Effect of Intramolecular High-Frequency Vibrational Mode Excitation

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Effect of Intramolecular High Frequency Vibrational Mode Excitation on Ultrafast Photoinduced Charge Transfer and Charge Recombination Kinetics Alexey E Nazarov, Vadim Yu Barykov, and Anatoly Ivanovich Ivanov J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.6b00539 • Publication Date (Web): 08 Mar 2016 Downloaded from http://pubs.acs.org on March 17, 2016

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The Journal of Physical Chemistry

Effect of Intramolecular High Frequency Vibrational Mode Excitation on Ultrafast Photoinduced Charge Transfer and Charge Recombination Kinetics Alexey E. Nazarov, Vadim Yu. Barykov, and Anatoly I. Ivanov∗ Volgograd State University, University Avenue 100, Volgograd 400062, Russia (Dated: March 7, 2016)

Abstract A model of photoinduced ultrafast charge separation and ensuing charge recombination into the ground state has been developed. The model includes explicit description of the formation and evolution of nonequlibrium state of both the intramolecular vibrations and the surrounding medium. An effect of the high frequency intramolecular vibrational mode excitation by a pumping pulse on ultrafast charge separation and charge recombination kinetics has been investigated. Simulations, in accord with experiment, have shown that the effect may be both positive (the vibrational mode excitation increases the charge transfer rate constant) and negative (opposite trend). The effect on charge separation kinetics is predicted to be bigger than that on the charge recombination rate but nevertheless the last is large enough to be observable. The amplitude of both effects falls with decreasing vibrational relaxation time constant but the effects are expected to be observable up to the time constants as short as 200 fs. Physical interpretation of the effects has been presented. Comparisons with the experimental data have shown that the simulations, in whole, provide results close to that obtained in the experiment. The reasons of the deviations have been discussed. Keywords: photoinduced electron transfer, vibrational excited states, vibrational relaxation, intramolecular reorganization, solvent relaxation, stochastic point-transition model

∗

Corresponding author. E-mail: [email protected]

1

ACS Paragon Plus Environment


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I.

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INTRODUCTION

Currently there is a growing interest in the study of the kinetics of electron transfer occurring from nonthermalized states. Typically, the kinetics of such reactions is rather complex, in contrast to thermal reactions, which are characterized by a single parameter – the rate constant.1–3 A quantitative description of processes proceeding from nonthermalized states requires the development of relevant theories. At the same time requirements for these theories are immeasurably higher because they have to reproduce complex kinetics, that can not be reduced to a few rate constants. Nevertheless, the complexity of the kinetics of nonthermal processes provides us certain benefits because its quantitative description demands improvement of current theories and their elaboration leads to a deeper understanding of the microscopic mechanisms not only nonequilibrium but also equilibrium reactions. Photoinduced charge transfer can proceed on time scale comparable with the relaxation times of the nuclear subsystem of the reactants and the surrounding medium.4–8 In such ultrafast processes the chemical transformation effectively competes with the nuclear relaxation.9–15 One of the manifestations of this competition is the spectral effect (a dependence of the charge transfer rate constant on the excitation pulse carrier frequency) that was observed in ultrafast charge recombination of excited donor-acceptor complexes in polar solvents.10,11,14 Supposing the charge recombination to proceed from a nonequilibrium state of the surrounding medium created by the pumping pulse and the relaxation of intramolecular high frequency vibrational modes to occur at considerably shorter time scales than both the medium relaxation and charge recombination, the spectral effect was rather well described in the framework of the stochastic multi-channel point-transition model.14 The excitation of the intramolecular high frequency modes by a pumping pulse is also expected to alter charge transfer kinetics when the charge transfer proceeds on the timescale comparable with that of the intramolecular vibrational relaxation. Although there are a lot of experimental evidences of manifestations of the excited states of high frequency vibrations in the charge transfer dynamics9,12,13,16–25 only recently first systematic investigation of the influence of the vibrational excited states on the charge transfer dynamics has been reported.26 These experiments have discovered that the excitation of high frequency vibrational modes can considerably change not only the forward electron transfer rate but also the rate of the backward electron transfer into the ground state.26 2


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Theoretical investigations of the effect of intramolecular high frequency vibrational modes excitation on photoinduced electron transfer also have been carried out.27–29 In particular, the excitation of high frequency vibrational modes was shown to increase the photoinduced electron transfer rate in the both regions of strong and weak exergonicity and to decrease at moderate exergonicity in the vicinity of the barrierless region.28,29 Ultrafast electron transitions can occur in various modes. The most important characteristics determining the mode are the couplings between electronic states, the environment coupling and the fluctuation spectrum of the environment. There is currently no universal method that could allow simulating the dynamics of electron transitions in all modes and broad time window. This has led to the creation of a huge number of theoretical approaches to solving this problem. The simplest approaches such as the Fermi’s golden rule and the Redfield theory cannot be used for description of the processes considered here. The Fermi’s golden rule is not applicable for description of ultrafast electron transfer because in such processes the electronic coupling is not weak. Redfield theory can describe the electron transition dynamics of a system when the environment coupling is significantly weaker than the coupling between electronic states. This requirement also is not fulfilled for a charge transfer in polar solvents where the interaction of the transferred charge with the environment is strong. The more rigorous and powerful methods such as the hierarchy equations of motion,30–33 the multi-layer multi-configuration time-dependent Hartree wave packet method,34–36 the quasi-adiabatic path integral method37,38 have been developed. They are aimed at fully quantum description of the dynamics of coupled electron-vibrational system with many degrees of freedom. These nonperturbative and non-Markovian approaches are able to adequately describe non-trivial and subtle dynamical features of open quantum electron-vibrational systems including long-lasting coherent dynamics observed in photosynthetic antenna complexes,39 mixing electronic and vibronic excitations due to resonance effects.40 A major limitation of their widespread use for the simulation of charge transfer dynamics is the computational expense that greatly increase when the long-term dynamics are required to be described. In the charge transfer such problems arise when the transfer dynamics in environments characterized by several strongly different relaxation times are studied. In this contribution we exploit the stochastic multi-channel point-transition model.14 At the heart of this approach lie the Marcus free energy surfaces. The motion along the surfaces 3



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reflects reorganization of the surrounding solvent and is fully characterized by the solvent relaxation function41 that can be borrowed from data of independent experiments. The solvent fluctuation frequency have to be rather low because they are classically described. Reorganization of intramolecular high frequency vibrational modes is quantum-mechanically described in the terms of the Huang-Rhys factors. The applicability conditions of the stochastic model are as follows. The first limitation of the model is caused by description of the reactant and product states in terms of populations. Such a description ignores the quantum coherence and it is applicable if the electronic coherence lifetime, τc , is much shorter than the reaction time constant, 1/ket . This condition requires the fulfillment of the inequality √ ~/ 2Erm kB T ≪ 1/ket ,42 where Erm is the solvent reorganization energy. This inequality is met when the interaction with the solvent fluctuations is strong and the temperature is high. For Erm = 1 eV and the room temperature the limitation leads to the inequality ket ≪ 1014 s−1 . The second limitation associates with usage of the diabatic basis. It requires the electronic couplings, V , to be small. Roughly they have to meet the condition V < kB T .43 The classical description of the solvent relaxation leads to the third limitation which requires the inequality ~/τi ≪ kB T to be met for each solvent relaxation mode, where τi is a solvent relaxation time. The main aims of this paper are (i) to develop a model of photoinduced charge transfer involving photoexcitation, charge transfer, and charge recombination stages in donor-acceptor pairs, (ii) to investigate the effect of excitation of intramolecular high frequency vibrational modes on the kinetics of photoinduced charge separation and charge recombination, and (iii) to compare the calculated kinetics with the experimental data.26

II.

THEORY AND COMPUTATIONAL DETAILS

The model considered here is presented in Figure 1. It includes three electronic states: the ground state (GS), a locally excited state (LES), a charge separated state (CSS), and their vibration repetitions. Interaction of the transferred charge with a polar solvent determines a part of key parameters of the electronic transitions. In particular, there is dynamic solvent effect that results in strong dependence of charge transfer kinetics on details of solvation dynamics.14,15,27,41,44–46 Electron-vibration interaction is of primary importance for charge transfer kinetics and it is a part of the model. This interaction is responsible for a depen4


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dence of the electron transfer kinetics on the excitation of high frequency vibrational modes. Figure 1 shows a few of possible states produced by excitation pulse and possible routes of their evolution: charge transfer from vibrational unrelaxed states proceeds in competition with its relaxation (vertical transitions) and motion of the wave packets to the equilibrium position (medium relaxation); charge transfer creates systems in a strongly nonequilibrium state which relaxes and, in parallel, nonthermal charge recombination into the ground state proceeds. The effect of high frequency vibrational mode excitation can be observed if the vibrational relaxation is sufficiently slow and charge transfer can compete with it. Typical timescale of these processes is several hundreds of femtoseconds. If the pumping and probing pulses have duration of the order of tens femtoseconds then the pump and charge transfer stages practically do not overlapped. It allows calculating a locally excited state formed by a pumping pulse and then using it as the initial state for photoinduced charge transfer. Further this approach is used. Real polar solvents are typically characterized by a nonexponential relaxation function.6,47–50 In this case charge transfer can be described in terms of multidimensional free energy surfaces corresponding to different electronic and vibronic states participating in the reaction. The relaxation function of a solvent, X(t), can be represented by a sum of a few (N ) exponential summands:47,50

X(t) =

N ∑

−t/τi

xi e

i=1

N ∑

xi = 1

(1)

i=1

Each summand is associated with a separate collective solvent coordinate Qi with relaxation time constant τi . The solvent coordinates form N -dimensional configuration space of the considered problem. The profiles of the Gibbs free energy in terms of the reaction coordinate Q are pictured in Figure 1. The diabatic free energy surfaces for the ground state, (l)

(n)

(m)

UGS , locally excited state ULES , and the charge separated electronic state UCSS with their 5



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vibrationally excited sublevels can be written as follows45 )2 √ 1 ∑( Qi − 2Eriex + ∆GGS + l~Ω 2 i=1 N

(l)

UGS =

1∑ 2 Q + n~Ω 2 i=1 i )2 N ( √ 1∑ = Qi − 2EriCS + m~Ω + ∆GCS 2 i=1

(2)

N

(n)

ULES =

(3)

(m)

(4)

UCSS

where ∆GGS is the free energy change for electronic transition from the locally excited to the ground state, ∆GCS is the free energy change of the photoinduced charge separation, l, n and m are the quantum numbers of intramolecular high frequency vibrational mode with the frequency Ω that must be ~Ω ≫ kB T with kB and T being the Boltzmann constant and the temperature, Eriex and EriCS are the reorganization energies of the i-th medium mode of the solvent at the stages of pump and charge separation, correspondingly. They are related ex CS to their weights in the relaxation function eq 1 as follows: xi = Eriex /Erm = EriCS /Erm , ∑ ∑ ex CS Erm = i Eriex and Erm = i EriCS are the total reorganization energies of the solvent at

the corresponding stage. There is a difference between the abbreviations CS and CSS. The abbreviation CS stands for the charge separation stage while the index CSS is used to indicate the charge separated state. In the framework of the stochastic point-transition approach,14,41,45 the temporal evolution of the system is described by a set of equations for the probability distribution func(n)

(m)

tions for the locally excited, ρLES (Q, t), the charge separated, ρCSS (Q, t), and the ground, (l)

ρGS (Q, t), states (n) ( ) ρ(n+1) ρ(n) ∑ ∂ρLES (n) (m) (n) (e) ˆ = LLES ρLES − knm ρLES − ρCSS + LES − LES (5) (n+1) (n) ∂t τ τv v m (m) ( ) ∑ ( ) ρ(m+1) ρ(m) ∑ ∂ρCSS (m) (n) (m) (g) (l) (m) (e) ˆ = LCSS ρCSS + knm ρLES − ρCSS + kml ρGS − ρCSS + CSS − CSS (6) (m+1) (m) ∂t τv τv n l (l) ) ρ(l+1) ρ(l) ∑ (g) ( (l) ∂ρGS (m) GS GS ˆ GS ρ(l) − =L k ρ − ρ (7) GS GS CSS + (l+1) − (l) ml ∂t τ τ v v m

ˆ and L ˆ CSS are the Smoluchowski operwhere Q is a vector with components Q1 , Q2 , ..., QN , L 6


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(n)

(l)

(m)

ators describing diffusion on the ULES , UGS , and UCSS free energy surfaces, correspondingly, [ ] N 2 ∑ ∂ ∂ 1 ˆ LES = 1 + Qi + kB T L τ ∂Q ∂Q2i i i i=1 ) ] [ ( N √ 2 ∑ 1 ∂ ∂ ˆ CSS = L 1 + Qi − 2EriCS + kB T τ ∂Qi ∂Q2i i i=1 ( [ ] ) N √ 2 ∑ ∂ 1 ∂ ˆ GS = L 1 + Qi − 2EriGS + kB T τ ∂Qi ∂Q2i i i=1

(8)

(9)

(10)

The model, eqs 5 – 7, includes single-quantum irreversible vibrational relaxation n → n − 1 (n)

(n)

proceeding with the rate constant 1/τv , where τv

(1)

= τv /n.51 (n)

Transitions between vibrational sublevels of the locally excited electronic state, ULES , (m)

(l)

charge separated state, UCSS , and the ground state, UGS , are described by the Zusman parameters14,41,42,45 ) ) 2πVg2 Fml ( (m) 2πVe2 Fnm ( (n) (m) (g) (l) δ ULES − UCSS , kml = δ UCSS − UGS , ~ ~  2 √ min(n,m) ∑ (−1)m−r ( S)n+m−2r  = exp {−S} n!m!  r!(n − r)!(m − r)! r=0

(e) knm =

(11)

Fnm

(12)

where Ve and Vg are the electronic couplings for transitions between LES and CSS and between CSS and GS, correspondingly, Fnm is the Franck-Condon factor for the transition between the vibrational sublevels n and m, the Huang-Rhys factor S is different for charge CS CR separation and charge recombination stages. They are S CS = Erv /~Ω and S CR = Erv /~Ω CS CR with Erv and Erv being the reorganization energy of the intramolecular high frequency

mode at the charge separation and charge recombination stages, correspondingly. The Zusman’s parameters, eq 11, determine the probability of an electronic transition at single passing of the system through the term crossing point (this limits the value of the electronic coupling). This probability must be small. But the stochastic motion leads to many repeated passes of the point (forward and backward) that can result in a large transition probability which can be independent of the value of the electronic coupling. The stochastic model accurately describes the statistics of such crossings. The problem was discussed in refs 42 and 52. The populations of all electronic states involved in the reaction are determined by the 7



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equations Pk (t) =

∑∫

(n) ρk (Q, t)

n

N ∏

dQi ,

(13)

i=1

where k = LES, CSS, or GS. The pump pulse with carrier frequency ωe is assumed to be of the Gaussian form { } t2 E(t) = E0 exp iωe t − 2 τe

(14)

and its duration, τe , is short so that the solvent is considered to be frozen during excitation. All high frequency vibrational modes are supposed initially to be in the ground state. This allows obtaining the following general expression for the normalized initial probability distribution function on the locally excited term10,53 { } ∑ e √ ex 2 2 (n) 2 ∑ e − Q [~δω 2E ] τ Q e i (n) e ri i ρLES (Q, t = 0) = APn exp − − 2~2 2kB T

(15)

S n e−S (16) n! √ ei = Qi − 2E ex , Pn is the Franck-Condon − n~Ω, Q ri Pn =

(n)

where ~δωe

ex = ~ωe − Erm + ∆GGS

factor at the excitation stage. The normalization factor A is determined from the equation ( ) ∫ ∑ ∏ ~ (n) N/2 ρLES (Q, t = 0) dQi = A(2πkB T ) Z=1 (17) στ e n i ex where σ 2 = (2Erm kB T ) + ~2 τe−2 is the dispersion of an electron-vibrational transition in the

absorption spectrum,

Z=

∑ n

Zn ,

 [ ]2  (n)    ~δωe  Zn = Pn exp −  2σ 2   

(18)

The probability of the population of the n-th vibrational sublevel of LES, Wn , is Wn =

Zn Z

(19)

The initial conditions for the charge separated and the ground state distribution functions, (m)

(l)

ρCSS (Q, t) and ρGS (Q, t), are: (m)

(l)

ρCSS (Q, t = 0) = 0, ρGS (Q, t = 0) = 0 8


(20)

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The set of eqs 5 – 6 with the initial conditions eq 15 – 20 is solved numerically by using the Brownian simulation method.10,54,55 The effect of excitation of the high frequency vibrational modes is expected for ultrafast charge separation occurring in femto- and picosecond time domain. The fluorescence and the internal conversion from the locally excited state are not considered in this paper since they proceed much slower than the charge separation. As well known the frequencies of the vibrations in the different electronic states can considerably differ. The frequency change influences on the Franck-Condon factors and can be important for large vibrational quantum numbers. We do not account for this effect as well as the Duschinsky effect because for a charge transfer stage the Huang-Rhys factor can be only roughly estimated and its uncertainty can lead to even larger uncertainty in the Franck-Condon factors. The diagonal anharmonicity in the vibrations is not considered due to the same reason. The non-diagonal anharmonizm being responsible for the vibrational redistribution and relaxation is partly accounted for in the terms of the vibrational relaxation time constant. It is well known that the resonance effects are important when the high-frequency modes participate in an electronic transition. It should be stressed that the stochastic model correctly accounts these effects. Electronic transitions are possible when the full energies of the initial and final electronic states are close each other. This guaranties the delta functions in eq 11. The full energy includes the energies of high frequency modes and the solvent modes with a continuous spectrum of their fluctuations. When the solvent reorganization energy is small the resonance effect manifests in oscillations of the electronic transition rate constant as a function of the free energy gap. This case is well described by the Redfield theory and its modifications. The stochastic approach is applicable in the opposite limit of strong solvent reorganization.42 In this limit the solvent can accept large amount of electronic energy that results in broad, hence strongly overlapping, vibrational resonances so that the vibrational structure cannot be seen. In fine, we describe the physical processes incorporated into the model. The excitation of the system is visualized as an appearance of a single wave packet on a vibrational sublevel √ ex is fulfilled. In this of the locally excited state if the inequality ~Ω ≫ σ ≃ 2kB T Erm case the probability Wn approaches to unity for corresponding vibrational level. In the opposite limit several wave packets simultaneously appear on different vibrational sublevels 9



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of the excited state (see Figure 1). Further several competing processes proceed. Among (n)

them: the vibrational relaxation n → n − 1 proceeding with the time constant τv , the wave (n)

packet motion to the bottom of the free energy surface ULES reflecting the medium relaxation, the charge separation occurring at the term crossing points that populates the vibrational sublevels of the charge separated state. Next, the systems created in the charge separated (m)

(n)

(m)

state at the term crossing points UCSS = ULES move to the UCSS term minimum undergoing, (m)

(m−1)

in parallel, quantum transitions between neighbor vibrational states UCSS → UCSS

and

nonthermal charge recombination into the ground state.

III.

RESULTS AND DISCUSSION

In this section the influence of vibrational mode excitation on the rate of photoinduced charge separation and following charge recombination into the ground state is investigated. Since both reactions proceed in nonthermal regime their kinetics can considerably deviate from exponential. In the general case they can be described by eqs 21, 22 dPLES (t) = −kf (t)PLES (t) dt dPCSS (t) = kf (t)PLES (t) − kb (t)PCSS (t) dt

(21) (22)

with time dependent rate constants of forward and backward charge transfer, kf (t) and kb (t), correspondingly. To compare the simulated and experimental data effective rate constants are introduced. They are determined by eqs 23, 24 ∫ ∞ 1 = dtPLES (t) kFET ∫0 ∞ 1 = dtPCSS (t) kBET 0

(23) (24)

For a quantitative description of the phenomenon the effect of vibrational mode excitation is exploited:

∗ kFET − kFET k ∗ − kBET χb = BET (25) kFET kBET and kFET , kBET stand for charge separation from the first excited vi-

χf =

∗ ∗ where kFET , kBET

brational state n = 1 of LES (S11 ) and from the ground vibrational state n = 0 (S10 ), correspondingly. We adopt the model parameters corresponding to donor-acceptor porphyrin-viologen complexes in water solutions.26 Porphyrin complexation has a little effect on the position of 10


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the Q-bands corresponding to transitions to different vibrational levels of the first excited state. Therefore, the free energy gap for this transition is fixed to ∆GGS = −2.02 eV for all complexes.26 In absorption spectra of these compounds a vibrational structure is well seen. This structure reflects an excitation of a vibrational mode of the porphyrin ring with the frequency of ~Ω ≃ 0.17 eV having a rather long relaxation time to compete with photoinduced charge separation. The clear vibrational structure means a small medium reorganization ex ex energy Erm . Hereinafter it is assumed to be Erm = 0.001 eV. This value of the reorganiza-

tion energy leads to population of the ground and the first excited vibrational levels of LES, correspondingly, with the probabilities close to unity by the pumping pulses with the carrier ex ex frequencies ~ωe = −∆GGS + Erm or ~ωe = −∆GGS + Erm + ~Ω and τe = 50 fs. So extremely ex small value of Erm is adopted to provide close values of the medium reorganization energies √ √ CS CR CS − ex ]2 . This for the charge separation, Erm , and the charge recombination Erm = [ Erm Erm

theory supposes full correlation between the reorganization of the medium at the stages of the excitation and the charge separation. In reality this correlation is weaker and can be described in terms of an angle between the reaction coordinate directions corresponding to the excitation and the charge separation stages.56–60 This aspect is not considered here and ex the drawback of the theory is corrected by a small value of Erm . Dynamical characteristics

of water are exploited for the description of the solvent dynamics. The weights of three relaxation modes are: x1 = 0.48, x2 = 0.20, x3 = 0.32, and their relaxation times are: τ1 = 0.026 ps, τ2 = 0.126 ps, τ3 = 0.880 ps.47 The variable parameters are the electronic CS couplings Ve and Vg , the medium reorganization energy, Erm , the reorganization energy of

high frequency vibrational mode Erv and its relaxation time constant τv . The results of the fitting of the forward photoinduced charge transfer rate constant, kFET , to experimental data are displayed in Figure 2. It presents a typical Marcus free energy gap law. The experimental data are those reported in ref 26. They were obtained for the complexes consisting of zinc(II) meso-tetrasulfonatophenylporphyrin (ZnTPPS4− ) and magnesium(II) meso-tetrasulfonatophenylporphyrin (MgTPPS4− ) as the electron donor and a series of viologens as electron acceptors: methyl viologen (MV2+ ), benzyl viologen (BV2+ ), 1-Methylbipyridine (MB+ ), 1,1′ -Dimethyl-2,2′ -bipyridine (DM2+ ), 1,1′ ,4,4′ -tetramethyl2,2′ -bipyridine (4,4′ -Me2 -DM2+ ), 1,1′ -Bis(cyanomethyl)-4,4′ -bipyridinium (MVCN2+ ) as electron acceptors.

The numbers stand for the following 1:1 porphyrin/viologen com-

plexes: 1 – ZnTPPS4− /DM2+ , 2 – ZnTPPS4− /MB+ , 3 – ZnTPPS4− /4,4′ -Me2 -DM2+ , 4 – 11



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ZnTPPS4− /MV2+ , 5 – MgTPPS4− /MV2+ , 6 – ZnTPPS4− /BV2+ , 7 – MgTPPS4− /BV2+ , 8 – ZnTPPS4− /MVCN2+ . The numbering of the complexes are the same as in ref 26. These numbers are used in the figures to specify the complex. CS In simulations the sum Erm + Erv was fixed at 0.85 eV that corresponds to the magnitude

of −∆GCS at which the rate constant kFET has a maximum. It is difficult to obtain the magnitudes of each summands from the fitting due to a weak dependence of the rate constant CS on specific values of Erm and Erv . We adopt the value Erv = 0.2 eV because the estimations

of the total reorganization energy of high frequency modes show that it lies in the range 0.2 – 0.4 eV.61 A typical value of the high frequency vibrational mode τv = 1 ps is used. The magnitude of Ve is determined from the fitting. All the parameters are determined for the charge transfer from the ground vibrational state, n = 0. The free energy gap dependence of the charge transfer rate constant from the first excited vibrational state is calculated without adjustable parameters. The difference in the kinetics of charge transfer from the ground and first excited vibrational states we discuss in terms of the effect of vibrational mode excitation, χf . The dependence of the effect on the driving force is presented in Figure 3. The figure shows that there are three regions: (a) ∆GCS > -0.58 eV with positive effect χf > 0 that reaches its maximum value χf = 0.87 at ∆GCS = −0.37 eV; (b) -1.01 eV < ∆GCS < −0.58 eV with negative effect reaching the minimum value χf = −0.42 at ∆GCS = −0.76 eV; (c) ∆GCS < −1.01 eV with the positive effect. In short, the effect is predicted to be positive in both regions of low and high exergonicity being negative in between. To gain an insight into physical mechanisms underlying the dependence let us first conCS sider the low exergonic reaction (−∆GCS < 0.58 eV). Here the inequality −∆GCS < Erm is

fulfilled. Obviously, for such parameters the charge separated state is predominantly formed in the ground vibrational state due to smaller activation energy (see Figure 1 that corresponds to these parameters). In this case the rate of charge separation from the first excited vibrational state S11 is larger than that from the ground state S10 due to two reasons: (i) the Franck-Condon factor is larger for vibrational excited state, F10 /F00 = S CS ≃ 1.2 > 1; (ii) the activation barrier for charge separation from S11 state is smaller. This results in a positive effect of vibrational excitation. The effect raises with decreasing −∆GCS because ∗ the difference between the heights of the activation barrier increases. In other words kFET

decreases with decreasing −∆GCS slower than kFET . When the charge separation rate be12


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comes smaller than the vibrational relaxation rate, 1/τv the effect χf starts to go down with decreasing −∆GCS . Indeed, for the ∆GCS = −0.37 eV the rate constant is kFET ≃ 1.0 ps−1 . For ∆GCS approaching to the positive area the charge separation becomes much slower than the vibrational relaxation and it occurs from the vibrational ground state independently from the initial vibrational state and the effect comes near to zero. In Figure 3 only the beginning of this trend is seen. CS In the neighborhood of the reaction rate constant maximum, −∆GCS = Erm + Erv , the

charge separation mainly proceeds through the sinks placed at the bottom of the free energy (n)

surface ULES . In this case the ratio of the charge separation rate constants from the excited, ∗ , and the ground vibrational states, kFET , is determined by the ratio of corresponding kFET

Franck-Condon factors

∗ kFET F1m∗ (m∗ − S)2 ∼ = kFET F0m∗ −1 m∗

(26)

where m∗ =

CS −∆GCS − Erm + ~Ω ~Ω

(27) (1)

is the number of the sink which is the nearest to the minimum of the free energy curve ULES . The area of ∆GCS where the negative effect is expected can be find out from the condition ∗ kFET /kFET = (m∗ − S)2 /m∗ < 1

(28)

From eq 28 one obtains 0.48 < m∗ < 2.87 that leads to 0.56 < −∆GCS < 0.97 eV. Despite the roughness of the evaluation it is perfectly consistent with the results of simulations (Figure 3). The zeros of the effect χf = 0 coincide with the edges of the negative effect and are determined by the equality (m∗ − S)2 /m∗ = 1. It is well known that eq 26 is applicable only in the nonadiabatic regime, nevertheless the predictions on its base are in a good agreement with the simulations despite large value of electronic coupling, Ve . This allows concluding that for such parameters the reaction in the inverted region proceeds in the nonadiabatic regime and the dynamic solvent effect41 is strongly suppressed. This is a consequence of the reduction of the effective electronic couplings by the Franck-Condon factors that for the most effective sinks are rather small (see eq 11).15 In the region of high exergonicity ∆GCS < −1.01 eV the effect, χf , becomes posi∗ tive and increases with increasing the exergonicity due to rise of the ratio kFET /kFET ∼

13



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F1m∗ /(F0m∗ −1 ). The increase continues while the charge transfer rate constant is greater than the vibrational relaxation rate, 1/τv . The maximum is reached at ∆GCS ≃ −1.4 eV (not shown in the Figure 3). For even larger exergonicity −∆GCS > 1.4 eV the rate of photoinduced charge separation becomes less than the vibrational relaxation rate and the effect, χf , decreases with increasing the exergonicity up to zero.29 Analysis of the data presented in Figure 3 allows concluding: (i) the experiment and simulations demonstrate that the effects, χf , can be both positive and negative, (ii) the magnitudes of the experimental and simulated effects are comparable, (iii) for the complexes 2 – 5 the signs of the experimental and simulated effects coincide, and (iv) for the complexes 1 and 6 the signs of the experimental and simulated effects are opposite. There are several reasons of last discrepancy. In the simulations all parameters of the complexes except for driving force are supposed to be invariable while in real complexes they can vary that results in considerable scattering experimental data. What concerns the complex 1, possibly, its driving force is incorrectly determined. This is indicated by the fact that its experimental values of forward and back charge transfer rates also differ considerably from those calculated. If its driving force were increased to −∆GCS ≃ 0.7 eV, the fitting would be much better for all rate constants and the effects. Another reason can be connected with rather strong nonexponentiality of the charge transfer kinetics in the nonequilibrium systems considered here. In this case the magnitude of the effective rate depends on its definition. These reasons can explain the discrepancies for all complexes except for complex 6. The discrepancy in this case is unclear. To show the scale of nonexponentiality of the charge separation kinetics, in Figure 4 the LES population, PLES (t), as a function of time for several values of the driving force, is pictured. The deviations of the curves from the straight lines are the measures of the nonexponentiality. Its magnitude is rather large and can result in considerable impact on the effect χf if the effective rates are differently defined. Moreover, the nonexponentiality extent of the kinetics of the charge separation from the ground and the excited vibrational states strongly differ that considerably strengthen the role of the definition of the effective rate constant. The results of the fitting of the charge recombination rate constant, kBET , as a function of charge recombination free energy, ∆GCR , to experimental data26 are pictured in Figure 5. Here ∆GCR = ∆GGS − ∆GCS . All experimental points are in the Marcus inverted region 14


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so that only the descended branch of the Marcus free energy gap law is seen. In the fitting ∗ a single parameter, Vg , has been varied. The rate constant kBET as well as the effect χf (see

Figure 6) are calculated without adjustable parameters. Let us first consider the mechanism of the effect of vibrational mode excitation on the charge recombination rate constant. For the low exergonic charge separation reaction the CS inequality −∆GCS < Erm is met and the forward electron transfer mainly results in popula-

tion of the ground vibrational level of the charge separated state independently of the initial (0)

vibrational state. The initial positions of the wave packets on the free energy surface UCSS are also close to each other. This leads to negligible effect in the area of large exergonicity of the charge recombination, −∆GCR . As it was mentioned above, in the neighborhood CS of the forward charge transfer rate constant maximum, −∆GCS = Erm + Erv , the charge

separation mainly proceeds through the sinks placed at the bottom of the free energy sur(n)

(m)

(m−1)

face ULES . Obviously the terms UCSS and UCSS (1)

are populated with approximately equal

(0)

effectiveness from the states ULES and ULES , correspondingly since these processes have the same energetics. Further the wave packets move towards the term minimum and the charge (m)

(l)

recombination occurs at every term crossings UCSS = UGS . This is a nonequilibrium recombination. It plays a very important role because the charge recombination proceeds on the timescale of the medium relaxation. For majority of the crossings the inequality Fml /Fm−1l−1 > 1 is met that results in larger recombination rate of the particles created by the decay of the vibrational excited state, S11 , and, hence, to the positive χb . In the region (m)

of low exergonicity, −∆GCS , the terms UCSS with large numbers m are populated, for which the inequality Fml /Fm−1l−1 < 1 is dominated that leads to the negative effect χb . The simulated data for the effect of vibrational mode excitation on the charge recombination kinetics much worse match the experimental data. Even the sign of the effect of vibrational mode excitation is incorrectly predicted for a majority of the complexes. Nevertheless, we may make some positive conclusions. The experimental data support the theoretical conclusion that the effect, χb , can be both positive and negative. Moreover, the magnitudes of the experimental and simulated effects are comparable. The most probable reason of the discrepancies is strong nonexponentiality of the charge recombination kinetics. To demonstrate the nonexponentiality of the charge recombination kinetics, in Figure 7 the dependence of the charge recombination rate constant, kb (t), on time for several values 15



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of the driving force, is pictured. It is calculated by eq 29 [ ] dPLES (t) dPCSS (t) kb (t) = − + /PCSS (t) dt dt

Page 16 of 36

(29)

that directly follows from eqs 21 and 22. The variation of the rate constant with time due to nonexponentiality is comparable with that due to the effect of the vibrational mode excitation. This means that different effective rate constant determination can lead to big differences in the magnitude of the effect. Another reason for the discrepancy between the theory and the experiment may be associated with the harmonic approximation that is used to calculate the Franck-Condon factors. Deviations of the calculated factors Fml from real ones can be great for large quantum numbers m and l. To conclude this section we consider the impact of the relaxation time of the high frequency vibrational mode, τv , on the effects χf and χb . The results of simulations are presented in Figures 8 and 9. The Figures demonstrate expected increase of the magnitude of both effects with increasing τv . An explicit dependence of χf on τv can be easily obtained supposing the populations of the first excited, S11 , and the ground, S10 , vibrational states to decay (due to charge transfer) exponentially with the intrinsic rate constants k1 and k0 , correspondingly. The result is29 χf =

[k1 (1 − w) − k0 ]τv 1 + (k0 + wk1 )τv

(30)

where w is the probability of nonthermal charge recombination into the state S10 when the charge separated state is produced from the vibrational excited state S11 . This equation predicts a monotonous raise of the absolute value of the spectral effect with increasing τv that is in a good accord with simulations.29 The dependence is saturated when the inequality (k0 + wk1 )τv ≫ 1 is fulfilled. The value of w can be analytically calculated62 and its CS magnitude is of the order of ten percent for the adopted parameters and −∆GCS < Erm .

In the Marcus inverted region the role of nonthermal recombination is negligible, and its probability, w, is poorly defined. Figures 8 and 9 shows that both effects for forward and backward electron transfer are observable for τv as short as 200 fs.

IV.

CONCLUSIONS

A model of photoinduced ultrafast charge transfer and charge recombination involving an excited vibrational state formation by a short laser pulse is developed. The model 16


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demonstrates a rather strong dependence of the charge separation and charge recombination kinetics on the intramolecular high frequency vibrational mode excitation. To quantify this dependence the effect of vibrational mode excitation is introduced. In accord with the experiment the simulations show that the effect can be both positive and negative. The magnitude of the effect on both the forward and backward reactions is expected to be rather large and can observable in experiments even when the vibrational relaxation is fast. One of the specific features of the considered ultrafast charge transfer is nonthermalized states of the medium and the intramolecular high frequency vibrations. The nonthermalized states are created by both the excitation pulse and the charge transfer so that both the charge separation and the charge recombination stages proceed in parallel with the nuclear relaxation. Such states manifest in considerable deviations of the charge separation and charge recombination kinetics from the exponential law that is in contrast to the thermal reactions. Kinetics of such reactions can be described in terms of time dependent rate constant. Nevertheless, it is often helpful to introduce a time independent effective rate constant that implies some kind of averaging. Since different types of averaging gives different values of the effective rate constant, the quantitative comparison of the results obtained in different studies is possible if only the same determination of the effective rate constants is exploited. This is especially important when such subtle phenomena as the effect of the high frequency vibration excitation are explored. In this paper the geometry of the donor-acceptor pairs is considered to be the same for all pairs of the ensemble and is invariant at the timescale of the charge separation and charge recombination. This approach is applicable for intramolecular charge transfer but can be also used for ultrafast charge transfer proceeding on subpicosecond timescale. Although the distribution of the donor-acceptor pairs over distances and orientations in solutions is very broad, the excitation selects a subensemble with largest electronic coupling and, hence, with narrow distribution. The geometrical changes on time of the order of a few picosecond are not large. Nevertheless, in real systems there is some dispersion of the electronic coupling, driving force, and reorganization energy. This difference also can results in differences between theoretical and experimental results. Here we investigate the photoinduced electron transfer kinetics within nonadiabatic model of electronic transitions. The obtained qualitative agreement with the experimental data evidences in favor of this model. Systematic explorations of the effect of the high frequency 17



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Page 18 of 36

vibration excitation on charge separation and charge recombination kinetics are required to clarify whether the considered theory is able to quantitatively describe a rather subtle features of ultrafast charge transfer reactions in nonthermalized systems.

ACKNOWLEDGMENTS

This work was supported by the Russian Foundation for Basic Research (Grants No. 14-03-00261 and No 15-43-02027).

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REFERENCES

1

Marcus, R. A. On the Theory of Oxidation-Reduction Reactions Involving Electron Transfer. I J. Chem. Phys. 1956. DOI: 10.1063/1.1742723.

2

Kuznetsov, A. M. Charge Transfer in Physics, Chemistry and Biology; Gordon & Breach: Amsterdam, 1995.

3

Jortner, J.; Bixon, M.; Eds. Electron Transfer: From isolated Molecules to Biomolecules. Advances in Chemical Physics; Wiley: New York, 1999. Vol. 106; Vol. 107.

4

Heitele, H.; Pöllinger, F.; Häberle, T.; Michel-Beyerle, M. E.; Staab, H. A. Energy Gap and Temperature Dependence of Photoinduced Electron Transfer in Porphyrin-Quinone Cyclophanes. J. Phys. Chem. 1994, DOI: 10.1021/j100081a028.

5

Akesson, E.; Walker, G. C.; Barbara, P. F. Dynamic Solvent Effects on Electron Transfer Rates in the Inverted Regime: Ultrafast Studies on the Betaines. J. Chem. Phys. 1991, DOI: 10.1063/1.460774.

6

Rosenthal, S. J.; Xie, X.; Du, M.; Fleming, G. R. Femtosecond Solvation Dynamics in Acetonitrile: Observation of the Inertial Contribution to the Solvent Response. J. Chem. Phys. 1991, DOI: 10.1063/1.461742.

7

Kandori, H.; Kemnitz, K.; Yoshihara, K. Subpicosecond Transient Absorption Study of Intermolecular Electron Transfer between Solute and Electron-Donating Solvents. J. Phys. Chem. 1992, DOI: 10.1021/j100199a039.

8

Seel, M.; Engleitner, S.; Zinth, W. Wavepacket Motion and Ultrafast Electron Transfer in the System Oxazine 1 in N,N-dimethylaniline. Chem. Phys. Lett. 1997, DOI: 10.1016/S00092614(97)00771-9.

9

Zimmermann, C.; Willig, F.; Ramakrishna, S.; Burfeindt, B.; Pettinger, B.; Eichberger, R.; Storck, W. Experimental Fingerprints of Vibrational Wave-Packet Motion during Ultrafast Heterogeneous Electron Transfer. J. Phys. Chem. B 2001, DOI: 10.1021/jp011106z.

10

Fedunov, R. G.; Feskov, S. V.; Ivanov, A. I.; Nicolet, O.; Pagès, S.; Vauthey, E. Effect of the Excitation Pulse Carrier Frequency on the Ultrafast Charge Recombination Dynamics of Donor-Acceptor Complexes: Stochastic Simulations and Experiments. J. Chem. Phys. 2004,

19



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 20 of 36

DOI: 10.1063/1.1772362. 11

Nicolet, O.; Banerji, N.; Pagès, S.; Vauthey, E. Effect of the Excitation Wavelength on the Ultrafast Charge Recombination Dynamics of Donor–Acceptor Complexes in Polar Solvents. J. Phys. Chem. A 2005, DOI: 10.1021/jp053221.

12

Wan, C.; Xia, T.; Becker, H. C.; Zewail, A. H. Ultrafast Unequilibrated Charge Transfer: A New Channel in the Quenching of Fluorescent Biological Probes. Chem. Phys. Lett. 2005, DOI: 10.1016/j.cplett.2005.06.101.

13

Kang, Y. K.; Duncan, T. V.; Therien, M. J. Temperature-Dependent Mechanistic Transition for Photoinduced Electron Transfer Modulated by Excited-State Vibrational Relaxation Dynamics. J. Phys. Chem. B 2007, DOI: 10.1021/jp070414f.

14

Feskov, S. V.; Ionkin, V. N.; Ivanov, A. I.; Hagemann, H.; Vauthey, E. Solvent and Spectral Effects in the Ultrafast Charge Recombination Dynamics of Excited Donor-Acceptor Complexes. J. Phys. Chem. A 2008, DOI: 10.1021/jp709587x.

15

Yudanov, V. V.; Mikhailova, V. A.; Ivanov, A. I. Reorganization of Intramolecular High Frequency Vibrational Modes and Dynamic Solvent Effect in Electron Transfer Reactions. J. Phys. Chem. A 2012, DOI: 10.1021/jp301837t.

16

Lin, Z.; Lawrence, C. M.; Xiao, D.; Kireev, V. V.; Skourtis, S. S.; Sessler, J. L.; Beratan, D. N.; Rubtsov, I. V. J. Modulating Unimolecular Charge Transfer by Exciting Bridge Vibrations. Am. Chem. Soc. 2009, DOI: 10.1021/ja907041t.

17

Lynch, M. S.; Van Kuiken, B. E.; Daifuku, S. L.; Khalil, M. On the Role of High-Frequency Intramolecular Vibrations in Ultrafast Back-Electron Transfer Reactions. J. Phys. Chem. Lett. 2011, DOI: 10.1021/jz200930h.

18

Delor, M.; Scattergood, P. A.; Sazanovich, I. V.; Parker, A. W.; Greetham, G. M.; Meijer, A. J. H. M.; Towrie, M.; Weinstein, J. A. Toward control of electron transfer in donor-acceptor molecules by bond-specific infrared excitation. Science 2014, DOI: 10.1126/science.1259995.

19

Delor, M.; Keane, T.; Scattergood, P. A.; Sazanovich, I. V.; Greetham, G. M.; Towrie, M.; Meijer, A. J. H. M.; Weinstein, J. A. On the mechanism of vibrational control of light-induced charge transfer in donor-bridge-acceptor assemblies. Nat. Chem. 2015, DOI:10.1038/nchem.2327.

20

Courtney, T. L.; Fox, Z. W.; Estergreen, L.; Khalil, M. Measuring Coherently Coupled Intramolecular Vibrational and Charge Transfer Dynamics with Two-Dimensional VibrationalElectronic Spectroscopy. J. Phys. Chem. Lett. 2015, DOI: 10.1021/acs.jpclett.5b00356.

20


Page 21 of 36

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


21

Lenzmann, F.; Krueger, J.; Burnside, S.; Brooks, K.; Grätzel, M.; Gal, D.; Ruëhle, S.; Cahen, D. Surface Photovoltage Spectroscopy of Dye-Sensitized Solar Cells with TiO2 , Nb2 O5 , and SrTiO3 Nanocrystalline Photoanodes: Indication for Electron Injection from Higher Excited Dye States. J. Phys. Chem. B 2001, DOI: 10.1021/jp010380q.

22

Shaw, L. E.; Langford, C. H. The photochemistry of Mo(CNPh)6 : Dissociative Photosubstitution and Evidence for Hot Electron Transfer. Coord. Chem. Rev. 2002, DOI: 10.1016/S00108545(02)00080-2.

23

Kallioinen, J.; Benkö, G.; Sundström, V.; Korppi-Tommola, J. E. I.; Yartsev, A. P. Electron Transfer from the Singlet and Triplet Excited States of Ru(dcbpy)2 (NCS)2 into Nanocrystalline TiO2 Thin Films. J. Phys. Chem. B 2002, DOI: 10.1021/jp0143443.

24

Petersson, J.; Eklund,M.; Davidsson, J.; Hammarström, L. Variation of Excitation Energy Influences the Product Distribution of a Two-Step Electron Transfer: S2 vs S1 Electron Transfer in a Zn(II)porphyrin-Viologen Complex. J. Am. Chem. Soc. 2009, DOI: 10.1021/ja900729j.

25

Petersson, J.; Eklund,M.; Davidsson, J.; Hammarström, L. Ultrafast Electron Transfer Dynamics of a Zn(II)porphyrin-Viologen Complex Revisited: S2 vs S1 Reactions and Survival of Excess Excitation Energy. J. Phys. Chem. B 2010, DOI: 10.1021/jp911686z.

26

Petersson, J.; Hammarsrtröm, L. Ultrafast Electron Transfer Dynamics in a Series of Porphyrin/Viologen Complexes: Involvement of Electronically Excited Radical Pair Products. J. Phys. Chem. B 2015, DOI: 10.1021/jp5113119.

27

Kichigina, A. O.; Ionkin, V. N.; Ivanov, A. I. U-Shaped Temperature Dependence of Rate Constant of Intramolecular Photoinduced Charge Separation in Zinc Porphyrin Bridge Quinone Compounds. J. Phys. Chem. B 2013, DOI: 10.1021/jp404222a.

28

Barykov V. Yu.; Ionkin, V. N.; Ivanov, A. I. Effect of Excitation Pulse Carrier Frequency on Ultrafast Photoinduced Charge Transfer Kinetics: Effect of Intramolecular High Frequency Vibrational Mode Excitation. J. Phys. Chem. C 2015, DOI: 10.1021/acs.jpcc.5b00005.

29

Barykov V. Yu.; Ivanov, A. I. Excited States of the High-Frequency Vibrational Modes and Kinetics of Ultrafast Photoinduced Electron Transfer. Rus. J. Phys. Chem. A 2016, DOI: 10.1134/S0036024416020060.

30

Tanimura, Y.; Kubo, R. Two-Time Correlation Functions of a System Coupled to a Heat Bath with a Gaussian-Markoffian Interaction. J. Phys. Soc. Jpn. 1989, DOI: 10.1143/JPSJ.58.1199.

21



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

31

Page 22 of 36

Str¨ umpfer, J.; Schulten, K. Open Quantum Dynamics Calculations with the Hierarchy Equations of Motion on Parallel Computers. J. Chem. Theory Comput. 2012, DOI: 10.1021/ct3003833.

32

Zhu, J.; Kais, S.; Rebentrost, P.; Aspuru-Guzik, A. Modified Scaled Hierarchical Equation of Motion Approach for the Study of Quantum Coherence in Photosynthetic Complexes. J. Phys. Chem. B 2012, DOI: 10.1021/jp109559p.

33

Kreisbeck, C.; Kramer, T. Long-Lived Electronic Coherence in Dissipative Exciton Dynamics of Light-Harvesting Complexes. J. Phys. Chem. Lett. 2012, DOI: 10.1021/jz3012029.

34

Wang, H.; Thoss, M. Multilayer formulation of the multiconfiguration time-dependent Hartree theory. J. Chem. Phys. 2012, DOI: 10.1063/1.1580111.

35

Manthe, U. A multilayer multiconfigurational time-dependent Hartree approach for quantum dynamics on general potential energy surfaces. J. Chem. Phys. 2008, DOI: 10.1063/1.2902982.

36

Vendrell, O.; Meyer, H.-D. Multilayer multiconfiguration time-dependent Hartree method: Implementation and applications to a Henon-Heiles Hamiltonian and to pyrazine. J. Chem. Phys. 2008, DOI: 10.1063/1.3535541.

37

Makri, N. Quantum Dissipative Dynamics: A Numerically Exact Methodology J. Phys. Chem. A 2008, DOI: 10.1021/jp980359y.

38

Nalbach, P.; Mujica-Martinez, C. A.; Thorwart, M. Vibronically coherent speed-up of the excitation energy transfer in the Fenna-Matthews-Olson complex. Phys. Rev. E 2015, DOI: 10.1103/PhysRevE.91.022706.

39

Engel, G. S.; Calhoun, T. R.; Read, E. L.; Ahn, T.-K.; Manˇcal, T.; Cheng, Y.-C.; Blankenship, R. E.; Fleming, G. R. Evidence for wavelike energy transfer through quantum coherence in photosynthetic systems. Nature 2007, DOI: 10.1038/nature05678.

40

Christensson, N.; Kauffmann, H. F.; Pullerits, T.; Manˇcal, T. Origin of Long-Lived Coherences in Light-Harvesting Complexes. J. Phys. Chem. B 2012, DOI:10.1021/jp304649c.

41

Zusman, L. D. Outer-Sphere Electron Transfer in Polar Solvents. Chem. Phys. 1980, DOI: 10.1016/0301-0104(80)85267-0.

42

Garg, A.; Onuchic, J.; Ambegoakar, V. Effect of Friction on Electron Transfer in Biomolecules. J. Chem. Phys. 1985, DOI: 10.1063/1.449017.

43

Barzykin, A.V.; Frantsuzov, P.A.; Seki, K.; Tachiya, M. Solvent effects in nonadiabatic electron– transfer reactions: theoretical aspects. Adv. Chem. Phys. 2002, DOI: 10.1002/0471231509.ch9.

22


Page 23 of 36

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


44

Yakobson, B. I.; Burshtein, A. I. Relaxation Hindrance in Nonadiabatic Cage Reactions. Chem. Phys. 1980, DOI: 10.1016/0301-0104(80)85057-9.

45

Feskov, S. V.; Ionkin, V. N.; Ivanov, A. I. Effect of High-Frequency Modes and Hot Transitions on Free Energy Gap Dependence of Charge Recombination. J. Phys. Chem. A 2006, DOI: 10.1021/jp063280z.

46

Yudanov, V. V.; Mikhailova, V. A.; Ivanov, A. I. Nonequilibrium Phenomena in Charge Recombination of Excited Donor-Acceptor Complexes and Free Energy Gap Law. J. Phys. Chem. A 2010, DOI: 10.1021/jp1072796.

47

Jimenez, R.; Fleming, G. R.; Kumar, P. V.; Maroncelli, M. Femtosecond Solvation Dynamics of Water. Nature 1994, DOI: 10.1038/369471a0.

48

Horng, M. L.; Gardecki, J. A.; Papazyan, A.; Maroncelli, M. Subpicosecond Measurements of Polar Solvation Dynamics – Coumarin 153 Revisited. J. Phys. Chem. 1995, DOI: 10.1021/j100048a004.

49

Gumy, J. C.; Nicolet, O.; Vauthey, E. Investigation of the Solvation Dynamics of an Organic Dye in Polar Solvents Using the Femtosecond Transient Grating Technique. J. Phys. Chem. A 1999, DOI: 10.1021/jp992265+.

50

Maroncelli, M.; Kumar, V. P.; Papazyan, A. A Simple Interpretation of Polar Solvation Dynamics. J. Phys. Chem. 1993, DOI: 10.1021/j100103a004.

51

Ivanov, A. I.; Ionkin, V. N.; Feskov, S. V. Acceleration of the Recombination of Photoexcited Donor-Acceptor Complexes with a High-Frequency Vibrational Mode. Russ. J. Phys. Chem. A 2008, DOI: 10.1134/S0036024408020295.

52

Frauenfelder, H.; Wolynes, P. G. Rate theories and puzzles of hemeprotein kinetics. Science 1985 DOI: 10.1126/science.4012322

53

Ivanov, A. I.; Belikeev, F. N.; Fedunov, R. G.; Vauthey, E. The Effect of Excitation Pulse Carrier Frequency on Ultrafast Charge Recombination Dynamics of Excited Donor-Acceptor Complexes. Chem. Phys. Lett. 2003, DOI: 10.1016/S0009-2614(03)00363-4.

54

Ionkin, V. N.; Ivanov, A. I. Independence of the Rate of the Hot Charge Recombination in Excited Donor-Acceptor Complexes from the Spectral Density of High-Frequency Vibrations. Chem. Phys. 2009, DOI: 10.1016/j.chemphys.2009.04.024.

55

Gladkikh, V.; Burshtein, A. I.; Feskov, S. V.; Ivanov, A. I.; Vauthey, E. Hot Recombination of Photogenerated Ion Pairs. J. Chem. Phys. 2005, DOI: 10.1063/1.2140279.

23



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

56

Page 24 of 36

Najbar, J.; Tachiya, M. Potential Energy Surfaces for Electron Transfer in a Supramolecular Triad System A1-D-A2 in a Polar Solvent. J. Phys. Chem. 1994, DOI: 10.1021/j100052a033.

57

Cho, M.; Silbey, R. J. Nonequilibrium Photoinduced Electron Transfer. J. Chem. Phys. 1995, DOI: 10.1063/1.470094.

58

Ando, K.; Sumi, H. Nonequilibrium Oscillatory Electron Transfer in Bacterial Photosynthesis. J. Phys. Chem. B 1998, DOI: 10.1021/jp982659l.

59

Fedunov, R. G.; Ivanov, A. I. Effect of the Excitation Pulse Frequency on the Ultrafast Photoinduced Electron Transfer Dynamics. J. Chem. Phys. 2005, DOI: 10.1063/1.1844371.

60

Khokhlova, S. S.; Mikhailova, V. A.; Ivanov, A. I. Three-centered Model of Ultrafast Photoinduced Charge Transfer: Continuum Dielectric Approach. J. Chem. Phys. 2006, DOI: 10.1063/1.2178810.

61

Gould, I. R.; Young, R. H.; Moody, R. E.; Farid, S. Contact and Solvent-Separated Geminate Radical Ion Pairs in Electron-Transfer Photochemistry. J. Phys. Chem. 1991, DOI: 10.1021/j100158a031.

62

Ivanov, A. I.; Potovoi, V. V. Theory of Non-Thermal Electron Transfer. Chem. Phys. 1999, DOI: 10.1016/S0301-0104(99)00197-4.

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Figure captions Figure 1. The scheme of the electronic states involved in photoinduced electron transfer with a few vibrational sublevels. The vertical arrows indicate the photoexcitation at different wavelengths, the vertical wavy lines visualize the intramolecular vibrational relaxation/redistribution, the solvent relaxation and electronic transition are shown with short arrows. The filled red and blue bells visualize a part of the ground state population transferred to sublevels of the locally excited electronic state. The gray bells point out the locations of the initially created bells some time later which they have occupied due to electronic transitions, vibrational and the solvent relaxation. Figure 2. Simulation of the dependence of the forward electron transfer rate constants, ∗ kFET and kFET , on the free energy gap, ∆GCS . The experimental data are pictured with

signs + and ×. Red and blue colors stand for initial excitation of the ground, n = 0, and the first excited, n = 1, vibrational levels of LES, correspondingly. The parameters are: Erv CS = 0.2 eV, Erm = 0.65 eV, Ve = 0.025 eV, Vg = 0.030 eV, τv = 1.0 ps. The numbers in the

Figure correspond to the numbers of the donor-acceptor complexes listed in the text. Figure 3. Dependence of the effect of vibrational mode excitation on the free energy gap, ∆GCS . The experimental data are pictured with the signs + and ×. The numbers of the complexes are also indicated. The parameters are the same as in Figure 2 caption. Figure 4. Semi-logarithmic plot of the LES population, PLES (t), as a function of time for several values of the driving force, ∆GCS . (1) −1.2 eV, (2) −0.8 eV, (3) −0.4 eV, τv = 1.0 ps. The rest parameters are the same as in Figure 2 caption. Solid and dashed lines stand for initial excitation of the ground, n = 0, and the first excited, n = 1, vibrational levels of LES, correspondingly. Figure 5. Fitting of the dependence of the backward electron transfer rate constants, ∗ kBET and kBET , on the free energy gap, ∆GCR , to the experimental data. The experimental

data are pictured with signs + and ×. Red and blue colors stand for initial excitation of the ground, n = 0, and the first excited, n = 1, vibrational levels of LES, correspondingly. The CS = 0.65 eV, Vg = 0.030 eV, Ve = 0.025 eV, τv = 1.0 ps. parameters are: Erv = 0.2 eV, Erm

Figure 6. Dependence of the effect of vibrational mode excitation for back electron transfer, χb , on the free energy gap, ∆GCR . The experimental data are pictured with signs + and ×. The numbers in the Figure correspond to the numbers of the donor-acceptor complexes listed in the text. The parameters are the same as in Figure 5 caption. 25



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Page 26 of 36

Figure 7. Dependence of the charge recombination rate constant, kb (t), on time for several values of the driving force, ∆GCR . (1) −0.823 eV, (2) −1.028 eV, (3) −1.130 eV, (4) −1.310 eV, τv = 1.0 ps. The rest parameters are the same as in Figure 5 caption. Solid and dashed lines stand for initial excitation of the ground, n = 0, and the first excited, n = 1, vibrational levels of LES, correspondingly. Figure 8. Influence of the vibrational relaxation time on the effect of vibrational mode excitation for the forward electron transfer, χf . (1) τv = 0.2 ps, (2) τv = 1.0 ps, and (3) τv = 5.0 ps. The rest parameters are the same as in Figure 2 caption. Figure 9. Influence of the vibrational relaxation time on the effect of vibrational mode excitation for the backward electron transfer, χb . (1) τv = 0.2 ps, (2) τv = 1.0 ps, and (3) τv = 5.0 ps. The rest parameters are the same as in Figure 5 caption.

26


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U

(1)

ULES (0)

U

LES

Free energy

Page 27 of 36

(1)

U

CS

(0)

(2)

U

UGS

CS

(1)

U

GS (0)

UGS

Reaction coordinate

FIG. 1. Nazarov, Barykov, Ivanov

27


Q


10

(6)

-1

kFET, ps

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(5)

n=1

(4) (1)

n=0

(3)

1 -1.2

-1

-0.8

∆GCS, eV


28


-0.6

(2) -0.4

Page 29 of 36

0.8 0.6 0.4

χf

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(6)

0.2 0

(3)

-0.2

(5)

(2)

(4)

-0.4 (1)

-0.6 -1.2

-1

-0.8

∆GCS, eV


29


-0.6

-0.4


1

(3)

PLES

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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(2) 0.1

(1)

0

0.5

1

t, ps


30


1.5

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10 n=1 (8)

kBET, ps-1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


n=0 (4) 1

(5) (6)

(1)

(3) (2)

0.1 -0.8

-1

-1.2

∆GCR, eV


31


-1.4

-1.6


(1)

0.2 0

χb

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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(5) (6)

-0.2

(4)

(3)

(2)

-0.4 -0.6

(8) -0.8

-1

-1.2

∆GCR, eV


32


-1.4

-1.6

Page 33 of 36

(1) 4 (2) 3

kb, ps-1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(3) 2 (4)

1 0 0.5

1

1.5

t, ps


33


2


1.6 (3) 1.2 0.8

(2)

χf

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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0.4 (1) 0 -0.4 -1.2

-1

-0.8

∆GCR, eV


34


-0.6

-0.4

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0.4 (3) (2)

0.2

χb

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0

(1)

-0.2

-0.4 -0.8

-1

-1.2

∆GCR, eV


35


-1.4

-1.6


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Effect of Intramolecular High-Frequency Vibrational Mode Excitation

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