5911
J. Phys. Chem. 1993,97, 5911-5916
Generalization of the Marcus Equation for the Electron-Transfer Rate M. Tachiya' National Institute of Materials and Chemical Research, Tsukuba, Ibaraki 305, Japan, and Department of Chemistry, University of Tsukuba, Tsukuba, Ibaraki 305, Japan Received: December 28, 1992; In Final Form: March 9, 1993
The Marcus equation for the electron-transfer rate is generalized to the form which is applicable to molecular models of the solvent. The rate constant is expressed in terms of the function 4(AV,), which describes the distribution of the electrostatic potential difference AV between donor and acceptor sites, produced by the surrounding fluctuating polar solvent molecules. This expression clearly shows that the functional dependence of the rate constant on the free-energy change of the reaction (the so-called energy gap law) reflects the potential difference distribution $ ( A n . The new expression reduces to the well-known Marcus equation, if it is combined with the potential difference distribution calculated on the basis of the dielectric continuum model. A clear physical explanation is given for the reorganization energy which appears as a parameter in the Marcus theory. The applicability of the new expression is not limited to the dielectric continuum model. The new expression can be used to improve the Marcus equation by combining it with the potential difference distribution calculated on the basis of a molecular model of the solvent.
1. Introduction The rate of electron transfer fromdonor to acceptor in a solvent is controlled by severalfactors such as the donor-acceptor distance, the free-energy change of the reaction, the polarity of the solvent, the dynamical property of the solvent, and the temperature. Marcus',* was the first to study theoretically how the rate depends on these factors. According to his theory, the effect of solvents enters into the rate expression only through the so-called reorganization energy, the physical meaning of which is not so clear. In the Marcus theory, the solvent is described by using the dielectric continuum model. Without doubt, the dielectric continuum model is an oversimplification. It does not take into account the molecularity of the solvent. Furthermore, it usually does not take into account the nonlinear effect, although it is in principle possible to take into account it even within the framework of the dielectric continuum model. In the present paper, we generalize the Marcus theory and derive a new general expression for the electron-transfer rate, which is applicable not only to the dielectric continuum model but also to more realistic molecular models of the solvent. By using this expression,westudy the effect of solvents on the electrontransfer rate. In section 2, we consider the fluctuation of local electrostatic potential produced by the surrounding fluctuating polar solvent molecules and express the rate constant in terms of the function 4(AV) which describes the distribution of the electrostatic potential difference AVbetween donor and acceptor sites. In section 3, we show for illustration how to calculate the distribution of the potential differenceon the basis of the dielectric continuum model. The calculation of the potential difference distribution on the basis of a molecular model of the solvent will be published in a forthcoming paper. Discussions are given in section 4. We show that the functional dependence of the rate constant on the free-energychange reflects the potential difference distribution function $(A?'). We also show that our general expression reduces to the Marcus equation, if it is combined with the potential difference distribution calculated on the basis of the dielectric continuum model.
* Sendcorrespondence to the National Instituteof Materialsandchemical Research. 0022-3654/93/2097-5911$04.00/0
2. Formulation of the Electron-TransferRate Consider neutral solutes A and B dissolved in a polar solvent as shown in Figure 1. Here the arrows stand for the permanent dipoles of randomly oriented solvent molecules. These dipoles produce the electrostatic potentials VAand VBat positions A and B. In our model, the solvent molecules are not frozen. They are allowed to move both translationally and rotationally. The electrostaticpotentialsfluctuate with time as the solvent molecules randomly move both translationally and rotationally. Let us consider the potential difference eAV between A and B sites, which is given by eAV = e( VA- VB)
(2.1) The potential difference also fluctuates with time as illustrated in Figure 2. In other words, the potential difference can take a variety ofvalues. Let 4(eAV)d(eAV)denote the probability that the potential difference will take a value between eAVand eAV + d(eAV)?.Hereafter, the probability density 4(eAV) is referred to as the potential difference distribution. To our knowledge, no one has ever thought of this distribution explicitly. However, this distribution turns out to play a very important role in electrontransfer reactions. The potential difference distribution depends on the charge statesof the solute pair. If A and Bare both neutral, the potential differencefluctuates around zero. On the other hand, if the solute pair is an ion pair, it fluctuates around a certain value which is not zero. We denote the potential difference distribution for the neutral pair (AB) by 4AB(eAV) and that for the ion pair (A+B-) by f$A+B-(eAV),reSpectiVe1)'. Now let us consider electron transfer from solute A to B. We first consider charge separation in which electron transfer occurs from a neutral donor to a neutral acceptor. In the initial state, A and B are both neutral, so the interaction energy between the solute pair and the solvent may be neglected as a first approximation. The energy of the system is then given by
EA, n(ri, rz, PZ, -..)rN, P N ) (2.2) Here II stands for the interaction energy between the permanent dipoles of the solvent molecules. r, and pi stand for the position and orientation of the ith dipole. In the final state, solutes A and B have a positive charge e and a negative charge -e, respectively, so the interaction energy between solute A (or B) and the solvent is given by eVA (or -eve). Therefore, the interaction energy 0 1993 American Chemical Society
Tachiya
5912 The Journal of Physical Chemistry, Vol. 97,No. 22, 1993
/ /
n
/
\n
w
a
-+ V
Figure 1. Solutes A and B dissolved in a polar solvent. The arrows stand for the permanent dipoles of randomly orientated solvent molecules. These dipoles produce the electrostatic potentials VAand VBat the positions of A and B.
A
-
e AV* poten'tial d i f f e r e n c e eAV Figure 3. Relation between the distribution of the potential difference eAV and the electron transfer. The full line stands for the equilibrium distribution. The electron transfer occurs at a specific value e A P of the potential difference. The electron transfer at e A P deforms the distribution from the equilibrium one, as schematically shown by the broken line. However, if the rateof fluctuation of the potential difference is sufficiently fast compared with the rate of electron transfer, the equilibrium distribution is maintained.
time t
Figure 2. Fluctuation of the potential difference eAVwith time 1. The coincidence of the initial state energy with the final state energy occurs when the potential difference takes a specific value e A P given by eq 2.4.
between the solute pair and the solvent is given by eAV, which is defined by eq 2.1. The interaction energy between the dipoles of the solvent molecules is still given by II (rl, PI, rz, PZ, ..., rN, pn). Accordingly, the energy of the final state is given by EA+B-
= n(rl,
P I , r2, P2,
rN, PN)
+ e'
IP - EA + eAV- Agz - E (2.3) where IP and EA stand for the ionization potential of A and the electron affinity of B, respectively. Agz stands for the solvation energy of the produced ion pair due to the electronic polarization of the solvent. R stands for the distance between A and B, and -e2/R is the Coulomb attractive energy between the ion pair. The potential differenceeAVfluctuates with time as shown in Figure 2. The final state energy (eq 2.3) fluctuates relative to the initial state energy (eq 2.2) as eAVfluctuates. The electron transfer occurs when the final state energy coincides with the initial state energy. The coincidence occurs when the fluctuating potential difference passes through a specific value given by eAv'
e.
-1P + EA + Agi + R
system will hop from the reactant surface to the product one is, according to the Landau-Zener theory,34 given by
P = 1- e x p ( - 2 ~ ~ ~ / h J & l )
(2.5)
where J stands for the interaction energy between the initial and the final states. In the case of electron transfer, this energy is often called the transfer integral. hE stands for the difference between the final state and the initial state energies, and Estands for its time derivative. In the present case, one obtains A,!? = eAV by noting eqs 2.2 and 2.3, so eq 2.5 leads to P((eAq) = 1 - exp(-2~J'/h(eAq)
(2.6) Since the rate constant is equal to the frequency that the system passes through the intersection region, multiplied by the probability that the system will hop to the product surface on passing, it is expressed as
kcs = S_~dAB(eAv')leAqP(leAq)U(eAt?d(eAi3 (2.7) If the transfer integral is small (nonadiabatic reactions), eq 2.6 is approximated by P(leAq) = 2rJ2/hleAq
(2.8)
In this case, the rate constant is expressed as
(2.4)
which is obtained by equating eq 2.3 to eq 2.2. Accordingly, the electron transfer occurs when the fluctuating potential difference passes through the specific value eAV. The frequency where the fluctuatingpotential difference passes through the value eAV is given by the probability density of having the potential difference equal to eAV, multiplied by the rate of change of the potential difference at eAV. In charge separation, the solute pair is initially a neutral pair, so the probability density to be used is that for the neutral pair. So if the rate of change of the potential difference at eAV is denot&d by eAV, the frequency of passing is given by 4AdeAV)leAV. Becauseof the stochastic natureof the fluctuationof the potential difference, the rate of change of the ptential difference has a distribution which is denoted by u(eAv). When thesystem passes through theintersectionregion between the reactant and the product surfaces, the probability that the
In the absence of electron transfer from A to B, we have the equilibrium distribution of the potential difference. In the presence of electron transfer, the distribution of the potential difference is in general deformed from the equilibrium one because the electron transfer occurs only at the specific value eAV of the potential difference (Figure 3). However, if the rate of fluctuation of the potential difference is sufficiently fast compared with the rate of electron transfer at eAY1, the equilibrium distribution of the potential difference is maintained even in the presence of electron transfer. The present paper deals with this situation. In this case, one can use the equilibrium distribution for +AB in eq 2.9. The other situation in which the rate of fluctuation of the potential difference is slow compared with the rate of electron transfer at eAV is also a current topi~.~-IIIn this case, the distribution of the potential difference is deformed from the
The Journal of Physical Chemistry, Vol. 97,No. 22, 1993 5913
Generalization of the Marcus Equation equilibrium one, and one cannot use the equilibrium distribution for AB in eq 2.9. In this case, the electron-transfer rate is influenced by the rate of fluctuation of the potential difference. This latter situation will be treated in a separate paper. Equation 2.9 leads to the following equation by noting eq 2.4
+ EA + Agz + -) e2 R
kcs = T~AB(-IP 2rJ2
(2.10)
It may be more convenient to express the above equation in terms of the free-energy change AG of the reaction. The free-energy change for charge separation is given by e' AGCS = IP - E A - A&- A & - E
(2.1 1)
where Ag: is the solvation energy of the produced ion pair due to the orientational polarization of the solvent. With the aid of eq 2.11, eq 2.9 is rewritten as (2.12) In the case of charge recombination which is the reversereaction of charge separation, the solute pair is initially an ion pair, so the potential difference distribution to be used is that for the ion pair. Accordingly, the rate constant is given by
which corresponds to eq 2.10 for charge separation. In terms of the free-energy change which is given by AGcR = -1P
+ EA + Ag: + Ag: + Ee'
(2.14)
the rate constant is expressed as 2rJ2
~ C = R 74A+B-(AGCR - Ag:)
(2.15)
In the case of charge shift which is described by A- + B A B-, the initial and final state energies are respectively given by
+
+
EA-B = n(rl,P I , r2, PZ,
e..,
rN9
PN) - EA(A) - eVA - Ah:(A) (2.16)
EAB-= W
I ,
PI,
r2,~
...,
2 , rN, PN)
-EA@) - eV, - Ah:@) (2.17)
where EA(X) is the electron affinity of solute X and Ah:(X) is the solvation energy of X- ion due to the electronic polarization. Therefore, the energy coincidence occurs for the potential difference given by eAv'
EA(B) - EA(A)
+ Ah:(B)
- Ah:(A) (2.18)
The potential difference distribution to be used is that for the case where solute A has a negative charge -e and solute B is neutral. Accordingly, the rate constant is given by 2?rJ2 kcsH = h @ A . B [ E A ( B ) - EA(A)
+ Ah:(B)
- Ah,"(A)] (2.19)
The free-energy change is given by AGC,, = EA(A) - EA(B)
+ Ah:(A)
- Ah:(B) + Ah:(A) - Ah:(B) (2.20)
where Ah:(X) is the solvation energy of X- ion due to the orientational polarization. With the aid of eq 2.20, the rate
constant is expressed as
Equations 2.12, 2.15, and 2.21 are the main results of the present paper. They express the rate constant in a very general form in terms of the function 4(eAV), which describes the distribution of the potential difference eAV between donor and acceptor sites, produced by the surrounding fluctuating polar solvent molecules. These equations are equivalent to the golden rule expression1*of the rate constant, in which 4 stands for the Franck-Condon-averaged density of states. These equations may be obtained more easily by using the golden rule. However, we took the present approach because it can be extended to include the case where the electron-transfer rate is influenced by the rate of fluctuationofthe potentialdifference. The relationshipbetween these expressionsand the Marcus equationZis discussed in section 4.
Equations 2.12, 2.15, and 2.21 are applicable not only to the dielectric continuum model but also to any molecular model of the solvent. These equations can be used to improve the Marcus equation by combining them with the potential difference distribution which is calculated on the basis of a molecular model of the solvent by using a method similar to those used by M a r ~ n c e l l i and I ~ ~ other p e o ~ l e . I ~ ~ g The idea of fluctuating energy levels of donor and acceptor with time evolution of the solvent is not new and is already discussed by Marcus and S ~ t i n . 'Friedman ~~ and Newtonl4b described the electron-transfer rate in terms of fluctuating electric field. However, their treatment is limited to the dielectric continuum model. Hopfiledl" described the electron-transfer rate in terms of the convolution of the electron-detachment spectrum of the donor and the electron-attachment spectrum of the acceptor. However, if the interaction of the donor and the acceptor with the solvent is important, the detachment spectrum of the donor and the attachment spectrum of the acceptor are correlated especially when the donor and the acceptor are close to each other, because they are both affected by the same solvent configuration around the donor and acceptor pair. In this case, it is not appropriate to describe the electron-transfer rate in terms of the convolutionof the detachment spectrum and the attachment spectrum. The transfer rate is more appropriately described in terms of the distribution of the potential difference. In the conventional theory, the electron-transfer rate is formulated in terms of the reaction coordinate, which is defined as the energy difference of two electronic states. Here we have formulated the electron-transfer rate without introducing the reaction coordinate. In our formulation, the potential difference eAV plays the same role as the reaction coordinate in the conventional theory. The electron-transfer rate has been calculated for more realistic models of the solvent by using the conventional theory.I3'+g In the conventional theory, the rate constant and its energy gap dependence are calculated in the following way. First, the distributionof the potential differenceis calculated by using Monte Carlo or molecular dynamics simulations. Then the free energies for two electronic states are calculated from the distribution of the potential difference, as a function of the energy difference of the two states, and thereaction coordinatediagram is constructed. The activation energy is calculated from the crossing point of the two free-energy curves. Finally, the rate constant and its energy gap dependence are calculated from the activation energy. In contrast, in the present theory, the rate constant and its energy gap dependence can be calculated in a much more straightforward way. According to eqs 2.12, 2.15, and 2.21, one can calculate the rate constant and its energy gap dependence immediately, once one calculates the distribution of the potential difference. As already stated, in our model, the solvent molecules are allowed to move both translationally and rotationally. In this
5914 The Journal of Physical Chemistry, Vol. 97, No. 22, I993
Tachiya
sense, our model is different from the model of ref 15 in which the solvent molecules are frozen.
With the aid of eq 3.8, the probability that the electrostatic potential will lie between VAand VA + dVA is expressed as
3. Calculation of the Potential Difference Distribution Here we show for illustration how to calculate the distribution of the electrostaticpotential difference on the basis of the dielectric continuum model in which the solvent is characterized by the static dielectric constant e, and the optional dielectric constant cop. The calculation of the potential difference distribution on the basis of a molecular model of the solvent will be published in a forthcoming paper.16 We assume that solutes A and B are spherical with radii a and b, respectively. They are separated by a distance R. First, we calculate the distribution of the electrostatic potential at the position of A. This can be done by using an approach first developed by us17-19 and Marcus2" and followed by other people.20bJJ In the dielectric continuum model, the geometrical configuration of solvent molecules is described by the polarization function P(r) which stands for the orientational polarization at a position r. A polarization field described by P(r) produces the electrostatic potential VA at A, which is given by
c$( V,) dVA = (27rkBT(eOp-'- ei1)/a)-'j2 X
According to this equation, the distribution of the local electrostatic potential is Gaussian. Maroncelli13* calculated the same distribution for acetonitrile on the basis of a molecular model of the solvent. The distribution of theelectrostatic potential differencebetween A and B can be calculated essentially in the same manner. The electrostatic potential difference is given by
AV=-spd;f(r)(---)dr r - r A
I.-
rAt3
r-rB
I.-
(3.10)
ret 3
where r A and rB stand for the positions of A and B, respectively. The polarization function P(r) which minimizes the free energy n, subject to the requirement of eq 3.10, can be determined by minimizing the following functional
On the other hand, the free energy of a polarization field described by P(r) is given
ll = 2?r(e0p1- e;1)-1J>a[P(r)]2dr
(3.2)
where we have neglected the screening effect of the electronic polarization. If this effect is taken into account, P(r) in eqs 3.1 and 3.2 is replaced by P(r)/cop20b However, the final result (eq 3.8) remains the same. Polarization fields described by different P(r)'s may produce the same electrostatic potential at A. Of these fields, thermodynamically realized is the one which minimizes the free energy n. According to Euler's variational principle,21the polarization function P(r) which minimizes the free energy n, subject to the requirement of eq 3.1, can be determined by minimizing the following functional
By minimizing Fwith respect to thevariation of P(r),one obtains
Substitution of eq 3.12 into eqs 3.2 and 3.10, followed by elimination of a,yields
(3.13) where
r-rA
r-rB
F = 2r(e0p1- C ; ~ ) - ' J , , [ P ( ~ )d]r~t
)
2
dr
(3.14)
which is approximated to a good approximation by22 where a is a constant to be determined. Minimization of F with respect to the variation of P(r) leads to the following Euler equation21
4* -1
cop
- (8
-lP(r) - CY-
r
=0
(3.15) With the aid of eq 3.13 together with eq 3.15, the distribution of the electrostatic potential difference is expressed as
(3.4)
143
[
1 exp -(4?rk~n)"~ 4kBTA
6AB(eAV
from which one obtains
(3.16)
where
(3.5) Substitution of eq 3.5 into eqs 3.1 and 3.2 yields
2
-1-
-
(3.17)
In the case where A and B have a positive charge e and a negative charge -e, respectively, the free energy of the system is given by
(3.7) By eliminating CY from eqs 3.6 and 3.7, one obtains
n = qeop-l
- -1 -1 vA2 (3.8) 2 Equation 3.8 gives the free energy of the polarization field which produces an electrostatic potential VAat the position of A.
where the second term on the right-hand side stands for the interaction energy between the charges and the polarization field. Following the previous treatment, one obtains for the distribution
The Journal of Physical Chemistry, Vol. 97, No.22, 1993 5915
Generalization of the Marcus Equation
3.16 or eq 2.15 with eq 3.19, one obtains ,A= 0 5 cV /,A.
1.0
,.--
'\
,'
L
I
I ,
/#
I
-3
-2
-I
t
(4.2) By noting that in the dielectric continuum model the solvation energy of the ion pair due to the orientational polarization is equal to X, the above two equations are unified into
e aV(eV)
Figure 4. Potential difference distribution 4(eAv) for the neutral pair (full lines) and for the ion pair (broken lines), calculated on the basis of the dielectric continuum model. The distribution is shown for several values of the parameter X which is given by eq 3.17.
of the potential difference
Figure4 shows the potentialdifferencedistributionfor a neutral pair (full lines) and for anion pair (broken lines). Thedistribution is shown for several values of the parameter A. According to eq 3.17, the parameter X depends on the polarity of the solvent and the geometrical configuration of the pair through top-'- cs-l and l / a l/b-2/R,respectively. Xincreaseswithincreasingpolarity e, and with increasing pair separation R. As X increases, the distribution becomes broader. Accordingly, the distribution becomes broader with increasing pair separation. The physical interpretation for this broadening is as follows. The electrostatic potentials at A and B are produced by the same polarization field, so they are correlated. As the pair separation increases, they are less correlated. In other words, they can take more different values. So the distribution of the potential difference becomes broader. The distribution also becomes broader with increasing polarity of the solvent. Figure 4 also shows that in the case of the neutral pair, the potential difference fluctuates around zero, irrespective of the value of A. However, in the case of the ion pair, the peak of the distribution shifts toward lower values of eAVwith increasing X. For charge shift, we need the distribution of the potential difference in the case where solute A has a negative charge -e and solute B is neutral. In this case, the free energy of the system is given by
+
Following the previous treatment, one obtains for the distribution of the potential difference
where (3.22) 4. Discussion
A. Relationship to tbe Marcus Theory. The Marcus theory1V2 for electron transfer is based on the dielectric continuum model, while our theory is applicablenot only to the dielectric continuum model but also to any model of the solvent. So our theory is a generalization of his theory. If one combines eq 2.12 with eq
k e -(417kgn)-It2 2rJ2
h
eXp[- (AG 4kBn + AI2]
(4.3)
This is the well-known Marcus equation.2 So our theory recovers the Marcus equation, if it is combined with the dielectric continuum model. In the case of charge shift, one obtains by combining eq 2.21 with eq 3.21
k,,,
= ?(4rkBTX)-'/' CXP[ -
X
(AGCSH- Ah:(A) + Ah:@) + P(A))' 4 k ~ n
] (4.4)
In the dielectric continuum model, the solvation energies of the A- ion and that of the B- ion due to the orientational polarization are given by ( 1/2)e2(c,ft - ss-I)/a and ( 1/2)e2(eop-' ss-l)/b, respectively. On the other hand, p(A) is given by eq 3.22. Accordingly, eq 4.4 is again reduced to the Marcus equation (eq 4.3). In the Marcus theory, the parameter X is called the reorganization energy. However, its physical meaning is not so clear. By reflecting how eq 4.3 has been derived from, for example, q 2.12, one realizes that the X's on the right-hand side of eq 4.3 stand for different physical quantities. The first one and the one in the denominator stand for the width of the Gaussiandistribution of the potential difference. In the case of charge separation, A in the numerator stands for the solvation energy of the produced ion pair due to the orientational polarization. When the nonlinear effect of solvent polarization is important, the former X cannot be defined because in this case the distribution of the potential difference is not Gaussian any more. However, the latter A can still be defined even in this case. The reorganization energy may alternatively be defined as half of the Stokes shift in charge-transfer spectra. The reorganization energy so defined and the X's in the Marcus equation are all equal in the linear response regime. However, they stand for different physical quantities and are not equal when the nonlinear effect is important. B. Energy Gap Law. The dependence of the electron-transfer rate on the free-energy change is often called the energy gap law. According to the Marcus equation, this dependence should be bell-shaped. That is, as AG is decreased,the rate constant should first increase, go through the maximum, and then decrease. The region where the rate constant decreases with decreasing AG is called the inverted region. The bell-shaped dependence of the rate constant on AG has been observed for charge shift23 and for charge recombination.** However, for charge separation, the rate constant remains constant25 even for very low AG. In other words, the inverted region has not been observed for charge separation. Somepeoplez6 attributed the reason for the lack of the inverted region in charge separation to the dielectric saturation effect. However, this explanation is not correct as shown in the following. Equation 2.12 shows that the functional dependence of the rate constant on AG reflects the distribution function c$(eAV) of
-
5916 The Journal of Physical Chemistry, Vol. 97, No. 22, 1993
Tachiya
the potential difference. In other words, the profile of the curve The simplest way to calculate the potential difference distriof the rate constant against AG should be the same as that of the bution is to use the dielectric continuum model. Our expression reduces to the well-known Marcus equation, if it is combined potential difference distribution. The curves of the potential with the potential difference distribution calculated on the basis difference distribution in Figure 4 were obtained on the basis of of the dielectric continuum model. So our theory is a generalthe dielectric continuum model, namely, by neglecting the ization of the Marcus theory. A clear explanation has been given dielectric saturation effect. The dielectric saturation effect is not important in a region around eAV = 0, where the solvent for the reorganization energy which appears as a parameter in the Marcus theory. The Marcus theory can be improved by polarization is very small. So even if the dielectric saturation combining our theory with the potential difference distribution effect is taken into account, the top region of the distribution calculated on the basis of a more realistic model of the solvent curve for the neutral pair is little a f f e ~ t e d . However, ~ ~ , ~ ~ the tail by using molecular dynamics simulations. The calculation of the region deviates downward from the original curve, because the potential difference distribution on the basis of a molecular model dielectricsaturation effect makes it difficult for a large magnitude of the solvent will be published in a forthcoming paper.16 of potential difference to be produced. As a consequence, the dielectric saturation effect causes a slight narrowing of the potential difference distribution curve and, accordingly, a slight References and Notes narrowing of the rate constant curve. In other words, the lack (1) Marcus, R. A. J . Chem. Phys. 1956, 24, 966. of the inverted region is by no means accountedfor by the dielectric (2) Marcus, R. A. Annu. Rev. Phys. Chem. 1964, 15, 155. saturation effect. Thishas already been pointed out e l ~ e w h e r e . ~ ~ * ~ * (3) Nikitin, E. E. TheoryofElementary AtomicandMolecularProcesses A new explanation for the lack of the inverted region has been in Gases; Clarendon: Oxford, 1974; pp 99-178. (4) Ulstrup, J. Charge Transfer Processes in Condensed Media; Springproposed in our recent paper.29 er: Berlin, 1979. C. Solvent Effect. According to eqs 2.10 and 2.13, the rate ( 5 ) Brunschwig,B. .%;Logan,J.;Newton, M. D.;Sutin, N. J . Am. Chem. constant depends on the solvent through the potential difference Soc. 1980, 102, 5798. (6) Newton, M. D.; Sutin, N. Annu. Rev.Phys. Chem. 1984,35, 437. distribution function 4(eAv) and the solvation energy Ag: of the (7) Zusman, L. D. Chem. Phys. 1980,49, 295. ion pair due to the electronic polarization. If one neglects the (8) Calef, D. E.; Wolynes, P. G. J . Phys. Chem. 1983, 87, 3387. very weak dependence of Agi on the solvent, the solvent effect (9) Sumi, H.; Marcus, R. A. J. Chem. Phys. 1986,84, 4894. on the rate constant is exclusively described by the potential (10) Rips, I.; Jortner, J. J . Chem. Phys. 1987, 87, 2090. difference distribution function. Assume that the rate constant (11) Onuchic, J. N.; Wolynes, P. G. J. Phys. Chem. 1988, 92, 6495. (12) Jortner, J. J. Chem. Phys. 1976,64, 4860. is plotted against AQ = -1P EA + Ag: e 2 / Rfor the same ( 1 3) (a) Maroncelli, M. J . Chem. Phys. 1991,94,2084. (b) Warshel, A,; solvent. Since the potential difference distribution function Parson, W. W. Annu. Rev. Phys. Chem. 1991,42,297. (c) Kuharski, R. A.; becomes broader with increasing polarity of the solvent, the rate Bader, J. S.;Chandler, D.; Sprik, M.; Klein, M. L.; Impey, R. W. J . Chem. constant curve is also expected to become broader with increasing Phys. 1988,69, 3248. (d) Marchi, M.; Chandler, D. J. Chem. Phys. 1991, 95,889. (e) Carter, E. A.; Hynes, J. T. J. Phys. Chem. 1989, 93, 2184. (f') polarity. In the case of charge separation, the peak of the rate Schulten,K.;Tesch,M. Chem. Phys. 1991,158,421. (g) Levy,R. M.;Belhadj, constant curve is expected to remain at the same position (AQ M.; Kitchen, D. B. J. Chem. Phys. 1991, 95, 3627. = 0), irrespectiveof the polarity, because the peak of the potential (14) (a) Marcus, R. A.; Sutin, N. Biochem. Biophys. Acta 1985,81I,265. (b) Friedman, H. L.; Newton, M. D. Faraday Discuss. Chem. Soc. 1986,74, difference distribution function for the neutral pair is always 73. (c) HopTield, J. J. Proc. Narl. Acad. Sci. U.S.A. 1974, 71, 3640. located at eAV = 0, irrespective of the polarity (see Figure 4). (15) Marcus, R. A. J . Phys. Chem. 1990, 94, 4963. This behavior has been observed e~perimentally.~~ On the other (16) Hilczer, M. J.; Tachiya, M. To be published. hand, in the case of charge recombination, the peak of the rate (17) Tachiya, M.; Tabata, Y.; Oshima, K. J . Phys. Chem. 1973,77,263. constant curve is expected to shift toward lower values of AQ (18) Tachiya, M.; Tabata, Y.; Oshima, K. J . Phys. Chem. 1973,77,2286. (19) Tachiya, M. J. Chem. Phys. 1974,60, 2275. with increasing polarity, because the peak of the distribution (20) (a) Marcus, R. A. Faraday Symp. Chem. Soc. 1975, 10, 60. (b) function for the ion pair shifts toward the lower values of eAV Eflima, S.;Bixon, M. J . Chem. Phys. 1976, 64, 3639. with increasing polarity. (21) Morse, P. M.;Feshbach, H. Methods of Theoretical Physics;
+
+
5. Concluding Remarks We have formulated the electron-transfer rate in terms of the fluctuation of local electrostatic potential in polar solvents and expressed the rate constant in a very general form by using the distributionof the potential differencebetween donor and acceptor sites, which is produced by the surrounding fluctuating polar solvent molecules. This expression shows that the functional dependence of the rate constant on the free-energy change (the so-called energy gap law) reflects the distribution of the potential difference. Since the energy gap law reflects the distribution of the potential difference in the solvent, the energy gap law should change from solvent to solvent. It is interesting to examine e~perimentally3'-~~ the energy gap law in a variety of solvents of different polarity.
McGraw-Hill: New York, 1953; pp 276-280. (22) Kharkats, Yu. I. Elektrokhim. 1973, 9, 881. (23) Miller, J. R.;Calcaterra, L. T.;Closs, G. L. J . Am. Chem.Soc. 1984, 106, 3047. (24) Gould, I. R.;Ege, D.;Moser, J. E.; Farid,S. J . Am. Chem.Sor. 1990, 112,4290. (25) Rehm, D.; Weller, A. Zsr. J . Chem. 1970, 8, 259. (26) Kakitani, T.; Mataga, N. J . Phys. Chem. 1985, 89, 8 . (27) Tachiya, M. Chem. Phys. Lett. 1989, 159, 505. (28) Tachiya, M. J. Phys. Chem. 1989, 93, 7050. (29) Tachiya, M.; Murata, S.J . Phys. Chem. 1992, 96, 8441. (30) Wasielewski, M. R. Presented at 1st International Symposium of
Graduate University for Advanced Studies. (31) Heitele, H.; Finckh, P.; Weeren, S.;Pollinger, F.; Michel-Byerle, M. E. J . Phys. Chem. 1989, 93, 5173. (32) Schmidt, J. A.; Liu, J.-Y.; Bolton, J. R.; Archer, M. D.; Gadzekpo, V. P. Y. J. Chem. Soc., Faraday Trans. 1 1989, 85, 1027. (33) Chen, J.-M.; Ho, T.-I.; Mou, C.-Y. J. Phys. Chem. 1990, 94, 2889.
