values was in thermal equilibrium in the potential V(X) initially at t = 0, with P(X; 0) given by Pe(X) t~ e x p [ with

fPe(X)

V( X)//kBT], d X = 1,

(3.1)

where the integration in X is performed in a region of - ~ to ~, and it is understood hereafter that the X integration is performed over this integration region without explicit writing of it on the integration sign. The survival probability of reactants remaining alive in the reactant state without reaction is given by

n(t)

=fe(x, t) dX,

with n(0) = 1.

(3.2)

The survival probability decreases as time t elapses, and the mean lifetime is given by fo t ' [ - d n ( t ' ) / d , ' ]

at ' = f0

constant obtained in the steady state, we prove in this section that the general expression is applicable more widely to the average rate constant k. The procedure for the proof is similar to that developed for the average rate constant of second-order reactions, but it is given with some duplication for later usage and also for avoiding reader's confusions caused by differences between first- and second-order reactions. It is reasonable to assume that no reaction take place in the limit of X-~ + 0% as k i ( X ) --) 0,

as X ~ ___oo.

(3.4)

It is possible, instead of (3.4), that k i ( X ) approaches a constant value, written as y, as X ~ _+~ where y represents a rate constant of nonreactive decay of reactants, as in natural decay of reactants in an excited state in excitation transfer. In this case, however, the constant term can be eliminated from ki(X) in (2.1) by considering P(X; t) e ~'t, instead of P(X; t). This corresponds to considering n(t)e w, instead of n(t), in calculating the average rate constant k by (3.3), as explicitly performed in Ref. [10]. Then, we can continue the analysis with the condition of (3.4). With (3.4), the boundary condition for the solution to (2.1) should be the natural one that there exists no diffusion current in the limit of X ~ ±oo, as --exp

~X

- -

P ( X ; t)

as X ~

+~,

~ kRT

n(t')dt'=--k -1, (3.3)

where the first equality is obtained by partial integration with the second equality in (3.2). The second equality in (3.3) defines the average rate constant k investigated in the present work. When the initial condition mentioned above is not satisfied, k obtained by (3.3) with this initial condition gives the rate constant in the steady state, as shown later. Although the true rate constant can be defined only in the steady state where n(t) decays single-exponentially, the average rate constant k can be defined by (3.3) irrespective of whether n(t) decays single-exponentially or not. Although the general expression (1.1) was proved in Ref. [ 13] intensively for the rate

~0,

(3.5)

where the last parentheses in the middle expression in (3.5) represents the diffusion current in (2.1), since (2.1) without the sink term [due to ki(X)] represents the equation of continuity with this diffusion current. Since the diffusion operator in (2.1), composed of partial-differential operators in X, is not self-adjoint, it is convenient to transform it into a self-adjoint form. To this end,

g( X) - P~( X) ,

with

f g( x) 2 d X =

1,

(3.6)

H. Sumi / Chemical Physics 212 (1996) 9-27

is introduced, where the second equality is ensured by (3.1). With this function g(X), let us rewrite the distribution function P(X; t) as

P(X; t ) = g ( X ) m ( X ; t),

(3.7)

introducing a new function m(X; t). Since P(X; t) satisfies (2.1), re(X; t)satisfies

am~at = - ( H + ki) m,

15

where the first equality is ensured by (3.7) while the second one by (3.12) with ( H + ki) -1 representing the inverse operator of H + k~. Since H of (3.9) is Hermitian, it satisfies Hlg)=0,

and ( g i l l = 0 .

(3.14)

The normalization relation for g(X) in the second equality in (3.6) is expressed as

(3.8)

( g i g ) = 1.

with

(3.15)

Let us define here a bra vector I p) by a2

n = -D

Ip)-~k(H+k~)-llg),

OX 2

+ k.------f 2 [

with ( g l p ) = l , (3.16)

r t--ST-

/

where the second equality is ensured by (3.13). Its coordinate representation is given by

dX

=kf0 re(X; t) dt, ~c

5

2

=

(3.9)

g(x)

where k i in (3,8) symbolizes k i ( X ) . Let us note here that H defined by (3.9) is a self-adjoint (that is, Hermitian) operator. The initial condition for P( X; t) corresponds to m(X; t ) ~ g ( X ) ,

as t--->0,

(3.10)

for m(X; t). The boundary condition for m(X; t) can be derived by applying (3.7) to (3.5), as

p(x) with

fg(x)p(x) dX= 1,

since re(X; t) is given by (3.12). Since (glki(H + = ( g ] ensured by the second equality in (3.14), the average rate constant k can be derived from I p) of (3.16) by ki) -1 = ( g [[1 - n ( H + k i ) - t ]

k= (g]k i]p) =

fk~(X)g(X)p(X) dX.

a

-~g(X)-'m(X;

m(X; t ) = e x p [ - ( H + k i ) t ] g ( X ).

(3.12)

In treating a Hermitian operator, it is convenient to employ Dirac's bra and ket vector representation introduced in quantum mechanics. For example, g(X) and m(X; t) are regarded as the coordinate (X) representation of ket vectors ] g) and Ira(t)) respectively. The bra vector of [ g ) is written as ( g 1, with which the average lifetime k-~ given by (3.3) with (3.2) is written as k-'

=fo (g]m(t))

(3.18)

t)--*O, as X ~ +~c. (3.11)

The formal solution to (3.8), satisfying (3.10) and (3.11), is given by

d t = (gl(H+k~)-~lg), (3.13)

(3.17)

Therefore, k is given by an average of the intrinsic reactivity ki(X) on g(X)p(X), which can then be called the average distribution function of X values in the reactant state. In parallel to this situation, in TST where it is a priori assumed that the distribution function is always maintained at the thermalized one Pe(X) of (3.1), the rate constant is given by

kTsT=fk,(X)Po(X) dX=(glk~lg),

(3.19)

where the second equality is ensured by (3.6). This is called the TST-expected rate constant. In order to obtain I p), it is convenient to introduce a projection operator L of

t-l-lg)(gl, satisfying L [ g) = 0 and L2 = L.

(3.20)

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H. Sumi / Chemical Physics 212 (1996) 9-27

We see from (3.14) that L and H commute with each other, as

In, L] -- HL - LH = 0.

(3.21)

Although H vanishes on I g ) as (3.14), its inverse operator H - ~ can be defined in a space projected out by L, since it annihilates a component proportional to I g). Then, we see

LI p) = H-IHLI p) = H-'LH I p)

where the first equality is ensured by (3.19) while the second one by the equality in the parentheses in (3.24). Eq. (3.25) is nothing but (1.1), which we wanted to obtain, with

kf ~- kTsT(glki I p ) / ( g l k i H - I L k i l p). The positivity of fied in Ref. [13].

(3.26)

kf given by (3.26) has been veri-

=H-IL[( H +k~)l p ) - k ~ l p)] =H-'L(kl g) - k il P)) = -H-ILki I p),

(3.22)

where the second equality is ensured by (3.21), the fourth one by (3.16), and the last one by the second equality in (3.20). The original definition of L in (3.20), on the other hand, enables us to rewrite LI p) as LI p) = I p) - I g ) ,

(3.23)

since ( g I p) = 1 from the second equality in (3.17). Then, equating the right-most expression of L I p) in (3.22) to the right-hand-side expression in (3.23), we get

I p) = I g) - H - ' L k i l P). (or, I g ) = I p) +H-tLki ] P ) )

Although H defined by (3.9) is diagonal in the coordinate X, its inverse operator H-1L is generally not so. Let us write the coordinate representation of H - l L as (X I H - l L I Y) with which the quantity in the denominator on the right-hand side of (3.26) is, for example, represented by f dX jdYg(X)ki(X) ( X I H-ILl Y ) k i(Y ) p( Y ). The explicit expression of (XIH-~LI Y) is given by (4.33) in Ref. [13]. With this ( X I H - I L I y ) the coordinate representation of (3.24) is

p ( X ) = gt X ) -

f(XIH-'LIY)k (Y)p(Y) dY,

(3.24)

Although I P) was defined with an unknown quantity k by (3.16), k has been eliminated in (3.24). It can be regarded as an equation to solve for ]p). This equation is the same as (4.32) in Ref. [13] for obtaining the rate constant in the steady state. It can be proved [9] that (3.24) has a unique solution. Therefore, k calculated by (3.18) coincides with the rate constant in the steady state, although originally k represents the average rate constant defined by (3.3) under the condition of initial thermalization in the reactant state. After obtaining the unique solution I p), whose coordinate representation is p(X), to (3.24), we can calculate the average rate constant k by (3.18). It can be rewritten as k = k~sv( g ]k i t p ) / ( g l k i ] g)

= kTs.r(glkil p) /[(glkilp)+(glkiH-~Lkilp)],

4. D e p e n d e n c e o f k r on the t h e r m a l i z a t i o n t i m e ~" in the reactant state

(4.1) which is the Fredholm's integral equation of the second kind. The kernel of this integral equation satisfies the boundary condition (3.11) for re(X; t), since an identity relation H( X [ H- IL[ Y) .= (X ] L ] Y) combined with (3.9) defining H gives d

dxg( X)-'( XIH-'LIY) oc

= D - ' g ( X)-2 f x g ( Z ) ( Z I L I Y ) dZ ~0,

as X ~ + ~ ,

(4.2)

because of ( g [ L = 0 for X ~ - ~. Then, the solution p(X) to (4.1) as well as g(X) satisfies the same boundary condition, as 0

(3.25)

-7--s.,g(X)-lp(X) ~ 0 ,

as X ~ _+~.

(4.3)

H. Sumi / Chemical Physics 212 (1996) 9-27

In treating an integral equation, it is convenient to cast it into a form with a symmetric kernel. To this end, let us transform g(X) into a function

17

from (4.8). The proof of the latter is given in Appendix A. Eigenfunctions of the kernel B(X, Y) are defined as satisfying

u ( X ) = g( X)!/ki( X)//kTsT , satisfying

f . ( x ) 2 dX =

x) 1,

=

dY,

s( x,

(4.4) for n = 1, 2, 3 . . . . .

where the second equality can be obtained from the definition of kTsT in (3.19). Similarly, p(X) is transformed into s(X) of s ( X ) -- p ( X ) Cki( X)/kTs T ,

(4.5)

with which (3.18) can be transformed into

k/kTsT = f . ( x ) s ( x ) dX.

(4.6)

Then, the boundary condition (4.3) for p(X) is converted to that for s(X) given by

Fs(X)=O,

with F -

lim

d

x--, += dX u( X)

-1

" (4.7)

which is satisfied by u(X), too. The two-variable function ( X I H - t L I Y ) is transformed into

Since this equation is homogeneous, called Fredholm's homogeneous equation [25], eigenfunctions q~,(X) can be found only for special values of /x, called eigenvalues. The eigenfunctions can be regarded as real, since so are those of H as noted in Appendix A. Eigenfunctions for different eigenvalues are orthogonal, as

f~o,(X)qJ,,(X) d X = 0 ,

(4.12)

d X = 6..,.,

- ~/k~(X)/kTs T ( X I H-ILl r)~/ki(Y)/kTs T . (4.8) This function satisfies the boundary condition of (4.7), as FB(X, Y) = 0, since ( X I H - ' L [ Y) satisfies (4.2). Moreover, B(X, Y) is real and symmetric with B(X, Y) = B(Y, X), since so is (X I H-~LI Y) from (3.21). With these functions, (4.1) can be transformed into an integral equation with a symmetric kernel given by B(X, Y), as

for n, m = 1,2, 3 . . . . .

(4.9)

The kernel B(X, Y) of (4.9) is positive definite. That is, for any real function f(X),

(4.13)

In the light of this equation, (4.11) means that the kernel can be expanded as

B( X, Y) = ~_,~bn(X)qJn(Y)/tz n.

(4.14)

n

Then, (4.10), expressing the positive definiteness of

B(X, Y), enables us to see that all the eigenvalues of the kernel are positive, as /x > 0 ,

f dX f drf(X)B(X,Y)f(Y)>O,

if /z, 4:/x,,.

When more than one eigenfunction is found for a single eigenvalue, orthogonality among these eigenfunctions is possible [25]. Then, normalizing the norm of any eigenfunction to unity, we can regard all the eigenfunctions of the kernel as constituting an orthonormal set satisfying

B(X,Y)

s(X) =u(X) -krsrfB(x, r)s(V) dr.

(4.11)

for n = 1 , 2 , 3 . . . . .

(4.15)

As shown in Appendix B, any real function satisfying the same boundary condition as (4.7) for s(X) can be expanded by the eigenfunctions, qJn(X) for n = 1, 2, 3 . . . . . of the kernel B(X, Y) and by ~b(X) of

(4.10)

is satisfied. This property of B(X, Y) comes from the same property of (X [ H- 1L I Y), as understood

i~( X) ~ ~

g( X)¢kTsT//ki( X)

=~

u( X)kTsT/ki(X),

(4.16)

H. Sumi / Chemical Physics 212 (1996) 9-27

18

with

expanded only by the eigenfunctions of the kernel B(X, Y) which is expressed as (4.14). Then, s(X) put on the left-hand side of (4.9) can be expanded as

satisfying O ' < 1,

0"-1 = - k T s T ( k i ( x ) - l > e ,

(4.17) where or was introduced for normalizing ~O(X). The second equality in (4.16) is obtained by (4.4), while ( f ( X ) ) e in (4.17) represents an average, with respect to X values, of a function f ( X ) on the thermally equilibrated distribution function, Pe(X) of (3.1), in the reactant state, as


(4.18)

The second inequality in (4.17) is ensured by Schwarz's inequality applied to kTsT(ki(X)-J>~ where kTsT = (ki(X)>~ with the notation of (4.18). The expansion is possible so long as ki(X) is a continuous nonvanishing function of X. This condition is assumed hereafter unless otherwise is explicitly mentioned. As noted also in Appendix B, the function @(X) normalized to unity is orthogonal to any ~b,(X) for n = 1, 2, 3 . . . . . as

f~b~(X)~b(X)dX=O,

for n = 1 , 2 , 3 (4.19)

Since u(X) of (4.4) satisfies the same boundary condition as (4.7) for s(X), it can be expanded by this complete set of functions, as

u(X) = ~

qJ(X) + Y'.u,q,(X),

(4.20)

n

with

u,= f ,(

x)

The coefficient q ~ for ~b(X) in (4.20) is ensured since (4.4) and (4.16) enable us to see fu(X) ~b(X) d X = v ~ f g ( X ) 2 d X = vr~. Since u(X) is normalized to unity as in (4.4), (4.22)

n

is satisfied among coefficients of expansion on the right-hand side of (4.20). On the right-hand side of (4.9), the first term can be expanded as (4.20), and the second term can be

~b(X) + ~s,q&(X).

(4.23)

n

The coefficient to ~,(X) can directly be checked by fs(X)~b(X) dX = ~ f g ( X ) p ( X ) dX = ~ which can be obtained from (4.5) and (4.16) together with the second equation in (3.17). The coefficient s n to ~b,(X) must be obtained in the procedure of solving the Fredholm's integral equation of the second kind of (4.9): Substitution of (4.14) and (4.23) to the second term on the fight-hand side of (4.9), followed by Y integration therein with the use of (4.13) and (4.19), enables us to see that the coefficient to ~b~(X) on the right-hand side there is given by U,-kTsTSn/IZ n where the first term is derived from u(X) in (4.9) expanded as (4.21). This coefficient is equated to the coefficient sn to ~,(X) on the lefthand side of (4.9), and we get Sn

= Un//( 1 -I- kTST///.Kn).

(4.24)

Multiplicating (4.20) and (4.23) side by side and performing the X integration enable us to see that k / k T s T of (4.6) is given by or plus the sum of u,s,. Then, (4.24) gives k / k T s T = or+ E U2n~ n / ( kTsT -b IZn),

(4.25)

n

for the average rate constant k. It can be expressed as (1.1) with the solvent-fluctuation-controlled rate constant k e. Then, it can be derived from (4.25) together with (4.22), as

dX, for n = 1 , 2 , 3 . . . . . (4.21)

Y'.u2, = 1 - o-,

s(X) = ~

kf =

or q"

Un tZn

Un

kxs T q-/d, n

kTs T -t- /L n

"

(4.26) Since B( X, Y) is proportional to ( X I H- IL I Y) in (4.8) and H is to the diffusion constant D in (3.9), all the eigenvalues /x,, with which B(X, Y) can be expressed as (4.14), are proportional to D. Since D is inversely proportional to the thermalization time r of the solvated-structure fluctuations in (2.3), all the eigenvalues /z, of the kernel B(X, Y) are inversely proportional to z, which is usually proportional to the solvent viscosity 7/. In the limit of

H. Sumi / Chemical Physics 212 (1996) 9-27

small 7-, therefore, we can neglect kTs T in comparison with /,, in (4.26). Then, the numerator on the right-hand side of (4.26) reduces to unity because of 2 (4.22), while the denominator to ~,u,/Ix,, which is equal to [dX fdY u(X)B(X, Y)u(Y) as ensured by (4.14) and (4.21). Rewriting it into an integration with g(X) and (XIH-IL[Y) by using (4.4) and (4.8), we get kf--~ k C (o~ 7"-1), in the small 7- limit (of k c >> kTST)

(4.27)

with

kC= k2TST/f

dX

f dY g( X)ki( X )

X ( X I H - ' L I Y ) k i ( Y ) g ( Y ).

(4.28)

The k c of (4.28) is the value of (3.26) for kf obtained by its perturbational expansion lowest-order in the reaction term ( s k i ( X ) ) since p(X) approaches g(X) in this order as understood from (3.24). As noted in (4.27), the small r limit where kf approaches k c is realized when k c >> kTsT [13]. This condition reflects the following situation: when the perturbational expansion in the reaction term is justified, the average rate constant k should approach the TST-expected rate constant kTsT. On the other hand, the perturbational expansion gives kf = kc as mentioned above, and it is when kf >> kTs T that k approaches kTsT in the general formula (1.1) for k. As also noted in (4.27), k c is proportional to 7-- 1 since (X I H - I L l Y) ot 7- in (4.28), and hence to the inverse of the solvent viscosity r/. Let us next investigate the large r limit of k c << kTsT. In this limit, /z~ (~x 7--1) can be neglected in comparison with kTsT in (4.26). Then, the denominator therein reduces to (1 - o')/kTs T as understood from (4.22). Since the second term in the numerator approaches zero as 7- increases, as shown later, kf given by (4.26) approaches a constant of kvs v t r / (1 -tr) SO long as o- does not vanish, being supplemented by a term decreasing with 7- in the large 7limit. The appearance of this constant term is more intelligible in terms of the mean lifetime k- 1 than in terms of kf, since (1.1) gives k -~ = (kTsT~) - 1 = < k i ( X ) -1 >e when kf =- kTST Or/(1 -- Or), with the use of (4.17) for or. This enables us to express this case as follows.

19

(I) W h e n e does not diverge, there appears in kf a constant term which gives k-' = (k~(X)-')e, in the large r limit of k c << kTs T.

(4.29)

Here, k~(X)-1 represents the lifetime at each X brought about by reaction with rate constant ki(X), when a diffusional change in populations does not take place. Eq. (4.29) means that the mean lifetime k-1 is given by the average of the lifetime ki(X)-1 at each X over the thermally equilibrated distribution of X values in the reactant state in the large r limit. Therefore, (4.29) is a natural result expected when diffusion is much slower than reaction due to slow solvent fluctuations, that is, in the limit of frozen solvents. It is instructive to point out here that in the opposite small 7- limit of rapidly fluent solvents, where diffusion is much faster than reaction due to fast solvent fluctuations, the rate constant k is given by kTs T of (3.19) which represents the average of the rate constant ki(X) at each X over the thermally equilibrated distribution in the reactant state. The tern2_approaching zero as 7- increases in kf is written as kf. Then we have k f = kTSTtY/(1 -- O') + k f ,

(4.30)

with 2

kf = (1 -- o')

-

Un/Zn I kTs T E kTST+ ].Ln,

(4.31)

n

in the large 7- limit. To evaluate kf, we must distinguish 2 = [ convergent; ~]nu,/*, ~ divergent;

Case A. Case B.

(4.32)

It is only in the case A that /z~(ct r - l ) can be neglected in kTsT +/z~ for kf, as follows. 2 n converges, kf is given by (A) When Y~,u~/z

7,r= ( l -

EU2o

7--'),

n

in the large 7- limit of k c << kTs T.

(4.33)

It decreases in proportion to ~'-1 with the increases of 7- (at the solvent viscosity r/).

20

H. Sumi / Chemical Physics 212 (1996)9-27

In the case B, /x. (at r -l ) cannot be neglected in kTS T -I" ~n in (4.32) however small r - l is, since the

jz. for large n become larger than kTsT even in this situation and they give an essential contribution in the n summation in (4.31) for 7~f, as shown below. (B) When E . u . 2/ z . diverges, we cannot help writing ke as (4.31) even in the large ~- limit of k¢ << kTsT. In this case, 7¢f decreases more slowly than ~-- 1 with the increases of r since the denominator on the right-hand side of (4.31) also decreases when the numerator decreases in proportion to r -1. It is this slower-than-r -~ decrease o f ]¢f that can be represented as the fractional (less-than-unity) power dependence on r - l . To show this explicitly, it is convenient to introduce a spectral function p ( t o ) = ]~u2.6( to - r/z.).

(4.34)

n

Since all the /z. are proportional to r - l , this function is independent of ~'. The condition for the case B in (4.32) means that although ftop(to)dto ( = rY'~nU2~/x.) diverges but fp(to) dto ( = 1 - or, from (4.22)) does not. These properties should originate from the behavior of p(to) in the large to limit: p(to) must decreases more rapidly than w - l , but more slowly than to-2 except a marginal case that ftop(to) dto shows only a very weak logarithmic divergence for p(to)at to-2. Then, in the large to limit p(to) must behave as p ( t o ) otto - l - ~ ,

with 0 < a < l ,

(4.35)

where a is a certain constant. The r dependence of the right-hand side of (4.31) given by

Eu2.

+

n

= fo toP(to)/(kTsTr+

tO) dto,

(4.36)

can be determined by an integration, convergent for 0
of to integration to zero there. In the case B, therefore, kf behaves as 7qatr -s,

with 0 < a < l ,

(4.38) showing a fractional (less-than-unity) power dependence on r - i ( ot -q- l ).

5. Condition for the fractional-power dependence The fractional (less-than-unity) power dependence of kf on 7-l is realized under the condition for the case B in (4.32). As shown in Appendix C, the n summation in the condition can be rewritten in a more intelligible form as

E u,2 tx,=D

~ ki(X)-'

n

= zr(krsxr)-~/sin[zr(1 - a)].

(4.37)

Divergence of the right-hand side as a ~ 1 in (4.37) originates in the artificial setting of the lowest bound

/(,ki(X)-') 2. e

(5.1) In the light of (5.1), we can summarize, as follows, the z dependence of the solvent-fluctuationcontrolled rate constant kf in the general expression (1.1) for the average rate constant k of solution reactions, where ~" represents the thermalization time of the solvated-structure fluctuations in the reactant state and is usually proportional to the solvent viscosity ~7. As shown in (4.30) for the large r limit of k C << kTs T with k c (oc r - 1) of (4.28), k is in general composed of two terms, a constant term given by kTsTtr/(l -- tr) with tr - I o t ( k i ( X ) - l ) e by (4.17), and another t e r m 7£f which approaches zero as r increases. In the case I that ( k i ( X ) - l ) e does not diverge, the constant term k T s T t r / ( 1 - g ) is nonvanishing since tr 4: 0. In terms of the average lifetime k - l , the constant term gives rise to k - l •,-.•. ( k i ( x ) - l ) e ,

fo~OO-~'/(kTsTr + to) dto

in the large r limit,

(5.2)

which is independent of r, describing the situation of frozen solvents. In the case II that ( k i ( X ) - l ) e diverges, on the other hand, the constant term disappears and kf becomes decreasing endlessly with the increase of r,

21

H. Sumi / Chemical Physics 212 (1996) 9-27

following kf. This case often occurs, as shown by explicit examples in the next section. In the case A that ([(d//dX)ki(X)-l]2)e// (ki(X)- ~)~ does not diverge, kf is given by

d

-I

kf=Pe(Xc)/(XclH-lLiXc),

e

/(ki( X ) - ' ) 2, (at ~.-l)

(5.3)

which decreases in proportion to r -1 . In the case B that ([(d/dX)k~(X)-~]2)e/ ( k i ( X ) - l ) ~ diverges, on the other hand, kf decreases in proportion to a fractional (less-than-unity) power of r - I ( a t r/- l), following (4.38). In this case B, it is more appropriate to express lcf as k f ~ v l - % -~,

with 0 < a < l .

(5.4)

where v represents the preexponential factor of the TST-expected rate constant kTs~, that is, the strength of reaction. The v dependence in (5.4) can be obtained by the dimensional analysis since only two terms, the diffusion constant D (at z -] ) and the intrinsic reactivity ki(X) (or v), are concerned with time in the starting equation (2.1) when kf ct r -~ must have a dimension of inverse time. The result of (5.4) was obtained by neglecting a very weak logarithmic divergence. As a special situation noted in sentences below (4.34), therefore, the ~" dependence weaker than r-~ with increasing ~-, obtained in the case B, can be ]¢f O[ T- I[log(

UT)] n ,

with n > 0.

value of X at Xc. In this case, kf can directly be obtained from (3.26) with (3.19) for kTsT, without solving (3.24) for p(X), as below. In the case III that ki(X) is proportional to 8(X - Xc), we get directly from (3.26)

(5.5)

instead of (5.4). This case in fact occurs, as shown by an explicit example in the next section. In combination of these cases, the solvent-fluctuation-controlled rate constant k t approaches zero with the increase of r in proportion to a fractional (lessthan-unity) power of r -~ (at r/-~) when both (k~(X)-~)e and ( [ ( d / d X ) k ~ ( X ) - ' ] 2 ) e / ( k i ( X ) - 1)~ diverge. The results in (5.1)-(5.5) have been obtained under an assumption that the intrinsic reactivity k~(X) is a continuous nonvanishing function of the coordinate X of the solvated-structure fluctuations. These results cannot be applied, however, when k~(X) is proportional to a delta function ~(X- Xc), that is, when reaction induced by fast intrasolute vibrational motions takes place only at a special

(otr -l)

(5.6)

in the entire region of r, where Pc(X) given by (3.1) represents the thermally equilibrated distribution function of X in the reactant state while (XIH-IL[ Y) is given by (4.33) in Ref. [13] as a function of X and Y. In this case, kf is proportional to r-~ as noted also in (5.6).

6. Discussion The r dependence of kf in the large r limit obtained in the previous section can be seen in more concrete forms by considering specific solution reactions. In intramolecular electron-transfer reactions in solution [19], when the potential V(X) in the reactant state is given by b X 2//2 quadratic in the entire region of X, the intrinsic reactivity ki(X) is given by k i ( X ) ot e x p [ - ( A o / / A i ) b ( X -

Xc)2//(2kBT)], (6.1)

peaking at a certain value X c of X, where Ao represents the energy of reorganization after reaction due to the solvated-structure fluctuations, called the outersphere reorganization energy, while A~ represents that due to intrasolute vibrational motions, called the inner-sphere reorganization energy. The Xc is related to the thermal activation energy AG* of kvs ~ by bX2c/2 = (1 + Ai//Ao)AG*. In this case, tr in (4.17) is given by

f

0,

for Ai < Ao 2A o A G * ) Ai - A° kaT

(6.2)

'

~, for Ai > Ao Since AG * is usually much larger than kBT, we can regard ~r as exponentially small. Then, we can neglect the constant term kxsT~r/(l - o-) in (4.30)

14. Sumi / Chemical Physics 212 (1996) 9-27

22

for the solvent-fluctuation-controlled rate constant kf, although the term is nonvanishing when A~ > Ao. We consider, therefore, that kf is composed only of the term 7~f which approaches zero with the increase of r ( = kBT/(bD)) in the large r limit. The r dependence of kf varies, dependent on whether ( [ ( d / d X ) k i ( X ) - l ] 2 ) e / ( k i ( X ) - l ) ~ diverges or not. It diverges when A J A 0 < 2, although we have ki(X)(X ~ ( X - X C) in the limit of Ai/A 0 0. Then we get the following classification in the ~" ( = kBT/(bD)) dependence of kf in the large ~" limit for intramolecular electron-transfer reactions. The case I of (5.2) can be neglected in usual cases that AG * >> kBT , and the reactions are effectively in the case II below (5.2), except in the limit of Ai/A o -'*0. The case A of (5.3) is realized for A~/Ao > 2. The case B of (5.4) is realized for Ai/A 0 < 2 but A i / A o ~: 0.

The case III of (5.6) is realized in the limit of Ai/A o ~ 0. The fractional (less-than-unity) power index a in (5.4) in the case B cannot be obtained with the present analysis. It will be shown in the forthcoming paper that a is given by

shown by Zwanzig [26] that the observation of c~ ,-~ 0.5 can be reproduced by assuming k(X)otX 2 for the potential V(X) also quadratic around X = 0 as X 2 in the reactant state. This result is consistent with the present analysis as shown below: since ( k i ( X ) -1 >e diverges in this case, the constant term does not appear in kf in the large r (ot r/) limit, as the case II below (5.2). Since ( [ ( d / d X ) ki( X )- l ]2 ) e / ( ki( X ) - l )ez also diverges, the reaction is in the case B of (5.4), and the solvent-fluctuationcontrolled rate constant k~ should behave as (1.2) with a < 1 in the large r limit. For the second-order reactions in solution, the criterions in (5.2) to (5.4) should be modified a little since the rate constant has a dimension of volume divided by time in second-order reactions while it has a dimension of inverse time in first-order reactions. It becomes apparent from the analysis in Ref. [9] that the quantity o- is determined by o- - I =

lim ( k i ( r ) > e ( k i ( r ) - l ) e ,

(6.4)

b - - - , oe

with

( ... )e =- fa br2 "..e -v(r)/kBr dr

a ~-I1 - A~/Ao I,

/ fabr2e - V~r)/%r dr,

when Ai/A o << 1 or 0 < 2 - Ai/A o << 1, (6.3) that is, in the neighborhood of the boundaries at Ai/A o = 0 or 2 for the case B~ When a ligand molecule (02 or CO) binds to heme proteins such as hemoglobins and myoglobins, it must get into the heme pocket inside the proteins, and the process is controlled by the speed of opening of the gate of the pocket [7]. In fact, it has been observed that the rate constant k with which a ligand molecule gets into the heme pocket from solvents decreases with the increase of the solvent viscosity 7/ in proportion to a fractional (less-than-unity) power of r/-1. In this case, the fractional power index is about 0.5. It is reasonable to consider that the reaction is in the solvent-fluctuation-controlled regime in the general formula (1.1) with (1.2) of the rate constant of solution reactions for kf << kTs T where (1.1) with (1.2) reduces to k = kf oc rl-". It has been

(6.5)

where U(r) represents the interaction potential between two molecules reacting with the intrinsic reactivity k~(r) at a mutual distance r in the diffusion-reaction equation (2.4) and a represents the smallest allowance for r called the encounter distance between the two molecules. Since the TST-expected rate constant for second-order reactions is given [9] by oc

kTST = 47r

rZki( r)e -v(r)/%r dr,

(6.6)

and U(r) ~ 0 as r ~ oo, the intrinsic reactivity ki(r) must approach zero more rapidly than r -3 as r ~ 0% so long as kTsT is finite. Then, the right-hand side of (6.4) diverges, and o-= 0 is obtained. Thus we see that kf in the general formula (1.1) for the rate constant always decreases endlessly in the large ~- (or r/) limit for second-order reactions in solution,

23

H. Sumi / Chemical Physics 212 (1996) 9-27

as the case II below (5.2). This is true irrespective of functional forms of ki(r) and U(r). The large ~limit is realized when k c (or r - ~) << kTs v where k c defined by (4.28) is determined by (XIH-ILI Y) which is given analytically by A(r, r') in (4.3) of Ref. [9]. The cases A or B in (5.3) or (5.4) are respectively realized when we have a nondivergence or a divergence of

lim b3<(dki(r)-~)21 / ( ki(r)-')2 dr

b-~

e

= b-~lim ~ - g . r

dr

a=(n-3)/(n-2),

dr

where the second equality is obtained since U ( r ) ~ 0 as r ~ ~. Since the right-hand side of (6.7) does not include U ( r ) in second-order reactions in solution, which case A or B is realized does not depend on U(r), and so does the fractional power index a ( < 1) in (1.2) obtained in the case B. In the case A, on the other hand, the fight-hand side of (5.3) describes kf which equals }f since ~ = 0, but it must be redefined by multiplying ~ b ( - 4 I r b 3 / 3 ) followed by taking the limit of b ~ 2 with the transformation used in (6.7). In second-order reactions in solution, the v ( ~ D -~ ) dependence of k f , approaching zero, in the large r limit does not depend on the functional form of the interaction potential U ( r ) between two reacting molecules, so long as the intrinsic reactivity ki(r) is not proportional to a delta function in r. To investigate the dependence, therefore, it is convenient to assume U(r)= 0. It is interesting here to investigate a case of k i ( r ) ot r - " ,

with n > 3,

FSrster's mechanism [27]. Likewise, n = 8 and n = 10 are realized in excitation-transfer reactions through the dipole-quadmpole interaction and the quadrupole-quadrupole interaction, respectively. For ki(r) of (6.8), the quantity in (6.7) diverges, and the diffusion-controlled part kf in the general formula (1.1) for the rate constant should become proportional to a fractional (less-than-unity) power of r-1 (¢x 77-i at D) in the large r limit, as the case B in (5.4) with k f = k f for ~r= 0, irrespective of the values of n. The fractional power index a in (5.4) has been determined in Ref. [10], as

(6.8)

since (6.8) can be realized in real systems, where n > 3 is necessary to ensure that kTs T of (6.6) is finite. In fact, n = 6 is realized in excitation-transfer reactions by the interaction through transition dipole moments between reacting molecules, that is, by

with n > 3 ,

(6.9)

which approaches zero as n approaches three. Although (6.9) was obtained for U(r) = 0 in Ref. [10], the present analysis have shown that (6.9) does not depend on the functional form of U(r) so long as U ( r ) ~ 0 as r --, oc. In electron-transfer second-order reactions in solution, it is reasonable to take ki(r )otexp(-r/d),

with d > 0 ,

(6.10)

which reflects an overlap of electron's wave functions, on reacting molecules, necessary to induce electron transfer between them. Also for ki(r) of (6.10), the quantity in (6.7) diverges, but it has been clarified in Ref. [I0] that (6.10) gives rise to an exceptional case of (5.5), instead of (5.4), since kf ( = kf since o - = 0) becomes proportional to r -I log ~- in the large z limit. In this case, kf decreases with increasing ~- only slightly more slowly than r - ~. Let us investigate also a case that the intrinsic reactivity k~(r) becomes proportional to a delta function in r. The delta function should be 6 ( r - a ) since it is when two molecules are closest at the encounter distance a that they react most rapidly. In this case, kf becomes proportional to ~--~ in the entire region of z, as the case III in (5.6). The explicit form of kf in this case for second-order reactions in solution has been given in Ref. [9] by oc

kf = 47rD/ f a r -2 ev{r)/aBrdr,

(6.11)

24

H. Sumi / Chemical Physics 212 (1996) 9-27

which is proportional to r-~ proportional to the diffusion constant D ( at 7/- l ). We have thus seen that the criterions for classifying the r dependence of kf in the large r limit, clarified in the present work, are consistent with examples in specific solution reactions. Moreover, the analysis with these criterions enabled us to generalize the results obtained for specific examples in second-order reactions in solution. These criterions will be effective in understanding the r dependence of the solvent-fluctuation-controlled rate constant kf in the general formula (1.1) for the average rate constant of solution reactions in the large r (or r/ct D - l ) limit. Finally, it seems worth mentioning that the diffusion-reaction equation (2.1) can be applied also to describe geminate rebinding of a ligand molecule to a heme iron inside the heme pocket in hemoglobin or myoglobin after photodissociation of the iigand (02 or CO) [28]. In this case, the coordinate X has been regarded as describing conformational fluctuations in protein projected onto the displacement of the iron from the mean heme plane. It has been argued for this system [29] that the survival probability in the unligated state shows a stretched exponential decay, expressed as e x p ( - y t t3) with constants Y and /3 ( < 1), because of almost-frozen distribution of various conformational substrates with slow-enough fluctuations among them. It has also been argued [30] that the index /3 ( < 1) of the stretched-exponential decay in the solid-like disordered system is related to the index a ( < 1), investigated in the present work, in the fractional-power dependence of the average rate constant on the solvent viscosity in the liquid-like disordered system. In fact, the deviation from unity of the both indices arises from a random distribution of conformational substrates in the reactant state. In the former system, the disorder can be regarded as almost frozen in, while in the latter system it can be regarded as fluctuating in time although smeared out in a long time scale. Since the former system is a special case of the latter one for slow-enough solvent-driven conformational fluctuations, it seems important to clarify the relation between these two fractional (less-than-unity) power indices. For second-order reactions by the intrinsic reactivity ki(r) of (6.8), for example, a is given by (6.9), while 13 = 3/n has been obtained [31].

Appendix A. Positive definiteness of ( X ]

H-1LIY) Let us consider a diffusion equation for a distribution function R(X; t)

OR(X;t) b ( a 1 dV(X)) =D-- - + - - - R( X; t), bt aX aX kBT dX (A.1) Which is obtained by removing the reaction term containing ki(X) in (2.1). Since this equation describes only the thermalization process in the reactant state, R(X; t) should approach the distribution function, Pe(X) of (3.1), thermalized there as time t elapses. Similarly to (3.7), R(X; t) is transformed into a(X; t)by

(A.2)

R( X; t ) = g ( X ) a ( X ; t).

Then, (A.1) is transformed into an equation corresponding to (3.8), as

(A.3)

Oa( X; t)/Ot = -Ha(X; t),

with the same H as given by (3.9). Since R(X; t) approaches Pe(X), equal to g(X) 2 by (3.6), as time t elapses, a(X; t) approaches g(X). Since R(X; t) should satisfy the same boundary condition as (3.5) for P(X; t), the same boundary condition as (3.11) for m(X; t) should be satisfied by a(X; t). Let us consider eigenfunctions of H under the same boundary condition as (3.11) for m(X; t). Since H is a Hermitian operator, all its eigenvalues are real numbers. Eigenfunctions can also be regarded as real in the present problem of classical diffusion. The eigenfunction associated with the nth eigenvalue 3,, is written as g,(X), as

Hg,(X)=y,g,(X),

for n = 1 , 2 , 3

.....

(A.4)

Its example is given by g(X) of (3.6) whose eigenvalue is zero as understood from (3.14). We consider, then, that g,(X) denotes an eigenfunction associated with nonvanishing eigenvalues. As in quantum mechanics, g(X) and g,(X) for n = 1, 2, 3 . . . . constitute a complete set in a space of functions satisfying the same boundary condition as (3.11) for m(X; t). Since a(X; t) is an element in

25

H. Sumi / Chemical Physics 212 (1996) 9-27

g(X)

this functional space, it can be expanded by and g,(X) for n = 1, 2, 3 . . . . as

fs(x,

a(X; t) = Cog(X ) + E c , g,(X)exp(--yJ), ii

(A.5)

where c o and c, for n = 1, 2, 3 . . . . are expansion coefficients. This a(X; t) in fact satisfies (A.3). As mentioned earlier, a(X; t) approaches g(X) as time t elapses. This means %>0,

for n = 1 , 2 , 3

.....

From (A.4), the eigenfunction (X I H - IL I Y) is given by

(A.6) expansion

( X I H - I L I Y ) = ]~_,g,(X)g,(Y)/y,,

of

(A.7)

n

where we should note that the summation on the right-hand side does not include a component conceming with g(X) since the component has been excluded by the projection operator L of (3.20). Since (A.6) is satisfied, we get from (A.7)

f d X f dYf(X)(XIH-tLIY)f(Y) =~

f ( X ) g ( X ) dX I T , > 0 ,

for any real function positive definite.

show this, let us note first that (4.8) defining enables us to see

(A.8)

f(X). That is, (X JH-JLI Y) is

Appendix B. A function orthogonal to all the eigenfunctions of B(X, Y) Since B(X, Y) is symmetric and satisfies the boundary condition FB(X, Y)= 0 with F defined by (4.7), B(X, Y) defines an operation of linear mapping within a space of real functions which satisfy this boundary condition. The second term on the right-hand side of (4.9) is an example of the linear mapping by B(X, Y). Let us remember here that in a functional space on which (X ] H - l L I Y) is operated, a state proportional to ] g ) has been excluded by the projection operator L of (3.20) in real functions satisfying the same boundary condition as (3.11) for re(X; t). A similar situation exists for the functional space on which B(X, Y) is operated: to

dr=0,

B(X, Y) (B.1)

irrespective of X. It is convenient, then, to introduce a function ~b(X) by (4.16). Eqs. (B.1) and (4.14) mean that 0 ( X ) is orthogonal to all the eigenfunctions of B(X, Y), as expressed explicitly by (4.19). In the functional space on which B(X, Y) is operated, therefore, a state proportional to ~b(X) has been excluded by the projection operator L of (3.20). Since g(X) has been excluded in the functional space on which (X I H-1LLY) is operated, eigenfunctions, gn(X) for n = l, 2, 3 . . . . in (A.7), of (X ]H-1LJY) can construct a complete set only when they are supplemented by g(X). That is, any real function satisfying the same boundary condition as (3.11) for m(X; t) can be expanded by this complete set. Similarly, since ~b(X) has been excluded in the functional space on which B(X, Y) is operated, eigenfunctions of B(X, Y) can construct a complete set only when they are supplemented by 0 ( X ) . To be more exact, any real function satisfying the same boundary condition as (4.7) for s(X) can be expanded by this complete set. This is possible so long as ki(X) is a continuous nonvanishing function of X since it connects (X [ H - i L I Y) and B(X, Y) in (4.8). It should be noted here that ~b(X) does not satisfy the same boundary condition as (4.7) for s(X), although all the eigenfunctions of B(X, Y ) do, since d [ u ( X ) - ' O ( X ) ] / d X ( = ~-~kTsTd[ki(X)-']/dX} does not necessarily vanish as X-~ _+oc. This type of inconsistencies often occurs in expansion by a complete set of functions at the boundaries for defining them. As a compromise, we can bear in mind that d[ki(X)-l ] / / d X vanishes at X -* _ ~, by modifying ki(X) a little bit, but not affecting the solution p ( X ) to (4.1), which is mainly determined by the bulk part of k~(X). 2 p, Appendix C. Summation of u11

Let us first define an operator B whose coordinate representation is given by B(X, Y) of (4.8). The inverse operator of B can be defined on a subspace which is spanned by all the eigenfunctions

26

H. Sumi / Chemical Physics 212 (1996) 9-27

of B(X, Y), introduced in Section 4, except ~b(X) of (4.16), since operation of B on ~b(X) vanishes as seen from (4.19). This subspace can be projected out by an operator A--- 1 - I q , ) ( ~ 0 I,

OCkTsT/ki(X),

(C.2)

with a form diagonal in the coordinate representation, where H represents the Hermitian operator of (3.9) which is also diagonal in the coordinate representation. Validity of (C.2) can be checked by ascertaining (ARA)B=B(ARA)=A, which can be verified by using B of (4.8) together with ACki(X)/kTs T L Cki(X)//kTs T L and ACkrsr/ki( X) L= ACkrsr/ki( X ) for L defined by (3.20) and A defined by (C.1) with I qJ) of (4.16). In fact, we can see =

ARAB = AR A ~ A R ~

=

= A ~

LH-' LH-'~ L~=A.

Since B(X, Y) can be expanded as (4.14), its inverse operator AR A can be expanded as

ARA = Y'. I ~.)/.t.(~. I,

(C.3)

n

where (~0. I and I qJ.) represent respectively the bra and the ket vector for an eigenfunction ~.(X) of B(X, Y). Then, (4.21) enables us to see that the left-hand side of (4.32) is equal to (ul ARAlu> where (ul and I u> represent respectively the bra and the ket vector for a function u(X) of (4.4). Since A is given by (C.1) and R[u)=0 for R given by (C.2), we get (ul ARAI u) = (ul ~O)2(~b I RI ~b). Here, (ul ~0>2 is equal to o- determined by (4.17) since u(X) and O(X) are respectively given by (4.4) and (4.16). Eq. (C.2) for R with (3.9) for H enables us to express (qJ IRis0> in terms of the thermal-average notation introduced in (4.18), as d

<4, I R I ~,5 =

-Dtrk~sTfk~(X)-'-d-~g( X):

-1 dX

= D kgST 5--2ki(X)-'

(C.1)

where (01 and I~b) represent respectively the bra and the ket vector for qt(X). The inverse operator of B can be expressed as ARA with an operator R. We can show that R is given by R = CkTsT//ki(X)

d ×--ki(X) dX

o, (C.4/

where the second equality is obtained by bearing in mind that dki(X)-l/dX vanishes in the limit of X ~ + ~ without affecting X-integrated terms, such as the right-most term in (C.4), in the same way as a compromise noted in the last paragraph in Appendix B concerning the eigenfunction expansion. Combining these results, we get (5.1) in the main text.

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