J. Phys. Chem. 1995,99, 1478-1483
1478
Comparison of Electron TransferDiffusion Models As Applied to Fluorescence Quenching Data Gordon L. Hug*,+and Bronislaw MarciniakT9t Radiation Laboratory, University of Notre Dame, Notre Dame, Indiana 46556, and Faculty of Chemistry, A. Mickiewicz University, 60-780 Poznan, Poland Received: September 29, 1994; In Final Form: November 12, 1994@ The fluorescence quenching data in solution compiled by Rehm and Weller (Zsr. J. Chem. 1970 8, 259) are analyzed systematically. Four different electron-transfer models are applied to the activated reaction step, and two different diffusion schemes are used to account for the transport of the reactants. The data are fit to the equations describing the combined activationltransport mechanisms. Two physical parameters are chosen from each combination for optimization in a nonlinear least-squares procedure. Other physical quantities needed to complete the calculation are estimated, assumed, or taken from measured quantities. The results are discussed and applied to the question of why, for charge-separation reactions in solution, there is an absence of a Marcus inverted region in the regime of high exothermicity in rate constant vs free energy correlations.
Introduction There has been concem since the Rehm-Weller paper' (describing fluorescence quenching rate constants for several typical donor-acceptor systems in acetonitrile solution) that the Marcus free energy r e l a t i ~ n ~was . ~ not valid because the quenching data did not show the predicted decrease of the electron-transfer quenching rate constant at large negative free energy change. Theoretical attempts4 to demonstrate plateau behavior of the Rehm-Weller data at large exothermicities of electron transfer showed an enhanced quenching rate constant in the Marcus inverted region but did not predict a plateau region at large exothermicities. Subsequently, the Marcus inverted region was nicely established by experimentP7 on systems having electron donors and acceptors that were hindered from diffusing relative to one another. The experimental studiess-l0 on back electron transfer of the radical ion pairs also showed the Marcus inverted region, but this again can be considered a hindered diffusion system. The classification of electron-transfer reactions vis-&-vis diffusion has already been discussed by Suppan.' The rationalization for the lack in an obvious inverted region in diffusion studies of electron-transfer reaction has generally been ascribed to at least four different reasons. One is that diffusion is usually the rate-limiting step.I2 The reactants cannot transfer an electron over a very large distance and must anive in the vicinity of each other. If the activated rate constant k,,, is much larger than the diffusion rate constant k ~ then , the observed rate constant kq will be approximately the same as the diffusion rate constant: l3
llk, = llk,
+ llkact
A second reason given for the lack of a good correlation of the electron-transfer rate constant with the free energy change of electron transfer is that there is a lack of a truly homogeneous series in particular chemical donors or acceptors. There is usually an implicit assumption that there are certain crucial parameters such as transmission coefficients and activation 'University of Notre Dame. 4 A. Mickiewicz University. Abstract published in Advance ACS Abstracts, January 1, 1995. @
0022-365419512099-1478$09.00/0
energies that remain constant throughout a particular correlation. It this is not true, then the correlation cannot hold quantitatively. The third reason for the inverted region not being evident in a rate constant vs free energy correlation is related to the reorganization energy A(R) being an increasing function of the separation distance R between the reactants (see, for example, eq 14 below). In the Marcus theory, the maximum rate constant (minimum activation energy) takes place when the free energy of reaction is equal to the reorganization energy (see eq 2). Thus, the predicted plots of logarithm of quenching rate constant vs free energy (parabolas in classical Marcus theory) shift to larger driving force (more negative free energy) when the reaction distance increases. Hence, with an acquisition of excess energy, the reactants may be able to react faster at these larger distances because the activation energy can be smaller than it would have been for that same reaction energy at a smaller reaction distance. Thus the reaction rate constants would be higher than those predicted by any fixed reaction radius. The fourth reason given for no inverted regions being seen in diffusional electron-transfer quenching reactions is that extra reaction channels become accessible at higher energies.12 For the highly exothermic electron-transfer Rehm-Weller data (aromatic hydrocarbons/tetracyanoethyleneor similar systems), it has been s ~ g g e s t e d ' ~ ' ~ that ~ ~ 'excited ~ ' ~ - ~states ~ of the radical ions might become accessible. The doublet states of radical ions can often be of fairly low energy and might be accessible from the singlet excited states in the Rehm-Weller data. However, the presence of excited states of radical ions was not directly observed using time-resolved absorption and fluorescence t e c h n i q ~ e s . ' ~ J ~The J ~ formation of a nonfluorescent exciplex was also suggested14J5as a possible mechanism of quenching in addition to electron transfer. Another reason that has been put forward for the lack of an inverted region is that the solvent configuration (orientational polarization) coordinate's frequency is much smaller when the solute is a neutral molecule compared to the analogous frequency when the solute is an ion.17 The source of this difference in frequencies is the partial dielectric saturation of polar solvents by the electric field of the solute ions, which restricts the orientational polarization coordinate to a narrower range and hence increases the effective frequency of the coordinate. The relative flatness of the solvent configuration 0 1995 American Chemical Society
Electron TransferDiffusion Models
J. Phys. Chem., Vol. 99, No. 5, 1995 1479
coordinate’s free energy curve for neutral molecules means that in photoinduced electron-transfer reactions that begin with neutral reactants and form ion pairs, the activation energy will be relatively insensitive to the driving force of the electrontransfer step. This picture was supported by Monte Carlo simulations.’* However, it has been argued that this frequency change was irrelevant since the important part of the free energy curves is in a range of the orientational polarization that is not close to the saturation value, and, thus, the effective “force constants” of the polarization coordinate should be the same in both the neutral reactants and the ionic products.lg An additional criticism” was made that inverted regions are also lacking in nonpolar solution where the orientational polarization should not be a factor. We will not consider this theory further in this paper. In previous work20,21 on electron-transferquenching of excited triplet states, we attempted to fit our data with combinations of various electron-transfer theories of the activated rate constant ka,, with various kinetic schemes. A particularly attractive kinetic scheme was recently presented by Tachiya and Murata22 based on a modified diffusion formalism. This allows for the activated rate constant to vary naturally with distance. With the help of such a scheme, it is possible to evaluate the various reasons listed above for the lack of an observed Marcus inverted region in electron-transfer reactions with diffusing reactants. This is the objective of this communication.
Method
(4) where kB, VDQ,h, T, v, = k ~ T / hare, respectively, Boltzmann’s constant, the interaction energy between the donor and quencher orbitals, Planck’s constant divided by 2n, the absolute temperature, and the classical nuclear frequency which is the reciprocal of the frequency factor in transition state the01-y.~~ In classical theories, ~ ~ 1.l The = forward electron-transfer rate constant in both the classical and semiclassical theories are written as products of the transmission coefficient, the classical frequency factor, and an exponential factor involving the free energy of activation kact
A&e1 = (AGel
+ A)2/4A
(2)
where A is the reorganization energy which will be described later. This formula will be referred to as the “quadratic” free energy relationship. A straightforward application of this relationship to the Rehm-Weller data did not work well; so several “asymptotic” ones were used in order to successfully account for the observed plateau in the Marcus inverted region. The original empirical formulation by Rehm and Weller’ was followed by based on Levine’s probability formulation, wherein
= ‘Gel
-AGel In 2 + AG*el(0)In[ 1 + exP( Acsel(o)
)]
(3)
Where AGSel(0) = 114 is the free energy of activation when there is no overall driving force for the reaction, namely, when AGel = 0 for the electron transfer. Adel(0) is called the intrinsic barrier.
= Kelv, exp{-AG*eJkBT)
(5)
Quantum-mechanical versions of electron-transfer reactions are based on an analogy to theories of radiationless transitions.?6 These theories use Fermi’s “Golden Rule” of time-dependent perturbation theory and deal with electronic transitions between an orbital on the donor and one on the acceptor. The version of the theoqf‘ that we use gives
iisw
I vDQ(rDQ) I
kel(rw) = 2n
The approach to the Rehm-Weller data that will be taken in the current work is to attempt to fit the data with several kinetic schemes coupled with several theoretical models of the outersphere electron-transfer process. Two kinetic schemes will be used. One is the original phenomenological scheme proposed by Rehm and Weller.’ The other is the diffusional kinetic scheme that has already been applied by Tachiya and Murata22 to the same Rehm-Weller data. The “activated” process, exclusive of diffusion, has sparked great interest in electron-transfer reactions in the past 30 years. In classical theories, the goal has been to relate the kinetics to the thermodynamics of reaction. This was achieved by Marcus within the transition-state theory of reaction kinetics. He showed that the free energy of activation A d e l could be related to the free energy change AGel of electron transfer at the reaction distance by
‘&e1
In semiclassical theories of electron transfer,24it is customary to write the transmission coefficient as
9-(e:
X
J-=o
- FAG,,
+ lS(rDQ) + W 4kB
TA~(rDQ)
]
~ V I ~
(6)
where A S ( ~ qis) the solvent reorganization energy when the interionic distance is ~ D QS, = AJhv is related to the original Huang-Rhys number,27 hv is the energy of a representative molecular vibration, and Av is the reorganization energy for the typical molecular vibration. This rate constant is a function of the distance of transfer which is shown explicitly in the formula. The equation is an approximation that holds in the limited temperature range where kBT is much larger than the energy of solvent vibrations but much smaller than the energy of the typical molecular vibration. The four models for the rate constant of the electron-transfer step can be summarized as follows. The first two models are classical models that share the feature of having an electronic factor ~~1 in eq 5 that is not characterized a priori. The first model to be considered has a nuclear factor in eq 5 that is defined by the Agmon-Levine free energy relationship (eq 3). The second model considered is the classical Marcus model with the quadratic free energy relationship (eq 2) defining the nuclear factor in eq 5. The third model to be considered in this communication is the semiclassical model which also uses the quadratic free-energy relationship, but the electronic factor in eq 5 is given by eq 4. This semiclassical form of the electronic factor in eq 5 pennits a fuller exploration of the variation of the rate constant on the reaction distance through the electronic matrix element as well as through the reorganization energy. The fourth model to be considered is the full quantum mechanical expression in eq 6. These four electron-transfer models will each be encapsulated into the Rehm-Weller kinetic mechanism: 1
D*
Q
kd
1
(D*-Q)
A ‘(D*-Q’) k-e~
5 ‘D + Q
(I)
1480 J. Phys. Chem., Vol. 99, No. 5, 1995
Hug and Marciniak
With the steady-state approximation for the intermediates, the quenching rate constant can be shown to be
9.0
where koel = KelkBT/his the preexponential factor in the forward electron-transfer rate constant and
U
Y
t
:i
8.0
.
(8)
7.0
.
The rate constants related to diffusive transport are calculated from the diffusion-controlled rate constant2*
6.0
ln(kellk-el)= -AGel/RT
0,
-0
(9)
where 7 is the solvent viscosity and from the Eigen equation29 for zero-charged reactants, i.e.
where ~-DQis the reaction radius, and NOis Avogadro’s number. As before kbt = k-d was used.21~30~31 The other “reaction” scheme is more physical in that it is based on the diffusion equation with a “closure” approximation. In this formalism the quenching rate constant is given by eq 1 with the second-order electron-transfer rate constant32
and with the diffusion-limited rate constant
kD = (D‘/4n)kac~/[~~+r~el(R’)R‘ dR’Jc+,bel(R)R2 dR
+
where R and R’ are dummy variables of integration, and the mutual diffusion coefficient is D’ = DD DQ. The integrations were carried out numerically with a Simpson’s rule algorithm together with Newton’s 3/8 rule or a combination of the two r ~ l e s . 3 ~The 3 ~ ~code for the integration was a double precision scientific subroutine package from Digital Equipment. The fitting routines, coupled with the integration routines, have been tested in several ways and have been applied to other data.21 For the current application, the routines were run as a test with the input parameters used by Tachiya and Murata22without the least-squares procedure being active. From the output of this test, it was possible to reproduce Figure 2 of the Tachiya and Murata paper.22 The basic strategy was to combine the four different theoretical electron-transfer models for the activated rate constant kact with the phenomenological chemical reaction scheme or with the physical diffusion scheme. The resulting equations were then used to fit the Rehm-Weller data. In all cases, only two parameters were allowed to vary. All other parameters were calculated or measured as described in this paper. In none of the combined models of electron transfer and transport were the quenching rate constants linearly dependent on the fitting parameters. Thus a nonlinear least-squares routine,35based on the Marquardt was used to find the optimized parameters for all the models. To get an idea of how sensitive the fits were to the optimized parameters and to obtain a rough measure for the “goodness” of fit, it was assumed that the raw data were good to &lo%; in other words 1 standard deviation
+
Rehm-Weller data Agmon-Levine lit
I
I
0
5,000
kd = 8000RT13~
l-
,
.
I
.
I
,
.
-5,000
I
.
I
.
.
.
.
I
,
,
.
-10,000 -15,000
.
l
.
.
.
.
-20,000 -25,000
AGel, cm-1 Figure 1. Nonlinear least-squares, two-parameter Ad,l(O) fit to the Rehm-Weller data with the Agmon-Levine free energy relationship and the phenomenological kinetics based on eq I.
was estimated to be *lo%. From this value, it was then possible to estimate, by the propagation of errors,35the standard deviation in the optimized parameters. These “errors” are listed along with the optimized parameters as one standard deviation. Values of a measure of the goodness of fit could also be computed on the same basis, and these are also reported.
x?,
Results The fit of the Rehm-Weller data to the classical kinetics with the Agmon-Levine free energy relationship are plotted in Figure 1. The two variable parameters were the intrinsic free energy barrier A&el(0) and the transmission coefficient ~ ~ These 1 . parameters, obtained from the nonlinear least-squares fit, are listed in Table 1 along with the estimated standard deviations. For these calculations, values of the diffusion rate constants and the redox potentials had to be computed. Redox potentials were taken from the Rehm-Weller paper,’ and the diffusion rate constants were computed from the equations in the Method Section. The calculation of the free energy
AG,, = F(E,, - Ere&- E,
+ AW
from the redox potentials (Eoxof the electron donor and Erd of the electron acceptor) was common to all fittings performed in the current work. Es is the energy of the excited singlet state, and Aw is the Coulombic interaction term taken to be -0.056 eV for acetonitrile solutions.’ The analogous fit, for the Marcus free energy relationship and the phenomenological kinetics, is shown in Figure 2. The two variable parameters were the reorganization energy I = 4A&,l(o) and the transmission coefficient ~ ~ The 1 . values of the optimized parameters are listed in Table 1. Additional parameters needed for this fit are the rate constants for diffusive transport and the redox potentials. The rest of the models employed were semiclassical and quantum mechanical in nature. They required a more complex set of auxiliary physical parameters to complete the calculations. When quantum-mechanical theories are employed, it is necessary to specify additional parameters of a molecular nature which must be assumed, calculated, and/or measured. The least complicated of these models is the semiclassical version of the Marcus theory with a phenomenological kinetics. All that is needed is a computation of the electronic factors from the transmission coefficient which was already optimized above. The most plausible way to do this is to calculate VDQ from the optimized through the use of eq 4 . For this it was first
J. Phys. Chem., Vol. 99, No. 5, 1995 1481
Electron TransferDiffusion Models
TABLE 1: Summary of Best-Fit Parameters to the Rehm-Weller Data electron-transfer modelkinetics Vmo/cm-l AQel(O)/cm-I Kel Agmon-Levineleq I 1140 f 30 0.026 f 0.004 Marcusleq I" 84 f 6 1700 f 30 0.32 f 0.04 QWeq 1 37 f 2 1100 f 30
D'/m2 s-'
xv2
13.7
semiclassical/diffusionb
4100 f 1600 semiclassicddiffusion' 1940 f 30 QWdiffusion 3750 f 50 a VDQO computed from best-fit parameter K ~ I . For this fit D' was fixed at 3.0 x
Ae
1.1 f 0.1 0.1) x 10-9 f 0.1) x 10-9 For this fit Ae was fixed as 1.
(2.8 (2.9
mz s-I.
23.5 22.5 9.2 9.1 9.2
11.0 L
10.0
-
9.0
-
8.0
-
7.0
-
P
Y
0)
0
6.0
1%
l 09.0 .OI
i
7.0
\
/ l
5,000
I%
0
.
.
I
.
l
I
-5,000
,
I
I
I
I
.
.
-10,000
l
l
-15,000
6.0 , . , I . . , .
-20,000
i /
I I I I I I , , . , 1 , , , , I , . . , I
-25,000
5,000
0
-5,000
A G ~cm-1 ~,
-10,000
-15,000
-20,000
, . , -25 300
A G ~cm-' ~,
Figure 2. Nonlinear least-squares, two-parameter (2 and Kel), fit to the Rehm-Weller data with the Marcus free energy relationship and the phenomenological kinetics based on eq I.
Figure 3. Nonlinear least-squares, two-parameter(AQel(0) and Vmo),
fit to the Rehm-Weller data with the quantum-mechanical model of the electron-transfer process coupled with the phenomenologicalkinetics based on eq I.
necessary to estimate the reorganization energy. The dielectriccontinuum model of the solvent was used together with the Bom formula for charging, giving
"'4"t /. l 09.0 .OI
where Ae is the electronic charge transferred, EO is the permittivity of free space, and n and cs are, respectively, the refractive index and the dielectric constant of the solvent. The molecular radii (Q and IQ) of the reactants are taken to be 0.3 nm. These values were also used by Tachiya and Murata.22 When the interaction distance QQ is set equal to the sum of the molecular radii, the interaction matrix element VDQ' = VDQ(Q I Q ) is given in Table 1. To complete a two-parameter fit with the quantum-mechanical model of electron transfer, even more physical parameters must be specified. Here we follow the traditional assumptions and estimations. Gould et al.37took the intramolecular reorganization energy A, equal to 0.25 eV, and the energy of a typical molecular vibration, hv, equal to 1500 cm-'. With these values and the Rehm-Weller kinetics, the interaction matrix element, VDQO,and the solvent reorganization energy, As,were optimized. The best-fit parameters are given in Table 1, and the fit is displayed in Figure 3. The final two models used diffusion kinetics, where the only parameter for the diffusion aspect of the model was the mutual diffusion coefficient. We used 3 x m2/s for the mutual diffusion coefficient which is the same as that used by Tachiya and Murata.22 The electronic interaction matrix element was taken to be38
+
0
VDQ(rDQ> = VDQ
e x ~ { - B ( r w - rD - rQWI
(15)
where is the characteristic inverse interaction distance for the exchange interaction which was set equal to 10 nm-1,39-42
Y
f Semldassld Tachiya-Murata k i n e I i i wlth varlwe e
7.0
5,000
0
-5,000
-10,000
-15,000
-20,000
-25,000
AGel, cm-I Figure 4. Nonlinear least-squares, two-parameter (Vmoand Ae), fit
to the Rehm-Weller data with the semiclassical model of the electrontransfer process coupled with the diffusion model for the transport of the reactants. D' is taken as 3 x m2 s-l. The radii of the reactants were again taken as 0.3 nm, but the interaction distance QQ was allowed to vary in both the formula for the interaction matrix element (eq 15) and in the solvent reorganization energy (eq 14). These supplementary assumed, calculated, and measured values, along with the redox potentials, are sufficient to complete the fitting procedure on the semiclassical model with the diffusion kinetic transport. When the electronic interaction matrix element, VDQ', and the amount of charge transferred, Ae, are used as the fitting parameters, the resulting best-fit values are given in Table 1, and the fit is shown in Figure 4. Since the charge transferred is reasonably close to one, it was deemed to be of interest to set Ae to 1 and to vary D', the mutual diffusion constant. That fit is shown in Figure 5 , and the parameters are listed in Table 1.
1482 J. Phys. Chem., Vol. 99, No. 5, 1995
Hug and Marciniak
l 09.0 .Ol
c r .
Y
-s? 8.0 .
Rehm-Weller dam Semiclassical Tachiya.Murala kine* wkh variable D
7.0 '
6 . 0 ~ " " 1 " " 1 " " 1 " " 1 " " " ~ " ~ 5,000 0 -5,000 -10,000 -15,000 -20,000 -25,000
AGel, cm-l
Figure 5. Nonlinear least-squares, two-parameter (VDQOand D'),fit to the Rehm-Weller data with the semiclassical model of the electrontransfer process coupled with the diffusion model for the transport of the reactants. Ae set to 1. 11.0,
10.0
-
9.0 0Y
-0
8.0
-
7.0
-
6.0
-
5,000
I
0
4.000
-
Rehm-Wellwdata Quantum-Mechanical model with Tachiya-Murala kinetics
-10,000
-15,000
-20,000
I
-25,000
AGel, cm-I
Figure 6. Nonlinear least-squares, two-parameter (VOQOand D'),fit to the Rehm-Weller data with the quantum-mechanicalmodel of the electron-transfer process coupled with the diffusion model of transport of the reactants. No additional assumption or estimation concerning supplementary physical quantities was necessary for computing the quantum-mechanical model with the physical diffusion kinetics. The fitting parameters were taken to be the electronic matrix element, V w o ,and the mutual diffusion constant, D'. The fit is shown in Figure 6, and the parameters are listed in Table 1.
Discussion The physical parameters obtained from the fitting procedures with the kinetics of eq I and the four different electron-transfer models appear to be reasonable. The intrinsic barriers for electron transfer, A@el(0) , are roughly all one half of the AGe1(O) computed from 1, (QQ = 0.6 nm)/4 = 2540 cm-' in the dielectric-continuummodel of the solvent. The transmission coefficients, ~ ~ obtained 1 , from fits, using both the AgmonLevine and the Marcus free-energy relationships, are somewhat less than 1, expected for adiabatic reactions. However, they are much greater than the transmission coefficients obtained from triplet state quenching by inorganic complexes that show strong nonadiabatic behavior.21 The electronic matrix elements from the semiclassical and the quantum-mechanical models do not agree among themselves as well as do the electronic matrix elements from these same models applied to the triplet-state quenching.21 In addition, the singlet quenching matrix elements are both larger than the corresponding matrix elements from triplet quenching data.21 The interaction matrix element of the
semiclassicalmodel is roughly equal to that assumed by Tachiya and Murata (100 cm-'), but this matrix element from our quantum-mechanical fit was one-third that assumed by Tachiya and Murata.22 Within the kinetics of eq I, the asymptotic free energy relationship of Agmon-Levine gives the most satisfactory fit. This has been a characteristic of solution kinetics that the asymptotic relationship gives more satisfactory fits to the data.'-43 However this relationship does not have as sound a theoretical basis as either the Marcus, semiclassical,or quantummechanical electron-transfer theories. This has prompted investigators to single out electron-transfer reactions occurring under diffusional kinetics as an anomaly." It can be seen from Figures 4-6 that the fits to the RehmWeller data using the diffusion kinetics are much better than the fits with the phenomenological kinetics of eq I. The fits that use diffusion kinetics give fits that visually appear excellent. These fits are even better than the fit in Figure 1 that used the Agmon-Levine relationship with phenomenological kinetics. Some of the physical parameters obtained from these fits using the diffusion kinetics are reasonable. In the semiclassical electron-transfer model used in Figure 5, for which it was assumed that D' was 3 x m2/s, the optimized charge transferred tumed out to be 1.1. In the other two fits that used diffusion kinetics, one of the fitting parameters was D'. In both these fits D' was found close to 3 x m2/s. In all three fits using the physical diffusion model, one of the two fitting parameters was the electronic matrix element, VDQO. This parameter seems to give values that are much too large, from to l/2 eV. With this survey of these major theoretical models for outersphere electron transfer coupled to the two kinetic models, it is now possible to look back at the various explanations of why the Marcus inverted region is not seen in the Rehm-Weller data. If the possibility is admitted that no extra quenching process is involved, then it can be seen from the first three figures that the quenching cannot be explained unless some significance is given to the asymptotic free energy relationship. The diffusion processes of eq I alone cannot simply "chop off' the Marcus parabola to give the plateau in the Marcus inverted region. It appears as though it is necessary to have both the diffusion process and the distance dependence of the reorganization energy in order to get an adequate fit to the entire set of RehmWeller data. Again this is assuming that there is only a single quenching process responsible for the entire set of data. The contrast between the first three figures and the last three figures is a strong indication that the distance dependence of the reorganization may play a major role in quenching events in the Marcus inverted region of electron-transfer reactions involving diffusion of the reactants. Diffusion alone with electron transfer occurring at a fixed reaction radius does not appear to be adequate to explain the observations. The excellent fits displayed in Figures 4-6 also demonstrate that it is possible to get fits in the plateau region without using asymptotic free energy relationships. The problem with the above tentative conclusions is the unreasonably large electronic matrix elements that are obtained as best fits from the physical diffusion kinetic schemes of Figures 4-6. These large interaction energies are not an intrinsic feature of the physical diffusion scheme because, with triplet energies and a smaller range of free energies, AGel, the exchange matrix element from the diffusion kinetics was very similar to those obtained from the various electron-transfer models in combination with the phenomenological kinetics.21
J. Phys. Chem., Vol. 99, No. 5, 1995 1483
Electron TransferDiffusion Models
In addition, Tachiya and Murata22 gave a plot of a curve computed with “reasonable” physical parameters that showed a plateau that tailed off at high energies. This is the hint as to what is causing the unusually large interaction energies in the fits displayed in Figures 4-6. It appears that even with the distance dependence of the reorganization energy there is a residual “chopping off’ of the Marcus parabola or Franck-Condon profiles in the quantum-mechanical models. The distance dependence of the reorganization energy can extend the plateau for a given diffusion coefficient. However, it appears that the Marcus curve will eventually decrease (see, for example, the figure in the Tachiya-Murata paper2*) due to high exothermicity or to the unfavorable Franck-Condon factors in the inverted region. Thus it is only through the excessively large exchange interaction that the rate constant can be compensated for the decreasing Franck-Condon factors in the Marcus inverted region. To form a plateau, the rate constant must be pushed high enough (for instance by an increase in the exchange matrix element) so that when it eventually falls off of the plateau, the plateau will extend far enough outward to account for the data. By extending the electron-transfer models with physical diffusion to its logical limit, it can be seen that the parameters so obtained are physically unreasonable. This points to the final reason for the failure to observe the Marcus inverted region in the Rehm-Weller data. That is that at large exothermicities there are other quenching processes involved. (Despite a lack of direct detection of the excited states of radical ions in highly exothermic quenching events,14 the results presented recently by Kikuchi et al.10v16indicated a generation of excited radical ions as a quenching process.) It can be seen from Figures 2 and 3 and the figure in the Tachiya and Murata paper22that the plateau at low and moderate exothermicities can be approximated without invoking an excessively large electronic interaction. If the data at large exothermicities can be ignored because of an unknown process, then even the electron-transfer theories in combination with eq I give a reasonable fit to the data. However, the fiie points of the fit, and the physical nature of the process, would profit from the physical diffusion theory. If the last five points of the Rehm-Weller data for AG < - 15 000 cm-’ are actually ignored, then the fitting procedure for the semiclassical model of electron transfer coupled with the Tachiya-Murata diffusion model led to VDQO 100 cm-’ with the fitting showing an inverted region. This supports the explanation that in addition to the electron transfer some extra quenching processes become accessible at large exothermicities (quenching processes leading to the formation of excited states of radical ions or exciplexes). Acknowledgment. The work described herein was supported by the Office of Basic Energy Sciences of the U.S. Department of Energy. This is Document No. “-3763 from the Notre Dame Radiation Laboratory. References and Notes (1) Rehm, D.; Weller, A. Isr. J. Chem. 1970, 8, 259-271. (2) Marcus, R. A. Discuss. Faruday SOC.1960, 29, 21-31.
(3) Marcus, R. A. Annu. Rev. Phys. Chem. 1964, 15, 155-196. (4) Kestner, N. R.; Logan, J.; Jortner, J. J. Phys. Chem. 1974, 78, 2148-2166. (5) Beitz, J. V.; Miller, J. R. J. Chem. Phys. 1979, 71, 4579-4595. (6) Miller, J. R.; Calcaterra, L. T.; Closs, G.L. J. Am. Chem. SOC. 1984,106, 3047-3049. (7) Wasielewski, M. R.; Niemczyk, M. P.; Svec, W. A.; Pewitt, E. B. J. Am. Chem. SOC. 1985, 107, 5562-5563. (8) Gould, I. R.; Ege, D.; Mattes, S. L.; Farid, S. J. Am. Chem. SOC. 1987,109, 3794-3796. (9) Mataga, N.; Asahi, T.; Kanda, Y.; Okada, T.; Kakitani, T. Chem. Phys. 1988, 127, 249-261. (10) Kikuchi, K.; Niwa, T.; Takahashi, Y.; Ikeda, H.; Miyashi, T. J. Phys. Chem. 1993, 97, 5070-5073. (11) Suppan, P. Top. Curr. Chem. 1992, 163, 95-130. (12) Closs, G. L.; Miller, J. R. Science 1988, 240, 440-447. (13) Noyes, R. M. In Progress in Reaction Kinetics; Porter, G., Ed.; Pergamon: New York, 1961; Vol. 1, pp 129-160. (14) Mataga, N.; Kanda, Y.; Asahi, T.; Miyasaka, H.; Okada, T.; Kakitani, T. Chem. Phys. 1988,127, 239-248. (15) Kikuchi, K.; Takahashi, Y.; Katagiri, T.; Niwa, T.; Hoshi, M.; Miyashi, T. Chem. Phys. Lett. 1991, 180, 403-408. (16) Kikuchi, K.; Katagiri, T.; Niwa, T.; Takahashi, Y.; Suzuki, T.; Ikeda, H.; Miyashi, T. Chem. Phys. Lett. 1992, 193, 155-160. (17) Kakitani, T.; Mataga, N. Chem. Phys. 1985, 93, 381-397. (18) Hatano, Y.; Saito, M.; Kakitani, T.; Mataga, N. J. Phys. Chem. 1988,92, 1008-1010. (19) Tachiya, M. Chem. Phys. Lett. 1989, 159, 505-510. (20) Marciniak, B.; Bobrowski, K.; Hug, G. L. J. Phys. Chem. 1993, 97, 11937- 11943. (21) Hug, G.L.; Marciniak, B. J. Phys. Chem. 1994, 98, 7523-7532. (22) Tachiya, M.; Murata, S. J. Phys. Chem. 1992, 96, 8441-8444. (23) Agmon, N.; Levine, R. D. Chem. Phys. Lett. 1977, 52, 197-201. (24) Brunschwig, B. S.; Logan, J.; Newton, M. D.; Sutin, N. J. Am. Chem. SOC.1980, 102, 5798-5809. (25) Glasstone, S.; Laidler, K. J.; Eyring, H. The Theory of Rate Processes; McGraw-Hill: New York, 1941. (26) Englman, R. Non-Radiative Decay of Ions and Molecules in Solids; North-Holland New York, 1979. (27) Huang, K.; Rhys, A. Proc. R. SOC. London, Ser. A 1950,204,406423. (28) Smoluchowski, M. 2. Phys. Chem. 1917, 92, 129-168. (29) Eigen, M. Z. Phys. Chem. (Lebzig) 1954, 203, 176-200. (30) Wilkinson, F.; Tsiamis, C. J. Am. Chem. SOC.1983, 105, 767774. (31) Marciniak, B.; Lijhmannsrtiben, H.-G. Chem. Phys. Left. 1988,148, 29-36. (32) Doi, M. Chem. Phys. 1975, 11, 115-121. (33) Hildebrand, F. B. In Introduction to Numerical Analysis; McGraw-Hill: New York, 1956; pp 71-76. (34) Zurmuehl, R. In Pruktische Muthematik h e r Ingenieure und Physiker; Springer: Berlin, 1963; pp 214-221. (35) Bevington, P. R. Dura Reduction and Error Analysis for the Physical Sciences; McGraw-Hill: New York, 1969. (36) Marquardt, D. W. J. SOC. Indust. Appl. Math. 1963, 11,431-441. (37) Gould, I. R.; Ege, D.; Moser, J. E.; Farid, S. J. Am. Chem. SOC. 1990, 112, 4290-4301. (38) Newton, M. D. Chem. Rev. 1991, 91, 767-792. (39) Logan, J.; Newton, M. D. J. Chem. Phys. 1983, 78, 4086-4091. (40) Closs, G. L.; Calcaterra, L. T.; Green, N. J.; Penfield, K. W.; Miller, J. R. J. Phys. Chem. 1986, 90, 3673-3683. (41) Penfield, K. W.; Miller, J. R.; Paddon-Row, M. N.; Cotsaris, E.; Oliver, A. M.; Hush, N. S. J. Am. Chem. SOC. 1987, 109, 5061-5065. (42) Sassoon, R. E.; Hug, G. L. J. Phys. Chem. 1993, 97, 7823-7832. (43) Scandola, F.; Balzani, V. J. Am. Chem. SOC.1979,101,6140-6142. JP942642C
