J. Phys. Chem. 1996, 100, 3005-3015
3005
Free Energy Gap Law under Diffusion Control A. I. Burshtein* Department of Chemical Physics, Weizmann Institute of Science 76100 RehoVot, Israel
E. Krissinel Institute for Water and EnVironmental Problems, 656099 Barnaul, Russia ReceiVed: April 21, 1995; In Final Form: August 11, 1995X
The free energy gap law in forward and back electron transfer (bimolecular and geminate) is examined. Both processes are specified by position-dependent rates of remote electron transfer. These rates are used within encounter theory to calculate the bimolecular rate constants of ionization and recombination as well as the quantum yield for photochemical ion separation. The free energy gap law is shown to be a property of kinetic-controlled reactions. In the diffusion-controlled limit this law for either forward or backward reaction constants is violated at the top of Marcus’ parabola. This conclusion pertains equally to an analogous quantity extracted from the separation quantum yield for geminate recombination.
Introduction Charge transfer in solutions is usually assisted by one or more vibrational modes of the surroundings linearly coupled to an electron. If the vibrational motions of heavy particles are considered classically, then the transfer occurs in (or above) the crossing points of the energy levels of a whole system. In a one-mode scheme presented in Figure 1 the transfer may be either from excited electronic level 1 to charge-transfer state 2 or from this state to a ground state 0. Both transitions are activated processes, but their Arhenius rates
W ) we-U/T (kB ) 1) have different preexponents and activation energies
U)
(∆G + λ)2 4λ
The latter depends quadratically on the free energy of the reaction ∆G which has its origin usually at the bottom of reactant well. The only parameter of this dependence, the reorganization energy λ, is actually the strength of the electronphonon interaction characteristic of the solvent. The parabolic free energy dependence of the activation energy is usually presented as ln W dependence against ∆G, which is known as a free energy gap law:
ln W ) ln w - (∆G + λ) /4λT 2
Essentially, it claims that the reaction is the fastest when it is activationless, at ∆G ) -λ (U ) 0), but for both smaller and larger ∆G the reaction is activated and hence slower. According to this law Marcus discerns two regions where electron transfer is activated:1 normal, where -∆G ) |∆G| < λ, and inVerted, where -∆G ) |∆G| > λ. For instance, in Figure 1 the transfer 1 f 2 is normal while the other, 2 f 0, is inverted. It was recognized very recently that these regions become markedly different in character and essential when ∆G and λ are considered as functions of interparticle distance.2,3 If the reaction is normal everywhere, the conventional presentation X
Abstract published in AdVance ACS Abstracts, January 15, 1996.
0022-3654/96/20100-3005$12.00/0
Figure 1. Level crossings in normal (1 f 2) and inverted (2 f 0) regions.
of W(r) as an exponentially decreasing “exchange” rate is semiquantitatively valid. Otherwise it is a nonmonotonous, bellshaped curve and possibly zero at the contact. This put a serious limitation on a contact approximation in the theory of diffusionaccelerated reactions coming from Smoluchowski. We have already demonstrated the difference within an analytically solvable model of a bell-shaped W(r),4 but here we use a nonmodel approach to back and forward electron transfer. The free energy gap law as a major prediction of a classical Marcus theory was experimentally checked many times. It was confirmed for intramolecular electron transfer and intermolecular transfer in solids but never in liquid solutions. It looks like the inverted region is not accessible for the forward (ionization) process while the normal region is not widely encountered in experimental studies of back electron transfer (geminate ion recombination). © 1996 American Chemical Society
3006 J. Phys. Chem., Vol. 100, No. 8, 1996
Burshtein and Krissinel
What actually happens is that it is not W that is actually studied in the case of bimolecular ionization but rather the reaction constant k. The latter may be significantly affected by diffusion, and therefore the k(∆G) dependence should be different from that predicted for W.5 Any quantitative characteristics of geminate recombination are even more indirectly related to W. No rate may be attributed to the kinetics of this process,6,7 nor can the quantum yield of charge separation be properly estimated within a naive “exponential model” which implies contact creation and recombination of ions. A much better description is provided by the “contact approximation” which also assumes contact recombination but leaves free a choice of initial charge distribution. Contact approach to a back electron transfer is a demerit that was overcome only recently. The unified theory of back and forward electron transfer8-10 eliminates the very necessity of the choice of initial ion distribution. Applying this theory, we will show here that the geminate recombination is not less affected by diffusion than bimolecular reactions. As a result there is a great diffusional distortion of what is effective quantitative measure of back electron transfer efficiency and is expected to follow the free energy gap law. The remainder of the paper is organized as follows: in section 1 we introduce the position-dependent rates of back and forward electron transfer and ascertain where are normal and inverted regions for each of them. In sections 2 and 3 we consider the bimolecular electron transfer as contact and remote. In the latter case the space distribution of photoionization products is found assuming ions are immobile and stable (section 4). In sections 5 and 6 we consider geminate ion recombination due to contact and remote back electron transfer, keeping initial charge separation as an arbitrary parameter. Finally, the unified theory of back and forward electron transfer is used to eliminate any arbitrariness and the very demand for initial ion distribution (section 7). In this section we show that the FEG law for geminate recombination is significantly violated at slow diffusion when the charge separation quantum yield is small. The results are summarized in section 8.
successfully employed by us to specify the initial charge distributions, the recombination kinetics, and the separation quantum yields at different viscosities.10 The separated ions can encounter the counterions originating from different pairs and react with them. In diluted solutions this bimolecular recombination is much slower than the preceding geminate reaction. As soon as the charge distribution becomes uniform, the bimolecular reaction
D+ + A- w D + A
is again a subject of encounter theory employing the same WR(r) as in the previous case. Hence, one may study separately WI(r) and WR(r) for bimolecular reactions 1.1 and 1.3, but for geminate reaction 1.2 both these rates are of equal importance because they contribute simultaneously to recombination kinetics and separation quantum yield.10 In most works cited above they were assumed to be exponential functions of interparticle distance, which is approximately a case for reactions in the normal region according to Marcus’ classification. However, the rates of highly exothermic reactions that fall into the inVerted region are not exponential at all. It goes through a maximum shifted from the contact and may be better interpolated by a bell-shaped curve used in ref 4. To avoid modeling of any kind, we prefer here to use WI(r) and WR(r) as they are in the conventional nonadiabatic perturbation theory of electron transfer with one assisting mode:3
WI(r) ) Wi(r)e-2(r-R0)/L
(1.4a)
WR(r) ) Wr(r)e-2(r-σ)/l
(1.4b)
Here R0 and σ are the closest approach distances for neutral reactants and ions while L and l are space decrements of exchange or “superexchange” interaction. Since the difference between them is small and insignificant, it will be ignored below by setting
R0 ) σ
1. Position-Dependent Rates Electron transfer is an efficient mechanism of energy (luminescence) quenching in solution. Following light excitation of an electron donor (D), the ionization proceeds according to the kinetic scheme11
D* + A w [D+‚‚‚A-]
Then the preexponents are given by the “golden rule” formulas of nonadiabatic perturbation theory (see ref 19)
Wi(r) )
xπVi2 xλT
e-(∆GI+λ) /4λT
(1.5a)
e-(∆GR+λ) /4λT
(1.5b)
2
(1.1)
where A is the electron acceptor. At sufficiently low concentration of acceptors, c, the differential encounter theory, developed in refs 12-15, best describes the quenching kinetics and quantum yield of luminescence provided the position-dependent ionization rate WI(r) is known.5-17 The ion pair produced by photoionization may either recombine to the ground state or separate according to the kinetic scheme18
[D‚‚‚A] W [D+‚‚‚A-] w D+ + A-
(1.3)
(1.2)
The geminate recombination should be considered together with a precursor reaction (eq 1.1) that generates the initial conditions for back electron transfer. The necessary extension of encounter theory to photoionization followed by geminate recombination was first proposed in refs 8 and 9 assuming the positiondependent recombination rate WR(r) to be also known. It was
Wr(r) )
xπVr2 xλT
2
Here Vi and Vr are off-diagonal matrix elements of nonadiabatic (exchange) interactions at the closest approach distance for neutral reactants and ions. These are essential parameters of the theory measured in inverse nanoseconds as well as ∆G, λ, and T (kB ) p ) 1). Although different in general, they may be (and will be) taken equal to simplify the analysis of an important particular case. The preexponential factors in eqs 1.5 are r-dependent due to both the reorganization energy λ(r) and the free energies of ionization and recombination
∆GI ) E(D+‚‚‚A-) - E(D*‚‚‚A) and ∆GR ) E(D‚‚‚A) - E(D+‚‚‚A-)
Free Energy Gap Law under Diffusion Control
J. Phys. Chem., Vol. 100, No. 8, 1996 3007
Figure 2. Energies of neutral reactants and the ion pair as functions of separation distance r. The decomposition of excitation energy into free energies of ionization and recombination is shown at an arbitrary point.
In our particular case the former is given by the relation
∆GI ) ∆GI(∞) - T
rc rc σ ) ∆Gi + T 1 r σ r
(
)
(1.6a)
while the latter is determined by a “conservation law” (Figure 2)
-∆GR - ∆GI ) E0
(1.6b)
Here rc ) e2/T is the Onsager radius for Coulomb attraction between ions determined by the static dielectric constant , ∆Gi is the ionization free energy at the contact distance, and E0 is the excitation energy of the electron donor (the difference between the energies of D* and D). If, for instance, ∆Gi is varied in a series of similar acceptors to check the free energy gap law, then ∆Gr ) -E0 - ∆Gi is simultaneously changing in the opposite direction provided E0 is fixed by using the same donor. To be more specific with λ(r), we restrict our consideration to the so-called outer-sphere reactions in polar solvents.20 In this case the reorganization energy is well-known to be
σ λ(r) ) λc 2 r
(
)
(1.7)
Here the contact reorganization energy
λc ) T
r0 - rc σ
(1.8)
where r0 ) e2/0T is determined by the “optical” dielectric constant 0 ≈ 2. As seen, λ(r) increases with particle separation so that λ(∞) ) 2λc, i.e. is twice its contact value. To get an impression about numbers, one can simply estimate for water ( ) 81) that λc ) 55 T and rc ) 7 Å at σ ) 5 Å. The free energy gap law for a contact pair follows from eqs 1.5 taken at r ) σ:
Wic ) wice- (∆Gi+λc) /4λcT
(1.9a)
Wrc ) wrce- (∆Gr+λc) /4λcT
(1.9b)
2
2
where
wic )
xπVi2
xλcT
wrc )
xπVr2
xλcT
It is usually visualized in a log plot where the above dependencies are considered as functions of contact free energy ∆Gc
Figure 3. Free energy gap law (ln W versus ∆G) for contact rates of ionization (Wic) and recombination (Wcr ) at wic ) wcr ) 103 ns-1, λ* ) 55, and E0 ) 3/2λc (A), 2λc (B), and 3λc (C). Different regions are marked by abbreviations: (NI) normal ionization, inverted recombination; (IN) inverted ionization, normal recombination; (NN) both normal; (II) both inverted.
) ∆G(σ) and are parabolas 2 ln Wic ) ln wic - (∆G* i + 1)
(1.10a)
2 ln Wrc ) ln wrc - (∆G* r + 1)
(1.10b)
where the reduced free energies ∆G* ) ∆Gc/λ and dimensionless reorganization energy λ* ) λ/T. Setting for simplicity Vi ) Vr ) V, i.e.
wic ) wrc ) wc )
xπV2 Txλ* c
we have two identical parabolas which are horizontally displaced from each other depending on the ratio E0/λc (Figure 3). The three different situations are possible at any r: (A) E0 < 2λ; (B) E0 ) 2λ; (C) E0 > 2λ. As seen from Figure 3 situation B is exceptional in the sense that both parabolas coincide. Therefore the ionization is normal (-∆GI < λ) while recombination is inverted (-∆GR > λ) and Vice Versa. In situation C the parabolas are split and their inverted branches overlap, so between the maxima both forward and back electron transfer occur in the inverted region. On the other hand, in situation A parabolas are shifted toward each other so that their normal branches cross and between the maxima both reactions are normal. By substituting different donor molecules in the same solvent, one can increase E0, keeping λ constant, and change the situation at contact from A to C, as is shown in Figure 3. But if the situation is C at contact, it may change to A with particle separation because λ is increasing with r at E0 ) const.
3008 J. Phys. Chem., Vol. 100, No. 8, 1996
Burshtein and Krissinel
Figure 4. Kinetic rate constants of ionization, k0, and recombination, k′0, as functions of ionization free energy (given in units of thermal energy) at λ* ) 55.
It is a common practice to consider only the quasi-exponential (“exchange”) rates of either forward or back electron transfer, thus addressing only the very exceptional normal-normal case (NN) of situation A. This pertains equally to our latest work,10 but now we overcome this restriction and take WI and WR as they are. We will mainly concentrate on the particular case where there is situation B at contact. As mentioned above the free energy gap law has been confirmed when either electron transfer in solids or intramolecular electron transfer was studied experimentally. In both cases the interparticle distance r is fixed and thus W(r) is actually measured. In liquids the situation is different in principle. The distance between the reactants is modulated by diffusion and the electron transfer occurs during rare encounters between them. This is a bimolecular reaction with a rate linear in concentration of partner c and the rate constant k which is not simply related to W(r). The only exception to this rule is a limit of kineticcontrolled reactions we will start with. 2. Electron Transfer in the “Gray Sphere” Model In the fast diffusion limit, where bimolecular reactions are controlled by electron transfer, the corresponding kinetic rate constants are
k0 ) ∫σ WI(r)4πr2 dr ∞
(2.1a)
Figure 5. Free energy gap law in situation B (E0 ) 2λc) for the kinetic rate constants of ionization (k0, a) and recombination (k′0, b), solid lines; the total rate constants ki, (a) and kr (b) at different diffusion rates, dashed lines; and the total rate constants in the contact approximation, dotted lines. wic ) wcr ) 103 ns-1, λ* ) 55, rc ) 7 Å, and σ ) 5 Å.
and V and V˜ are the reaction Volumes. Neglecting the rdependence of Wi and Wr, one may show that10
V ) 4π(σ2L + 2σL2 + 2L3)
These rough estimates of kinetic constants are semiquantitatively valid within normal regions, where -∆Gi, -∆Gr < λ, and only qualitatively in inverted regions, when these inequalities are reversed. Relations 2.2 gave promise that the free energy dependence of Wci (∆Gi) is duplicated in that of the ionization rate constant, k0(∆Gi), as well as Wrc(∆Gr) in the recombination kinetic constant, k′0(∆Gr); in other words,
ln k0(∆Gi) ) B + Wic(∆Gi)
for ionization and
k′0 ) ∫σ WR(r)erc/r4πr2 dr ∞
(2.1b)
for recombination. The free energy dependencies of these constants are shown in Figure 4. When they are depicted in logarithm plots in Figure 5, they have a bell shape similar to that shown for contact rates in Figure 3. Although asymmetric, they reproduce qualitatively the free energy gap law for the following reasons. If the reactions are normal at contact distance, then WI(r) and WR(r) quasi-exponentially decrease with r from the very beginning and the integrals in eqs 2.1 may be roughly approximated as
k0 ) WI(σ)V ) WicV
(2.2a)
k′0 ) WR(σ)erc/σV˜ ) k˜0erc/σ
(2.2b)
k˜0 ) WrcV˜
(2.3)
and
where
V˜ ) 4π(σ2l + 2σl2 + 2l3)
ln k′0(∆Gr) ) B′ + Wrc(∆Gr)
where B ) ln V and B′ ) ln V˜ + rc/σ are independent of free energies. These quite general expectations had not been verified. Starting from the pioneering work of Rehm and Weller,11 a lot of experiments have shown that ionization reaction constants in liquids do not follow the free energy gap law, although it is peculiar to Wci . There were many attempts to attribute this discrepancy to some microscopic reasons like nonlinear interaction with an assisting mode,21 generation of vibrationally excited products (multichannel reaction),22 etc. However, the simplest and more reasonable explanation given by Marcus and Siders5 is that the measured reaction constant is not identical to the kinetic constant k0. The real constant takes proper account of diffusion and may be either kinetic or diffusional. In the latter case one should not expect that it follows the free energy gap law. To find out what is actually measured, one may resort to the gray sphere model23 that generalizes the classical Smoluchowski approach to bimolecular reactions in solutions. Within the model the time-dependent reaction constant
kI(t) ) k0n(σ,t)
(2.4)
Free Energy Gap Law under Diffusion Control
J. Phys. Chem., Vol. 100, No. 8, 1996 3009
is defined via the pair distribution function n(r,t) of D* and A. The latter satisfies the diffusional equation
D ∂ ∂ n˘ ) 2 r2 n r ∂r ∂r
(2.5)
where D is the encounter diffusion coefficient. The reaction is assumed to occur in contact and is introduced by k0 in the boundary condition
|
∂n ) k0n(σ,t) ∂r σ
4πσ2D
(2.6)
The generalized reaction constants are
k′0 ) k˜0erc/σ and k′D )
4πrcD ˜ 1 - e- rc/σ
In solvents of very high polarity (rc/σ , 1), they reduce to their ˜ . In water (rc ) usual values for neutral reactants, k˜0 and 4πσD 7 Å) for σ ) 5 Å, the magnitude of the recombination constants is slightly higher than that of ionization (Figure 5) because of the concentration of partners in the shallow Coulomb well. The effect should be much more pronounced in nonpolar solvents. 3. Remote Charge Separation in Encounter Theory
The initial distribution is
n(r,0) ) 1
(2.7)
In the kinetic-controlled limit this distribution does not change significantly with time and the measured rate constant kI ≈ k0 follows the free energy gap law at least qualitatively. However, at low diffusion kI(t) initially follows n(σ,t), and after the encounter time τe ) σ2/D it reaches its stationary value, given by
In practice the electron transfer is never at contact but proceeds with the rates in eqs 1.4 at any separation distances. Moreover, the r-dependence of the rates is different for different regions, normal and inverted. This fact of great importance should be and may be taken into account rigorously by applying differential encounter theory to reaction 1.1.14,15 Within this theory the time-dependent ionization rate constant is defined differently than in eq 2.4:
kI(t) ) ∫σ WI(r) n(r,t)4πr2 dr ∞
k0kD ki ) kI(∞) ) k0 + kD
(2.8)
where kD ) 4πσD is the diffusional rate constant independent of ∆Gi. If k0 is much greater than kD, then the measured reaction constant ki f kD also becomes ∆Gi-independent. If the maximum kinetic constant k0(λ) . kD, then the top of a free energy law parabola, with the highest reaction constants, is cut by a plateau ki ) kD that presents the “black sphere” limit (diffusion-controlled reaction). This is shown in Figure 5a, where ln ki(∆Gi) dependencies for different D are presented by dotted lines. Essentially the same thing takes place for a back electron transfer in the contact approximation. The time-dependent recombination rate constant
kR(t) ) k˜0n˜ (σ,t)
(2.9)
is defined via the pair distribution function n˜ (r,t) of D+ and A-. The latter is a solution of the diffusional equation, taking into account the Coulomb attraction:
n˜˘ )
D ˜ ∂ 2 r /r ∂ - r /r r e c e c n˜ ∂r r2 ∂r
(2.10)
where D ˜ is an encounter diffusion coefficient for ions. The contact boundary condition is
˜ 4πσ2D
(
(2.14)
∂n˜
rc + 2n˜ ∂r r
)|
) k˜0n˜ (σ,t)
(2.11)
To use this definition, one has first to solve the auxiliary kinetic equation for a pair distribution function of reactants which also differs from its contact analogue (eq 2.5):
n˘ ) -WI(r)n +
D ∂ 2∂ r n r2 ∂r ∂r
(3.2)
Since the position-dependent rate in eqs 3.1 and 3.2 accounts for the reaction wherever it happens, the reflecting boundary condition
|
∂n )0 ∂r σ
(3.3)
must be substituted for an absorbing one. Only the initial distribution of reactants remains the same:
n(r,0) ) 1
(3.4)
Actually n(r,t) is the distribution of excitations around a single acceptor with a hole at the center that becomes deeper and wider with time. The widening of the hole ceases at t > τe ) RQ2/D, when its radius reaches the maximum value RQ. This is an effective radius for quasi-stationary ionization with a rate constant given by
ki ) kI(∞) ) 4πRQD
(3.5)
r)σ
The initial distribution of ions remains the same:
n˜ (r,0) ) 1
(2.12)
As a result the stationary recombination constant kr ) kR(∞) is given by a relation similar to eq 2.8:24
kr )
(3.1)
k′0k′D k′0 + k′D
(2.13)
An essential fact is that RQ is D-dependent. If, for instance, the ionization is everywhere normal, WI(r) is approximately exponential (Wi ≈ const) and the corresponding effective radius is given by RQ ) σ + L/2 ln(γ2WiL2/4D).14 For sufficiently small D the similar behavior is inherent to a bell-shaped WI(r) as well. Contrary to what was found in ref 25 the effective radius of remote transfer is not only larger than σ but monotonically increases with decreased diffusion.4 The encounter theory remains valid until RQ is much less than the average distance between acceptors. This is actually the only limitation of the binary approximation: cRQ3 , 1.
3010 J. Phys. Chem., Vol. 100, No. 8, 1996
Burshtein and Krissinel
Within the binary approximation the recombination of ions may be treated the same way using the reaction constant of back electron transfer
kR(t) ) ∫σ WR(r) n˜ (r,t)4πr2 dr ∞
(3.6)
The auxiliary kinetic equation for the pair distribution function of ions n˜ is a generalization of its contact analogue (eq 2.10):
n˜˘ ) -WRn˜ +
D ˜ ∂ 2 r /r ∂ - r /r r e c e c n˜ ∂r r2 ∂r
(3.7)
with the reflecting boundary condition
(
∂n˜
rc + 2n˜ ∂r r
)|
)0
(3.8)
r)σ
and the homogeneous initial distribution of ions
n˜ (r,0) ) 1
(3.9)
The quasi-stationary recombination is attained at time t > ˜ , and the corresponding rate constant is (R′Q)2/D
˜ kr ) kR(∞) ) 4πR′QD
(3.10)
The effective recombination radius R′Q(D ˜ ) depends on ion diffusion in just the same way as RQ(D). The back electron transfer is kinetic-controlled until R′Q < rc and diffusioncontrolled when R′Q > rc > σ. To convert the kinetic regime to the diffusional one should either hamper diffusion or accelerate the electron transfer. It has been just the latter approach which has led to experimental verification of the free energy gap law. The electron transfer at a given diffusion is the fastest at the top of the quasi-parabola representing the law. If it is too fast, the top of a curve is cut by a plateau of the diffusion-controlled reaction, as shown by the dashed lines in Figure 5. As a matter of fact the plateaus are never so flat as in the contact approximation (dotted lines), but the difference in curvature is not well pronounced in a logarithm plot. The difference between remote and contact transfer is more readily seen from the RQ(∆Gci ) and R′Q(∆Grc) dependencies shown for the same conditions in Figure 6. In the diffusion-controlled region the ionization radius exceeds the contact distance and the recombination one exceeds the Onsager radius although by no more than 2 or 3 Å. Nonetheless, this must be considered seriously because under diffusion control the distributions of the reaction products are centered at these radii. 4. Distribution of Ion Pairs After pulse excitation of the sample, the total number of excited donors N obeys the conventional kinetic equation of differential non-Markovian encounter theory:
N˙ ) -kI(t)cN - N/τD
(4.1)
where τD is the lifetime of the excitation and kI(t) was defined in eq 3.1. If there is no recombination, the total amount of ion pairs P produced in the reaction should obey the following equation:
P˙ ) kI(t)cN
(4.2)
Figure 6. Free energy dependence of effective radius of ionization (RQ) and recombination (R′Q) at D ) D ˜ ) 10-6 cm2/s and R0 ) σ ) 5 Å. The rest parameters are the same as in Figure 5.
The initial conditions to the set of kinetic equations (eqs 4.1 and 4.2) are
N(0) ) 1, P(0) ) 0
(4.3)
At these conditions N(t) falls to zero while P(t) monotonically increases, approaching the quantum yield of ionization
ψ ) P(∞) which is less than 1 at any finite τD. The distribution of ions m(r,t) also changes with time. It is connected with P(t) by the relation
P(t) ) c∫σ m(r,t)4πr2 dr ∞
(4.4)
If the ions are immobile, the kinetic equation for m(r,t) may be deduced from eqs 4.2, 4.4, and 3.1:
m˘ ) WI(r) n(r,t) N(t)
(4.5)
It should be stressed that this equation is valid only for WR ) D ˜ ) 0. Initially m(r,0) ) 0, but the final distribution of ions is
m0(r) ) m(r,∞) ) WI(r)∫0 n(r,t) N(t) dt ∞
(4.6)
The shape of the normalized distributions given by
f0(r) )
m0(r)
∫m0(r) d3r
(4.7)
and the ionization quantum yield given by
ψ ) c∫σ m0(r)4πr2 dr ∞
(4.8)
have already been investigated in the static (D ) 0), diffusioncontrolled, and kinetic regimes of ionization provided WI(r) is exponential, i.e. normal.10 It was shown that f0(r) reproduces the shape of WI(r) in the kinetic-controlled limit while diffusioncontrolled ionization results in a bell-shaped ion distribution with a maximum situated at r ) RQ > σ.
Free Energy Gap Law under Diffusion Control
J. Phys. Chem., Vol. 100, No. 8, 1996 3011 broadened and shifted away a little but then returns back and finally coincides in shape with WI(r). When recombination is resonant (∆Gci + 2λc ) 0 ) ∆Grc), the ionization rate has already decreased to an extent that the electron transfer becomes kinetic-controlled again. The shape of ion distributions at low (1), intermediate (2), and highly exothermic (3) ionization essentially affects the kinetics of subsequent geminate recombination. In radiation chemistry these distributions may be prepared initially, at lower temperature, and serve as real initial conditions when recombination is turned on by a temperature jump. The situation in photochemistry is different, since the back and forward electron transfers proceed simultaneously and the recombination starts before the formation of f0(r) is completed. Nonetheless, the peculiarities of the mutual arrangement of ionization and recombination layers and the ion generation zone are very important for understanding the results of the unified theory of back and forward electron transfer presented in the last section. 5. Geminate Recombination in the Contact Approximation
Figure 7. Position-dependent rates of ionization, WI(r) (dashed lines), and recombination, WR(r) (dashed-dotted lines), and the normalized distribution of stable and immobile ions, f0(r) (solid lines), at different ionization free energies at contact ∆Gic. When they are too small, they are magnified (×100, ×103, etc.). The distance is given in Å, ˜ ) and |∆Gic| increases from the top down and from left to right. (D 10-7 cm2/s; the rest of the parameters are the same as in Figure 6.)
Roughly speaking these distributions serve as initial conditions for subsequent geminate recombination. Their shapes should be compared with WR(r) to learn where charges are created and where they can recombine. This is all the more important for study of the free energy gap law, which is only half normal. In the inverted region ionization is never at contact even in the kinetic-controlled regime because WI(r) itself is a bell-shaped curve with a maximum shifted from the contact. When keeping D ) const and varying ∆Gci , one goes from normal and kinetic-controlled ionization via the diffusion control regime at the top of the parabola to kinetic control again but in the inverted region with everything changing simultaneously: WI(r), WR(r), and f0(r) (Figure 7). From the analysis of this figure one may reach the following conclusions: (1) The resonant ionization (∆Gci ) 0) is everywhere normal and kinetic-controlled. Therefore, f0(r) reproduces the shape of a quasi-exponential WI(r) while the recombination rate WR(r) has a bell shape with a maximum shifted 2 Å away from contact. Most of the ions are born at the closest approach distance, but this is an inverted region for recombination where it proceeds too slowly. Hence, in this case ions must move outside to recombine in the middle of a remote zone for back electron transfer. (2) As we are discussing situation B (see Figure 3), the back and forward electron transfers at contact distance become activationless simultaneously, at ∆Gci ) -λc ) ∆Grc. Their rates coincide and decrease monotonically although nonexponentially with distance. The ionization is controlled by diffusion, and the products are concentrated near RQ, 2.5 Å away from contact. In this case the oppositely charged ions have to move toward each other to recombine in a contact reaction zone. (3) With further increase of |∆Gci |, ionization becomes noncontact. In the beginning the distribution of the ions is
The conventional but naive model of geminate recombination implies that ions are created in contact and also recombine at contact with the rate k-e or separate with the rate ksep ) 3rcD/ σ3[exp(rc/σ) - 1].18,26 The model is known as exponential because it predicts an exponential decrease of the survival probability
R(t) ) (1 - φ)e-(k-e+ksep)t + φ
(5.1)
where φ is the quantum yield of separation:
φ)
1 1 + k-e/ksep
(5.2)
A much better approach to the problem is given by the contact approximation similar to that used in the theory of bimolecular reactions. It does not assume ions to be created at contact but keeps their initial separation (r0) or distribution (f0(r0)) free. However the recombination remains contact and has the same rate k˜0 that appears in eqs 2.3 and 2.11. The survival probability
R(t) ) ∫σ f(r,t)4πr2 dr ∞
(5.3)
is expressed in terms of a normalized pair distribution function f(r,t), which obeys the diffusional equation
f˙ )
D ˜ ∂ 2 r /r ∂ -r /r rec e c f ∂r r2 ∂r
(5.4)
and the boundary condition6,27
˜ 4πσ2D
(
)|
∂f rc + f ∂r r2
) k˜0f(σ,t)
(5.5)
σ
They are identical to eqs 2.10 and 2.11 of bimolecular contact theory. However, the initial condition for geminate recombination is different in principle. Assuming the initial separation of ions to be r0, one must take
f(r,0) ) δ(r - r0)/4πrr0
(5.6)
As shown in ref 6, even at contact creation of ions (r0 ) σ), the exponential kinetic law (eq 5.1) is never reproduced. The very existence of k-e is impossible. The simple chemical
3012 J. Phys. Chem., Vol. 100, No. 8, 1996
Burshtein and Krissinel
kinetics appropriate for quasi-stationary bimolecular reaction cannot be extended to a geminate recombination. Not only does its kinetics differ from exponential but the diffusion dependence of the characteristic time of recombination (separation) is quite the opposite to what was obtained within the “exponential model”. The only property that may be compared is the separation quantum yield. In the contact approximation, this is6,27
6. Remote Geminate Recombination The appropriate description of remote recombination is given by the equation similar to eq 3.7
f˙ ) -WR(r)f +
D ˜ ∂ 2 r /r ∂ - r /r rec e c f ∂r r2 ∂r
(6.1)
It has the same (reflecting) boundary condition
φ(r0) ) 1 -
1 - exp(-rc/r0)
x 1 - exp(-rc/σ) 1 + x
(5.7)
where
∂ - rc/r f(r,t)|σ ) 0 e ∂r
(6.2)
but the initial condition for geminate recombination is different: r /σ k′0 k˜0(e c - 1) x) ) k′D 4πrcD ˜
with the definitions given to k′0 and k′D in eq 2.14. Under the assumption that ions were born in contact, the separation quantum yield
φ(σ) )
1 1+x
(5.9)
is essentially the same as in the exponential model because x coincides with k-e/ksep, provided
k˜0 ) k-e4πσ3/3
f(r,0) ) f0(r)
(5.8)
The initial distribution of ions f0(r) is free. If the particles are initially separated by the distance r0, it coincides with that given in eq 5.6. More reasonably it may be one of those found in section 4. The arbitrariness of the initial condition is the only demerit that remains in this approach proposed in refs 6 and 7. It follows from eqs 5.3 and 6.1 that
R˙ ) -∫WR(r)f d3r
(5.11)
is actually x ) k-e/ksep ) k′0/k′D and should reproduce Wrc-dependence on ∆Grc. If this is true, the product
XD ˜ ) k′0
1 - e-rc/σ ≡ xD ˜ 4πrc
(6.5)
f(r,t) ) ∫p(r,r′,t) f0(r′) d3r′
(6.6)
where
as well as k˜0, has the same ∆Grc-dependence as Wrc. To compare it with the free energy gap law, it was widely acknowledged that the measurable quantity
1 -1 φ
R ) 1 - ∫0 dt′ ∫WR(r) f(r,t′) d3r t
3V˜ l ≈ Wrc k-e ) Wrc σ 4πσ3
X)
(6.4)
After integration we find
(5.10)
With account of eq 2.3 one may ensure that
(6.3)
(5.12)
is a universal function, independent of diffusion (viscosity) and reproducing the free energy dependence peculiar to the kinetic rate constant of recombination k′0. As a matter of fact this is not true for two reasons: (1) Substituting eq 5.7 into eq 5.11, one can show that X * x if the ions were initially separated (r0 * σ). The difference is not unduly great but may increase if, on the contrary, the contactborn ions should recombine in the distance as in the case of resonant ionization (remote recombination) shown in Figure 7. (2) The contact approximation as a whole is not valid at D ˜ f 0. This can be seen from the fact that limxf∞ φ(r0) * 0 at r0 * σ while the separation quantum yield should be 0 when the ions are immobile. To eliminate these objections, one has to consider the back electron transfer as a remote process specified by its position-dependent rate WR(r) rather than by an absorbing boundary condition.
is determined by the Green function p(r,r′,t) of eq 6.1. As shown in ref 7 the real separation quantum yield φ j ) R(∞) f 0 at D ˜ f 0, as it should. Therefore it differs from the contact estimate (eq 5.7) in the slow diffusion region, and the larger r0 - σ, the more significant the difference becomes. Hence the exponential model prediction (eq 5.12) is expected to be right in the opposite case, when diffusion is so fast that the reaction is controlled by electron transfer. In what follows we will confirm this expectation for the particles created at contact distance. Restricting ourselves to a kinetic-controlled recombination, we may neglect the first term on the right-hand side of eq 6.1, thus reducing it to eq 5.4, and use the Green function of the latter in eq 6.6 instead of p(r,r′,t). This substitution is presented by the equality
˜ τ) at x , 1 p(r,r′,t) ≈ p0(r,r′,D
(6.7)
In this approximation we obtain from eq 6.5
R)1-
1 ∫WR(r) d3r ∫0D˜ tdτ ∫p0(r,r′,t) f0(r′) d3r′ D ˜
Correspondingly the separation quantum yield φ j ) R(∞) is averaged over the initial distribution and given by the relation
φ j )1-
1 ∫WR(r) d3r ∫p˜ 0(r,r′,0) f0(r′) d3r′ ) 1 - Xh (6.8) D ˜
where p˜ 0(r,r′,s) ) ∫0∞ p0(r,r′,τ) exp(-sτ) dτ is a Laplace transform of the Green function. Although eq 6.8 is valid only
Free Energy Gap Law under Diffusion Control
J. Phys. Chem., Vol. 100, No. 8, 1996 3013
for X h , 1, it gives a desirable result for the product
X hD ˜ ) ∫WR d3r ∫p˜ 0(r,r′,0) f0(r′) d3r′
(6.9)
which is diffusion-independent and hence universal, as in eq 5.12. When the initial separation of ions is r0, then eq 5.6 must be used for f0(r). Assuming ions were created in contact (r0 ) σ), we obtain
˜ ) ∫WR(r) p˜ 0(r,σ,0) d3r X0D where p˜ 0(r,σ,0) is a well-known
The solution of eqs 3.2 and 4.1 must be found and used in eq 7.1 to obtain P(t), defined in eq 4.4. Alternatively, the latter may be presented in the following form:8,10
P(t) ) c∫σ d3r WI(r)∫0 Ω(r,t-t′) n(r,t′) N(t′) dt′ (7.4) ∞
Here Ω(r′,t) is the fraction of ions that survived at time t provided they were initially separated by the distance r′. This quantity obeys the equation conjugate to eq 6.1:
D ˜ ∂ ∂ Ω˙ ) -WR(r)Ω + 2e- rc/r r2erc/r Ω ∂r ∂r r
(6.10)
function:6
[ () ]
rc 1 p˜ 0(r,σ,0) ) exp -1 4πrc r
t
(7.5)
with the reflecting boundary condition
(6.11)
∂ Ω(r,t)|σ ) 0 ∂r
(7.6)
Substituting this function in eq 6.10, we obtain
[ () ]
k′0 - k˜0 rc 1 ˜ ) WR(r) exp - 1 d3r ) X 0D ∫ 4πrc r 4πrc
and the initial condition
Ω(r,0) ) 1
(6.12)
If recombination is also at contact, then according to eq 2.2b k′0 ) k˜0 exp(rc/σ). In this particular case, expression 6.12 reduces to that given by eq 5.12, thus confirming the expectation ˜ must reproduce the free that the product XD ˜ as well as X0D energy dependence of the kinetic constant k′0 ∝ Wrc(∆Grc). On the other hand this is not necessarily so if recombination is not a kinetic-controlled reaction. In particular, at the top of a quasi-parabola presenting the free energy gap law, where the recombination is the fastest, it may be controlled by diffusion, so eq 6.7 is no longer valid. Then the top of the curve will be essentially deformed, as happens with bimolecular reaction constants. In the next section we will prove this statement.
In general, the photoionization kinetics presented by P(t) is qualitatively different from the recombination one, R(t). It consists of ascending and descending branches. Initially P(t) increases due to ionization but begins to decrease as soon as recombination starts to prevail. After passing the maximum (Pmax e ψ), the surviving probability goes down to a constant value, which is the photoseparation quantum yield P(∞) ) φ. This kinetics has been discussed in detail in our previous works,8,10 and now we need to concentrate attention only on the φ dependence on free energy. Taking into account that Ω(r,∞) ) φ(r), we obtain from eq 7.4 the following formula:
φ ) c∫σ φ(r) d3r WI(r)∫0 n(r,t′) N(t′) dt′ ) ∞
∞
c∫σ φ(r) m0(r) d3r (7.8) ∞
7. Recombination after Photoionization The arbitrariness of the initial conditions is the sole but essential demerit of the preceding approach. Moreover, the distributions of ions found in section 4 may be used as real initial conditions only if ionization is so fast that it has been completed before the recombination actually starts. On the other hand, when both reactions have comparable rates, they may not be considered separately. One should take simultaneously into account the back and forward electron transfer, summing on the right-hand side of the following equation the arrival term from eq 4.5 and departure terms from eq 6.1:
m˘ ) WIn(r,t) N(t) - WRm +
D ˜ ∂ 2 r /r ∂ - r /r r e c e c m (7.1) ∂r r2 ∂r
where m0(r) is the initial distribution of ions defined in eq 4.6. With account of eqs 4.7 and 4.8 we may rewrite it as follows:
φ ) ψφ j
(7.9)
where ψ is a quantum yield of ionization defined in eq 4.8 while
φ j ) ∫σ φ(r) f0(r)4πr2 dr ∞
(7.10)
is the separation quantum yield averaged over the initial ion distribution f0(r). Now we see that φ and φ j may be identified only in the case
τD ) ∞
The solution must be obtained with the reflecting boundary condition
∂ - rc/r m(r,t)|σ ) 0 e ∂r
(7.7)
(7.11)
when according to eqs 4.6, 3.1, 4.1, and 7.11 we have
(7.2)
ψ ) c∫m0(r) d3r ) c∫0 kI(t) exp(-c∫0 kI(t′) dt′) dt ) ∞
t
-∫0 N˙ (t) dt ) 1 ∞
and the initial condition
m(r,0) ) 0
(7.3)
The latter is qualitatively different from those used for f(r,t) because ions have not been prepared initially but are generated in the course of the photoionization described in section 3.
In the opposite case ψ ≈ ckiτD , 1, and for kinetic-controlled ionization ki ≈ k0(∆Gci ), so ψ is free energy dependent as well as φ j . To avoid this complication, we have already restricted ourselves to the simplest case (eq 7.11) when calculating the
3014 J. Phys. Chem., Vol. 100, No. 8, 1996
Burshtein and Krissinel
Figure 8. Quantum yield of photoseparation φ j against recombination free energy at contact ∆Gcr for different diffusion rates: (a) D ˜ ) 10-7 cm2/s; (b) D ˜ ) 10-6 cm2/s; (c) D ˜ ) 10-5 cm2/s. Figure 10. Same as in Figure 9 but for wic ) 10wcr ) 103 ns-1. The exponential model expectation is presented by curve d.
the reaction zone. The deviations from the kinetic curve become smaller as diffusion increases. However, even in the limit of infinitely fast diffusion, when X h ) X0, they both differ from x ∝ Wrc (dotted line) in the highly exothermic region. This is due to remote electron transfer in this region that makes an essential difference between the real kinetic constant (eq 2.1b) and its contact estimate (eq 2.2b). 8. Discussion
Figure 9. Free energy dependence of the averaged product X hD ˜ at the same diffusion as in Figure 8 [(a) D ˜ ) 1 Å2/ns; (b) D ˜ ) 10 Å2/ns; (c) D ˜ ) 100 Å2/ns] in comparison with its contact-born estimate X h 0D ˜ from eq 6.12 (solid line) and the exponential model expectation X hD ˜ ∝ Wcr (∆Gcr ) (dotted line d); wic ) wcr ) wc ) 103 ns-1.
initial distributions. In this particular case we may define X h as in equality 5.9
φ)φ j)
1 1+X h
(7.12)
and then use it for comparison with the exponential model or contact approximation. In particular we need to check whether the product X hD ˜ is actually diffusion-independent and reproduces the free energy gap law peculiar for the kinetic constant k′0, as was predicted in eqs 5.12 and 6.12. From the free energy dependence of the quantum yield, shown in Figure 8, one can conclude that the separation is hindered when diffusion decreases. This effect is more pronounced in the NI region (at the left), where, according to Figure 7, ions are born near contact and screened from outside by the remote recombination layer. On the contrary (at the right) they separate much more easily, since the situation is opposite (IN) and ions are created far from the contact where they recombine. Therefore many of them escape, having no time to visit the contact zone of active recombination. The free energy dependence of the quantity X h ) 1/φ j - 1 is so significantly affected by diffusion that is hardly amenable to theoretical treatment. To straighten out the data, one should better plot the product X hD ˜ in the same coordinates together with its kinetic estimate (eq 6.12) for contact-born pairs (solid curve in Figure 9). It is clear that all curves coincide in the lowest part of the figure, where recombination is slow and therefore kinetic-controlled. Evidently, this is not the case at the top of a curve where the limiting stage is the diffusion of ions toward
The distortion of the free energy gap law is more pronounced when the electron transfer is faster. However, at wc > 103 ns-1, one should take into account in the normal region the saturation of electron transfer at short distances due to the dynamical solvent effect. It results in the following generalization of the assumed shape of the transfer rate28,29
W)
w e-U/T 1 + wτ
In polar solvents τ is a longitudinal time τL ) τD0/ related to the Debye relaxation time τD. If w(σ) ) wc exceeds τ-1, the preexponent does not depend on r near the contact, where wτ . 1. This kind of transfer saturation was shown to decrease with an increase of diffusion accompanied by a corresponding increase of 1/τ.30 Therefore the saturation does not affect the bimolecular rate constants in the kinetic-controlled limit but may correct the diffusional rate constant if wc is actually so high. The diffusional contribution to ionization and recombination may be different. The experimental data on back electron transfer in solutions qualitatively reproduce the expected bellhD ˜ depenshaped dependence of k-e(∆Grc), which is actually X dence on ∆Grc.18,31,32 This was usually considered as a verification of the free energy gap law. As was shown, this appears to be the case if X h , 1 and correspondingly 1 - φ j , 1, which is to say that the recombination is kinetic-controlled. On the other hand the fastest ionization should be diffusion-controlled, as there is a diffusional plateau cutting the top of the Marcus parabola (Figure 5). These conflicting demands can be met only if wci . wrc. For illustration of this statement, we calculate the separation quantum yield for wrc ) 10-1wci ) 100 ns-1 and present the results in Figure 10. With this parametrization, the forward electron transfer remains distorted by diffusion, as shown in Figure 5, while the back transfer follows the free energy gap law much better than it did in Figure 9, where wrc ) wci ) 1000 ns-1.
Free Energy Gap Law under Diffusion Control However, the recombination is not necessarily kineticcontrolled even though the free energy dependence of X hD ˜ looks like a bell-shaped curve. When φ j < 0.05, as in ref 33, this is a straight indication of diffusion-controlled recombination to which the exponential model is inapplicable. Parabolic interpolation of similar data done in ref 34 leads only to imitation of the free energy gap law. As seen from Figure 9, the location of the maximum is quite different and the whole curve is very sensitive to diffusion when φ j , 1 (ke/ksep . 1). In conclusion, it must be mentioned that the back electron transfer may be slow for a different reason. If the excited donor is in the triplet state, the generated ion pair is also a triplet and its recombination into the ground state is strongly prohibited by spin-selection rules. Such a system was experimentally studied in ref 31, and the free energy gap law was reported to be verified within the frame of the exponential model. The real situation is much more complex because the theory should be generalized to take into account the spin evolution of the ion pair due to a superfine interaction or an external magnetic field. The latter was actually shown to affect significantly the separation quantum yield and reaction products. The magnetic field effects are worthy of special attention, and we hope to address them in our next publication. Acknowledgment. E.K. would like to thank the Fundamental Research Foundation of Russia for Grant 93-03-5038, which partially supported this research. References and Notes (1) Marcus, R. A. J. Chem. Phys. 1956, 24, 966; 1965, 43, 679. (2) (a) Brunschwig, B. S.; Ehrenson, S.; Sutin, N. J. Am. Chem. Soc. 1984, 106, 6859. (b) Burshtein, A. I.; Morozov, V. A. Chem. Phys. Lett. 1990, 165, 432. (3) Burshtein, A. I.; Frantsuzov, P. A.; Zharikov, A. A. Chem. Phys. 1991, 155, 91. (4) Burshtein, A. I.; Frantsuzov, P. A. J. Lumin. 1992, 51, 215. (5) Marcus, R. A.; Siders, P. J. Phys. Chem. 1982, 86, 622. (6) Burshtein, A. I.; Zharikov, A. A.; Shokhirev, N. V.; Spirina, O. B.; Krissinel, E. B. J. Chem. Phys. 1991, 95, 8013. (7) Burshtein, A. I.; Zharikov, A. A.; Shokhirev, N. V. J. Chem. Phys. 1992, 96, 1951. (8) Burshtein, A. I. Chem. Phys. Lett. 1992, 194, 247.
J. Phys. Chem., Vol. 100, No. 8, 1996 3015 (9) Dorfman, R. C.; Fayer, M. D. J. Chem. Phys. 1992, 96, 7410. (10) Burshtein, A. I.; Krissinel, E.; Mikhelashvili, M. S. J. Phys. Chem. 1994, 98, 7319. (11) Rehm, D.; Weller, A. Isr. J. Chem. 1970, 8, 259. (12) (a) Tunitskii, N. N.; Bagdasar’yan, Kh. S. Opt. Spectrosc. (USSR) 1963, 15, 303. (b) Kilin, S. F.; Mikhelashvilli, M. S.; Rozman, I. M. Opt. Spectrosc. (USSR) 1964, 16, 576. (c) Vasil’ev, I. I.; Kirsanov, B. P.; Krongaus, V. A. Kinet. Katal. 1964, 5, 792. (d) Steinberg, I. Z.; Katchalsky, E. J. Chem. Phys. 1968, 48, 2404. (e) Wilemski, G.; Fixman, M. J. Chem. Phys. 1973, 58, 4009. (13) Pilling, M. J.; Rice, S. A. J. Chem. Soc., Faraday Trans. 2 1975, 71, 1563. (14) Doktorov, A. B.; Burshtein, A. I. SoV. Phys. JETP 1975, 41, 671. (15) Kipriyanov, A. A.; Doktorov, A. B.; Burshtein, A. I. Chem. Phys. 1983, 76, 149. (16) (a) Szabo, A. Chem. Phys. 1989, 93, 6929. (b) Eads, D. D.; Dismer, B. G.; Fleming, G. R. J. Chem. Phys. 1990, 93, 1136. (c) Song, L.; Dorfman, R. C.; Swallen, S. F.; Fayer, M. D. J. Phys. Chem. 1991, 95, 3454. (17) Burshtein, A. I.; Kapinus, E. I.; Kucherova, I. Yu.; Morozov, V. A. J. Lumin. 1989, 43, 291. (18) (a) Gould, I. R.; Moser, J. E.; Armitage, B.; Farid, S.; Goodman, J. L.; Herman, M. S. J. Am. Chem. Soc. 1989, 111, 1917. (b) Asahi, T.; Mataga, N. J. Phys. Chem. 1989, 93, 6575. (c) Grampp, G.; Hetz, G. Ber. Bunsen-Ges. Phys. Chem. 1992, 96, 198. (19) Burshtein, A. I.; Georgievskii, Yu. J. Chem. Phys. 1994, 100, 7319. (20) Dogonadze, R. A.; Kuznetsov, A. M. Chemical Reaction Kinetics in Polar SolVents; VINITI: Moscow, 1973; in Russian. (21) Kakitani, T.; Mataga, N. J. Phys. Chem. 1985, 89, 8. (22) Jortner, J.; Bixon, M. J. Chem. Phys. 1988, 88, 167. (23) Collins, F. C.; Kimball, G. E. J. Colloid Sci. 1949, 4, 425. (24) Ovchinnikov, A. A.; Timashev, S. F.; Belyv, A. A. Kinetics of Diffusion Controlled Chemical Processes; Nova: Commack, New York, 1989. (25) Tachiya, M.; Murata, S. J. Phys. Chem. 1992, 96, 8442. (26) Kikuchi, K.; Niwa, T.; Takahashi, Y.; Ikeda, H.; Miyashi, T. J. Phys. Chem. 1993, 97, 8070. (27) Hong, K. M.; Noolandi, J. J. Chem. Phys. 1978, 68, 5163. (28) Zusman, L. D. Chem. Phys. 1980, 49. (29) Yakobson, B. I.; Burshtein, A. I. Chem. Phys. 1980, 49, 385. (30) Burshtein, A. I. J. Chem. Phys. 1993, 98, 4711. (31) Levin, P. P.; Pluznikov, P. F.; Kuzmin, V. A. Chem. Phys. Lett. 1988, 147, 283. (32) Kakitani, T.; Yoshimori, A.; Mataga, N. In Electron Transfer in Inorganic, Organic, and Biological Systems; Advances in Chemistry Series 228 Bolton, J. R., Mataga, N., McLendon, G. Eds.; p 45. (33) Gan, H.; Leinhos, U.; Gould, I. R.; Whitten, D. G. J. Phys. Chem. 1995, 99, 3566. (34) Gould, I. R.; Farid, S. J. Phys. Chem. 1993, 97, 3567.
JP951134F
