Reorganization of Intramolecular High Frequency Vibrational Modes

Article pubs.acs.org/JPCA

Reorganization of Intramolecular High Frequency Vibrational Modes and Dynamic Solvent Effect in Electron Transfer Reactions Vladislav V. Yudanov, Valentina A. Mikhailova, and Anatoly I. Ivanov* Volgograd State University, University Avenue 100, Volgograd 400062, Russia ABSTRACT: The possibility of the multichannel stochastic model to adequately describe all principal regularities observed in thermal electron transfer kinetics has been demonstrated. The most important are as follows: (i) the model predicts the solvent controlled regime in the Marcus normal region and its almost full suppression in the Marcus inverted region as well as a continuous transition between them in the vicinity of the activationless region; (ii) the suppression of dynamic solvent effect (DSE) is principally caused by the reorganization of high frequency vibrational modes; (iii) an additional factor of the DSE suppression stems from fast solvent relaxation component; (iv) in the inverted region, the multichannel stochastic model predicts the apparent activation energy to be much less than that calculated with Marcus equation. The exploration of the multichannel stochastic model has allowed one to conclude that the reorganization of high frequency vibrational modes can (i) raise the maximum rate constant above the solvent controlled limit by 2 and more orders of magnitude, (ii) shift the rate constant maximum to larger values of the free energy gap, and (iii) approach the electron transfer kinetics to the nonadiabatic regime.

I. INTRODUCTION In the early 1980s theoretical predictions of a dynamic solvent effect (DSE) on electron transfer (ET) kinetics were made.1−4 Theory of DSE allowed bridging between nonadiabatic and solvent controlled regimes and answering the question when solvent relaxation, rather than microscopic electronic processes, constitutes the rate-determining step for ET.2 In the framework of the stochastic point transition model the ET rate constant is described by eq 1.12 ⎡ E ⎤ Vel 2 π ket = exp⎢ − a ⎥ (1 + g )ℏ ErmkBT ⎣ kBT ⎦ (1.1)

These predictions were partly confirmed in a series of experimental studies.5−12 Later, however, systematic studies of both the intramolecular and intermolecular ET have revealed that the ET rate can be 50−1000 times larger than 1/τL.13−26 Moreover, these experimental results demonstrated a weak ET rate dependence on solvent relaxation time that also contradicts the theoretical predictions. The analysis has shown that the first series of experiments dealt with slow ET proceeding in Marcus normal region whereas the second ones corresponded to fast ET occurring in activationless or inverted region.27 These results indicated that there are differences between mechanisms of ET proceeding in the normal and inverted regions. Bixon and Jortner27 pointed out that the theories of solventinduced ET28−39 with proper modifications to account for the role of intramolecular modes, is applicable only in the Marcus normal region where diffusion on the initial potential surface to the term intersection is required. At the same time, these theories do not describe correctly the activationless and inverted-region processes because the effects of medium relaxation dynamics will only be manifested provided that the microscopic ET rates are sensitive to the details of the distribution in the initial vibronic manifold.27 In reality, there is a weak excess energy dependence of the microscopic ET rates.40 This implies that the depletion dynamics of the reactant manifold is practically invariant with respect to medium dielectric relaxation dynamics. In other words, in activationless and inverted regions the rates of ET are not limited by solvent dynamics and can be appreciably faster than previously expected, being determined by the nonadiabatic radiationless

where Vel is the electronic coupling, ℏ is the Planck constant, kB is the Boltzmann constant, T is the temperature, Erm is the medium reorganization energy, g=

⎤ 2πVel 2τL ⎡ 1 1 + ⎥ ⎢ ℏ ⎣ |ΔG + Erm| |ΔG − Erm| ⎦

(1.2)

is the Zusman parameter determining crossover from the Golden Rule to the solvent control, ΔG is the reaction free energy, τL is the longitudinal dielectric relaxation time, and

Ea =

(ΔG + Erm)2 4Erm

(1.3)

is the activation energy. There are three important conclusions emerging from these studies: (i) for solvent controlled limit, g ≫ 1, the ET rate is inversely proportional to the longitudinal relaxation time of the solvent, τL; (ii) there exists a saturation of ET rate as the electronic coupling parameter, Vel, is increased; (iii) the upper limit for the ET rate is achieved in the activationless region where it is close to 1/τL for polar solvents with Erm ≈ 1 eV. © 2012 American Chemical Society

Received: February 24, 2012 Revised: April 2, 2012 Published: April 2, 2012 4010

dx.doi.org/10.1021/jp301837t | J. Phys. Chem. A 2012, 116, 4010−4019

The Journal of Physical Chemistry A

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transition theory.40 Because in this treatment the dynamics of motion on the free energy surfaces were not considered, ET kinetics in the normal and inverted (including activationless) regions should be described by using different models. One more important factor was pointed out in refs 37 and 38. The dynamic solvent effect in ET was proposed to be weakened not only because of the participation of the intramolecular high frequency modes but also because of ultrafast solvation in polar solutions. This proposition was initiated by observation that in most solvents the solvation dynamics are nonexponential and contain at least two vastly different time scales.41−46 The calculations have shown that an increase of the amplitude of ultrafast solvation component can significantly enhance the rate of barrier crossing and can markedly weaken the dependence of the rate on effective solvent polarization relaxation time.47 In these models the reorganization of intramolecular vibrations was approximated by a single effective mode. In reality, ET is accompanied by the reorganization of a number of intramolecular vibrational modes. For example, the combination of information contained in the resonance Raman scattering spectra and in the stationary charge transfer absorption bands allows obtaining the frequencies and the Huang−Rhys factors of all quantum vibrational modes involved in the charge recombination of excited donor−acceptor complexes.48−52 A few realizations of this approach have shown that ten or so quantum modes really participate in the charge recombination.49,53 Participation of a number of high frequency vibrational modes in ET results in an appearance of a large number of sinks on the reactant free energy surface. The density of the sinks is correlated with the Franck−Condon density used in ref 27. Evidently, when there is a high density of sinks on the reactant free energy surface in the reaction zone, necessity of diffusion delivery disappears. This extreme was investigated in a series of papers reviewed in ref 27. Because both extremes (profound DSE and its almost full suppression) were observed in the experiments, generalization of the theory to be able to bridge these extremes is needed. The theory should include (i) reorganization of a number of high frequency vibrational modes and (ii) reorganization of the solvent with several relaxation time scales. Recently, the stochastic point-transition approach2,3 has been generalized to multilevel systems.47,53,54 The multichannel stochastic model has demonstrated its capability to quantitatively reproduce experimental data on ultrafast charge transfer kinetics, including dependence of the kinetics on the spectral characteristics of the pump pulse (the carrier frequency of excitation pulse) and solvent relaxation time scales. The adequacy of this model was tested for a number of charge transfer reactions in excited donor−acceptor complexes, in photoinduced charge separation reaction from the second excited state in zinc−porphyrin derivatives.53,55−59 The main focus here is to demonstrate the possibility of the multichannel stochastic model to adequately describe all principal regularities observed in thermal ET kinetics. To show this, the influence of both the reorganization of a number of high frequency vibrational modes and ultrafast solvation component on thermal ET kinetics in the framework of the multichannel stochastic model is investigated. For quantitative treatment of the phenomenon, we introduce the concept of DSE magnitude and discuss its dependence on the model parameters. This should allow us to present a description of solvent controlled regime in the Marcus normal region and its

almost full suppression in the Marcus inverted region as well as a continuous transition between them in the vicinity of the activationless region in the framework of single ET model.

II. MULTICHANNEL STOCHASTIC MODEL For description of charge transfer kinetics a minimal model including two electronic states: the reactant, |R⟩, and the product, |P⟩, states is employed (Figure 1). The nonadiabatic

Figure 1. Schematic representation of the free energy curves for the multichannel ET reaction for the normal (A) and inverted (B) regions. The additional free energy curves corresponding to excitation of the high frequency intramolecular vibrational modes of the product are shown as blue lines. The initial reactant state distribution is pictured by a red line.

approach, in particular, the Fermi’s Golden Rule is not applicable for kinetic modeling of ET in the solvent controlled regime. The stochastic point-transition approach2 generalized to multilevel systems is well suited to this aim.53,54 The model was earlier described elsewhere in detail.53−55 So we only outline it to introduce the designations used hereafter. The real solvents with several relaxation time scales41−43 are described in the framework of the Markovian approximation, that is, the solvent relaxation function, X(t), is written as a sum of exponentials N

X (t ) =

∑ xie−t /τ

i

(2.1)

i=1

where xi = Eri/Erm, τi, and Eri are the weight, the relaxation time constant, and the reorganization energy of the ith medium mode, respectively, Erm = ∑iEri is the solvent reorganization energy, and N is the number of the solvent modes. The diabatic free energy surfaces for the electronic states in terms of the reaction coordinates Qi (ith coordinate corresponds to the i-th medium mode) may be written as follows47,54 N

UR =

∑ i=1

Q i2 4Eri

N

UP(n⃗) =

∑ i=1

(2.2)

(Q i − 2Eri)2 4Eri

M

+

∑ nαℏΩα + ΔG α=1

(2.3)

where ΔG is the reaction free energy and nα (nα = 0, 1, 2, ... and α = 0, 1, 2, ..., M) is the quantum number of the αth high frequency intramolecular mode with the frequency Ωα. The vector n⃗ has M components (n1, n2, ..., nα, ..., nM). This approach supposes a division of the vibrational degrees of 4011



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Table 1. Parameters of High Frequency Vibrational Modes for Charge Transfer in Hexamethylbenzene/Tetracyanoethylene Complex50,53 α

1

2

3

4

5

6

7

8

9

10

ℏΩα, eV Sα

0.0558 0.1176

0.0672 0.0129

0.0744 0.0239

0.1188 0.0706

0.1602 0.2196

0.1722 0.0863

0.1782 0.0353

0.1923 0.6941

0.1947 0.0706

0.2755 0.2862

freedom into the low and the high frequency modes. The first group is described classically whereas the second in the quantum-mechanical terms (ℏΩα > kBT). In the framework of the stochastic point-transition approach,2,53,54 the temporal evolution of the system is described by a set of equations for the probability distribution functions for the reactant state, ρR(Q,t), and for the nth ) sublevel of the product state ρ(n⃗ P (Q,t) ∂ρR ∂t

= L̂ R ρR −

∂ρP(n⃗) ∂t

∑

k n(⃗ Q)(ρR − ρP(n⃗)) (2.4)

n1, n2 ,..., nM

= L̂ PρP(n⃗) − k n(⃗ Q)(ρP(n⃗) − ρR ) +

∑ α

−

∑ α

factors, Sα = Ervα/ℏΩα and Ervα are the Huang−Rhys factor and the reorganization energy of the αth high frequency vibrational mode, respectively. Erv = ∑αErvα is the total reorganization energy of the high frequency vibrational modes. A singlequantum mechanism of high frequency mode relaxation is adopted and the transitions nα → nα − 1 are supposed to 60 (nα) (1) α) proceed with the rate constant 1/τ(n vα where τvα = τvα /nα. To specify the initial conditions, we assume the systems to be initially in the reactant state distributed along the reaction coordinates according to the Boltzmann law N

ρR (Q,t =0) =

1 τv(αnα

ρ (nα⃗′ ) + 1) P

i=1

N

∑ i=1 N

L̂ P =

∑ i=1

∂ ∂ 2 ⎤⎥ 1 ⎡⎢ + ⟨Q i 2⟩ 1 + Qi τi ⎢⎣ ∂Q i ∂Q i 2 ⎥⎦ ∂ ∂ 2 ⎤⎥ 1 ⎡⎢ + ⟨Q i 2⟩ 1 + (Q i − 2Eri) τi ⎢⎣ ∂Q i ∂Q i 2 ⎥⎦

(2.6)

∏ α

Sαnα e−Sα nα!

∑ ∫ ρj(n⃗)(Q,t ) ∏ i=1

dQ i (2.10)

where the index runs j = R, P. The effective rate constant and the effective time of the reaction are determined by eq 2.11

(2.7)

keff −1 = τeff =

∫0

∞

PR (t ) dt

(2.11)

The model accounts for the local reversibility of electron transfer that can be adequately described only if the intramolecular vibrational relaxation or the vibrational redistribution is also taken into consideration.63 The intramolecular vibrational redistribution is well-known to proceed on the time scale of ∼100 fs.64 In simulations the value τ(1) vα = 100 fs is set for all vibrational modes.65 With the exception of a very limited number of systems only the total reorganization energy of the quantum vibrational modes can be estimated from the available experimental data and, as a consequence, such a reorganization is customarily described in terms of a single effective mode. Although there is a principle possibility to determine the frequencies and the Huang−Rhys factors of all quantum vibrational modes involved in the charge recombination of excited donor−acceptor complexes, up to now there are only a few realizations of this approach.48−52 Because for the vast majority of charge transfer processes there are no Raman resonance scattering data, a problem arises how to imitate such a spectrum. A possible solution of this problem was suggested in ref 56. For a fixed number of high frequency vibrational modes the charge transfer rate was shown to depend weakly on the vibrational spectral density provided the total reorganization energy, Erv, is constant. This implies that a universal spectral density with

2πVn⃗ 2 2πVn⃗ 2 δ(UP(n⃗) − UR ) = δ(z − z n†⃗ ) ℏ ℏ Fn⃗ =

⎪

(2.9)

n1, n2 ,..., nM

with = 2ErikBT being the dispersion of the equilibrium distribution along the ith coordinate. ) In eqs 2.4−2.5 the quantity ρ(n⃗ P (Q,t) is the probability distribution function for the product state with the excitation of nα vibrational quanta for αth intramolecular modes. The vector n⃗′α differs from n⃗ only by the number of vibrational quanta for αth mode n⃗′α = (n1, n2, ..., nα + 1, ..., nM). So, the model accounts for the reorganization of a number of intramolecular high frequency vibrational modes that generally leads to the vibrational sublevels of both the reactant and product states. However, vibrational sublevels of the reactant state may be omitted from the consideration owing to their fast relaxation. Figure 1 shows a number of vibrationally excited sublevels of the product state (blue lines). Electron transitions between the reactant state and a vibrational sublevel of the product state are described by the Zusman parameters

Vn⃗ 2 = Vel 2Fn⃗

⎪

⎪

N

Pj(t ) =

⟨Qi2⟩

k n⃗ =

⎪

and all quantum modes are in the ground state due to their fast relaxation. The system of eqs 2.4−2.5 with the initial condition eq 2.9 is solved numerically using the Brownian simulation method.61,62 The populations of the reactant and product states are calculated with the equation

(2.5)

where Q stands for the vector with components Q1, Q2, ..., QN and L̂ R and L̂ p are the Smoluchowski operators describing diffusion on the U R and U P(n⃗ ) free energy surfaces, correspondingly L̂ R =

⎧ Q i2 ⎫ ⎬ exp⎨− 2 ⎩ 2⟨Q i ⟩ ⎭ 2π ⟨Q i 2⟩ 1

ρP (Q,t =0) = 0

1

ρP(n⃗) τv(αnα)

∏

(2.8)

where z = ∑iQi is the collective energetic reaction coordinate, z†n⃗ = Erm + ΔG + ∑αnαℏΩα are the points of the intersection of ) terms U(n⃗ P and UR. Fn⃗ is the product of the Franck−Condon 4012



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variable total reorganization energy may be exploited as a good approximation for any donor−acceptor complex. As universal spectral density we accept the high frequency vibrational spectrum of a donor−acceptor complex consisting of hexamethylbenzene as the electron donor and tetracyanoethylene as the electron acceptor.56 The spectrum of this complex involves 10 active vibrational modes. The values of the spectral parameters (frequencies and Huang−Rhys factors) are given in Table 1.50,53

ξ=

τeff (τL + δτL) − τeff (τL) τeff (τL)δτL /τL

(3.1)

Here δτL is the medium relaxation time variation. One should emphasize that the quantity ξ does not depend on δτL if τeff is a linear function of τL. Obviously, in the solvent controlled regime when the rate is inversely proportional to the medium relaxation time, ξ is close to unity. In the opposite extreme when the rate is practically independent of the medium relaxation time the inequality ξ ≪ 1 is fulfilled. In the region of eq 1.1 applicability (high activation barrier), the magnitude of DSE is given by

III. SIMULATION RESULTS AND DISCUSSION A. Dynamic Solvent Effect. In this subsection a suppression of solvent controlled regime in ET by reorganization of intramolecular high frequency vibrational modes is explored. At first a much used Debye model of polar medium with single relaxation time is considered. The simulations of the ET kinetics are fulfilled in the framework of a stochastic model incorporating the reorganization of either one or ten intramolecular high frequency vibrational modes as well as without the reorganization. For the model with single effective high frequency vibrational mode we use the following values of parameters: the effective frequency ℏΩ = 0.17 eV and Erv = 0.3 eV. To apprehend how the number of high frequency modes influence the DSE magnitude, the total reorganization energy Erv = ∑αℏΩαSα is fixed, that is, Erv = 0.3 eV. The spectral parameters of the model involving 10 active vibrational modes are stated in Table 1. All simulations are performed at the temperature, kBT = 0.025 eV, and the solvent relaxation is supposed to be exponential with single time constant τL. The dependence of ET effective time, τeff, on the longitudinal relaxation time is found to be very close to a linear one. This dependence for the activationless region is shown in Figure 2.

ξ=1−

1 1 + g (τL)

(3.2)

where g(τL) is determined by eq 1.2. This equation shows the DSE magnitude to be determined by a dimensionless parameter a=

Vel 2τL ℏErm

(3.3)

So we may expect that a variation of the model parameters should result in a minor alteration of ξ magnitude if the value of a is kept constant. Dependencies of the DSE magnitude on the reaction free energy are pictured in Figure 3. In the calculations the values τL = 0.5 ps and δτL = 1.5 ps are used. A rather large δτL = 1.5 ps is needed to get ξ with sufficient precision. For illustration, the dependencies of ξ(ΔG) for a model without high frequency mode reorganization are presented in frame A. Solid lines are obtained with eq 3.2, and the symbols present the results of numerical calculations. The analytical and numerical results are seen to coincide only in the region of large values of |ΔG + Erm| where eq 3.2 is applicable. In the vicinity of activationless region there are considerable deviations because there eq 3.2 is wrong. It should be emphasized that the shape of ξ(ΔG) closely mimics the well-known behavior (turned over) of the frequency factor of the rate constant.66,67 This result directly emerges from eq 3.2 because the term 1/(1 + g) reflects the dependence of the frequency factor on ΔG. The figure demonstrates a monotonic fall of the DSE magnitude with decrease of both the electronic coupling (frame A, circles and triangles) and the solvent relaxation time scale (stars). To get a feel for this result, we notice that for ΔG = −Erm the sink is placed at the point of the reactant term minimum where the center of gravity of the particle distribution is located. In this case the particles have to travel a minimum distance to react and the frequency factor of the rate constant and ξ reach a maximum and a minimum, correspondingly. A variation of ΔG shifts the sink from the center of gravity of the particle distribution and the particles, in average, have to run a larger distance that results in a reduction of the frequency factor and a rise of ξ. For larger deviation from the point ΔG = −Erm, the reaction approaches the nonadiabatic regime (reduction of g) and ξ starts to go down. One can see that the distance between two maxima of ξ increases with the rise of electronic coupling Vel. This trend is kept in the models with the reorganization of high frequency modes (Figure 3, frames B and C). The influence of high frequency vibrational mode reorganization on the DSE magnitude is demonstrated in Figure 3 (frames B and C). In the area of strong exergonicity, −ΔG > Erm + Erv, this influence is of particular importance. A mechanism of this effect stated earlier in detail27,37,38 consists

Figure 2. ET effective time, τeff, dependence on the solvent relaxation time, τL. A number of active high frequency modes is specified near the corresponding lines. The solvent model with single relaxation time scale τL is used. The values of the energy parameters are Vel = 0.08 eV, Erm = 0.5 eV, ΔG = −0.5 eV (curve 0), and ΔG = −0.8 eV, Erv = 0.3 eV (curves 1 and 10).

Two conclusions are readily apparent from these data. An increase of the number of active high frequency modes leads to (i) crucial decreasing ET effective time and (ii) weakening the dependence of τeff on τL. These results indicate the DSE to be profoundly destroyed by the reorganization of a considerable number of high frequency modes even in a solvent with single relaxation time. As a quantitative measure of the DSE we introduce a ratio of effective reaction time change with increase of the medium relaxation time to effective reaction time change which is directly proportional to τL 4013



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number of active high frequency vibrational modes. The shape of curves pictured in frames B and C resembles that presented in frame A, but the branch in the high exergonicity region is strongly suppressed by the reorganization of the high frequency modes. This suppression enhances with rising the number of active high frequency modes. In ref 56 the rate constant of nonthermal charge recombination was shown to depend weakly on the vibrational spectral density. This conclusion also appeared to be valid for thermal ET. The calculations performed for considerably different vibrational spectrum (all Huang−Rhys factors are equal each other) with the same Erv demonstrate a minor influence of vibrational spectral density on DSE magnitude, ξ, (violet symbols in frame D, Figure 3). Notwithstanding that the DSE magnitude is calculated with a specific vibrational spectral density, we may expect the conclusions obtained to be of high degree of generality. To gain some insight into the regularities obtained in this subsection for multichannel stochastic model, hereafter we discuss the distributions of sink efficiencies and the particles on the reactant free energy surface. B. Distribution of Product Quantum Yield over Reaction Coordinate. The coarse grain probability distributions of electronic transitions, Y(z), over the reaction coordinate z = ∑iQi are shown in Figure 4 for a number of values of reaction free energy. This quantity is determined by eq 3.4 N

Y (z ) =

∑

∫

dt

∫ ∏ dQ i k n(⃗ Q)[ρR (Q,t ) i=1

n ∈Δz

−

ρP(n⃗)(Q,t )]

(3.4)

Here the sum runs all sinks located on an interval from z − Δz to z and Δz = 2kBT. This quantity shows the relative yield of product created in reaction coordinate area Δz. For comparison, the thermal probability distribution for particles on the reactant free energy curve is pictured. Figure 4 (frame A) shows that for weak exergonic reactions only the sinks placed in the vicinity of the lowest term intersection point are operative. This is why the reorganization of high frequency modes slightly influences on DSE magnitude, ξ. For −ΔG = Erm (frame B) all sinks are located to the right of UR term minimum and a half of the particles has to diffuse to reach the sinks. So, in this region a considerable DSE is predicted. For larger values of the free energy gap (frames C and D), the distributions of the sink efficiencies and the particles are close to each other. In these cases the particles can react without diffusional motion and that results in ξ close to zero. For still greater values of −ΔG (frames E and F), the DSE magnitude increases because of the relative displacement of distributions of the sink efficiencies and the particles that again necessitates the particles to diffuse to the reaction zone. The nonzero value of ξ evidences that the nonadiabatic regime has not been actualized yet. For the parameters adopted, this regime is reached at larger values of the free energy gap. C. Influence of Fast Component of Solvent Relaxation on DSE. An important outcome is that the DSE is weakened not only by the reorganization of the high frequency vibrational modes but also due to ultrafast medium solvation.47 One should note that in the model with two relaxation time scales of the medium τeff = τeff(τ1,τ2) is a function of both time scales. Because the fast time scale, τ1, is practically solvent independent

Figure 3. Free energy gap dependence of dynamic solvent effect magnitude, ξ, for a few numbers of active high frequency modes M: 0 (A), 1 (B), 10 (C, D). The solvent model with single relaxation time scale τL = 0.5 ps is used. The values of the rest parameters are δτL = 1.5 ps, Erm = 1.0 eV, Erv = 0.3 eV, Vel = 0.02 eV (red), Vel = 0.05 eV (black), and Vel = 0.08 eV (blue). The solid lines and symbols represent analytical (eq 3.2) and numerical results, correspondingly. For comparison, ξ is pictured for solvent model with relaxation time scale τL = 0.19 ps, Vel = 0.02 eV, and δτL = 1.81 ps (frame A, dashed line and symbols “star”). In frame D the violet symbols present ξ magnitude numerically calculated with the same parameters but with modified vibrational spectrum, Sα = 0.2015 for all modes; the values of Ωα are given in Table 1.

of the following. The excitation of high frequency modes in the product state can effectively open up new reaction channels (Figure 1). For strong exergonic ET reaction the high density of these additional sinks is located in the vicinity of the reactant term minimum where the particles are distributed (Figure 1 B). In this case the ET reaction can occur as an activationless reaction with much greater rate than that in the Marcus normal region (Figure 1 A). That is why the necessity of diffusion delivery of the particles to the reaction zone disappears with growth of exergonicity, −ΔG. This results in extremely weak ET rate dependence on solvent dynamics that corresponds to ξ → 0. This trend becomes stronger with the increase of the 4014



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Figure 5. Dependence of dynamic solvent effect magnitude, ξ, on the weight of the fast component of solvent relaxation, x1. The number of active high frequency modes is specified near corresponding lines. The values of the parameters are Vel = 0.05 eV, Erm = 1.0 eV, Erv = 0.3 eV, ΔG = −1.0 eV (dashed and black solid lines), and ΔG = −1.3 eV (blue and red symbols).

Figure 4. Coarse grain probability distributions of electronic transitions, Y(z), over the reaction coordinate, z, for a number of values of reaction free energy (marked on each frame). The solvent model with single relaxation time scale τL = 0.5 ps is used. Number of active high frequency modes M is equal to 10. The values of the rest parameters are Erm = 1.0 eV, Erv = 0.3 eV, and Vel = 0.05 eV. The red line represents the thermal distribution of the particles (arbitrary units).

in contrast to the time scale of the slow component, τ2, which is proportional to the solvent viscosity and may vary in a wide range, the DSE magnitude for the solvents with two relaxation time scales is defined as ξ=

τeff (τ1 ,τ2+δτ2) − τeff (τ1 ,τ2) τeff (τ1 ,τ2)δτ2/τ2

(3.5)

The dependence of ξ on x1 for −ΔG = Erm + Erv is pictured in Figure 5. The results show a monotonic decrease of DSE magnitude with increasing the weight of the fast component of solvent relaxation, x1. There are at least two reasons for such a behavior. First, an increase of x1 leads to a decrease of the effective relaxation time of the solvent that results in a decrease of ξ as it is seen from eq 3.2, second, a decrease of x2 weakens the effect of τ2 variation on the rate constant. Figure 6 demonstrates DSE magnitude dependence on the free energy gap for a solvent model with two relaxation time scales. The fast component decreases DSE magnitude while the qualitative behavior of the curves ξ(ΔG) remains practically invariant. In particular, an increase of the weight of the fast relaxation component, x1, from 0.3 to 0.5 leads to a decrease of the DSE magnitude by a factor of 1.5. D. Free Energy Gap Law. It is well-known that the reorganization of the high frequency vibrational modes

Figure 6. Free energy dependence of dynamic solvent effect magnitude, ξ, for a few numbers of high frequency modes M: 0 (A, D), 1 (B, E), 10 (C, F). The solvent model with two relaxation time scales τ1 = 0.19 ps and τ2 = 0.5 ps is used. The left and right halves of the figure show the data obtained with x1 = 0.3, x2 = 0.7, and x1 = x2 = 0.5, correspondingly. The values of the rest parameters are δτ2 = 1.5 ps, Erm = 1.0 eV, Erv = 0.3 eV, Vel = 0.08 eV (blue), Vel = 0.05 eV (black), and Vel = 0.02 eV (red).

modifies the free energy gap law,27 increasing the rate constant in the high exergonicity region and suppressing it in the low exergonicity region. Nevertheless, in the nonadiabatic limit the bell-shaped form of the rate constant dependence on the reaction free energy with a maximum at ΔG = −Erm − Erv is retained. In this subsection an influence of both the diffusional delivery of the particles to effective sinks and the reorganization of the high frequency vibrational modes on the free energy gap 4015



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(iii) increasing the electronic coupling shifts the position of the rate constant maximum to larger free energy gap values, −ΔG. The first two findings are a direct consequence of appearance of the sink multitude in the reaction zone. As a result, the average distance of diffusion of the particles becomes much shorter and ET kinetics approaches the solvent independent regime. To understand the third finding, it is necessary to account for the following regularities: (i) increasing the free energy gap leads to sink density rise and to a decrease of specific sink efficiency (due to Franck−Condon factors, eq 2.8) in the reaction zone; (ii) the most powerful sinks can be saturated (i.e. their efficiencies become independent of electronic coupling). For this reason, strengthening electronic coupling shifts the rate constant maximum to larger free energy gaps. Thus, the rate constant can achieve its maximum at −ΔG considerably larger than for Erm + Erv. It should be pointed out that the stochastic model with a single high frequency mode predicts a plateau on the dependence of the rate constant on free energy gap (Figure 7, frame A) for strong electronic coupling. This implies that the solvent controlled regime is achieved due to the large distance between the neighbors sinks. This distance decreases with an increase of the active high frequency mode number and the plateau can disappear (frame B). Moreover, the ascending and descending branches of the curve approach to straight lines. It should be noted that the stochastic model is applicable only for sufficiently small values of electronic coupling. The limits of applicability of the model with single sink can be roughly estimated with inequality Vel ≤ kBT.68 For the model with the reorganization of high frequency modes Vel should be replaced with the effective electronic coupling for a specific sink, n⃗, which is equal to Veff = (Vel2Fn⃗)1/2. For Vel = 0.2 eV (violet lines) the condition Veff ≤ kBT is fulfilled only in the far inverted region. In the vicinity of the rate constant maximum this inequality is invalid. So, the violet curves demonstrate only the trend predicted by the multichannel stochastic model. E. Suppression of Solvent Controlled Regime in Inverted Region. Figure 8 shows the effect of the electronic coupling on the ET rate constant, keff, for Vel varying in the interval from 0.02 to 0.08 eV. These results allow us to conclude (i) only in the Marcus normal region, −ΔG < Erm + Erv, does the effective rate constant, independently of the reorganization of high frequency modes, demonstrate a saturation with electronic coupling increase and (ii) in the inverted region, −ΔG > Erm + Erv, the reorganization of high frequency modes eliminates the saturation. This effect becomes stronger with the increase of both the number of active high frequency modes and free energy gap, −ΔG. The origin of these regularities is rather transparent. The reorganization of a number of high frequency modes leads to the replacement of a single powerful sink by a vast number of weak sinks. In the inverted region the value of Veff is sufficiently small for majority of the sinks located in the vicinity of the reactant term bottom where the particles are distributed. So each of the sinks operates in the nonadiabatic regime that leads to quadratic dependence of the rate constant on the magnitude of Vel (frames E and F). F. Apparent Activation Energy. Within the original Marcus theory, the bell-shaped dependence of the rate constant is a direct consequence of activation energy variation with the reaction free energy (eq 1.3). This equation shows a fast rise of the activation energy in both the normal and inverted Marcus

Figure 7. Dependence of effective ET rate constant, keff (s−1), on the free energy gap, −ΔG. A number of high frequency modes M: 1 (frame A), 10 (frame B). The solvent model with single relaxation time scale τL = 0.5 ps is used. The values of the parameters are Erm = 0.5 eV, Erv = 0.3 eV, Vel = 0.2 eV (violet), Vel = 0.08 eV (blue), Vel = 0.05 eV (black), and Vel = 0.02 eV (red). The solid lines and symbols represent analytical (eq 3.6) and numerical results, correspondingly. For comparison, the value of log(1/τL), τL in s, is shown by the horizontal dashed line.

law is investigated. The results of numerical calculations are presented in Figure 7 (symbols). For comparison, the ET rate constant determined by eq 3.6 k th =

Vel 2 ℏ

π ErmkBT

⎡

∑ ⎢∏ nα

⎢⎣

α

e−SαSαnα ⎤ ⎥ nα! ⎥⎦

⎡ (ΔG + E + ∑ n ℏΩ )2 ⎤ rm α α α ⎥ × exp⎢ − ⎢⎣ ⎥⎦ 4ErmkBT

(3.6)

is displayed by solid lines. This equation is a straightforward generalization of the well-known expression.36 The reorganization of high frequency vibrational modes results in the following findings: (i) the maximum of ET rate constant can considerably exceed the solvent relaxation rate, 1/τL; (ii) increasing the number of active high frequency vibrational modes enhances this trend; 4016



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the reactant and the product states. The reorganization of high frequency modes considerably modifies the ET mechanism. In particular, the competing electronic transitions corresponding to different channels (given number of vibrational quanta created in the product state) proceed over different free energy barriers and the concept of the activation energy becomes uncertain. This should result in non-Arrhenius dependence of the rate constant on the temperature. Nevertheless, the calculations demonstrate a nearly linear dependence of logarithm of the effective rate constant keff on the inverse temperature in a wide interval (Figure 10) that

Figure 10. Temperature dependence of ET rate constant keff (s−1). The number of high frequency modes M = 10. The values of the parameters are Erm = 1.0 eV, Erv = 0.3 eV, Vel = 0.05 eV, τ1 = 0.19 ps, τ2 = 0.50 ps, and x1 = x2 = 0.5. The values of the free energy gap, −ΔG (eV), are specified near corresponding lines.

Figure 8. Dependence of effective ET rate constant, keff, on the electronic coupling, Vel. The number of high frequency modes M is equal to 0 (red), 1 (black), and 10 (blue). The values of the rest parameters are Erm = 0.5 eV and Erv = 0.3 eV. The solvent model with single relaxation time scale τL = 0.5 ps is used. The values of the free energy gap, −ΔG (marked on each frame). For comparison, the value of 1/τL in s−1 is shown by the horizontal dashed line.

assigns some meaning to activation energy Ea in the model considered. At least this allows us to introduce the apparent activation energy determined by eq 3.7 Eã = kB

regions (dashed line in Figure 9). The Marcus activation energy is directly determined by the activation barrier height between

ln keff (T2) − ln keff (T1) 1/T1 − 1/T2

(3.7)

This quantity does not associate with the height of real free energy barrier, but it is a much used characteristic of the temperature dependence of the rate constant. For the apparent activation energy calculation, the values of the effective rate constant keff at the temperatures kBT = 0.025 and 0.030 eV are used. In Figure 9 the free energy gap dependence of Ẽ a is pictured for multichannel stochastic model (red symbols). For comparison, the apparent activation energy calculated with eq 3.6 is also presented (blue line). As one could expect, in the normal region the reorganization of high frequency modes results in weak deviation of Ẽ a from that predicted by the Marcus equation (1.3). On the other hand, the multichannel stochastic model predicts negative apparent activation energy in a larger part of the inverted region. It becomes positive only at the edge of experimentally accessible region of the free energy gap. The calculations in this subsection are performed within the supposition that all model parameters are temperature independent. It is well-known that the medium reorganization energy and the reaction free energy vary weakly with the temperature. Their temperature alteration can slightly change the apparent activation energy. In the inverted region this change is especially minor due to the weak dependence of Ẽ a on these quantities. On the other hand, the medium relaxation time τL depends heavily on the temperature. The temperature dependence of 1/τL ∼ exp(−Ea′ /kBT) includes a factor with an

Figure 9. Free energy gap dependence of the activation energy. The solid blue and dashed black lines represent analytical results calculated with eq 3.6 and eq 1.3, correspondingly. Numerical results are plotted with red symbols. The number of high frequency modes M is equal to 10. The values of the parameters are Erm = 1.0 eV, Erv = 0.3 eV, Vel = 0.05 eV, τ1 = 0.19 ps, τ2 = 0.50 ps, and x1 = x2 = 0.5. 4017



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activation energy E′a. In the solvent controlled regime for −ΔG < Erm, the quantity E′a has to be added to the activation energy obtained in this section.2 In the inverted region, the apparent activation energy, Ẽ a, depends weakly on τL (small ξ); hence, it should be weakly sensitive to a temperature variation of τL.

ACKNOWLEDGMENTS We are indebted to Dr. S. Feskov and Dr. V. Ionkin for the use of their software. This work was supported by the Ministry of Education and Science of the Russian Federation (contract 14.740.11.0374) and the Russian Foundation for Basic Research (Grant No. 11-03-00736).

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IV. CONCLUDING REMARKS The multichannel stochastic model was shown to be well approved for description of ultrafast nonthermal ET kinetics.47,53,55,57−59,69,70 In this paper we have given evidence of the possibility of the model to adequately describe all principal regularities observed in thermal ET kinetics. The most important of them are itemized here: • The model predicts a solvent controlled regime in the Marcus normal region and its almost full suppression in the Marcus inverted region as well as a continuous transition between them in the vicinity of the activationless region. To quantitatively describe this phenomenon, the DSE magnitude, ξ, is introduced. The reorganization of high frequency vibrational modes effectively suppresses DSE in the Marcus inverted region whereas it plays a minor role in the normal region. The suppression becomes stronger with the increasing number of active vibrational modes. • Additional factor of the DSE suppression stems from fast solvent relaxation component. • The dependence of ξ on the free energy gap reflects the behavior of the frequency factor of the rate constant. • The reorganization of high frequency vibrational modes raises the maximum rate constant above the solvent controlled limit, 1/τL, by 1 order and 2 orders of magnitude for the models with single and ten active high frequency modes, correspondingly. One more important finding is a considerable shift of the rate constant maximum to larger values of the free energy gap, −ΔG > Erm + Erv, with strengthening electronic coupling. One should point out that the ascending and descending branches of the free energy gap law are straightened due to the reorganization of the large number of high frequency vibrational modes. • In the Marcus inverted region, the reorganization of high frequency vibrational modes approaches the ET to the nonadiabatic regime. As a result, a quadratic dependence of the rate constant on the electronic coupling is kept up to considerable larger values. That is why the solvent controlled regime in the inverted region can become inaccessible. • In the inverted region, the multichannel stochastic model predicts the activation energy to be much less than that calculated with Marcus equation (1.3). Moreover, it is extremely small (|Ea| < kBT) in the inverted region for all experimentally accessible values of the free energy gap. • The models with single and many active high frequency vibrational modes predict quantitatively different ET kinetics. This indicates that the model with the single active mode has a rather limited area of applicability.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest. 4018



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Reorganization of Intramolecular High Frequency Vibrational Modes

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