- Email: [email protected]
The effect of solvent relaxation dynamics on outer-sphere electron transfer
Volume 133, number 5
30 January 1987
CHEMICAL PHYSICS LETTERS
THE EFFECT OF SOLVENT RELAXATION DYNAMICS ON OUTER-SPHERE ELECTRON TRANSFER
Ilya RIPS and Joshua JGRTNER School of Chemistry, Tel Aviv University, 69978 Tel Aviv, Israel Received 16 September 1986; in tinal form 24 November 1986
We analyse outer-sphere electron transfer (ET) in a dielectric medium, which is characterised by a continuous distribution of dielectric relaxation times. An explicit expression for the averaged ET rate is derived in the particular case of the Davidson-Cole dielectric susceptibility function. A quantitative account of recent experimental data for ET in the Ru(phen#+ methylviologen*+ system in glycerol is provided.
1. Introduction
Recent experimental studies of photochemical [ I], electrochemical [ 21 and thermal [ 31 electron transfer (ET) in polar solvents have established that the ET rate in the adiabatic limit is determined by the longitudinal dielectric relaxation time, TV,of the solvent [ 4-6 1. Kosower and Huppert [ 1 ] have demonstrated that the intramolecular ET rates in aliphatic alcohols are inversely proportional to rL, while McGuire and McLendon [ 31 have shown that intermolecular ET rates in glycerol are proportional to r~‘.‘. Concurrently, theoretical studies of the polar solvent dynamics on the ET rates are emerging [7-l 71. The competition between electronic processes and the medium dielectric relaxation has been addressed by Zusman [ 71 using a stochastic Liouville equation method, by Friedman and Newton [ 111 and by Sumi and Marcus [ 141 utilising the mean-first-passage-time approach, and by us [ 17 ] adopting the real-time path integral formalism. An explicit expression for the ET rate, which combines features of both the adiabatic and the non-adiabatic limits, was derived [ 7,11,14,17 ] for a system which is characterised by the following: (1) The dynamics of the reaction coordinate corresponds to a classical diffusion process. (2) The solvent is characterised by a Debye relaxation function for the dielectric susceptibility,
t(w)=~,+(~,-t,)/(l+ior),
(1)
with a single Debye relaxation time r. Here E,and t, are the static and the optical dielectric constants, respectively. The ET rate, k(r), which depends on the dielectric relaxation time, assumes the form [ 7-l 71 k(~)=kNAICAD(T)I[lCNA+IC*D(r)]
)
(2)
where kNA is the non-adiabatic ET rate, kN*= [21cV”M(4nErkT)1’2]
eXp(
-E,/kT)
,
(3)
with Vbeing the electronic coupling, E, the reorganisation energy of the solvent and EA = ( AE- E,) 2/4E, the activation energy, with AE being the (free) energy gap. km(z) in eq. (2) is the adiabatic ET rate [7-l 71 kAD(r) =Alz, with [ 171 112
exp( -E,IkT)
.
(4)
The r dependence of the ET rate can be recast in the explicit form (5) From eq. (5) it is apparent that kNAIkAD( T) serves as the adiabaticity parameter. For kNA/kAD( 7) 9 1 oneexpectsthatkarc’, wherer,=(e,/e,)r [7-171. The validity of this result has been experimentally
0 009-26 14/87/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
411
30 January 1987
CHEMICAL PHYSICS LETTERS
Volume 133, number 5
established [ 1 ] for intramolecular ET in a series of alcohol solvents which, to a good approximation, are characterised by a Debye relaxation spectrum [ 181. However, the Debye spectrum, eq. (1)) constitutes an exception rather than a rule in real chemical systems. The dielectric relaxation of the vast majority of polar solvents is characterised by a continuous distribution of dielectric relaxation times [ 181. It is therefore not surprising that the effect of solvent dynamics on intermolecular ET in rigid glycerol [ 31, which does not correspond to a simple Debye spectrum [ 181, exhibits serious deviations from relation ( 5 ) . The same state of affairs is also expected to prevail for adiabatic ET in glasses and polymers. In this note we consider ET in a dielectric medium, which is characterised by a continuous distribution of dielectric relaxation times, deriving an explicit expression for the ET rate in such a microscopically inhomogeneous medium.
2. Electron transfer in a microscopically inhomogeneous medium We consider ET in a polar medium, which is characterised by the dielectric relaxation time distribution function, g(r) . An exact expression for g( T) can be obtained from the explicit analytic form for the complex dielectric susceptibility function S(o) [ 181,
co
J’(t)
=Jdr g(r) exp[ -6Wl
,
(7)
0
where k(r) is the rate in the case of the simple Debye spectrum, eq. (5). The time evolution can be expressed in terms of the cumulant expansion F(t) = exp
2 ( - 1)” $E”
( n=l
*
(8)
, >
where K .=-.(d”ldt”)
(exp[ -k(r)t]),=,
.
(9)
Here ( ) denotes averaging over the distribution g(r). The lowest cumulants in eq. (9) are K,=
(lOa)
Kz=(k2(~))-<4~))2
(lob)
The time evolution of the ET kinetics is non-exponential, as is appropriate for dynamics in a microscopically inhomogeneous medium. We shall consider the short-time ET rate, k, which is given by rG=K,, eq. (10a). Under these circumstances the experimentally observable ET rate, k, is given by the statistical average ~ Idr g(r)
k(7)
,
(11)
0
g(r)=[1/2x(e,-E,)T]
xilio
{Im[e( -o+i/r)]
-Im[Qo+i/r)]}
. (6)
Extensive information concerning approximate and empirical expressions for the distribution function, g(r) , is available [ 18 1. At this stage we invoke a basic assumption regarding the inhomogeneity of the medium. We assume that the microscopic environment of each solute molecule is characterised by a single dielectric relaxation time r, while the distribution of the values of T is given by g(r), eq. (6). The time evolution of the ET process DA+D+A-, i.e. the population probability of the initial state DA, is given by the statistical average 412
where k(t) is the rate in the case of the simple Debye spectrum, eq. (5). As a particular example, we consider a medium characterised by the Davidson-Cole (DC) dielectric relaxation spectrum [ 19,201, with the complex dielectric susceptibility function EI(~)=~,+(e~-t,)/(l+ic0r~)~.
(12)
In expression (12) 7. is the characteristic relaxation time, which corresponds to the upper limit of the distribution of 7 values, while 0
Volume 133, number 5
CHEMICAL PHYSICS LETTERS
the DC relaxation spectrum in a closed form. The relaxation-time distribution function in this case is givenby [18]
30 January 1987
From this result two conclusions emerge. First, the dependence of the ET rate on the characteristic relaxation time, To, is I?= cc@
=o (ZBTO). (13) The most important feature of this distribution is the large portion of short relaxation times. Su~titution of eqs. ( 5) and (13 ) into eq. (11) results in the averaged rate for the DC distribution EDC
=A sin(M) ET0
I’
dxxfl-’
o (1 -x)@(x+ l/X) ’
(14)
which leads to the simple expression P=P(l+JE4)--B
)
(15)
where we have introduced R= kNA/km( To)= kNAzdA.The averaged rate can, therefore, be recast in a physically transparent form kDC= kNA[1+ kNAIkAD(ro)] -fl .
(16)
The following implications of this result are apparent: (A) The effective adiabaticity parameter .%= kNA/kAD( To), which with the use of eqs. (3) and (4) assumes the form #z= (e,k,)(47c
I+olftE,)
(17)
and determines the “transition” from the non-adiabatic to the adiabatic limit. (B) In the non-adiabatic limit ( X Q 1) kDc %kNA recovers the well-known golden rule result. (C) The adiabatic limit is characterised by X % 1, so that the rate determining step involves dielectric relaxation of the solvent. In this limit (18)
Accordingly, we expect that in the adiabatic limit Iz”: < kN* , revealing a retardation of the ET rate by the slow rates from the solvent dielectric relaxation. (D) An explicit expression for the adiabatic rate is fFm=
2zP (Ls)Bexp(-EJk7’). fi(4xE,kT)“2 ea, 4Pro
(19)
(20)
and is weaker than in the simple Debye relaxation case. Secondly, the adiabatic rate depends upon the electronic coupling, RDCcc IXW-8)
(21)
This dependence of the ET rate on Vis considerably weaker than in the non-adiabatic limit, eq. (3). This I/ dependence in the adiabatic limit can readily be traced to the contribution of the short relaxation times, for whkh the ET rate is still non-adiabatic, so that a partial dependence of & on Vsurvives after the averaging process. Relations (20) and (2 1) can be compared directly with experiment.
3. Com~son
with experiment
McGuire and McL.endon [ 31 studied ET reactions between Ru( phen,) 2+ homologues and methylviologen2+ dispersed in r&id glycerol in the temperature range 180-253 K. Glycerol was actually the Srst solvent to which the DC expression for r”(o), eq. ( 12)) was applied [ 19,201, In the temperature range studied by McGuire and McLendon the value of the r. parameter varies by about seven orders of magnitude. They found that the ET rate varies with r. approximately as kccz,-0.6 .
(221
A brief inspection of the Davidson and Cole results for glycerol [ 19,201 shows that the value of the j3 parameter exhibits a very weak temperature dependence from B=O.608 at T=233 K to j?=O.SS at T= 198 K Thus, the McGuire-Mc~ndon experimental result is in very good agreement with the qrediction of eq. (20). Since the values of ~~for glycerol in this temperature range are very large (varying from 1.15~10-~ s at T=233 K to 0.96 s at T=l98 K), there is good reason to believe that the effective adiabaticity parameter .#‘, eq. (17), exceeds unity and the adiabatic limit is realised. In the McGuire-McLendon experiment [ 31, intermolecular ET among randomly distributed 413
Volume 133, number 5
CHEMICAL PHYSICS LETTERS
donors and acceptors has been explored, and configurational averaging over the dist~bution of Vvalues has to be performed. Accordingly, these data [3] cannot be utilised to establish the dependence of the ET rate on V, eq. (2 1 ), or to explore the deviations of the ET kinetics from exponential decay at long times, which is in accord with eq. (8). A direct experimental test of our theory should rest on an experimental study of ET in bridged systems where V is fixed.
4, Concluding remarks In this note we have presented an extension of diffusive ET [7] to situations in which the solvent is characterised by a continuous distribution of dielectric relaxation times. Our approach, which rests on statistical averaging, is valid when the dist~bution of the dielectric relaxation times originates from inhomogeneity of the medium. For ET in a medium characterised by the simple Debye relaxation spectrum, the reaction coordinate dynamics represents either uniform motion in the t~sition-site theory limit, or classical diffusion [ 71, provided that the upper limit of the frequency spectrum is properly accounted for [ 171. In the situation of a continuous distribution of dielectric relaxation times the dynamics should be described in terms of the continuous-time random walk approach [ 2 11. The problem will thus be reduced to the solution of the stochastic Liouville equation with the time-dependent diffusion function replacing the diffusion constant. The solution of this problem in the mean-first passage approximation will be of considerable interest.
414
30 January 1987
References [ 11D. Huppert, H. Kanety and E.M. Kosower, Faraday Discussions Chem. Sot. 74 (1982) 161; E.M. Kosower and D. Huppert, Chem. Phys. Letters 96 (1983) 433. [2] M.J. Weaver and T. Gennett, Chem. Phys. Letters 113 (1985) 213; T. Gennett, D.F. Mimer and M.J. Weaver, J. Phys. Chem. 89 (1985) 2787. [3] M. McGuire and G. McLendon, J. Phys. Chem. 90 (1986) 2549. 141 M. Friihlich, Theory of dielectrics (Clarendon Press, Oxford, 1958), ch. 3. [ 51 S. Mozumder, J. Chem. Phys. 50 (1969) 3153. [6] L. Onsager, Can. J. Chem. 55 (1977) 1819. [ 71 L.D. Zusman, Chem. Phys. 49 (1980) 295. [ 81 LV. Alexandrov, Chem. Phys. 5 1 (1980) 449. [9] M.Ya. Gvchinnikova, Tear. Eksp. Khim. 17 (1981) 651 (English transl. Theor. Exp. Chem. 17 (1982) 507). [lo] G. van der Zwan and J.T. Hynes, J. Chem. Phys. 76 (1982) 2993. [ 111 H.L. Friedman and M.D. Newton, Faraday Discussions Chem. Sot. 74 (1982) 73. [ 121 L.D. Zusman, Chem. Phys. 80 (1983) 29. [ 131 D.F. Calef and P.G. Wolynes, J. Phys. Chem. 87 (1983) 3387. [ 141 H. Sumi and R.A. Marcus,J. Chem. Phys. 84 (1986) 4894. [ 151 J.T. Hynes, J. Stat. Phys. 42 (1986) 149. { 161 J.T. Hynes, J. Phys. Chem. 90 (1986) 3701. [ 171 I. Rips and J. Jortner, J. Chem. Phys., submitted for publication. [ 181 C.J.F. Bottcher and P. Bordewijk, Theory of electric polarization, Vol. 2 (Elsevier, Amsterdam, 1978) ch. 9. [19] D.W. Davidson and R.H. Cole, J. Chem. Phys. 19 (1951) 1484. [ 201 D.W. Davidson and R.H. Cole, J. Chem. Phys. 18 ( 1950) 1417. [21] E.W. Montroll and G.H. Weiss, J. Math. Phys. 6 (1965) 167; H. Scher and E.W. Montroll, Phys. Rev. B12 (1975) 2455; G.H. Weiss and R.J. Rubin, Advan. Chem. Phys. 52 (1983) 363.
30 January 1987
CHEMICAL PHYSICS LETTERS
THE EFFECT OF SOLVENT RELAXATION DYNAMICS ON OUTER-SPHERE ELECTRON TRANSFER
Ilya RIPS and Joshua JGRTNER School of Chemistry, Tel Aviv University, 69978 Tel Aviv, Israel Received 16 September 1986; in tinal form 24 November 1986
We analyse outer-sphere electron transfer (ET) in a dielectric medium, which is characterised by a continuous distribution of dielectric relaxation times. An explicit expression for the averaged ET rate is derived in the particular case of the Davidson-Cole dielectric susceptibility function. A quantitative account of recent experimental data for ET in the Ru(phen#+ methylviologen*+ system in glycerol is provided.
1. Introduction
Recent experimental studies of photochemical [ I], electrochemical [ 21 and thermal [ 31 electron transfer (ET) in polar solvents have established that the ET rate in the adiabatic limit is determined by the longitudinal dielectric relaxation time, TV,of the solvent [ 4-6 1. Kosower and Huppert [ 1 ] have demonstrated that the intramolecular ET rates in aliphatic alcohols are inversely proportional to rL, while McGuire and McLendon [ 31 have shown that intermolecular ET rates in glycerol are proportional to r~‘.‘. Concurrently, theoretical studies of the polar solvent dynamics on the ET rates are emerging [7-l 71. The competition between electronic processes and the medium dielectric relaxation has been addressed by Zusman [ 71 using a stochastic Liouville equation method, by Friedman and Newton [ 111 and by Sumi and Marcus [ 141 utilising the mean-first-passage-time approach, and by us [ 17 ] adopting the real-time path integral formalism. An explicit expression for the ET rate, which combines features of both the adiabatic and the non-adiabatic limits, was derived [ 7,11,14,17 ] for a system which is characterised by the following: (1) The dynamics of the reaction coordinate corresponds to a classical diffusion process. (2) The solvent is characterised by a Debye relaxation function for the dielectric susceptibility,
t(w)=~,+(~,-t,)/(l+ior),
(1)
with a single Debye relaxation time r. Here E,and t, are the static and the optical dielectric constants, respectively. The ET rate, k(r), which depends on the dielectric relaxation time, assumes the form [ 7-l 71 k(~)=kNAICAD(T)I[lCNA+IC*D(r)]
)
(2)
where kNA is the non-adiabatic ET rate, kN*= [21cV”M(4nErkT)1’2]
eXp(
-E,/kT)
,
(3)
with Vbeing the electronic coupling, E, the reorganisation energy of the solvent and EA = ( AE- E,) 2/4E, the activation energy, with AE being the (free) energy gap. km(z) in eq. (2) is the adiabatic ET rate [7-l 71 kAD(r) =Alz, with [ 171 112
exp( -E,IkT)
.
(4)
The r dependence of the ET rate can be recast in the explicit form (5) From eq. (5) it is apparent that kNAIkAD( T) serves as the adiabaticity parameter. For kNA/kAD( 7) 9 1 oneexpectsthatkarc’, wherer,=(e,/e,)r [7-171. The validity of this result has been experimentally
0 009-26 14/87/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
411
30 January 1987
CHEMICAL PHYSICS LETTERS
Volume 133, number 5
established [ 1 ] for intramolecular ET in a series of alcohol solvents which, to a good approximation, are characterised by a Debye relaxation spectrum [ 181. However, the Debye spectrum, eq. (1)) constitutes an exception rather than a rule in real chemical systems. The dielectric relaxation of the vast majority of polar solvents is characterised by a continuous distribution of dielectric relaxation times [ 181. It is therefore not surprising that the effect of solvent dynamics on intermolecular ET in rigid glycerol [ 31, which does not correspond to a simple Debye spectrum [ 181, exhibits serious deviations from relation ( 5 ) . The same state of affairs is also expected to prevail for adiabatic ET in glasses and polymers. In this note we consider ET in a dielectric medium, which is characterised by a continuous distribution of dielectric relaxation times, deriving an explicit expression for the ET rate in such a microscopically inhomogeneous medium.
2. Electron transfer in a microscopically inhomogeneous medium We consider ET in a polar medium, which is characterised by the dielectric relaxation time distribution function, g(r) . An exact expression for g( T) can be obtained from the explicit analytic form for the complex dielectric susceptibility function S(o) [ 181,
co
J’(t)
=Jdr g(r) exp[ -6Wl
,
(7)
0
where k(r) is the rate in the case of the simple Debye spectrum, eq. (5). The time evolution can be expressed in terms of the cumulant expansion F(t) = exp
2 ( - 1)” $E”
( n=l
*
(8)
, >
where K .=-.(d”ldt”)
(exp[ -k(r)t]),=,
.
(9)
Here ( ) denotes averaging over the distribution g(r). The lowest cumulants in eq. (9) are K,=
(lOa)
Kz=(k2(~))-<4~))2
(lob)
The time evolution of the ET kinetics is non-exponential, as is appropriate for dynamics in a microscopically inhomogeneous medium. We shall consider the short-time ET rate, k, which is given by rG=K,, eq. (10a). Under these circumstances the experimentally observable ET rate, k, is given by the statistical average ~ Idr g(r)
k(7)
,
(11)
0
g(r)=[1/2x(e,-E,)T]
xilio
{Im[e( -o+i/r)]
-Im[Qo+i/r)]}
. (6)
Extensive information concerning approximate and empirical expressions for the distribution function, g(r) , is available [ 18 1. At this stage we invoke a basic assumption regarding the inhomogeneity of the medium. We assume that the microscopic environment of each solute molecule is characterised by a single dielectric relaxation time r, while the distribution of the values of T is given by g(r), eq. (6). The time evolution of the ET process DA+D+A-, i.e. the population probability of the initial state DA, is given by the statistical average 412
where k(t) is the rate in the case of the simple Debye spectrum, eq. (5). As a particular example, we consider a medium characterised by the Davidson-Cole (DC) dielectric relaxation spectrum [ 19,201, with the complex dielectric susceptibility function EI(~)=~,+(e~-t,)/(l+ic0r~)~.
(12)
In expression (12) 7. is the characteristic relaxation time, which corresponds to the upper limit of the distribution of 7 values, while 0
Volume 133, number 5
CHEMICAL PHYSICS LETTERS
the DC relaxation spectrum in a closed form. The relaxation-time distribution function in this case is givenby [18]
30 January 1987
From this result two conclusions emerge. First, the dependence of the ET rate on the characteristic relaxation time, To, is I?= cc@
=o (ZBTO). (13) The most important feature of this distribution is the large portion of short relaxation times. Su~titution of eqs. ( 5) and (13 ) into eq. (11) results in the averaged rate for the DC distribution EDC
=A sin(M) ET0
I’
dxxfl-’
o (1 -x)@(x+ l/X) ’
(14)
which leads to the simple expression P=P(l+JE4)--B
)
(15)
where we have introduced R= kNA/km( To)= kNAzdA.The averaged rate can, therefore, be recast in a physically transparent form kDC= kNA[1+ kNAIkAD(ro)] -fl .
(16)
The following implications of this result are apparent: (A) The effective adiabaticity parameter .%= kNA/kAD( To), which with the use of eqs. (3) and (4) assumes the form #z= (e,k,)(47c
I+olftE,)
(17)
and determines the “transition” from the non-adiabatic to the adiabatic limit. (B) In the non-adiabatic limit ( X Q 1) kDc %kNA recovers the well-known golden rule result. (C) The adiabatic limit is characterised by X % 1, so that the rate determining step involves dielectric relaxation of the solvent. In this limit (18)
Accordingly, we expect that in the adiabatic limit Iz”: < kN* , revealing a retardation of the ET rate by the slow rates from the solvent dielectric relaxation. (D) An explicit expression for the adiabatic rate is fFm=
2zP (Ls)Bexp(-EJk7’). fi(4xE,kT)“2 ea, 4Pro
(19)
(20)
and is weaker than in the simple Debye relaxation case. Secondly, the adiabatic rate depends upon the electronic coupling, RDCcc IXW-8)
(21)
This dependence of the ET rate on Vis considerably weaker than in the non-adiabatic limit, eq. (3). This I/ dependence in the adiabatic limit can readily be traced to the contribution of the short relaxation times, for whkh the ET rate is still non-adiabatic, so that a partial dependence of & on Vsurvives after the averaging process. Relations (20) and (2 1) can be compared directly with experiment.
3. Com~son
with experiment
McGuire and McL.endon [ 31 studied ET reactions between Ru( phen,) 2+ homologues and methylviologen2+ dispersed in r&id glycerol in the temperature range 180-253 K. Glycerol was actually the Srst solvent to which the DC expression for r”(o), eq. ( 12)) was applied [ 19,201, In the temperature range studied by McGuire and McLendon the value of the r. parameter varies by about seven orders of magnitude. They found that the ET rate varies with r. approximately as kccz,-0.6 .
(221
A brief inspection of the Davidson and Cole results for glycerol [ 19,201 shows that the value of the j3 parameter exhibits a very weak temperature dependence from B=O.608 at T=233 K to j?=O.SS at T= 198 K Thus, the McGuire-Mc~ndon experimental result is in very good agreement with the qrediction of eq. (20). Since the values of ~~for glycerol in this temperature range are very large (varying from 1.15~10-~ s at T=233 K to 0.96 s at T=l98 K), there is good reason to believe that the effective adiabaticity parameter .#‘, eq. (17), exceeds unity and the adiabatic limit is realised. In the McGuire-McLendon experiment [ 31, intermolecular ET among randomly distributed 413
Volume 133, number 5
CHEMICAL PHYSICS LETTERS
donors and acceptors has been explored, and configurational averaging over the dist~bution of Vvalues has to be performed. Accordingly, these data [3] cannot be utilised to establish the dependence of the ET rate on V, eq. (2 1 ), or to explore the deviations of the ET kinetics from exponential decay at long times, which is in accord with eq. (8). A direct experimental test of our theory should rest on an experimental study of ET in bridged systems where V is fixed.
4, Concluding remarks In this note we have presented an extension of diffusive ET [7] to situations in which the solvent is characterised by a continuous distribution of dielectric relaxation times. Our approach, which rests on statistical averaging, is valid when the dist~bution of the dielectric relaxation times originates from inhomogeneity of the medium. For ET in a medium characterised by the simple Debye relaxation spectrum, the reaction coordinate dynamics represents either uniform motion in the t~sition-site theory limit, or classical diffusion [ 71, provided that the upper limit of the frequency spectrum is properly accounted for [ 171. In the situation of a continuous distribution of dielectric relaxation times the dynamics should be described in terms of the continuous-time random walk approach [ 2 11. The problem will thus be reduced to the solution of the stochastic Liouville equation with the time-dependent diffusion function replacing the diffusion constant. The solution of this problem in the mean-first passage approximation will be of considerable interest.
414
30 January 1987
References [ 11D. Huppert, H. Kanety and E.M. Kosower, Faraday Discussions Chem. Sot. 74 (1982) 161; E.M. Kosower and D. Huppert, Chem. Phys. Letters 96 (1983) 433. [2] M.J. Weaver and T. Gennett, Chem. Phys. Letters 113 (1985) 213; T. Gennett, D.F. Mimer and M.J. Weaver, J. Phys. Chem. 89 (1985) 2787. [3] M. McGuire and G. McLendon, J. Phys. Chem. 90 (1986) 2549. 141 M. Friihlich, Theory of dielectrics (Clarendon Press, Oxford, 1958), ch. 3. [ 51 S. Mozumder, J. Chem. Phys. 50 (1969) 3153. [6] L. Onsager, Can. J. Chem. 55 (1977) 1819. [ 71 L.D. Zusman, Chem. Phys. 49 (1980) 295. [ 81 LV. Alexandrov, Chem. Phys. 5 1 (1980) 449. [9] M.Ya. Gvchinnikova, Tear. Eksp. Khim. 17 (1981) 651 (English transl. Theor. Exp. Chem. 17 (1982) 507). [lo] G. van der Zwan and J.T. Hynes, J. Chem. Phys. 76 (1982) 2993. [ 111 H.L. Friedman and M.D. Newton, Faraday Discussions Chem. Sot. 74 (1982) 73. [ 121 L.D. Zusman, Chem. Phys. 80 (1983) 29. [ 131 D.F. Calef and P.G. Wolynes, J. Phys. Chem. 87 (1983) 3387. [ 141 H. Sumi and R.A. Marcus,J. Chem. Phys. 84 (1986) 4894. [ 151 J.T. Hynes, J. Stat. Phys. 42 (1986) 149. { 161 J.T. Hynes, J. Phys. Chem. 90 (1986) 3701. [ 171 I. Rips and J. Jortner, J. Chem. Phys., submitted for publication. [ 181 C.J.F. Bottcher and P. Bordewijk, Theory of electric polarization, Vol. 2 (Elsevier, Amsterdam, 1978) ch. 9. [19] D.W. Davidson and R.H. Cole, J. Chem. Phys. 19 (1951) 1484. [ 201 D.W. Davidson and R.H. Cole, J. Chem. Phys. 18 ( 1950) 1417. [21] E.W. Montroll and G.H. Weiss, J. Math. Phys. 6 (1965) 167; H. Scher and E.W. Montroll, Phys. Rev. B12 (1975) 2455; G.H. Weiss and R.J. Rubin, Advan. Chem. Phys. 52 (1983) 363.