Exothermic Rate Restrictions in Long-Range Photoinduced Charge

2778

J. Phys. Chem. A 2010, 114, 2778–2787

Exothermic Rate Restrictions in Long-Range Photoinduced Charge Separations in Rigid Media Paulo J. S. Gomes, Carlos Serpa,* Rui M. D. Nunes, Luis G. Arnaut, and Sebastiaõ J. Formosinho Chemistry Department, UniVersity of Coimbra, 3004-535 Coimbra, Portugal ReceiVed: NoVember 13, 2009; ReVised Manuscript ReceiVed: January 19, 2010

Glycerol/methanol (9:1) mixtures at 255 K behave as rigid media for photoinduced electron transfers that take place within a few hundred nanoseconds. This media also provides enough polarity and plasticity to accommodate charge separations with reaction free energies ranging from +3 to -34 kcal/mol. The distance dependence of the electron transfer rates from electronically excited aromatic hydrocarbons to nitriles in this medium is accurately described by an exponential decay constant of 1.65 per angstrom. These photoinduced electron transfers display, for the first time in charge separations between independent electron donors and acceptors, a free-energy relationship with a maximum rate followed by a decrease in the rate for more exothermic reactions. According to this free-energy relationship, Franck-Condon factors are maximized at ∆G0 ≈ -15 kcal/mol. It is suggested that the inverted region observed for these first-order photoinduced charge separations originates from a slower increase of their reorganization energies with ∆G0 than that of the analogous second-order photoinduced charge separations, for which inverted regions have never been clearly observed. Introduction The transfer of an electron from a donor to an acceptor is one of the simplest chemical reactions and is certainly one of the most fundamental processes in basic sciences. It is implicated in a wide variety of primary processes covering many areas of physics, chemistry, and biology.1,2 The rates of elementary second-order electron transfers (ET) in solution increase with their exothermicities until they reach ∆G0 ≈ -10 kcal/mol and then become controlled by diffusion,3-6 with minor subtleties.7-10 This contrasts with the free-energy dependence of first-order charge shifts or charge recombinations, which have approximately the form of an inverted parabola, with an “inverted region” characterized by slower rates for higher exothermicities.11-16 A similar freeenergy dependence was also recently found in charge separations in covalently linked donor-acceptor systems.17,18 Such inverted regions were first predicted by Marcus half a century ago19 and influenced current ET theories.5,20-27 Although these are among the most intensively studied elementary reactions, the diffusion control observed for second-order photoinduced charge separations (PCS) at reaction energies where the rates of analogous first-order reactions are in the inverted region continues to challenge theoretical models. Trivial solutions to this challenge, such as the presence of competing reaction channels at higher energies or changes in the nature of the reactants, lack experimental support in prototypical PCS such as those involving aromatic hydrocarbons and nitriles.28-30 It is generally accepted that inverted regions are not observed in second-order PCS because their onset is displaced to experimentally inaccessible driving forces. Thus, the understanding of the different free-energy relationships exhibited by first- and second-order ETs relies on the explanation of why the inverted region in the latter reactions is displaced to much higher driving forces. Marcus proposed that the reorga* To whom correspondence should be addressed. E-mail: serpasoa@ ci.uc.pt.

nization energies of ET reactions increase with the donoracceptor separations and that very exothermic second-order ETs occur at larger donor-acceptor distances.31 It was even argued that bimolecular ET in acetonitrile takes place preferably when donor and acceptor are separated by distances as large as 12 Å at ∆G0 ) -60 kcal/mol.32 Alternatively, it was proposed that the reorganization energies of ET reactions increase with |∆G0|5,33 and that such an increase is less accentuated in the case of charge recombinations in contact radical ion pairs (CRIP) or exciplexes.30 This work presents an experimental test to these alternative explanations using very well characterized electron donors and acceptors in a rigid and polar matrix. It is shown that first-order PCS between independent donors and acceptors exhibit an inverted region that closely resembles that of charge recombinations in CRIPs. Moreover, the rigidity of the media precludes the formation of CRIP and lends support to the hypothesis that the shape of free-energy relationships depends on the ability to form donor-acceptor complexes as primary products. Long-distance ETs are controlled by two factors: the Franck-Condon factor (FCF), associated with the molecular structures of electron donor and acceptor, and the electronic coupling (V), associated with the overlap of the electronic wave functions of donor and acceptor. According to the golden rule of quantum mechanics, the ET rate constant is given by21,34,35

kgr )

2π 2 |V |(FCF) p

(1)

The FCF depends on the free energy of the reaction, increasing as the reactions become exothermic, maximizing for sufficiently exothermic reactions, and eventually decreasing for very exothermic reactions (the “inverted region”). The electronic coupling has an exponential dependence on the donor-acceptor distance, V2 ) V02 exp[-β(r - r0)], where r0 is the donor-

10.1021/jp9108255  2010 American Chemical Society Published on Web 02/05/2010

Photoinduced Electron Transfers in Rigid Media

J. Phys. Chem. A, Vol. 114, No. 8, 2010 2779

CHART 1: Structures, Reduction or Oxidation Potentials, and Singlet State Energies (in eV)a

a

The lifetimes in nanoseconds were measured in 9:1 glycerol:methanol mixtures, and the effective radii in Å were calculated with Connolly surfaces.

acceptor contact distance and β the distance decay factor. Early experimental evidence for the exponential distance dependence was presented by Miller, based on the scavenging kinetics of electrons trapped in an organic glass by dispersed electron scavengers.36 Using pulse radiolysis, Miller and co-workers produced the biphenyl anion in MTHF at 77 K, followed its decay in the presence of organic electron acceptors, and also provided evidence for an inverted region in very exothermic charge-shift reactions.11,37 The value of β obtained by Miller for MTHF glass, β ) 1.2 Å-1,11 was later revised to β ) 1.7 Å-1, and β ) 1.2 Å-1 was reported for toluene glass.38 Aqueous glasses were found to have β ) 1.68 ( 0.07 Å-1 at 77 K,39 and glycerol at low temperatures has β ) 1.4 Å-1.40 These β values tend to be slightly higher than those measured for donor and acceptor covalently linked by rigid nonconjugated hydrocarbon bridges, β ) 1.0-0.9 Å-1.41,42 Long-range ETs measured in ruthenium-modified proteins have distance decay factors between those of nonconjugated hydrocarbon bridges and MTHF glass.38,43 The study of long-range ET in rigid media remains one of the most versatile methods to investigate their distance and free energy dependences and is a method of renewed scientific interest and continuing research.38-40,44-47 Experimental approaches to elucidate both distance and freeenergy dependences of ET reactions are restricted by various practical considerations. The assessment of the distance dependence of PCS requires long-lived excited states and donors and acceptors dispersed in a rigid and polar medium with sufficient plasticity to stabilize the nascent charges. The free-energy dependence of the FCF must be studied with a very homogeneous series of reactants that cover both nearly isothermic and rather exothermic reactions. The choice of donor-acceptor systems is limited by the fact that rigid media at low temperatures slows down the reactions, precluding the study of endothermic reactions within the lifetimes of the singlet states. Such stringent constraints justify the absence of studies on PCS in rigid media. This work finds a compromise between such constraints in the study at -18 °C of a series of aromatic hydrocarbons, as electron donors, and nitriles, as electron

acceptors, in 9:1 mixtures of glycerol (ε ) 51 at -18 °C) and methanol (ε ) 41 at -18 °C). This series of donors and acceptors is very well characterized (see Chart 1) and covers a range of free energies from +3 to -34 kcal/mol in 9:1 glycerol: methanol mixtures, and their FCF can be related to results obtained in fluid media.30,48 The high viscosity of glycerol at room temperature, η ) 0.945 Pa s,49 is only reduced by onethird in 9:1 glycerol/methanol mixtures.50 Assuming that this factor also applies at lower temperatures, we estimate η ) 33 Pa s 9:1 for glycerol/methanol at -18 °C. The upper limit for the translational distance, Rt ) (2Dat)1/2, accessible in our systems, can be obtained from the diffusion coefficient, given by the Stokes-Einstein expression

Da )

kBT 6πηar

(2)

which is Da ) 0.002 Å2 ns-1, and the 322 ns lifetime of pyrene gives Rt ) 1 Å for fumaronitrile. Solvation dynamics is a picoseconds process even for glycerol,51 and equilibrium solvation is rapidly established with the environmental flexibility of the media. Thus, the molecular systems of Chart 1 in 9:1 glycerol/methanol matrices at -18 °C meet the requirements necessary to investigate the distance and free-energy dependence of PCS and bridge the gap between studies of first-order charge recombinations in ion pairs and of second-order PCS in solution. Here we show that the similarity of free-energy relationships in first-order charge recombinations and PCS in rigid media has profound consequences in our understanding of ET reactions. Experimental Section Naphthalene, pyrene, benzo[ghi]perylene, and coronene (Aldrich, 99%+) were zone-refined. Benzo[a]pyrene (Aldrich 99,8%) and acenaphthene (Aldrich g99%) were used as received. Fumaronitrile (Aldrich, 98%) was vacuum sublimed; methacrylonitrile (Aldrich, 98%) and 1,4-dimethoxybenzene

2780

J. Phys. Chem. A, Vol. 114, No. 8, 2010

Gomes et al.

(Aldrich g99%) were vacuum distilled to remove stabilizer. Methanol (Aldrich, Chromasolv for HPLC 99.9%) and glycerol (Aldrich, 99%+) were used as received. The appropriate amounts of donor and acceptor were initially dissolved in methanol, using 1/10 of the flask capacity, and then glycerol was added to the final volume. The solution was repeatedly shaken and submitted to ultrasound for 30 min. Absorption and luminescence spectra were recorded with Shimadzu UV-2100 and SPEX Fluoromax 3.22 spectrophotometers, respectively. Fluorescence lifetimes were measured with a previously described time-correlated single-photon counting (SPC) apparatus using IBH nanoLEDs (281, 310, 339, 373 nm) as excitation sources, Philips XP2020Q photomultiplier with excitation and emission wavelengths selected with Jobin-Ivon H20 monochromators, and a Canberra Instruments time-toamplitude converter and multichannel analyzer.52 The data analysis followed the procedure described by Gray and Winkler for long-range ET in rigid glasses,38,39 with the adaptations required for the analysis of shorter lifetimes with SPC. Briefly, the distance-dependent rate constant has the form

k(R) ) kET0 exp[-β(r - r0)]

(3)

Figure 1. Fluorescence of Bpe in the presence of fumaronitrile in 9:1 glycerol/methanol at 255 K, for various fumaronitrile concentrations.

where kET0 is the rate constant at the fluorophore-quencher contact distance r0, and β is the distance decay factor. Assuming the continuum limit for a fixed quencher distribution around a central fluorophore, the ratio of the fluorescence intensity decay I(t) and its value at time zero I(t ) 0) is38,39,53,54

[Q] ( I(tI(t)) 0) ) ) - τt - ( 132.12 )∫

ln

0

r0

∞

{1 - exp(-kET0t ×

exp[-β(r - r0)])}r2 dr (4) where τ0 is the fluorophore lifetime in the absence of quencher, the constant 132.12 is the factor needed to convert from molar to molecular units, and r0 is the excluded distance given by the sum of the donor and acceptor radii. In order to decouple the unknown parameters kET0 and β, it was proposed to scale the fluorescence decay intensities by the corresponding fluorescence quantum yields.39 We followed this procedure and corroborate its success. However, the value of I(t ) 0) is critical to obtain accurate quantum-yield-normalized decay kinetics. In view of the short lifetimes of some of the systems addressed in this work and of the use of SPC to measure such lifetimes, it was necessary to convolute the theoretical decay with the instrumental response (obtained from the scatter of a Ludox solution) to simulate the observed decay. Thus, the fitting procedure consisted in the following steps: (i) the lifetime (τ0) of the fluorophore decay is measured in the absence of quencher and its time origin identified; (ii) the decay of the fluorophore in the presence of a given quencher concentration is scaled by the ratio of fluorescence quantum yields in the absence and presence of quencher, I/I0, measured by steady state fluorescence; (iii) eq 4 is employed to generate a model decay with the value of τ0 and trial values of kET0 and β; (iv) the model decay is iteratively convoluted with the instrumental response and compared with the experimental function, using 256 data points starting at the beginning of the instrumental response; (v) the values of kET0 and β are systematically changed to minimize the residuals; (vi) the presence of other local minima in the fitting is investigated, initiating the fitting procedure with substantially different initial kET0 and β values.

Figure 2. Fluorescence decays obtained by single photon counting, for pyrene 1.1 × 10-5 M in the absence and in the presence of fumaronitrile. The instrument response is shown as a thin line.

Results and Discussion Steady-state fluorescence data was used to calculate relative fluorescence quantum yields for all fluorophores in the presence of different concentrations of quencher. Figure 1 illustrates typical results. Such data also inform on the formation of excimers and, consequently, on the rigidity of the solutions. We verified that a pyrene concentration of 0.01 M in 9:1 glycerol/ methanol at -18 °C does not lead to observable pyrene excimer emission. Single photon counting was employed to measure the fluorescence lifetimes in our experimental conditions. Figure 2 illustrates a monoexponential decay obtained in the absence of quencher and the loss exponentiality observed in the presence


J. Phys. Chem. A, Vol. 114, No. 8, 2010 2781 TABLE 1: Reaction Energies, Critical Distances, Rate Constants at Contact, and Distance Decay Factors of Photoinduced Charge Separations Measured in Glycerol/ Methanol (9:1) Mixtures at -18 °C

Figure 3. Perrin plots for the relative steady-state fluorescence intensity of the fluorophores in the presence of various concentrations of quenchers in 9:1 glycerol/methanol mixtures at 255 K.

of various concentrations of a quencher. Our matrices are airequilibrated, but oxygen quenching has only a minor effect in the measured lifetimes (see Chart 1), which is consistent with negligible molecular diffusion on the scale of our experiments. The rigidity of our matrices during the ET process can also be assessed with the Perrin model.55,56 This model assumes an “active sphere” of radius Rc defining the borders of the reactive volume. When (Rc - r0) > β and β > 1 in eq 3, there is a steep dependence of the rate on the donor-acceptor distance, and averaging over the critical distance Rc occurs over a narrow range of distances. Under these conditions, the observed steadystate fluorescence as a function of the quencher concentration takes the form

(

I0 [Q] ) exp I 3/4πRc3

)

system

∆G0 (kcal/mol)

Rc (Å)

kET0/1010 (s-1)

β (Å-1)

Py + MN Np + MN Cn + FN Bpy + FN Bpe + FN Py + FN Np + FN Ac + FN dmB+ FN

3.3 -3.0 -8.5 -12 -17 -20 -26 -28 -34

6.46 9.32 9.12 11.16 10.95 10.37 6.87 7.53 5.89

0.00308 2.81 40.1 276 314 172 139 31.2 7.5

1.65 1.63 1.64 1.66 1.68 1.63 1.64 1.65 1.68

where g is equal to 1.9. Figure 4 shows plots of this function for the decays of the longer-lived fluorophores, Py and Cn, for various concentrations of FN. The intensities of the fluorescence decays were scaled by the relative intensities measured in the steady state, which are a good measure of their relative fluorescence quantum yields. The scaling was applied to all the points after I(t ) 0), which was taken as the first channel after the maximum scatter intensity, but the first 10 ns were not included in the plot to avoid the instrumental response. The linearity of the plots for the first 400 ns of the decays confirms the validity of the approximations involved in deriving eq 6, namely, the random distribution of the quenchers and the rigidity of the matrix in the time frame of the experiment. It must also be emphasized that there is a perfect overlap for all the concentrations of the same system. Previous attempts to apply eq 6 to luminescence decays in rigid matrices succeeded in producing linear plots with the same slope but failed to yield a common intercept.38 The success obtained with our systems offers the perspective to extract reliable values for both β and kET0. The slopes of the plots represented in Figure 4 are the reciprocal of β, and from the intercept it is also possible to extract the value of kET0. However, the linear fit of the data depends on the value selected for I(t ) 0), which is convoluted with the instrumental response. Additionally, the convolution of the instrumental response extends over the first nanoseconds of the decays and limits the amount of data that can be treated

(5)

where the concentration of acceptor is in molecules per Å3 and Rc is in Å. The linearity of the Perrin plots in Figure 3 indicates that the approximations involved in deriving eq 5 are verified, namely, that the solution is rigid and long-range ET is taking place. Table 1 presents the corresponding critical distances. Note that at the critical concentration there is on average exactly one acceptor molecule within a distance Rc from one given donor molecule. Finally, the random distribution of the quenchers and the rigidity of donor-acceptor distances in the time window of the experiment can be confirmed by plotting the effective tunneling distance (Reff) as a function of ln(t)11,38

Reff(t) ) -

I(t) t ln + ( 132.12 [Q] ) [ ( I(t ) 0) ) τ ] 1/3

1/3

0

1 1 ) r0 + ln(gkET0) + ln(t) β β

(6)

Figure 4. Effective tunneling distances in ET through 9:1 glycerol/ methanol mixtures at 255 K.

2782


Gomes et al.

Figure 5. Fluorescence decays in 9:1 glycerol/methanol mixtures at 255 K of the systems (A) 1,4-dimethoxybenzene (τ ) 3.22 ns) and its quenching by fumaronitrile, [dmB] ) 2.6 × 10-5 M and [FN] ) 0.050 or 0.074 M; (B) acenaphthene (τ ) 42 ns) and its quenching by fumaronitrile, [Ac] ) 2.3 × 10-5 M and [FN] ) 0.107 or 0.165 M; (C) coronene (τ ) 297 ns) and its quenching by fumaronitrile, [Cn] ) 1.5 × 10-5 M and [FN] ) 0.078 or 0.142 M. The lines were calculated with eq 6 using the values of kET0 and β presented in the plots.

with this procedure. Rather than exploring the linearization of eq 4 implicit in eq 6, we developed a procedure to account for the instrumental response and obtain better estimates of β and kET0. The data analysis described a procedure to fit the entire fluorescence decay at various quencher concentrations. This procedure scales the fluorescence decays by the relative fluorescence intensities measured in the same conditions in the presence of various concentrations of quenchers, and the convolution with the excitation pulse profile provides fits to the fluorescence decays at very early times. Figure 5 shows representative data for fluorophores with short, intermediate, and long singlet lifetimes. The values of β that best reproduce the experimental decays are presented in Table 1. They are conveniently decoupled from the values of kET0. The Supporting Information shows how small variations in β and kET0 affect the quality of the fitting and can be used to ascertain the uncertainty in these parameters. The values of β in glycerol/ methanol matrices, β ≈ 1.65 Å-1, are remarkably close to that obtained for aqueous glasses, β ) 1.68 ( 0.07 Å-1.39 Whereas the values of β are approximately constant for all the systems studied, there are large variations in the values of kET0. Before such variations are interpreted, it must be emphasized that the fitting of β and kET0 also depends on the value of r0. The sizes of the molecules in Chart 1 were estimated using Connolly surfaces.57 The radii of spheres with the same volume as the objects obtained are presented in the chart, rD and rA. The value of r0 was defined as the fluorophore-quencher contact distance. In practice, it is calculated as the sum of the radii of fluorophore and quencher, r0 ) rD + rA. This is equivalent to estimating the distance (r - r0) in eq 3 as the donor-acceptor edge-to-edge distance.58 This method gives physically meaningful values for r0 that are independent of the fitting of the fluorescence decays. The structural homogeneity of the systems employed in this work guaranties that the changes in kET0 can only be assigned to the dependence of the FCF on the reaction energy. In view of the large values of Rc, it is appropriate to take the reaction free-energy of long-distance ET as the energy released in the formation of the free ions from the singlet state of the fluorophore, that is,

∆G0 ) Eox - Ered - E*

(7)

This ignores a small electrostatic stabilization that depends on the donor-acceptor distance, -e2/(εr). Given the values of the electronic charge (e2 ) 14.4 eV Å) and of the dielectric constant ε ≈ 50, this term contributes with only 1 kcal/mol stabilization for shortest critical distance found in this work. Chart 1 presents the oxidation potentials of the aromatic hydrocarbons measured in acetonitrile vs SCE,59,60 the reduction potentials of the nitriles also in acetonitrile vs SCE,61,62 and the singlet state energies in polar solvents.63 The free energies obtained with eq 7 are given in Table 1. Figure 6 shows the free-energy dependence of kET0. This parameter is a reaction frequency, expressed in units of s-1, and corresponds to the rate constant when the donor-acceptor distance is extrapolated to r0 (or when the quencher concentration is extrapolated to infinity). Thus, kET0 is the PCS rate between donor and acceptor in contact, which is the opposite of the charge recombination rate (kCR) in a contact radical ion pair (CRIP), photoinduced charge separation

D* · A 98 thermal charge recombination

D•+ · A•- 98 D · A

The initial and final electronic states of the two processes are not identical and the two reactions may have different electronic factors. However, this should not affect their freeenergy dependence, except perhaps for a minor difference between the FCF of D* and that of D. Thus, the maxima of kET0 and kCR should occur at the same value of ∆G0, when structurally related reactants exchange an electron in media with the same dielectric constant and refractive index. Table 2 gathers literature data on the least exothermic charge recombinations in CRIP formed by direct excitation of charge transfer bands in acetonitrile (ε ) 36 and nD ) 1.34)20 available in the literature for aromatic hydrocarbons as donors and tetracyanoethylene (TCNE) as acceptor. Interestingly, the “normal region” of these reactions has never been observed.18,65,66 There are uncertainties of a few kcal/mol in the estimates of their ∆G0, but it is nevertheless clear that these ultrafast charge recombination rates overlap with the values of kET0 obtained in this work for -12 e ∆G0 e -26 kcal/mol.


J. Phys. Chem. A, Vol. 114, No. 8, 2010 2783 of the molecular structures of donor and acceptor and from the reorganization of the solvent around the nascent charges, λ ) λv + λs. Marcus has shown that when the solvent is treated as a dielectric continuum, the solvent reorganization energy is given by20,70

(

λs ) e2

)(

1 1 1 1 1 + 2rD 2rA r n 2 ε D

)

(8)

where r is the center-to-center separation distance, as in eqs 3 and 4. Using ε ≈ 50 and nD ≈ 1.48 for 9:1 glycerol:methanol at -18 °C and for Py + FN we obtain λs ) 30.5 kcal/mol when the molecules are at the critical distance r ) Rc ) 10.4 Å and λs ) 22.7 kcal/mol when the molecules are in contact (rD ) 3.85 Å, rA ) 2.83 Å, r ) r0 ) rD + rA). Various methods have been employed to estimate the reorganization energies of the molecular bonds. In the framework of Marcus theory, the equation proposed by Sutin is frequently employed71

Figure 6. Free-energy dependence of photoinduced charge separation rate constants extrapolated to contact in rigid 9:1 glycerol:methanol matrices (circles) at 255 K. The error bars reflect the range of kET0 values that acceptably fit the fluorescence decays at the various quencher concentrations. Charge recombinations in charge-transfer complexes from aromatic hydrocarbons to TCNB adsorbed on porous glass at 77 K (diamonds)64 and to TCNE in acetonitrile (triangles, from data in Table 2), and from aromatic hydrocarbons to FN in CRIP formed in isopropyl ether at 253 K (squares)48 are shown for comparison. The line represents golden rule calculations with eqs 1 and 9, using T ) 255 K, V0 ) 150 cm-1, pωv ) 1500 cm-1 and the values of λ indicated in the plot, in kcal/mol. The dashed line illustrates compensation effects between λs and λv.

TABLE 2: Representative Literature Data on Charge Recombination Rates Following the Excitation of Charge Transfer Bands of Ground-State Electron Donor-Acceptor Complexes between Aromatic Hydrocarbons and TCNE in Acetonitrile electron donor

∆G0 (kcal/mol)

Eox - Ered (kcal/mol)

k0ET/1012 (s-1)

ref

perylene perylene pyrene pyrene hexamethylbenzene naphthalene

-14 -13 -21 -27 -35 -29

-14 -14 -21 -21 -25 -30

3.3 3 2 3.4 1.6 1

29 67 67 68 69 67

Figure 6 presents the rates in Tables 1 and 2 together with the charge recombination rates of ion pairs produced by excitation of charge-transfer complexes between aromatic hydrocarbons and 1,2,4,5-tetracyanobenzene (TCNB) adsorbed on porous glass at 77 K64 and of CRIP formed between Py or Np and FN in isopropyl ether at -20 °C.48 These systems seem to share the same Franck-Condon factors, as expected from their structural similarity. According to the free-energy relationship obtained for PCS in glycerol:methanol matrices, we expect the “normal region” of charge recombinations in CRIP to be observable for ∆G0 g -10 kcal/mol in polar media. This explains why this region has never been observed in such reactions: total reorganization energy in polar media is much lower than expected from Marcus theory and the FCF is maximized at ∆G0 ≈ -15 kcal/mol. According to Marcus theory, the reorganization energy is partitioned between contributions arising from the reorganization

1 2

λv )

∑ pωi∆i2

(9)

i

where the normal mode displacement ∆i is calculated as the difference between the equilibrium bond lengths in reactant and product, and ωi is the vibrational frequency of the bond. This model gives λv ) 6.4 and 6.7 kcal/mol for the TCNE0/- and naphthalene0/+, respectively,72,73 and leads to λv ≈ 6.5 kcal/mol for their cross-reaction. As mentioned above, the singlet state of naphthalene may give a different FCF than its ground state, but the difference is certainly small. The total reorganization energy given by eqs 8 and 9 is λ ) λs + λv ≈ 22.9 or 37 kcal/ mol, depending on whether the calculations are made for the molecules in contact or at the critical distance of 10.4 Å, respectively. The discrepancy between the calculated reorganization energies and the experimental λ ≈ 15 kcal/mol can be ascribed to an overestimate of λs by eq 8, in view of the evidence that the dielectric continuum model overestimates the solvent reorganization energy in polar solvents.5,6,10,30,64 The poor description of solvent fluctuations by the dielectric continuum model was also recently exposed in the context of atomic ET reactions.74 In order to understand this discrepancy, we first resort to golden rule calculations of ET rate constants. The FCF is the vibrational overlap integral between initial and final states. Within the displaced harmonic oscillator approximation, and when the donor and acceptor are represented by one single high frequency, this overlap is given by75

FCF )

1

∞

∑e

√4πσ2 n)0

n

-S S

n!

[

exp -

[∆G0 + λs + npωv]2 4σ2

]

(10)

where S ) λv/pωv is the electron-vibration coupling strength for a distinct vibration mode ωv, and σ2 is the width of the Gaussian spectral functions. It became common to employ a semiclassical approximation and write σ2 ) λskBT,20,76 although Kuzmin rightfully pointed out that the assignment of the broadening of the vibronic bands to thermal fluctuations of the medium polarization ignores various kinds of intermolecular interactions besides the medium reorganization that also contribute to inhomogeneous broadening of vibronic levels.26 A

2784


Gomes et al.

typical value for the high frequency employed to represent aromatic hydrocarbons and nitriles is pωv ) 1500 cm-1. This frequency and the reorganization energies calculated with eqs 8 and 9 suffice to calculate the Franck-Condon factors. The comparison between golden rule and experimental rates additionally requires a value for the electronic coupling. The rates presented in Figure 6 were calculated with the electronic coupling at donor-acceptor contact V0 ) 150 cm-1, which is in the range employed by other authors to optimize the agreement between golden rule calculations and the experimental charge recombination rates in CRIP.14,30,77 Golden rule calculations with the distance-dependent solvent reorganization energies given by eq 8 and the molecular reorganization energy of eq 9 predict that the onset of the inverted region should be observed at ∆G0 < -35 kcal/mol, in conflict with the data in Figure 6. This conflict can be resolved by using the typical values of pωv, V, and λv, if λs is reduced to 8 kcal/mol. Table 1 shows that the critical distance Rc accompanies the rate constant kET0, and that the faster PCS occur over a wider range of distances in rigid polar matrices. However, the rates measured in 9:1 glycerol:methanol matrices share the same reorganization energy as charge recombinations in CRIP measured in acetonitrile or in porous glass. Furthermore, the rates of systems with Rc from 5.89 to 11.16 Å follow the same free-energy relationship, although the donor-acceptor distance dependence of eq 8 anticipates reorganization energies 8.6 kcal/ mol apart. Thus, our data excludes a significant dependence of λs on the donor-acceptor distance, and the absence of an inverted region in second-order ET cannot be ascribed to increased solvent reorganization at large donor-acceptor separations. An alternative explanation for the absence of inverted regions in PCS is based on the increase of the reorganization energy with the driving force of the reaction.5,30,33,67,78 This explanation also accounts for the mild energy gap dependences of charge recombinations in CRIP and their small solvent dependence.30,77,79,80 At present, the only quantitative formulation for this effect is that of the intersecting-state model (ISM).5 ISM calculates FCF on the basis of an averaged highfrequency vibration of an oscillator representing the reactants and another one representing the products. Each reactant and product is represented by an oscillator with a characteristic frequency (or force constant, fr and fp), bond length (lr and lp), and bond order (n‡). For example, benzene and alkyl benzenes are characterized by a frequency of 1500 cm-1 (or force constant 1.15 × 103 kcal/(mol Å2)), bond length of 1.35 Å, and bond order of 1.5.30,48 It is a reasonable approximation to take the properties of the oscillator representing the reactants as the average of the properties of donor and acceptor and similarly for the products. The sum of the bond extensions of reactant and product oscillators from equilibrium to transition state configurations is given by

d)

{

}

1 + exp(√2n‡∆G0/Λ) a′ ln (lr + lp) ‡ 2n 1 - [1 + exp(√2n‡∆G0/Λ)]-1 (11)

where a′ is a constant (a′ ) 0.156). Given that second- and first-order ET between aromatic hydrocarbons and nitriles are described by the same molecular parameters [n‡ ) 1.8, lr + lp ) 2.7 Å, f ) fr ) fp ) 1.15 × 103 kcal/(mol Å2)], the reaction rates of the two types of reactions must be identical when ∆G0

) 0 and differ for exothermic reactions insofar as they may have different values of the parameter Λ. This coupling parameter regulates the dissipation of ∆G0 by the accepting modes. The separation d between reactant and product oscillators, their force constants, and the reaction energy ∆G0 determine the energy of the crossing point, ∆G‡. The classical ISM reaction rate is then

kISM ) ν exp(-∆G‡/RT)

(12)

where ν is the reaction frequency. Figure 7 illustrates how Λ modulates the free-energy dependence of ET reactions. An infinite value of Λ leads to d ) (lr + lp)a′ ln(2)/n‡, which is independent of ∆G0. In the case of Λ ) ∞, the free-energy dependence of ET reaction rates is the Marcus inverted parabola. A finite value of Λ leads a measurable increase of d with ∆G0, i.e., the reorganization energy increases with the exothermicity of the reaction. A smaller value for Λ leads to larger the increase of λ with ∆G0. For the molecular systems addressed in this work, the increase in λ with ∆G0 when Λ approaches 70 kcal/mol produces an increase in the rates of the reactions with ∆G0 < -70 kcal/mol, which has been experimentally observed.30 As we further reduce Λ from 70 to 40 kcal/mol, the predicted free-energy relationship changes from the double-inverted region of Λ ≈ 70 kcal/mol to the Rehm-Weller plateau of diffusion-controlled ET rates.3 The difference between the free-energy relationships of secondversus first-order PCS may be explained by a greater sensitivity of the former to the increase of λ with ∆G0 (lower Λ). Figure 8 shows that ISM calculations with the parameters previously employed for CRIP formed between aromatic hydrocarbons and nitriles, with the obvious exception of the temperature, also fit very well the new data in rigid media at low temperature when tunneling corrections are included as described in detail elsewhere.48 The free-energy dependence of first-order ET reactions takes the form of a double inverted region when the data in heptane and cyclohexane is included in the plot.30 The excellent agreement between ISM calculations with earlier parameters and the new experimental data strongly supports the concept that the reorganization energy increases with the driving force of ET reactions. As mentioned before, the increase of the reorganization energy with the driving force of the ET reaction is associated with the finite value of Λ. Its physical meaning can be best appreciated by comparison with the role of S in eq 10. When S ) 0, eq 10 takes the form of the classical parabolic Marcus free-energy relationship, whereas the asymmetry of the inverted region increases when S takes increasing larger values.81 Within the displaced harmonic oscillator model,27 the electron-vibration coupling strength can be expressed in terms of the reduced displacement

(

(lr - lp) S) 2σv

)

2

1 f(l - lp)2 2 r ) pω

(13)

which is the change of the equilibrium configuration of the vibrational mode expressed in units of the root-mean-square displacement at the zero-point energy.82 An increase in the displacement between initial and final states leads to an increase in the asymmetry of the inverted region. ISM and the golden rule give similar free-energy relationships if S is allowed to increase with |∆G0|. Figure 9 shows that the


J. Phys. Chem. A, Vol. 114, No. 8, 2010 2785

Figure 7. ISM reaction coordinate for ET reactions and predicted free-energy relationships. The sum of reactant and product bond extensions from equilibrium to transition state configurations (d) increases with the exothermicity of the reactions. (Left) For very large Λ, d is independent of ∆G0 and a symmetrical Marcus inverted region should be observed. (Center) With Λ ) 70 kcal/mol, the modest increase in d allows for the occurrence of the inverted region, but for ultraexothermic reactions this increase is accentuated and brings the inverted region to an end, originating a double inverted region. (Right) With Λ ) 40 kcal/mol, the increase of d maintains the intersection of reactants and product curves close to the minimum of the former over an extended range of ∆G0, and a diffusion plateau is expected.

rates of ultraexothermic charge recombinations in various solvents30,48 and of PCS in glycerol:methanol matrices can be reproduced by golden rule calculations when λv increases from 6.5 kcal/mol for weakly exothermic reactions to 25 kcal/mol for the most exothermic charge recombinations in this series. The concept of energy-dependent reorganization energies stimulates new thinking about free-energy relations in secondand first-order ET reactions. The experimental data suggests that the reorganization energy increases more rapidly with the driving force in second-order PCS than in first-order ETs. These observations invite further studies before a quantitative answer can be found. Our present knowledge of ET reactions is only sufficient to offer a qualitative explanation. From the viewpoint of ISM, a more negative ∆G0 may increase the displacement of high-frequency modes and/or increase the population of other vibrational modes. The empiricism of the coupling parameter Λ masks the oversimplification of the ISM reaction coordinate to one average high-frequency mode representing reactants and products. A more detailed description of the reaction coordinate of more exothermic reactions should involve more vibrational modes. The absence of an inverted region in second-order PCS suggests that additional modes contribute to the reorganization energy at lower exothermicities than in first-order reactions. The identification of such modes remains speculative but it is tempting to assign the low-frequency donor-acceptor stretch present in exciplexes

formed by second-order PCS as one of the additional accepting modes. It was shown that the intercomplex stretch is a good energy-accepting mode,83 and it seems to be particularly well suited to accommodate the energy of second-order PCS leading to exciplexes. Channeling the reaction energy to exciplex modes increases the reorganization energy of the reaction, displaces the inverted region to increased exothermicities, and leads to faster rates in the inverted region. In the perspective of ISM, the inverted region observed at relatively low driving forces for PCS in rigid matrices can be ascribed to the absence of intercomplex stretch mode. According to the golden rule formulation with only one high-frequency mode, employed to derive eq 10, the increase in the reorganization energy with the driving force of the reaction can be assigned to an increase in S.82 However, in a real chemical system, several high- and medium-frequency modes are coupled to the ET reaction, and a multimode approach is preferable.84 Increasing the number of mediumand high-frequency modes leads to an increase in the asymmetry of the inverted region,84 that is, to an increase in the ET rates in the inverted region. The consonance between ISM and the golden rule is achieved when more “nonactive” vibrational modes become “active” (increase in the number of terms of eq 9) as |∆G0| increases.

2786


Figure 8. Same as Figure 6 but the line now represents ISM calculations at 255 K with eqs 11 and 12 and tunneling corrections as in ref 48, using ν ) 4 × 1012 s-1, n‡ ) 1.8, lr + lp ) 2.7 Å, fr ) fp ) 1.15 × 103 kcal/(mol Å2), Λ ) 70 kcal/mol, and a total reduced mass for Np and FN of 96.5 amu.

Gomes et al. methanol at -18 °C versus glycerol at -23 °C, may have slightly different distance decay coefficients, which may account for part of the difference. However, we believe that the accurate data analysis employed in this work and the consistency between the results obtained for nine independent systems indicate that the distance dependence of ET reactions in glycerol:methanol matrixes is higher than previously reported and approaches that measured in aqueous glasses at 77 K, β ) 1.68 ( 0.07 Å-1.39 The free-energy dependence of photoinduced charge separations between electronically excited aromatic hydrocarbons and nitriles arrested in polar matrixes exhibits a maximum rate at ∆G0 ≈ -15 kcal/mol. The solvent reorganization energy derived from this free-energy dependence is ca. 3 times lower than that expected from the dielectric continuum model. Moreover, the solvent reorganization energy does not depend to an appreciable extent on the donor-acceptor separation. Our data show that distance-dependent reorganization energies cannot be invoked to explain the diversity of free-energy relationships observed in electron transfer reactions. Although the interpretation of free-energy dependences of ET reactions remains largely phenomenological, the emerging view is that the close contact between the products of secondorder PCS may provide additional accepting modes that increase their reorganization energies and maintain high rates at large driving forces. On the other hand, first-order PCS between reactants arrested in rigid matrices, charge recombinations in ion pairs, and intramolecular electron transfers in rigid donor-spacer-acceptor systems lack the intercomplex stretch of the exciplex formed in second-order PCS, have reorganization energies that increase more slowly with the driving force of the reaction, and yield distinct inverted regions. Acknowledgment. We thank Fundaçaõ para a Cieˆncia e Tecnologia (Portugal) and FEDER (European Union) for financial support through project no. PTDC/QUI/70637/2006. P.J.S.G. and R.M.D.N. acknowledge FCT for fellowships SFRH/ BD/31478/2006 and SFRH/BD/24005/2005. Note Added after ASAP Publication. This article was published ASAP on February 5, 2010, with an error in eq 4. The correct version was reposted on February 10, 2010.

Figure 9. Free-energy dependence of high-frequency modes viewed through the golden rule for ET between aromatic hydrocarbons and fumaronitrile. Golden rule calculations (thin lines) employed T ) 25 °C, V ) 150 cm-1, pωv ) 1500 cm-1, λs ) 6 kcal/mol and the values of S indicated in the plot, and ISM calculations (thick line) are as in the previous figure. Triangles, CRIP charge recombinations formed in heptane at room temperature;30 squares, CRIP charge recombinations in isopropyl ether at 253 K;48 circles, PCS extrapolated to contact in rigid 9:1 glycerol:methanol matrices at 255 K, measured in this work.

Conclusions The average distance decay obtained in this work, β ≈ 1.65 Å-1, is higher than that previously reported for glycerol at low temperatures, β ≈ 1.4 Å-1.40 The two media, 9:1 glycerol:

Supporting Information Available: Fluorescence spectra of aromatic donors in the presence of suppressor in 9:1 glycerol/ methanol at -18 °C, for various suppressor concentrations and corresponding Perrin plots. Fluorescence decays in 9:1 glycerol/ methanol matrices at -18 °C of the aromatic donors and its quenching by the suppressor. Experimental data fits using monoexponential and nonexponential decay functions. Examples of fits using inadequate parameters and their impact on the quality of fit. This information is available free of charge via the Internet at http://pubs.acs.org. References and Notes (1) Fox, M. A. Chem. ReV. 1992, 92, 365. (2) Arnaut, L. G. J. Photochem. Photobiol. A: Chem. 1994, 82, vii. (3) Rehm, D.; Weller, A. Isr. J. Chem. 1970, 8, 259. (4) Niwa, T.; Kikuchi, K.; Matsusita, N.; Hayashi, M.; Katagiri, T.; Takahashi, Y.; Miyashi, T. J. Phys. Chem. 1993, 97, 11960–11964. (5) Formosinho, S. J.; Arnaut, L. G.; Fausto, R. Prog. React. Kinet. 1998, 23, 1–90. (6) Nelsen, S. F.; Weaver, M. N.; Pladziewicz, J. R.; Ausman, L. K.; Jentzsch, T. L.; O’Konek, J. J. J. Phys. Chem. A 2006, 110, 11665–11676. (7) Creutz, C.; Sutin, N. J. Am. Chem. Soc. 1977, 99, 241. (8) Chen, J.-M.; Ho, T.-I.; Mou, C.-Y. J. Phys. Chem. 1990, 94, 2889. (9) Turro´, C.; Zaleski, J. M.; Karabatsos, Y. M.; Nocera, D. G. J. Am. Chem. Soc. 1996, 118, 6060.

Photoinduced Electron Transfers in Rigid Media (10) Rosspeintner, A.; Kattnig, D. R.; Angulo, G.; Landgraf, S.; Grampp, G. Chem.sEur. J. 2008, 14, 6213–6221. (11) Miller, J. R.; Beitz, J. V.; Huddleston, R. K. J. Am. Chem. Soc. 1984, 106, 5057–5068. (12) Miller, J. R.; Calcaterra, L. T.; Closs, G. L. J. Am. Chem. Soc. 1984, 106, 3047–3049. (13) Mataga, N.; Asahi, T.; Kanda, Y.; Okada, T.; Kakitani, T. Chem. Phys. 1988, 127, 249–261. (14) Gould, I. R.; Moody, R.; Farid, S. J. Am. Chem. Soc. 1988, 110, 7242–7244. (15) Guldi, D. M.; Asmus, K.-D. J. Am. Chem. Soc. 1997, 119, 5744– 5745. (16) Serpa, C.; Arnaut, L. G. J. Phys. Chem. A 2000, 104, 11075–11086. (17) Mataga, N.; Taniguchi, S.; Chosrowjan, H.; Osuka, A.; Kurotobi, K. Chem. Phys. Lett. 2005, 403, 163–168. (18) Mataga, N.; Chosrowjan, H.; Taniguchi, S. J. Photochem. Photobiol. C: Photochem. ReV. 2005, 6, 37–79. (19) Marcus, R. A. Faraday Discuss. Chem. Soc. 1960, 29, 21. (20) Marcus, R. A.; Sutin, N. Biochim. Biophys. Acta 1985, 811, 265. (21) Kestner, N. R.; Logan, J.; Jortner, J. J. Phys. Chem. 1974, 78, 2148. (22) Brunschwig, B. S.; Logan, J.; Newton, M. D.; Sutin, N. J. Am. Chem. Soc. 1980, 102, 5798. (23) Kuznetsov, A. M. J. Phys. Chem. 1992, 96, 3337–3345. (24) Small, D. W.; Matyushov, D. V.; Voth, G. A. J. Am. Chem. Soc. 2003, 125, 7470–7478. (25) Li, X. Y.; Fu, K.-X. J. Comput. Chem. 2004, 25, 500–509. (26) Kuzmin, M. G.; Soboleva, I. V.; Dolotova, E. V. J. Phys. Chem. A 2007, 111, 206–215. (27) Arnaut, L. G.; Formosinho, S. J.; Burrows, H. D. Chemical Kinetics; Elsevier: Amsterdam, 2007; p 549. (28) Mataga, N.; Kanda, Y.; Asahi, T.; Miyasaka, H.; Okada, T.; Kakitani, T. Chem. Phys. 1988, 127, 239–248. (29) Page`s, S.; Lang, B.; Vauthey, E. J. Phys. Chem. A 2004, 108, 549– 555. (30) Serpa, C.; Gomes, P. J. S.; Arnaut, L. G.; Formosinho, S. J.; Pina, J.; Seixas de Melo, J. Chem.sEur. J. 2006, 12, 5014–5023. (31) Marcus, R. A.; Siders, P. J. Phys. Chem. 1982, 86, 622–630. (32) Kakitani, T.; Matsuda, N.; Yoshimori, A.; Mataga, N. Prog. React. Kinet. 1995, 20, 347–381. (33) Arnaut, L. G.; Formosinho, S. J. J. Mol. Struct. (THEOCHEM) 1991, 233, 209–230. (34) Van Duyne, R. P.; Fischer, S. F. Chem. Phys. 1974, 5, 183. (35) Efrima, S.; Bixon, M. Chem. Phys. Lett. 1974, 25, 34. (36) Miller, J. R. J. Chem. Phys. 1972, 56, 5173–5183. (37) Beitz, J. V.; Miller, J. R. J. Chem. Phys. 1979, 71, 4579–4595. (38) Wenger, O. S.; Leigh, B. S.; Villahermosa, R. M.; Gray, H. B.; Winkler, J. R. Science 2005, 307, 99–102. (39) Ponce, A.; Gray, H. B.; Winkler, J. R. J. Am. Chem. Soc. 2000, 122, 8187–8191. (40) Strauch, S.; McLendon, G.; McGuire, M.; Guarr, T. J. Phys. Chem. 1983, 87, 3579–3581. (41) Closs, G. L.; Calcaterra, L. T.; Green, N. J.; Penfield, K. W.; Miller, J. R. J. Phys. Chem. 1986, 90, 3673–3683. (42) Paddon-Row, M. N.; Oliver, A. M.; Warman, J. M.; Smit, K. J.; Haas, M. P.; Oevering, H.; Verhoeven, J. W. J. Phys. Chem. 1988, 92, 6958–6962. (43) Gray, H. B.; Winkler, J. R. Proc. Natl. Acad. Sci. U.S.A. 2005, 102, 3534–3539. (44) Krongauz, V. V. J. Phys. Chem. 1992, 96, 2609. (45) Swallen, S. F.; Weidemaier, K.; Tavernier, H. L.; Fayer, M. D. J. Phys. Chem. 1996, 100, 8106–8117. (46) Weidemaier, K.; Tavernier, H. L.; Swallen, S. F.; Fayer, M. D. J. Phys. Chem. 1997, 101, 1887–1902.

J. Phys. Chem. A, Vol. 114, No. 8, 2010 2787 (47) Rae, M.; Fedorov, A.; Berberan-Santos, M. N. J. Chem. Phys. 2003, 119, 2223–2231. (48) Serpa, C.; Gomes, P. J. S.; Arnaut, L. G.; Formosinho, S. J.; Seixas de Melo, J. ChemPhysChem 2006, 7, 2533–2539. (49) Marcus, Y. The Properties of SolVents; Wiley Series in Solution Chemistry; J. Wiley & Sons: Chicester, 1998; Vol. 4. (50) Rangel-Zamudio, L. I.; Rushforth, D. S.; van Duyne, R. P. J. Phys. Chem. 1986, 90, 807–811. (51) Kambhampati, P.; Son, D. H.; Kee, T. W.; Barbara, P. F. J. Phys. Chem. A 2000, 104, 10637–10644. (52) Seixas de Melo, J.; Fernandes, P. F. J. Mol. Struct. 2001, 565, 69– 566. (53) Blumen, A. J. Chem. Phys. 1980, 72, 2632–2640. (54) Blumen, A.; Manz, J. J. Chem. Phys. 1979, 71, 4694–4702. (55) Inokuti, M.; Hirayama, F. J. Chem. Phys. 1965, 43, 1978–1989. (56) Guarr, T.; McGuire, M.; Strauch, S.; McLendon, G. J. Am. Chem. Soc. 1983, 105, 616. (57) Connolly, M. L. Science 1983, 221, 709. (58) Arnaut, L. G.; Formosinho, L. G. J. Photochem. Photobiol. A: Chem. 1996, 100, 15–34. (59) Pysh, E. S.; Yang, N. C. J. Am. Chem. Soc. 1963, 85, 2124. (60) Lui, H. T.; Xu, J. K.; Zhou, W. Q.; Pu, S. Z.; Wei, Z. H.; Shen, L.; Luo, M. B. Acta Polym. Sinica 2007, 554–558. (61) Lu¨cking, K.; Rese, M.; Sustmann, R. Leibigs Ann. Chem. 1995, 1129. (62) Tyssee, D. A.; Baizer, M. M. J. Org. Chem. 1974, 39, 2819. (63) Murov, S. L.; Carmichael, I.; Hug, G. L. Handbook of Photochemistry, 2nd ed.; Marcel Dekker, Inc.: New York, 1993. (64) Miyasaka, H.; Kotani, S.; Itaya, A.; Schweitzer, G.; De Schryver, F. C.; Mataga, N. J. Phys. Chem. B 1997, 101, 7978–7984. (65) Gould, I. R.; Farid, S. Acc. Chem. Res. 1996, 29, 522–528. (66) Vauthey, E. J. Photochem. Photobiol. A: Chem. 2006, 179, 1–12. (67) Asahi, T.; Mataga, N. J. Phys. Chem. 1991, 95, 1956–1963. (68) Wynne, K.; Reid, G. D.; Hochstrasser, R. M. J. Chem. Phys. 1996, 105, 2287–2297. (69) Wynne, K.; Hochstrasser, R. M. AdV. Chem. Phys. 1999, 107, 263. (70) Marcus, R. A. J. Chem. Phys. 1956, 24, 966–978. (71) Sutin, N. Ann. ReV. Nucl. Sci. 1962, 12, 285. (72) Rosokha, S. V.; Newton, M. D.; Head-Gordon, M.; Kochi, J. K. Chem. Phys. 2006, 324, 117–128. (73) Jakobsen, S.; Mikkelsen, K. V.; Pedersen, S. U. J. Phys. Chem. 1996, 100, 7411. (74) Bragg, A. E.; Cavanagh, M. C.; Schwartz, B. J. Science 2008, 321, 1817–1822. (75) Serpa, C.; Arnaut, L. G.; Formosinho, S. J.; Naqvi, K. R. Photochem. Photobiol. Sci. 2003, 2, 616. (76) DeVault, D. Q. ReV. Biophys. 1980, 13, 387–564. (77) Vauthey, E. J. Phys. Chem. A 2001, 105, 340–348. (78) Gould, I. R.; Noukakis, D.; Gomez-Jahn, L.; Goodman, J. L.; Farid, S. J. Am. Chem. Soc. 1993, 115, 4405–4406. (79) Asahi, T.; Ohkohchi, M.; Mataga, N. J. Phys. Chem. 1993, 97, 13132–13137. (80) Benniston, A. C.; Harriman, A.; Philp, D.; Stoddart, J. F. J. Am. Chem. Soc. 1993, 115, 5298–5299. (81) Bixon, M.; Jortner, J. AdV. Chem. Phys. 1999, 106, 35–202. (82) Bixon, M.; Jortner, J. J. Phys. Chem. 1991, 95, 1941–1944. (83) Wynne, K.; Galli, C.; Hochstrasser, R. M. J. Chem. Phys. 1994, 100, 4797. (84) Bixon, M.; Jortner, J.; Cortes, J.; Heitele, H.; Michel-Beyerle, M. E. J. Phys. Chem. 1994, 98, 7289–7299.

JP9108255

Exothermic Rate Restrictions in Long-Range Photoinduced Charge

Recommend Documents