Bimolecular photo-induced electron transfer enlightened by diffusion

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Bimolecular photo-induced electron transfer enlightened by diffusion

ANGULO NUNEZ, Gonzalo Manuel, ROSSPEINTNER, Arnulf

Abstract

Photochemical electron transfer between freely diffusing molecules has been studied extensively. Here, we try to elucidate how much these works have contributed to the understanding of electron transfer. To this end, we have revisited the work performed in the experimental and theoretical areas of concern from the beginning of the 20th century up to the present day. We present a critical look at the major contributions and compile the current picture of a variety of phenomena around electron transfer in solution. This is based on two main developments, besides the theory of Marcus: encounter theories of diffusion and laser techniques in time-resolved spectroscopy.

ANGULO NUNEZ, Gonzalo Manuel, ROSSPEINTNER, Arnulf. Bimolecular photo-induced electron transfer enlightened by diffusion. Journal of Chemical Physics , 2020, vol. 153, no.

040902

DOI : 10.1063/5.0014384

Available at:

http://archive-ouverte.unige.ch/unige:141044

Disclaimer: layout of this document may differ from the published version.

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Cite as: J. Chem. Phys. 153, 040902 (2020); https://doi.org/10.1063/5.0014384 Submitted: 19 May 2020 . Accepted: 18 June 2020 . Published Online: 29 July 2020 Gonzalo Angulo , and Arnulf Rosspeintner

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Bimolecular photo-induced electron transfer enlightened by diffusion

Cite as: J. Chem. Phys.153, 040902 (2020);doi: 10.1063/5.0014384 Submitted: 19 May 2020•Accepted: 18 June 2020•

Published Online: 29 July 2020

Gonzalo Angulo^1,a) and Arnulf Rosspeintner^2,b) AFFILIATIONS

1Institute of Physical Chemistry, Polish Academy of Sciences, Kasprzaka 44/52, 01-224 Warsaw, Poland

2Department of Physical Chemistry, University of Geneva, 30 Quai Ernest-Ansermet, CH-1211 Geneva 4, Switzerland

Note:This paper is part of the JCP Special Topic on 65 Years of Electron Transfer.

a)Author to whom correspondence should be addressed:gangulo@ichf.edu.pl

b)Electronic mail:arnulf.rosspeintner@unige.ch

ABSTRACT

Photochemical electron transfer between freely diffusing molecules has been studied extensively. Here, we try to elucidate how much these works have contributed to the understanding of electron transfer. To this end, we have revisited the work performed in the experimental and theoretical areas of concern from the beginning of the 20th century up to the present day. We present a critical look at the major contributions and compile the current picture of a variety of phenomena around electron transfer in solution. This is based on two main developments, besides the theory of Marcus: encounter theories of diffusion and laser techniques in time-resolved spectroscopy.

Published under license by AIP Publishing.https://doi.org/10.1063/5.0014384., s

I. INTRODUCTION

It is customary in many fields of research dealing with chemical reactivity to say that the analysis of an experiment was “obscured by diffusion.” This may be due to two major reasons: either the experimental conditions were not appropriate or broad enough to record the full dynamics of the reaction, or the models used for the analysis did not consider diffusion in a proper way. However, nowadays, we count on both the technologies to interrogate down to very short times and the theories that account for diffusion in most of the imaginable scenarios. This has not been the case over many years, particularly those during which both theory and experiment were being developed, and misconceptions around electron transfer have grown to stay. It is time to critically revise these works in light of our current capabilities and set a new picture for photoinduced bimolecular electron transfer reactions. As we shall see in this perspective, several authors have previously forwarded the point of view defended here but been unable to convince the whole community.

Regarding electron transfer, much was learned from the field of pulse radiolysis, electrochemistry, and electron paramagnetic res- onance.^1–3 The celebrated Miller–Closs experiment in which the

Marcus “inverted region” was unambiguously observed was of the first type.⁴On the other hand, that of photochemistry, the Rehm–

Weller experiment⁵ cast a shadow for over four decades over the applicability of Marcus’s ideas⁶ to freely diffusing reactants. The effect was invigorating, nonetheless, as many theoreticians and experimentalists made efforts to solve a major paradox: Was there no Marcus inverted region for charge separation in bimolecular photoinduced reactions? Why did the quenching of organic fluorophores in acetonitrile (ACN) seem to be still diffusionally con- trolled at free energies as large as−2 eV? However, what does diffusion control mean in the first place? Does diffusion completely mask the peculiarities of the reaction? In other words, as soon as the intrinsic reaction rate constant becomes larger than that of the diffusional approach of the reactants does it become impossible to say anything about how the intrinsic reaction proceeds? The answer, we know now, is no. Categorically not. We will justify this assertion in detail.

The history of the development of the theories of diffusion as we use them now is a long one. It begins with the seminal work of Smoluchowski in 1918.⁸ Intended for a problem different than that of our concern, coagulation in the colloidal solutions of Zsig- mondy, he combined the concepts of Brownian motion developed

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by himself,⁹Perrin,¹⁰Einstein,¹¹and Langevin,¹²with an infinitely fast coagulation probability when the particles collided. The task was not at all a simple one in view of the—at the time—dominant percep- tion of chemical reactivity in terms of the gas phase collision theory.

As a matter of fact, until the 1930s, a clear distinguishing concept between the reactions in both aggregation states was not available:

the cage effect. Rabinowitch and Wood—with the help of a mechan- ical model (Fig. 1)—explained that in liquid solutions, the collisions cluster in time, defining an encounter.⁷ This does not necessarily mean that the reactants get trapped in a fixed geometry but rather that for a given time, they follow similar fluctuations of the solvent and re-collide several times before separating.

When applying this idea to the study of electron transfer, an additional difficulty arose quite soon: the reaction does not seem to be constrained to the contact between the reactants. The spatial overlap of the orbitals involved in the reaction—or in other words, the coupling between the reactants’ and products’ electronic clouds—decays over a given distance, a tunneling length, which can be obtained in the first approximation from the Gamow theory.¹³Thus, at the same time that the reactants move, the electron transfer can take place with different probabilities depending on the distance between the reactants. This was not incorporated until many years after Smoluchowski in what we now call the differential

FIG. 1. Rabinowitch’s exemplary experiment showing the difference between (a) gas and (b) liquid phases. In blue, a central post that when hit by a metallic ball (red) closes a circuit lighting a bulb. The table was shaken to mimic thermally driven Brownian motion. The lower panels (a) and (b) correspond to the two shown densities of particles. They recorded the number of times the bulb was illuminated over time. As can be seen, while at low densities (gas), the bulb illuminated ran- domly, at high ones (liquid), it did so in clusters. The picture has been adapted from Ref.7.

encounter theory, obtained by several groups independently. More- over, to make things worse, several other quantities entering in the Marcus expression also depend on the separation distance between the reactants, such as the solvent reorganization energy and even the free energy of the reaction.

In photochemical experiments, the electron transfer reaction between two species is followed by monitoring the quenching of luminescence or the transient absorption signal of the excited state. The reaction occurs over a given time, as long as there are molecules in the excited state. During this reaction time, the products can either separate or recombine back to the ground state after a consecutive electron transfer step. The situation is even more complex than that of the forward reaction as despite having been born on average at a given distance, there is a distribution of them in space, which changes over time. Under such conditions, the kinetics become quite complex and coupled to the previous step. Trying to extract the electron transfer parameters for recombination from the experiments is an unavoidably difficult task, and simplified models with very rough assumptions have been the most widely used among experimentalists over decades, with few exceptions.^14–17

An added complication arises from the fact that the interaction between aromatic organic reactants in solution seems to be quite strong in general. This leads to the appearance of mixed states, called exciplexes, electronic donor acceptor complexes, and others. Depending on the conditions of the experiment—polarity of the medium, concentration of the reactants, excitation wavelength, and free energies of different states—instead of observing an electron transfer reaction tractable by the non-adiabatic Marcus theory,⁶ something completely different may be observed.

We will try to go through most of these issues and define clear conditions to separate the different cases and set the path for the application of the proper model in each of them. We will also address some limitations of the theories of electron transfer and diffusion and expose those issues that in our opinion are still to be solved. Although, as mentioned above, the experiments were triggered by the theoretical prediction of the “inverted region” and they, in turn, stimulated new theories in electron transfer and in diffusion, currently, there seems to be a divorce between both communities impeding further advances.

The idea of this perspective is to go through the history of bimolecular photoinduced electron transfer in liquid solution with emphasis on the importance of material diffusion. We have aimed to limit references to those which explicitly focused on the developments of experiments and/or theoretical concepts dedicated to this particular problem and, most certainly, have missed some.

II. RATE CONSTANTS ARE NOT CONSTANTS

In most undergraduate chemistry courses, only classical chemical kinetics are taught. There, one of the assumptions is that each step in a complex chemical reaction can be characterized by a rate constant. Therefore, if a diffusional step (D) is to be considered because the intrinsic reaction (R) is fast enough to compete with it, the two processes are considered to be consecutive, and the overall observed rate constant is given by

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1 kobs = 1

kD

+ 1 kR

. (1)

At most, the diffusion rate constant,kD, is provided from the result of Smoluchowski as

kD=4πσD, (2)

where σ is a distance of collision and D is a mutual diffusion coefficient. Thus, the link with collision theory of gases is deceiv- ingly straightforward, and the concepts are internalized easily.¹⁸The same is taught for the opposite case, i.e., when the products orig- inate from a molecular complex or a molecule experiencing bond cleavage. Now, the diffusional rate constant is of first order and acquires different forms depending on the presence or absence of an electrostatic attraction.

Although appealing due to its simplicity, the former picture is only a crude approximation and utterly wrong if diffusion influ- enced reactions are examined in a little more detail. The reason is that under the assumptions of classical formal kinetics, each of the steps is independent, Markovian. In other words, the rate constants do not depend on their history at all. The inadequacy of this assumption was already identified by Smoluchowski more than a century ago.⁸He realized that, in the presence of diffusion, the rate constant is no longer a constant but depends on time. The reason for this is that the reaction starts—triggered, for example, by a light pulse or a temperature jump providing sufficient energy to alter the equilibrium of a solution—first for those pairs of reactants that are closest to each other. Thus, in the next time snap, the distribution of reactants differs from the previous one and so on as time runs. Eventually, diffusion and reaction equilibrate and a steady gradient of reactants is reached. The chosen way to solve such a complex problem was to imagine a single central immo- bile reactant mimicking all the molecules of the same species, sur- rounded by many of the other kind, approximated by points and moving with the mutual diffusion coefficient, the sum of the diffusion coefficients of each species. Of course, such a picture requires the central reactant to be present in a much lower concentration than the other. Additionally, interactions between the molecules of the reactant in excess ought to be negligible to keep the illusion of their point-like behavior. This is what is called the target problem (cf.Fig. 2).

The result obtained by Smoluchowski is applicable to reactions that happen infinitely faster than diffusion and only when the reactants are in contact. His result, much used over the years, is a rate coefficient, no longer a constant, that depends on time and reflects the non-Markovian nature of diffusion-assisted reactions,

k(t) =4πσD(1 + σ

√πDt). (3)

The famous equation(2)can be easily recognized as the long time asymptote of this expression. Many have wondered how appropriate this treatment is for chemical reactions.^19–21Indeed, mathematically, the formulation is exact, meaning that under the assumptions made, it could be obtained from non-equilibrium many-particle statistical theory.^22,23Later on, we will see extensions to this model in various aspects.

In the decade following Smoluchowski’s groundbreaking paper, several laboratories started finding evidence that fluorescence

FIG. 2. Transition from the many pair description to the target model and the most common types of the targets. In the central panel, the target representation of the problem summarizes all the pairs’ trajectories from the left (colors are conserved).

In the rightmost panel, the different kinds of targets, of inner boundary conditions, are shown, corresponding to the Smoluchowski (all contacts are reactive), Collins–

Kimball (not all contacts are reactive), and differential encounter theory (contacts are simply reflected and the reactivity may expand beyond the contact) from top to bottom, respectively.

quenching was not tractable using the simple classical kinetic models. The Perrins proposed, what is now called, a sphere of action around molecules in which quenching can take place in order to explain the optimal concentration to observe fluorescence in the absence of quenchers.²⁴A few years later, Jette and West performed quenching experiments on the fluorescence of quinine sulfate and fluorescein using various inorganic salts as quenchers.²⁵ Unfortu- nately, their data were not presented in the anamorphosis invented by Stern and Volmer almost a decade earlier,²⁶given by

I0

I(c)=1 +κτ0c. (4) I0is the steady-state intensity of emission andτ0is the lifetime of the fluorophore, in a solution without a quencher.I(c) is the same intensity but in solutions with a concentration,c, of the quencher.

κis the Stern–Volmer rate constant. In such a representation, the so-called Stern–Volmer plot, deviations from linearity may be a hint of non-Markovian behavior, a rate coefficient that depends on time, provided ground state complex formation is excluded.²⁷Nev- ertheless, presented in such a way, Jette and West’s data show a non-linearity.

Around the same time, major advances were made in the laboratory headed by Sergey Vavilov. Using several examples, among which were also some systems studied by Jette and West, they analyzed the quenching data using the sphere of action model.^28–30A few years later, one of Vavilov’s students, Sveshnikov, improved the theory and reached an important conclusion: the time decay of the

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fluorescence in the presence of a quencher cannot be exponential if diffusion is assisting the reaction.^31–33 In fact, they only had to wait 20 years to confirm this effect experimentally using a fluorescence modulation setup.³⁴This leads us to a second consequence of the non-Markovian character of the Smoluchowski result. The decay of the fluorophore population, upon diffusion-assisted quenching, follows a non-exponential law of the following form:

N(t) =exp(−t

τ0−c∫₀^tk(t^′)dt^′). (5)

III. FIRST STEPS OF QUENCHING BY ELECTRON TRANSFER

However, which role does electron transfer play in all of this?

In fact, electron transfer was proposed in these years as a possible excited state reaction by Baur³⁵and later by Weiss and Fischgold.³⁶ However, one had to wait several decades until a solid experimental confirmation was obtained. Meanwhile, the theory of diffusion continued to develop.

One of the limitations of Smoluchowski’s approach is the lack of long distance forces between the target, the fluorophore in our case, and the quenchers. Debye—in 1942—derived a still widely used expression for the separation of charged particles in the stationary state.³⁷ Only four years later, Montroll included it in the time dependent expression of the rate coefficient, more specifically, in the diffusional operator.³⁸ Another limitation was removed by Collins and Kimball in 1949.³⁹As mentioned above, in the case of Smoluchowski, the reaction event was considered to be infinitely fast. Collins and Kimball solved that problem, and the intrinsic reaction rate constant, kR, appears in their model. The reaction was, however, still thought to be operative only at contact. As a matter of fact, Eq.(1)is also the long time limit of their theory. A theory combining these two effects, namely, that of a Coulombic potential and a finite reaction probability at contact, was later developed by Weller in 1957.⁴⁰

This was not the last determining contribution of Weller to the field. In 1961, his laboratory confirmed that electron transfer was indeed a mechanism of quenching by monitoring the transient absorption signal of the radical anion of perylene after photoexcita- tion in the presence ofN,N-dimethylaniline.⁴¹It is understandable that it took so long after the first indirect evidence found by Baur as the flash photolysis technique was developed by Porter and co- workers only after World War II.⁴²At the time of Weller’s findings, Marcus had already published his theory, which itself was also some- how prompted by developments linked to the war, through the study of isotopic separation.⁴³In a simple version, already including a pre- exponential factor due to Levich and Dogonadze,^44,45the expression can be written as follows:

w(r) = V(r)²

̵h

√ π

λ(r)kBTexp[−(λ(r)+ΔG(r))²

4λ(r)kBT ], (6) whereV is the coupling matrix element or coupling constant,λis the solvent reorganization energy, andΔGis the free energy of the reaction. It is important to keep in mind that this theory is applicable in principle to non-adiabatic reactions only, meaning that the

coupling term,V, must be small. The usual limit given to its value is half the thermal energy,kBT. Otherwise, the equation is not applicable in this form.^46–48We will later see an extension of it, which expands its use to large values ofV.

Already, at this time, further photochemical experiments revealed the complexity of electron transfer in the excited state. First, the detection of products with a spin multiplicity different from that of the original reactants revealed a spin change mechanism in the photogenerated radical pairs.⁴¹Second, when changing the polarity of the solvent, new emission bands were observed, attributed to what became known as exciplexes or complexes of partial charge separation in the excited state.⁴⁹All the ingredients for a strong discussion about the mechanism of electron transfer were present. On top of this, a problem became apparent: which is the free energy of a reaction in the excited state? Weller proposed a now abundantly used recipe: add the excitation energy acquired by the fluorophore upon

FIG. 3. Marcus electron transfer probability expanded in the distance between reactants for several free enthalpies (top panel). The probability of reaction has been normalized to its volume and multiplied by 4πr². The black dashed and solid lines correspond to the free enthalpies used for charge separation and charge recombination inFigs. 4,9, and10, respectively. Note how the reaction probability expands and its maximum shifts away from the contact as the free energy becomes more negative. The lower panels show the distance dependence of the coupling matrix element; the free energy, as calculated from Eq.(7); and the reorganization energy, from the continuum model of Born,⁶respectively. For the free energy, we have represented the difference with respect to the value at infinite separation. The equation used to obtain the uppermost panel is Eq.(20). For a full list of parameters, see theAppendix.

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excitation to the free energy of the reaction.⁵⁰ This is the Weller equation,⁵¹

ΔGcs(r) =E(D⁺/D) −E(A/A⁻) −E00+kBTrC

r, (7) where theEs are the reduction potentials times the Faraday constant [all are written in the direction of reduction, so usuallyE(D⁺/D) is positive andE(A/A⁻) is negative];E00is the excitation energy, taken positive;⁵²rCis the Onsager radius (see theAppendix); andris the center to center distance between the reactants. Thus, an exergonic, i.e., spontaneous, reaction gives a negativeΔGcs.

From Eqs.(6)and(7), it becomes quite clear that there are several terms that depend on the distance between the reactants (cf.

Fig. 3). This means that electron transfer can happen at distances larger than merely at contact. One of them is of paramount importance: the coupling between the states,V. At that time, the 1960s, this was an issue that had yet to be addressed. However, in the field of resonant energy transfer, as explained by Förster theory⁵³or for exchange interaction, in agreement with the description of Dexter,⁵⁴ it was well known that the reactivity was distance dependent. Obvi- ously, the target model had to be adapted as both Smoluchowski’s and Collins and Kimball’s expressions are only applicable for contact reactions.

IV. ELECTRON TRANSFER: NOT A CONTACT REACTION

Apparently, the first to introduce this distance dependence in diffusion–reaction equations of the Smoluchowski type were Tunitskii and Bagdasar’yan in 1963.⁵⁵ Four and five years later, respectively, Yokota and Tanimoto⁵⁶and Steinberg and Katchalski⁵⁷ arrived independently at similar equations explicitly for the energy transfer reaction. Wilemski and Fixman, in 1973, on the other hand, proposed a method to include distance dependence by making use of a series of step functions for the reactivity.⁵⁸However, none of these models were sufficiently well-grounded until Doktorov and Burshtein provided a clear and consistent picture of the problem in 1975.⁵⁹

In this evolution of the target problem (cf.Fig. 2), we have passed from a target that absorbs every particle approaching (Smolu- chowski) through a partially absorbing (Collins and Kimball) to a completely opaque and perfectly reflecting sphere (Doktorov and Burshtein). This theory became known as differential encounter theory. Its name is owed to the combination of having the mathematical form of a differential equation and being devised for the encounter problem in solution—in contrast to the collision gas phase problem. This is better appreciated when written down explicitly. In the case of Smoluchowski, the diffusion equation for the normalized pair distribution function,ρ^R, in a purely centrosymmetric problem is given by

∂ρ^R(r,t)

∂t =DLρˆ ^R(r,t), (8) where the diffusion operator, ˆL, is the divergence of the gradient of the pairs, the Laplace operator,

Lˆ= 1 r²

∂

∂rr²∂

∂r. (9)

Equation(8)is solved with perfectly absorbing boundary conditions, assuming a distant unaltered supply of reactants. This means

ρ^R(σ,t) =0, (10a)

ρ^R(∞,t) =1. (10b)

This way, Eq.(3)can be obtained. In the case of Collins and Kimball, the incorporation of a contact reactivity,kR, means altering the inner boundary condition to account for its finite probability,

4πDr²∂ρ^R(r,t)

∂r ∣

r=σ =kRρ^R(σ,t). (11) This is fine for contact reactions such as proton transfer. However, if a distance-dependent reactivity is needed, as in the case of energy or electron transfer, the diffusion equation [Eq.(8)] itself must be altered to include it, becoming a reaction–diffusion equation

∂ρ^R(r,t)

∂t =DLρˆ ^R(r,t) −w(r)ρ^R(r,t), (12) which is to be solved with a reflective boundary condition that excludes the volume of overlap between the fluorophore and the quenchers,

∂ρ^R(r,t)

∂r ∣

r=σ=0. (13)

Finally, the rate coefficient is expressed as the integral over space of the product of the reactivity and the solution of the reaction–

diffusion equation for the pair distribution function,

k(t) =4π∫_σ^∞w(r)ρ^R(r,t)r²dr. (14) Figure 4 shows how the rate coefficient changes from an initial value equal to the initial reactivity,k0, which contains the details of the intrinsic one, to an asymptotic value,k∞, which can always be expressed as Eq.(2)but with a modified radius different from the contact one. This radius, the quenching radius in this context, RQ, can be either smaller than the contact radius, if diffusion is not the controlling step, or reach values much larger at high viscosities.

The same is true—an increase ofRQ—with the increase in the space expansion of the reactivity (cf.Fig. 7).

Expression(14)can now be inserted into the kinetics of the decaying fluorophore, i.e., Eq.(5). As a result, this kinetics becomes nonexponential due to the higher values ofk(t) at shorter times than at longer and as long ask(t) has not reachedk∞. This is called the transient effect (cf.Figs. 9and14). When integrated over time, as is done in steady-state measurements of fluorescence, the Stern–

Volmer plot gains in curvature with an increase in the quencher concentration. In fact, the Stern–Volmer constant is not a constant anymore but becomes concentration dependent,κ(c) (cf.Fig. 14).

At low quencher concentrations, the Stern–Volmer plot is almost linear, especially at low viscosities, and its slope is related to the

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FIG. 4. The rate coefficient as calculated from differential encounter theory [Eqs.(12)–(14), withΔGcs=−1.1 eV and parameters in theAppendix] for different viscosities [multiples of the one of acetonitrile (ACN)]. The transient effect of the kinetics of the fluorophore is nothing else than this change from the value at short times,k0, determined by the intrinsic reaction probability, to the stationary value at long times,k_∞. Note also that the stationary stage is actually never reached.⁶⁰

Laplace transform of Eq. (14) with the fluorescence lifetime as argument. The increasing curvature is a consequence of an increased probability of observing a reaction event at shorter times, i.e., when k(t) is larger. All these equations have been demonstrated to be exact for this problem, as mentioned above, and are equivalent to pair treatments, even though they show such a strong resemblance to macroscopic Fickian diffusion.^21,22

V. SOLVENT STRUCTURE AND THE HYDRODYNAMIC EFFECT

What are the shortcomings of the differential encounter theory, as expressed above? There are many, although one of the most important is related to the fact that solvents are simply not ideal continuous media as they are composed of molecules. On the other hand, can diffusion in a microscopic reaction only be characterized by a bulk macroscopic quantity? This leads us to the next developments in the theory.

The second question of the previous paragraph was addressed as early as in 1966 by Friedman⁶¹ and later refined by Deutch and Felderhof⁶² in 1973.⁶³ In a nutshell, the diffusion coefficient is reduced when the reactants approach as they start to feel each other’s influence on the surrounding solvent molecules (seeFig. 5).

For example, the expression of Deutch and Felderhof for this hydrodynamic effect is given by

D(r) =D(1−3σ

4r), (15)

whereDdenotes the bulk mutual diffusion coefficient.

The other microscopic effect tries to grasp the molecularity of the solvent. Imagine a given solvent molecule or, in general, a point in the space inside a solution. The probability to find another solvent molecule at a given distance is not a flat line equal to one but is higher near this very same point. This reflects certain solvent structure due to the finite size of the molecules and depends on the interaction potential between the solvent molecules.⁶⁵How does this

FIG. 5. Representative distance dependencies of the diffusion coefficient,D(r), and the solvent pair distribution function,g(r). The former has been calculated using Eq.(15). The latter was obtained using the parameters of acetonitrile with the Percus–Yevick approximation.⁶⁴

structure affect our diffusion–reaction problem? Basically, it is like introducing a potential between the reactants,

v(r) = −kBTlng(r). (16) This can be combined with an electrostatic potential as introduced by Debye, if the reactants are charged. Its effect is to enhance the probability of finding a quencher molecule in the vicinity of the target, sometimes by more than a factor of two with respect to the bulk (seeFig. 5). Whether or not this effect is included explains many deviations observed in several experiments we will go through. To our knowledge, the first theoreticians to introduce bothD(r) and g(r) in a reaction–diffusion equation in this field were Sipp and Voltz in 1985.⁶⁶This passed unnoticed for experimentalists for one decade.

VI. THE PARADOXICAL REHM–WELLER EXPERIMENT In the meanwhile, a key experiment was performed in the laboratory of Weller in the late 1960s: together with his student Rehm, they endeavored to vary the free energy of electron transfer in the quenching of fluorescence of aromatic hydrocarbons. The result is well known (cf.Fig. 6). After an increase in the rate constant as this energy got more negative (more exergonic), a plateau attributed to the diffusion limit was reached.⁶⁷According to a simple logic based on the Marcus expectation for an inverted region, increasing the energy further should lead to a decrease in the rate constant.⁵This was not the case.

Moreover, this is not the only paradox posed by the famous Rehm–Weller experiment. Each of the three regions into which the plot can be divided (rising branch, diffusional plateau, and inverted

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FIG. 6. Top panel: oversimplified horizontally flipped Jabło ´nski diagram for the energy levels of the products (ion pairs) with respect to the ground,S₀, and excited state energy,S1, of the fluorophores. Data were taken from Rehm and Weller (•),⁵Kikuchi and co-workers (•),⁶⁹and Grampp and co-workers (○).⁷⁰The Rehm–Weller plot in linear (middle panel) and logarithmic (lower panel) scales.

The dashed line is the Marcus expectation according to Eq.(6). The dotted line is the result of the Rehm–Weller interpolation formula.⁶⁷ The colored regions correspond to the three regions of Kikuchi:⁷¹red, excited state complexes and reversibility; white, true non-adiabatic electron transfer; and green, ground state complexes and possible excited state ions.

region) has its problems.⁶⁸In the rising branch, the rate constants are larger than one would expect for reversible reactions as the low free energy would suggest. Indeed, one can fit a simple irreversible electron transfer rate to them. Is the reversible step suppressed or is there another reaction channel opening up? The “plateau” should not be categorized as such. As we will see later, taking into account all the distance dependent ingredients in the Marcus rate constant, it should be, albeit weakly, rising as the average quenching distance should increase with the free energy of the reaction.

Some previous observations realized in Weller’s laboratories could already, at that time, provide explanations for the antinomies of both the ascending branch and the plateau. In 1963, together with his student Leonhardt, Weller reported the emission of exciplexes in apolar solvents for reactions with relatively small exergonicities on the one hand and non-linear Stern–Volmer plots and little or no exciplex emission in polar solvents on the other hand.⁴⁹This may be taken as the first evidence of changes in the electron transfer distance with the solvent polarity (or free energy).

At the time, however, the absence of the Marcus inverted region, i.e., the decrease in the reaction rate with decreasing free energy (more negative ΔG), caught almost all the attention. The explanation provided by Rehm and Weller was the existence of

excited photoproducts, i.e., electronically excited radicals in these charge separation reactions. Many contributions from both the theoretical and experimental fields may have been prompted by their results. For example, Ulstrup and Jortner,⁷²and Efrima and Bixon⁷³ extended the Marcus expression to include excited vibrational states of the products to account for the retardation in the appearance of the inverted region upon increasing the free energy (multichannel transfer). However, even this extension to the theory did not lift the mystery surrounding the Rehm–Weller experiment.

VII. OBSERVATIONS OF THE TRANSIENT EFFECT Is diffusion responsible for this? If it is important, the transient effect had to be observed. Already, in the 1950s, Sveshnikov reported it from fluorescence modulation measurements.³³ In the 1970s, Nemzek and Ware used single-photon counting and stationary measurements, arriving to a partial success in the explanation of their experiments:^74,75the Collins–Kimball model could be fitted to the decays, but an additional ground state complex had to be invoked to explain the extra curvature of the Stern–Volmer plots of the steady-state quenching data (which contain the entire information—

albeit in the integrated form). However, the reactions studied by them were not clearly identifiable as electron transfer reactions.

In 1975, Chuang and Eisenthal reported transient absorption experiments on the reaction between anthracene and N,N- diethylaniline following the charge separation monitoring the products in the picosecond regime.⁷⁶ They even tried to explain the non-exponential kinetics with the use of the Collins–Kimball model, although deviations from it were apparent. They attributed this to the use of the truncated version of the Collins–Kimball expression (its long time approximation, closely resembling the Smoluchowski equation) on the one hand and to the importance of rotational effects on the other hand.

So, clearly something that would accelerate the reaction at short times—too short for their time resolution—was missing in the model. A distance dependence of the reaction under investigation?

Only four years later, Yoshihara and co-workers indeed showed in solid matrices of ethanol that electron transfer was occurring at distances larger than contact.⁷⁷It should be noted that pulse radiolysis experiments had already shown this in the early 1970s.^78–80The first use of a distance dependent rate coefficient in the context of photoinduced bimolecular electron transfer arrived by the hand of the group of Fleming.⁸¹Their impressive results could be well explained, although a full congruence between all the measurements eluded them (vide infra).

The distance dependence was therefore later used as an argument to explain the lack of the inverted region in the Rehm–Weller plot.⁸²In summary, the impression was that diffusion distorts the results and “obscures” the possibility to extract electron transfer rate constants.

VIII. THE QUEST FOR THE INVERTED REGION

The inverted region remained elusive in bimolecular reactions of photoexcited molecules, despite Creutz and Sutin reporting vestiges of it in 1977 upon studying the quenching of ruthenium complexes in 0.5M sulfuric acid solutions.⁸³ However, the first

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clear observation of the inverted region had to come from another field: again pulse radiolysis. In a series of papers, Miller and co- workers found it first in rigid matrices and later in liquid solution.^4,84 Moreover, at almost the same time, Wasielewski and co- workers observed the inverted region in the recombination of photoinduced products for intramolecular electron transfer, i.e., linked reactants.⁸⁵These findings further nurtured the mystery surrounding the Rehm–Weller experiment. What is then to blame? Diffusion, charge separation, multichannel transfer, excited ions, all of them, none?

Several workers indeed found that the quenching radius of the reaction increased with the viscosity of the solvent (seeFig. 7).^86–89 This points again to the essentials of diffusion models incorporat- ing electron transfer theory being correct as we shall see. In 1989, almost 14 years after Eisenthal’s first attempt to do so, the group of Peters rediscovered the usefulness of the Collins–Kimball model, in the analysis of ps-resolved transient effects in photoinduced electron transfer reactions.⁹⁰Major efforts were made by the group of Fleming to study the adequacy of several models for studying fluorescence quenching using a combination of up-conversion, single- photon counting, and steady-state spectroscopy.^81,91,92They arrived at the disappointing conclusion that none of the employed models (not even differential encounter theory) could explain all the measurements in a coherent manner. However, not all effects that we have so far mentioned, i.e.,g(r) andD(r), were considered. Murataet al.⁹³in 1995 extractedVand its decay lengthL[see Eq.(6)] in a relatively viscous solvent, although they obtained a rather small value most likely due to the limited time resolution of their time-correlated single-photon counting and also without includingg(r) andD(r).⁹⁴ In fact, when the group of Fayer introduced these effects one year later and explicitly studied quenching with an almost full Mar- cus expression, the coherence between their few experiments was excellent.⁹⁶Fayeret al.basically studied two fluorophore/quencher systems, namely, rubrene/duroquinone and rhodamine 3B/N,N- dimethylaniline, showing a striking difference: the coupling matrix element,V, in the former was very small (about 1/20th ofkBT) while in the latter very large (up to 1.5 timeskBT).^14,15,97This huge difference in a key quantity for the applicability of non-adiabatic electron

FIG. 7. Experimental dependence of the quenching radius,R_Q, with free energy (right) and with viscosity (left) using data from Kikuchi and co-workers (•)⁶⁹ and Grampp and co-workers (○).^70,95The red circled data point, measured in acetonitrile, is common for both panels.

transfer theory can be traced back to the molecular structure of these chromophores. While rubrene has four large phenyl groups protect- ing the electron transfer active center, rhodamine 3B is almost bare.

This allows for the quencher to approach the latter much closer, and therefore, the retrieved coupling matrix elements obtained were larger.

The development and increased use of laser technologies opened the possibility to explore electron transfer in many laboratories. In the 1980s, several groups embarked on studying it by means of laser flash photolysis. While reports about the products and their recombination up until this point had been relatively scarce, all of a sudden, a considerable amount of data became available. We will mention a few of these reports, which were to provide what is now a quite common point of view. The group of Mataga worked with both transition metal complexes and organic fluorophores.^98–104 They were able to find the full parabolic behavior for charge recombination, as predicted by the Marcus theory. Others, such as Gould and Farid, constrained their studies to a narrower range of free energies in the inverted region for the recombination^105,106but dedicated time and efforts to investigate the role of molecular size,¹⁰⁷ steric and isotope effects,¹⁰⁸as well as molecular charge¹⁰⁹or external pres- sure.¹¹⁰Major differences between these two groups concerned not only the available time resolution of their setups (ps in the case of Mataga and tens of ns for Gould and Farid) and the data analysis but also the availability of excitation wavelengths. In the latter case, they were selectively exciting in spectral regions free or not of additional bands appearing when the quencher was added. In doing so, they claimed to distinguish two sorts of reactions or, in their inter- pretation, two sorts of ion pairs distinguished by the inclusion of a solvent molecule in between them. Borrowing a concept from the study of salt dissociation,^111–114they used these apparent species to fit classical formal kinetics to the data to extract information about the electron transfer step, an approach that unfortunately has sur- vived until today. Interestingly, they also investigated the charge recombination dynamics of various systems using spectral line shape analysis of exciplex emission, i.e., by definition strongly coupled systems.^115,116Simultaneously, in Fribourg, Haselbach and co- workers performed transient photocurrent measurements to extract the yield of the ions, also observing the inverted region in their recombination data.¹¹⁷ Kuzmin and Levin reported the observations of all regions (inverted, barrierless, and normal), albeit for the recombination of ion pairs stemming from the quenching of triplet states.¹¹⁸Löhmannsröben and co-workers also provided additional evidence for the inverted region.¹¹⁹Many differences are found in these and later works in the conclusions about electron transfer.

A major difference is to be found in the values of the coupling matrix element,V, which for similar chemical systems vary con- siderably, as well as the values of the free ion efficiency (vide infra, Fig. 8).

Surprisingly, while in charge separation or quenching in general, the Rehm–Weller plot (at the time still poorly understood) does not show traces of the inverted region, quite the opposite was observed for the recombination. Moreover, a common feature in most of these works is the finding of smallVs, as expected for expression(6)to be applicable. This is, as we shall see later (cf. Sec.IX), only a consequence of using a contact approximation for all the reaction steps. Indeed, it is quite interesting to note that while the same reactants show large couplings forming complexes, low values are

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FIG. 8. Compilation of the experimental free ion efficiencies,φE, as a function of the difference in the redox potentials. The data were taken from Refs.69,120, and 121(•),119(▲),122(▽),123(△), and124(○). Note that the data of Ref.122 were corrected according to the suggestion from Ref.125. The regions according to Kikuchi⁷¹are shown analogously toFig. 6. An additional region has been added in blue to show the appearance of the recombination to the triplet state of the fluorophore,T1. The triplet energies were taken from the above cited papers, where available.

found if the conditions of the measurements are changed. How to understand that the same fluorophores in different solvents show sometimes exciplexes and sometimes not? Or when combined with different quenchers of similar nature, they show strongly and sometimes weakly coupled systems? A careful examination of the experimental conditions under which the different experiments were performed leads to the conclusion that both the quencher concentration and the excitation wavelength play a major role. Thus, the electron transfer parameters extracted in these works are most likely biased by the use of inadequate models. It was, in fact, only later that Kikuchi reorganized the concepts concerning the Rehm–Weller plot allowing for a proper and consistent reasoning about electron transfer and associated phenomena.⁷¹

A quantity susceptible of further analysis, regardless of the models used in the original studies of products’ recombination, is the free ion efficiency,φE(cf.Fig. 8).¹²⁶It is obtained from kinetics of the signal attributed to the products. If the measurement is performed by means of a time-resolved technique in the sub-ps range, this is the long time asymptotic value of the kinetics.¹⁰¹In flash photolysis measurements, with worse time resolution, it is the initial value. In some cases, such as the papers by the group of Gould and Farid, a scavenger reacting with one of the products was used because of its spectroscopic properties.¹²⁷Additional caution must be taken: many of the reports do not provide full spectra, while a proper separation of the bands and removal of the relaxation dynamics of the species in polar solvents are highly insightful, but not simple.^16,128The typi- cal way to analyzeφE(or also direct product time-traces) in order to obtain the rate constant of electron transfer (charge recombination) is the so-called exponential model,

φE= ksep

krec+ksep

. (17)

In this model, all processes are characterized by rate constants and diffusion is no exception: krec andksep stand for the recombination and separation rate constants, respectively. This is plainly wrong, especially for a reaction that is already in the preceding step, the quenching, and distance dependent, as we have seen.⁶⁸This leads to a time dependent population of the products that are created at different distances. Moreover, they also recombine at different distances. How can all this be characterized by a single rate constant?

We will show what the proper treatment looks like and compare it to the exponential model and its predictions later on.

It is quite interesting to note that the most reliable measurements free of many of the complications seen above were not performed with the latest technology of the 1980s and 1990s but with a flash photolysis apparatus conceived in the late 1960s.¹²⁹The group of Kikuchi developed a method to extract the ion efficiency using such a technique that allowed them to work at low quencher concentrations (in the millimolar range), thus eliminating a major problem usually encountered by the presence of additional absorption bands. Correcting their transient data with the in situ measured fluorescence, they reported a large series of measurements system- atically investigating the effect of molecular charge, solvent polarity, or viscosity.69,121,130–132Moreover, the insight gained by this group made them recognize what—in our opinion—should be a dominant paradigm in the field of photoinduced electron transfer: the existence of different regions in the Rehm–Weller plot, very much in line with what was advanced in this text when we mentioned the Rehm–Weller paradoxes (cf.Fig. 6). According to Kikuchi, in a region up to −0.4 eV, the presence of exciplexes (admixture of the locally excited state and the ion pair) is unavoidable (even in acetonitrile123,133,134). Decreasing the free energy leads to a region in which full intermolecular electron transfer can be studied. Finally, at very large free energies, again complexes—now mixing the locally excited fluorophore with the excited ions or the ion pair with the ground state—appear, serving to obscure the results. This is quite logical if we consider that in most of the cases, the coupling, V, between organic molecules in solution seems to be rather large all along the Rehm–Weller plot, in contrast to the findings of most of the reports listed in this section. As soon as the energy levels of the reactants, in any of the states (ground or excited) and that of the products, are close, they mix, leading to the appearance of new bands of absorption and emission. Recently, there have been reports exemplifying this in detail.^135,136

IX. UNIFIED DESCRIPTION OF QUENCHING AND RECOMBINATION

Of all the previously mentioned works of the 20th century, the only set of data, which were fully analyzed using reaction–diffusion models including the structure of the solvent and the hydrodynamic effect on both the quenching and the recombination, were those of Fayer’s laboratory.^14,96,137 Indeed, these results show very good congruence even with the steady-state measurements. Many additional aspects regarding electron transfer could be discussed, thanks to this appropriate treatment of diffusion and a relatively consistent version of the Marcus expression [see Eq.(20)]. However, the

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number of systems and conditions studied left margin for further improvements.

We have touched on a difficulty in the treatment of the data of recombination due to the changing concentration profiles of quenchers near the fluorophore. Two groups independently developed the model needed for a full theoretical description of the irreversible geminate electron transfer and the product recombination: that of Fayer himself¹³⁸ and that of Burshtein.¹³⁹ They basically arrived at the very same expressions. This extension of the theory unifying both aspects became known as unified encounter theory (from now on, we will simply refer to this as encounter theory). In addition to the reaction–diffusion equation for the fluorophore–quencher pairs, another coupled partial differential equation describes the time evolution of the product pairs,

∂ρ^R(r,t)

∂t = [LˆR−wR(r)]ρ^R(r,t),

∂ρ^P(r,t)

∂t = [LˆP−wP(r)]ρ^P(r,t)+wR(r)ρ^R(r,t)N(t), Lˆx= 1

r²

∂

∂rr²Dx(r)e^−v^x^(r)∂

∂re^v^x^(r).

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Here, ρ^x stands for the distribution function of reactants (R) and products (P), wx stands for the reactivities, and ˆLx denotes the respective diffusion operators and the interparticle potentials divided bykBT,νx. If no Coulombic attraction is present, the latter is equal to Eq.(16); otherwise, the Coulombic potential divided bykBT,rC/r, is added. Note that in the second equation, in addition to the diffusion (first term on the rhs) and the recombination term (second term on the rhs), a creation term for the ions is introduced (third term on the rhs). In this term, the population of the fluorophore is included, accounting for the totally different problem that represents passing from a set of fluorophore–quencher pairs to single product pairs. From the second equation, the population of the products is simply calculated as the space integral of its solution, P(t) =4πc∫_σ^∞r²ρ^P(r,t)dr. (19) A simulation with these equations is shown inFig. 9. The kinetics of the fluorophore, N(t), are clearly not simple and deviate from

exponentiality, decaying faster at short times. Those of the products are rather complex and cannot be explained with a simple exponential decay convolved with the derivative of the kinetics of the fluorophore, as often assumed in the analysis using the exponential model. The quantum yield of free ions (ϕE=φEϕI, withϕIbeing the yield of the quenching reaction) is obtained from the value at long times of its population kinetics if, as in the calculation with Eqs.(18) and(19), it is referred to the total amount of excited fluorophore.

The profiles of the pair distribution function for the reactants in the left panel show theg(r) structure at short times. Be aware that the inner reflective boundary condition has to be applied to the flux, not to the derivative ofρ^R, and therefore, at short times, the solution of Eq.(18)is not flat at contact. The profile for the product pair distribution function shows how they separate with time. Already, at very short times, there are ions not only at contact but also at larger separation distances.

We have seen that in the literature, there is some dispersion for the electron transfer coupling constants, and we also noted the conditions under which they were obtained. In two recent works, Castner and co-workers¹⁴⁰as well as Scherer and Tachiya¹⁴¹found that the coupling is indeed a dynamic quantity that can take on very large values. In other words, there are configurations between the reactants—distances in the models provided—which lead to situa- tions outside the applicability of the non-adiabatic Marcus expression as reproduced above and usually used. The works of Gould and Farid represent an exception as they recognized this problem.¹⁰⁶A better expression than Eq.(6)with a broader applicability is nev- ertheless available, if instead of using the pre-exponential factor of Eq.(6), one uses the full expression for the Landau–Zener terms for the crossing over the transition state.^106,142In addition to this, one must also include the solvent relaxation time,τL. At large couplings, there is the risk that the time response of the solvent becomes the rate limiting factor for the elementary electron transfer act.^46,143This is due to the fact that the reorganization of the solvent requires some time to accommodate the changes in the electronic structure during the reaction and cannot respond instantaneously, in agreement with the Born–Oppenheimer approximation. Here, we will follow Zus- man’s approach,¹⁴⁴although many groups contributed to this term and less crude approximations are available.¹⁴⁵The final expression reads

FIG. 9. Numerical calculations with the encounter theory of the spatial profiles of the pair distribution function for the reactants (central left) and products (central right). The latter have been normalized to their total integral at each time. The leftmost panel shows the normalized decay of the excited fluorophore, while the rightmost exhibits the evolution of the total amount of products, normalized with respect to the reactants. Colors represent the same moments in time. The solvent is acetonitrile,c= 0.1 mol/l, free energy for charge separation is−1.1 eV, and that for charge recombination is−1.9 eV. The full set of parameters for this calculation are provided in theAppendix.

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wm(r) = ∑

n

1 τs

Aexp{−(ΔG(r)+(n−m)̵hω+λ(r))²

4kBTλ(r) }, (20a)

A=⎧⎪⎪⎪

⎨⎪⎪⎪⎩ 1−B

2−B normal region, i. e.,ΔG+(n−m)̵hω+λ>0, (1−B)B inverted region, i. e.,ΔG+(n−m)̵hω+λ<0,

with B=exp(−τsU(r)Fm,n), (20b)

U(r) =V²e^{−2(r−σ)/L}

√ π

̵h²kBTλ(r), (20c)

Fm,n=exp(−S)m!n!⎡⎢

⎢⎢⎢⎣

min(m,n)

∑j=0

(−1)^n−j(√ S)^m+n−2j j!(m−j)!(n−j)!

⎤⎥⎥⎥

⎥⎦

2

with S= λq

̵hω. (20d)

This expression is suited for the transfer between an arbitrary vibrational levelmin the reactants to a set ofnlevels in the products.

ωstands for the high frequency vibrational and harmonic quantum mode.Sis the Huang–Rhys factor, andλqis the reorganization energy associated with this mode.τs is directly proportional toτL

(see theAppendix). The rest of the quantities have been defined in previous equations. A result of this expression with the parameters compiled in theAppendixis shown inFig. 3; it has been used for the calculations ofFigs. 4,9, and10. The multichannel character for

FIG. 10. Calculations using the encounter theory of the free energy dependence of the efficiency of free ion production,φE[the long time limit value ofP(t) in Fig. 9]. Several concentrations of the quencher, as indicated in the legend, have been simulated in acetonitrile. The parameters are common to those inFig. 9.

Note that we have conserved the axis scales and dimensions ofFig. 8. The upper panel showsk_recobtained from the data in the lower panel and Eq.(17), i.e.,krec

=ksep(_φ¹−1)(see theAppendixfor further details). In addition, the thick gray lines show a calculation ofkrecwith Eq.(20)without any distance dependence.

the reaction as developed by Efrima and Bixon,⁷³and Ulstrup and Jortner⁷²is also included. It is important to realize that the preexpo- nential factor for the crossing probability is different in the normal and inverted regions due to the differences in the geometry of the state mixing. Anyway, for single-channel transfer, at low values of V, expression(6)is recovered if the relaxation time of the solvent is very fast.

We are now in a position to compare the expectations from the theory with the results from those several groups that obtained free ion efficiencies with certain confidence. A calculation reveals that for the usual conditions met in laboratory experiments, i.e., large couplings for electron transfer, the free ion efficiency is a function of the free energy changes with quencher concentration (cf.Fig. 10).

In contrast, in the exponential model, no such change is expected as none of the rate constants is concentration dependent. Indeed, the dispersion of the data reported inFig. 8may be due to the different conditions of the experiments, i.e., different concentrations of the quencher.

From Eq. (17) and using the expression for ksep derived by Debye³⁷(see theAppendix), it seems possible to extract the recombination rate constant and from it, assuming a contact reaction or single distance, the electron transfer coupling matrix element.¹⁴⁶In doing so, most of the groups obtained rather small values ofV, well in the applicability of the Marcus expression [Eq.(6)] (V<kBT/2).¹⁴⁷ For example, the highest values forV in meV (kBT= 25 meV at room temperature) obtained by several groups are 5 (Ref.69), 3 (Ref.

122), 4 (Ref.101), 2 (Ref.106), and 3 (Ref.119). When exciting a charge transfer band, Gould and Farid obtained a much larger value of 100 meV.¹⁰⁶

However, this is a self-confirmed prediction, since the problem was treated as one limited to a single reaction layer, forcing an

“averaged” value of the coupling being obtained, depending on the experimental conditions such as quencher concentration and excitation wavelength, different from the actual contact coupling. This is the case in the two methods found in the literature for extractingkrec, the abovementioned and the direct fit to the ion kinetics. We have emulated this method to extract akrecfrom theφEobtained in the simulations with encounter theory (cf.Fig. 10) as if these were experiments. It can be seen that the values forkrecnow strongly depend on the quencher concentration, which is by definition contradictory.

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Moreover, an attempt to reproduce the so-obtained krec using Eq.(20), taking only contact quantities, is also shown inFig. 10. In doing so, the best match to the simulatedkrecrequires much smaller V than those used in the simulation of their kinetics (10 meV vs 75 meV used in the encounter model; seeTable IIin theAppendix).

This indicates that theVvalues extracted from the experiment, as it was conducted by the groups referred to in the text, do not reflect its contact value and underestimate the magnitude of the interaction. Also shown inFig. 10are calculations at another viscosity, ten times that of acetonitrile. The obtainedkrecvalues differ from those in acetonitrile, which, in principle, is not dependent on this parameter.

We have not tried a direct comparison with the data inFig. 8 because of the many unknowns from the experimental reports. In some cases, either the concentrations or the excitation wavelength is not provided. Besides, while in the calculation, we have not taken into account any additional channels, such as the recombination to the triplet state of the fluorophore, in the region of most interest—

the central one—uncontaminated by complexes, the reports do not always provide ion efficiencies corrected for the presence of this channel.

This is not the only manifestation of the failure of the classical formal kinetics approach. At the turn of the century, in collaboration with Burshtein, the group of Grampp in Graz studied the viscosity dependence ofφE. They found that it varies at odds with the exponential model but is perfectly in accordance with the encounter theory. The latter can account for the distance dependence of electron transfer together with the distribution of reactants and products in space, which, in turn, change over time. Turning Eq.(17) upside down, one can extract a recombination efficiency, which in the case of the exponential model is independent of viscosity. How- ever, the encounter theory prediction in the normal Marcus region, confirmed by the experiments of one of us, is that at a given viscosity, this efficiency passes through a maximum value. This is quite straightforward to understand in the case studied in Refs.148and 149 for whichΔGcs ∼ −0.6 eV and ΔGcr ∼ −2.2 eV: in the Mar- cus normal region, the reaction probability expands in space almost exponentially, while in the inverted region, it shows a maximum shifted away from the contact (seeFig. 11). This is due to the change of sign in the sum of the free energy and the reorganization energy, and the spatial dependence of the latter (seeFig. 3). In the reaction studied in dimethylsulfoxide–glycerol mixtures,¹⁵⁰ the charge separation lies in the normal region and the recombination lies in the inverted region. Thus, at low viscosities, the charges are created mostly inside the recombination layer¹⁵¹and in order to escape have to cross it “dying” in an attempt to do so. As the viscosity increases, the quenching radius increases, as observed experimentally by many before (e.g.,Fig. 7), and at some point, the ions are created outside the layer, being free to escape with marginal recombination [com- parew(r) inFig. 3]. Unfortunately, in this study, the time resolution was not very good, and neither were the theoretical details introduced by Fayer considered. Later, in collaboration with the group of Vauthey, we could repeat and extend the measurements to confirm the original observations.¹⁶Moreover, full kinetics of the geminate processes were recorded and satisfactorily compared not only to the encounter theory but also to the exponential model. Evidently, the latter failed even to account for the free ion efficiency, never mind reproduced the full dynamics.

FIG. 11. Schematic depiction of the spatial distribution of the regions of charge separation in the normal region and charge recombination in the inverted region.

Note that the regions are not scaled to the actual reactivity distributions.

X. A SOLUTION FOR THE REHM–WELLER PARADOX Now, let us return to the Rehm–Weller plot (Fig. 12). The mystery—the paradoxes—seem to persevere despite various early

FIG. 12. The Rehm–Weller plot, as inFig. 6(gray dots), plus the recently acquired data with sub-ps time-resolved techniques. The extracted intrinsic rate constants, k₀(black filled circles in both panels) from the latter are shown both in linear and logarithmic scales. In the logarithmic scale (bottom panel), the corresponding Stern–Volmer rate constants at low concentrations,κ0, corresponding to the same systems as the black filled circles, are shown too (open circles). Despite not all systems being explained with the very same parameters—owing to their slightly different chemical nature—an average Marcus parabola can be drawn usingV= 72 meV andL= 1.29 Å.¹⁵⁶The spikes are the result of the Franck–Condon factors in the multichannel transfer version of the electron transfer theory [Eq.(20)].