Non-equilibrium effects in ultrafast photoinduced ...

Journal of Photochemistry and Photobiology C: Photochemistry Reviews 29 (2016) 48–72

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Journal of Photochemistry and Photobiology C: Photochemistry Reviews journal homepage: www.elsevier.com/locate/jphotochemrev

Review

Non-equilibrium effects in ultrafast photoinduced charge transfer kinetics Serguei V. Feskov, Valentina A. Mikhailova, Anatoly I. Ivanov ∗ Volgograd State University, University Avenue 100, Volgograd 400062, Russia

a r t i c l e

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Article history: Received 6 June 2016 Received in revised form 17 October 2016 Accepted 8 November 2016 Available online 22 November 2016 Keywords: Higher excited states Intramolecular reorganization Solvent relaxation Stochastic point-transition model

a b s t r a c t Modern laser-based spectroscopy has provided methods for detection ultrafast photochemical transformations occurring on the timescale of intramolecular and solvent reorganization. Such processes usually proceed in non-equilibrium regime, in parallel with nuclear relaxation, and often manifest strong deviations from the Kasha–Vavilov rule. In particular, they offer a possibility to control the yield of photoinduced electron transfer (ET) by using different excitation wavelengths. In the last decade the non-equilibrium charge transfer (CT) processes have attracted considerable interest from the scientific community due to their determining role in photosynthesis, dye-sensitized solar cells and various molecular electronic devices. Non-equilibrium of nuclear (intramolecular and solvent) degrees of freedom can be created by a pump pulse or by photoreaction itself at some of its stages. In this review both situations are considered and illustrated by examples in which non-equilibrium effects are pronounced. It is shown that ultrafast charge recombination in photoexcited donor–acceptor complexes and photochemical processes in donor–acceptor1 –acceptor2 molecular compounds proceed predominantly in non-equilibrium (hot) regime. It is important that kinetics and product yields of these reactions demonstrate regularities that considerably differ from that observed in thermal reactions. Among them, the lack of the Marcus normal region in the free energy gap law for charge recombination of the excited donor–acceptor complexes, extremely low quantum yields of the thermalized charge separated states in ultrafast CT from the second excited state of the donor are most known. Although there have been many efforts to clarify microscopic mechanisms of non-equilibrium photoreactions by using ultrafast time-resolved spectroscopy techniques, control of the rate and efficiency of photoinduced charge transfer reactions is still an open challenge. One of the most important applications here is a suppression of ultrafast charge recombination in CT systems, formed either by direct optical excitation or by the preceding ET step. In these systems charge recombination is often regarded as undesirable process, leading to the loss of energy and selectivity of photoreaction. In this review some strategies of ultrafast charge recombination suppression are discussed. The non-equilibrium effects are interpreted from a unified point of view in context of the multichannel point-transition stochastic model. This approach demonstrates similarities and differences in ET mechanisms in various donor–acceptor molecular systems and allows formulating general regularities inherent to these phenomena. We believe that new advances in this research area will not only help to discover new fundamental information about these regularities, but will also have impact on many emerging technologies where ultrafast CT plays the central role. © 2016 Published by Elsevier B.V.

Contents 1. 2.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 Models of electron transfer in polar solvents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 2.1. Effects of intramolecular vibrational modes in equilibrium and non-equilibrium ET kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

∗ Corresponding author. E-mail address: [email protected] (A.I. Ivanov). http://dx.doi.org/10.1016/j.jphotochemrev.2016.11.001 1389-5567/© 2016 Published by Elsevier B.V.

S.V. Feskov et al. / Journal of Photochemistry and Photobiology C: Photochemistry Reviews 29 (2016) 48–72

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2.2. Stochastic model of ultrafast photoinduced CT in multi-mode polar solvents. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .53 Non-equilibrium effects in charge recombination of the excited donor–acceptor complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.1. The spectral effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.2. Free energy gap law in charge recombination kinetics of excited DACs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.3. Dynamic solvent effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.4. Resume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 Charge recombination in excited donor–acceptor complexes with two charge transfer absorption bands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.1. Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.2. Determination of electronic transition parameters from the absorption spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.3. Fitting the CR kinetics in excited donor–acceptor complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.4. Resume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 Effect of intramolecular high-frequency vibrational mode on ultrafast photoinduced CT and CR kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 5.1. Effect of intramolecular high-frequency vibrational mode excitation on the CT kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 5.2. The mechanism of the effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 5.3. Influence of the vibrational relaxation time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 5.4. Resume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 Ultrafast kinetics of S1 , S2 , and charge separated state populations in electron transfer quenching of the S2 state of directly linked Zn–porphyrin–imide dyads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 6.1. Similarities and differences between ET quenching of the S1 and S2 states. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .60 6.2. Fitting to ultrafast kinetics of S1 and S2 state populations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 6.3. Role of reorganization of high-frequency intramolecular vibrational modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 6.4. Kinetics of the charge separated state population . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 6.5. Influence of the dyad geometry on the quantum yield of ultrafast charge separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 6.6. Resume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 Fitting to the two hump kinetic curve of the charge separated state population in rigid dyads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 7.1. Modeling the charge separation and recombination kinetics in rigid dyads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 7.2. What factors control the width of the peak in the femtosecond time interval? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 7.3. Resume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 Intramolecular charge separation from the second excited state: suppression of hot charge recombination by electron transfer to the secondary acceptor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 8.1. The model and the mechanism of ultrafast CR suppression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 8.2. Influence of D–A1 –A2 system parameters on the hot CS quantum yield . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 8.3. Role of spatial geometry of the triad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 8.4. Resume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 Conclusion and perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

S. Feskov is currently a Professor at the Institute of Mathematics and Information Technologies, Volgograd State University (VolSU). After receiving his master degree in Radiophysics at VolSU he joined the group of Prof. A.I. Ivanov (1995) to study magnetic effects in spin-selective charge transfer reactions and kinetics of ultrafast processes in photoexcited molecular systems. He obtained PhD (2000) and habilitation (2012) degrees in Chemical Physics at VolSU. From 2006 to 2009 he joined several times the group of Prof. A.I. Burshtein at The Weizmann Institute of Science (Rehovot, Israel) as a visiting researcher. Besides photochemistry his research interests include numerical simulations, software development and high-performance computing.

A. Ivanov is Professor of physics at Volgograd State University. He graduated from Bashkirian State University, department of physics (Ufa, Russia) in 1973. Before joining the Volgograd state University in 1982 he worked in Institute of chemistry of Russian academy of science (Ufa) as a researcher where he received his PhD in physics. He obtained habilitation degrees in Chemical Physics from Physical-Technical Institute of the Russian Academy of Sciences. His first scientific topics were theory of elementary act of chemical reactions, nonradiative transitions in polyatomic molecules, and electron capture by polyatomic molecules in gas phase. Since the mid-nineties his interests evolved in the direction of the theory of ultrafast charge transfer in solutions.

1. Introduction V. Mikhailova graduated from the Department of Physics of Volgograd State University (Volgograd, Russia) in 1985, got her candidate degree at the Physical-Technical Institute named after E. K. Zavoisky of the Kazan Scientific Center of the Russian Academy of Sciences (1992), specialized in Chemical Physics, and doctoral degree (1992) at the Institute of Chemical Physics named after N. N. Semenov of the Russian Academy of Sciences (Moscow, Russia). Her doctoral thesis was devoted to the development of nonstationary models of photoinduced electron transfer reactions in condensed media. Her research interests are related to the theory of electron transfer in molecular donor–acceptor systems. During recent years, her interests included studies of the dynamics of ultrafast photochemical processes in such systems placed in liquid solutions. At the moment, she is a Head of the Department of Theoretical Physics and wave processes at Volgograd State University.

Charge transfer (CT) is one of the most widespread chemical reaction, and its study is essential in chemistry [1–8], physics [9], biology [1,3,10], molecular electronics [11,12], and other areas. It is important both to fundamental science and to diverse technological applications [11]. Currently several promising applications in the area of molecular electronics are actively developed. Among them are molecular sensors, photonics, electrocatalysis, solar energy conversion and storage [13–17]. Despite vast advances in study of CT by both experimental and theoretical methods, some problems still exist in understanding its detailed mechanism and controlling its efficiency. One of the problems is a large number of regimes in which charge transport

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Fig. 1. Cuts of the free energy surfaces of the ET reactant state (black) and the ET product state at different values of the reaction free energy gap, −G, corresponding to the Marcus normal region (red), barrierless reaction (green), and the Marcus inverted region (blue). The initial equilibrium distribution of particles on the reactant surface is shown with gray dashed line.

can occur. In various regimes dependencies of the CT rate on the controlling parameters can strongly differ, and be even opposite. This creates difficulties in the CT control. As an example we point out a modern family of photovoltaic devices, dye-sensitized solar cells, in which ultrafast charge recombination considerably limits their efficiency [12]. Obviously, to design a targeted search for ways of its control it is necessary to know in which regime such reaction proceeds and to understand its mechanism. Modern era of electron transfer studies started in 1956 when Rudolph Marcus presented his theory of electron transfer reactions in polar solutions. This triggered a large amount of experimental and theoretical explorations of the problem. The Marcus theory connected the ET rate, kET , with two main parameters of the reaction — the free energy change, G, and the reorganization energy of the surrounding medium (polar solvent), Erm , as follows [18]

kET = A exp

(G + Erm ) − 4Erm kB T

2

(1)

where A is the pre-exponential factor, T is the bath temperature, and kB is the Boltzmann constant. The exponent in Eq. (1) is the usual Arrhenius factor but with activation energy quadratic in the free energy gap G. This is actually the famous Marcus free energy gap (FEG) law for ET reactions in polar solvents. According to this law, the reaction is slow in regions of strong and weak exergonicity, but fast in between, reaching maximum at G =− Erm . In other words, the ET rate demonstrates a bell-shaped FEG dependence. In the weak exergonicity region, −G < Erm , the rate increases with growing −G. This region was called Marcus normal region (MNR) (see Fig. 1). But in the strong exergonic region, −G > Erm , the inverse trend is predicted. It is the so-called Marcus inverted region (MIR). Decrease of the ET reaction rate with growing exergonicity, −G, in the inverted region was the most surprising for chemists. That time linear FEG dependencies for chemical rates were prevailing [19–21] which led to expectation of monotonic rise of kET with −G. Series of experiments by Rehm and Weller on bimolecular ET fluorescence quenching in various donor–acceptor systems strongly enhanced the doubt on the existence of MIR [22]. These experiments have demonstrated that the rate of bimolecular photoinduced charge separation (CS) increases with the reaction exergonicity up to the values corresponding to the reactant

diffusion rate, and remains diffusion-limited even at −G > Erm , where inversion is predicted. Only in the mid-80s experiments on intramolecular charge shift reactions appeared that straightforwardly confirmed not only existence of the MIR but also a bell-shaped free energy gap dependence representing the entire Marcus FEG law [23,24]. The latter has been shown to be inherent not only to intramolecular charge shift, but also to intramolecular charge recombination (CR) [25], intermolecular CR [26–28], intermolecular charge shift [29,30], intramolecular charge separation [31], and charge separation in hydrogen-bonded complexes [32]. Recently it was discovered that intrinsic, diffusion free, rate constant of bimolecular photoinduced electron transfer reactions obtained from the fluorescence quenching experimental data clearly demonstrates presence of both the normal and inverted regions [33]. However, geminate CR in the excited donor–acceptor complexes (DACs) is one of the reactions for which the bell-shaped FEG dependence was not observed so far. These reactions show rather unexpected dependence of the rate constant on the reaction free energy gap: logarithm of the CR rate constant increases monotonically, almost linearly, with decreasing −G. In other words, the Marcus normal region in these reactions is absent [34,35]. Consider now ultrafast CR in DACs in more detail. This reaction proceeds according to the scheme: D+ A−

ϕ

−→

D+ + A−

h ↑↓ kCR DA Photoexcitation of the ground-state DAC, (DA), by a short laser pulse in the CT band (h) leads to formation of the contact radical-ion pair, (D+ A− ), and triggers two competing processes: intermolecular CR with the rate kCR and formation of free ions in the bulk with the quantum yield ϕ. Although geminate CR in excited DACs is one of the simplest chemical reactions, its mechanism is still not completely clarified. A few mechanisms have been proposed to explain the absence of MNR in excited DACs. According to Ref. [35], CR kinetics cannot be explained in the framework of Marcus nonadiabatic theory and, hence, alternative mechanisms should be invoked. It was supposed that CR in excited DACs is controlled by reorganization of intramolecular and intracomplex vibrational high-frequency modes, so that solvent plays a minor role here [35]. This means that the theory of nonradiative transitions in polyatomic molecules predicting nearly linear dependence of the logarithm of the rate constant on the reaction driving force is applicable for description of the CR kinetics. Another explanation for the absence of the MNR is connected with non-equilibrium initial vibrational state of the DAC formed by a short excitation pulse [36]. This explanation however encountered two problems [37]: (i) too large electronic couplings are needed to get a good fit; (ii) the model predicts strong temporal dependence of the CR rate constant in contrast to experimental data [35,38–40]. Later the adiabatic model of CR in excited DACs has been proposed [37] assuming that electronic coupling between the donor and acceptor compounds is strong and, hence, CR proceeds as a transition between the excited and ground adiabatic states. Suggesting that the reaction proceeds in stationary regime, the experimentally observed free energy dependence of the rate constant was reproduced in Ref. [37]. The weakness of this interpretation is that typical solvent relaxation time is of the same order or even longer than that of CR for reactions in the area of low exergonicity and, therefore, the model has to include an explicit description of nuclear relaxation [41].


The generalized model including description of relaxation of slow solvent modes also predicts rather strong time dependence of the CR rate constant [41]. Here adiabatic states of a DAC, ± , are presented by linear combinations of the ground (neutral) and the excited (ionic) states [37] ± = c1± (AD) + c2± (A− D+ )

(2)

where the diabatic states, (AD) and (A− D+ ), are pure neutral and pure radical-ion pair states, respectively. Ionicity of the upper adiabatic state is thus determined as |c2+ |2 . In DAC, short excitation pulse transfers the system vertically to the upper adiabatic term. As a result, the initial position of the system after photoexcitation is far from the upper term minimum and ionicity is close to unity. Solvent relaxation brings the system to the term minimum where ionicity is close to 0.5. This result is in contrast with experimental data showing gradual increase of the excited DAC ionicity with time [42]. Another obvious drawback of the adiabatic and nonstationary models is a complete disregard of reorganization of intramolecular high-frequency vibrational modes, in contrast to the mechanism proposed in Ref. [35]. It is well known that active intramolecular vibrations can strongly increase the rate of both highly exergonic thermal electron transfer [43–47] and weakly exergonic non-equilibrium CR [47–49]. Besides, quantitative description of CR kinetics requires consideration of real spectrum [50,51] which typically involves 5–10 active high-frequency vibrational modes. Moreover, relaxation of the excited vibrational states can be also significant [49]. The multi-channel stochastic model involving explicit description of non-equilibrium excited state formation by a pump pulse and accounting for reorganization of intramolecular highfrequency vibrational modes, their relaxation as well as relaxation of solvent with several timescales can well reproduce the free energy gap dependence of the CR rate constant [52] observed in experiments [35,40,53]. The model also provides correct description of nonexponential CR kinetics in the excited DACs. The model suggests that most excited DACs recombine at non-equilibrium (non-thermal or hot) stage when the wave packet passes a number of term crossings with vibrationally excited sublevels of the ground electronic state in the area of low and moderate exergonicity. The multi-channel stochastic model can also explain the absence of the MNR in DACs and nearly exponential decay of the excited state population. However, this result is obtained for large electronic couplings when CR is faster than solvent relaxation. For smaller values of electronic couplings this decay involves two stages: (i) fast initial non-equilibrium stage, and (ii) subsequent much slower thermal decay. To the best of our knowledge there is the only report on observation of the two-stepped decay of the excited state population in DACs — comprising of 1,2,4-trimethoxybenzene (TMB) as an electron donor and tetracianoethylene (TCNE) as an electron acceptor in acetonitrile (ACN) [53]. These experimental data as well as nearly exponential decay in slower solvents, valeronitrile (VaCN) and octanonitrile (OcCN), were well reproduced with the multi-channel stochastic model [54]. The problem of non-equilibrium CR in excited DACs is most relevant to the issue of CS from the second excited electronic state of the donor molecule, S2 [31,55–71]. Investigations of both intermolecular [60,61] and intramolecular [59,62,63,67,68] ET from the S2 state invariably show two regularities: (i) the S2 state decay is immediately followed by the S1 state population increase and (ii) the CS state population is considerably smaller than the initial population of the S2 state. These patterns suggest that CR into the first excited state, S1 , is so fast and efficient that the larger part of ionic products recombines at the stage of nuclear relaxation [59,67,72,73].

51

To highlight a profound analogy between CR in excited DACs and CR in photoinduced CS from the S2 state, schemes of electronic transitions starting from the first and the second excited states are shown in Fig. 2. The S2 state populated by a laser pulse decays to produce initially either the S1 state (due to internal conversion) or the charge separated state. At first, the intramolecular degrees of freedom of a molecule in the CS state and the surrounding medium are highly non-equilibrium. Relaxation of the nuclear subsystem and CR into the S1 state proceed here in parallel. Charge separation from S1 after its equilibration can proceed in thermal regime to produce the CS state again. The last can also decay to form the ground state, S0 . This chain of CT reactions results in the two-humped kinetic curve of the CS state population discovered in experiments [67,68]. Another important manifestation of nuclear non-equilibrium in ultrafast photoinduced reactions is the spectral effect — dependence of the CT rate constant on the excitation pulse wavelength. This effect was observed in ultrafast CR of excited DACs and in photoinduced ET reactions in polar and nonpolar solvents [51,53,74,75]. Assuming the CR to proceed from a non-equilibrium state of solvent polarization created by a pumping pulse and relaxation of intramolecular high-frequency vibrational modes to occur at considerably shorter time scales than both solvent relaxation and CR, the spectral effect in excited DACs was rather well described in the framework of the stochastic multi-channel point-transition model [51]. Computer simulations show that ultrafast ET can proceed in various regimes. The most important characteristics determining the regime are the coupling energies between electronic states, the solvent reorganization energy, the environment fluctuation spectrum and the intramolecular reorganization energy for high-frequency vibrational modes. Currently there is no universal method that could allow simulating ET kinetics in all regimes and a broad time window. This has led to creation of a large number of theoretical approaches to solving this problem. The simplest approaches such as the Fermi Golden Rule and the Redfield theory cannot however be used for description of the processes considered here. The Fermi Golden Rule is not applicable for description of ultrafast electron transfer because electronic couplings in such processes are not weak. The Redfield theory can describe electronic transitions properly only if coupling to the environment is significantly weaker than coupling between electronic states. This condition is not met for CT in polar solvents where interaction of the transferred charge with environment is strong. More rigorous methods such as the hierarchy equations of motion [76–79], the multilayer multi-configuration time-dependent Hartree wave packet method [80–82], the quasi-adiabatic path integral method [83,84] are aimed at fully quantum description of the dynamics of coupled electron-vibrational systems with many degrees of freedom. These nonperturbative approaches adequately describe non-trivial and subtle dynamical features of open quantum electron-vibrational systems including long-lasting coherent dynamics observed in photosynthetic antenna complexes [85], mixing electronic and vibronic excitations due to resonance effects [86]. A major limitation of their widespread use is computational expenses that increase greatly when the long-term dynamics are calculated in solvents with several strongly different relaxation times. In this contribution we use the stochastic multi-channel pointtransition model [51]. This model is based on the Marcus concept of parabolic free energy surfaces constructed in space of solvent polarization coordinates. Motion of particles along these surfaces reflects reorganization of solvent and is fully characterized by the solvent relaxation function [87]. Solvent fluctuation frequencies have to be low because they are described classically. Reorganization of intramolecular high-frequency modes is described

52


Fig. 2. Cuts of the free energy surfaces for the first excited electronic state S1 , second excited state S2 , and the charge separated state CSS typical for ET from the S1 state (panel A) and from the S2 state (panel B). Dashed lines are vibrational sublevels of the corresponding electronic states.

quantum-mechanically in terms of vibrational sublevels and Franck–Condon factors. The applicability conditions of the stochastic model are as follows. First limitation is caused by description of reactant and product states in terms of classical distribution functions. Such description ignores quantum coherence and is applicable if the electronic coherence lifetime, c , is much shorter than the reaction time, 1/kET . This condition requires the inequality / 2Erm kB T 1/kET to be fulfilled, where is the Planck constant. This inequality is met when interaction with solvent is strong and the temperature is high. For Erm = 1 eV and the room temperature this limitation leads to estimate kET 1014 s−1 . Second limitation is associated with the use of diabatic basis. This requires the electronic coupling energy, Vel , to be small. Roughly it should meet the condition Vel < kB T [88]. However, it should be noted that reorganization of high-frequency vibrational modes can considerably weaken this requirement. Classical description of solvent relaxation leads to the third limitation which requires the inequality / i kB T to be met for each solvent relaxation mode, where i is the solvent relaxation time. The aim of this review is to analyze the non-equilibrium effects in photoinduced ultrafast CT which have been studied for the last 15 years with emphasis on the past five years. We restrict our attention to the two reaction types, in which nonequilibrium effects have been explored in more details. These reactions are: ultrafast CR processes in excited DACs and photoinduced intramolecular ET reactions from the second locally excited electronic state of the donor. In both reactions non-equilibrium CR plays a central role. The CR is often an unwanted process leading to a loss of selectivity of desirable reactions [67] or an energy waste [12]. Here we discuss the methods of suppression of ultrafast CR. This review do not touch a large area of bimolecular charge transfer reactions in solution. The ET fluorescence quenching can also proceed as ultrafast reaction in non-equilibrium regime [4,5,8,89–93]. Spatial diffusion of reactants and products in viscous solvents brings a lot of new aspects to the bimolecular CT kinetics.

2. Models of electron transfer in polar solvents In this section we summarize development of theoretical approaches to ET description. Most important achievements of ET theory are briefly described and the general stochastic model of photoinduced electron transfer is presented. 2.1. Effects of intramolecular vibrational modes in equilibrium and non-equilibrium ET kinetics Original Marcus theory described solvent reorganization during ET in terms of classical mechanics. Later the ET theory was modified to account for solvent reorganization at quantum level in the Fermi Golden Rule limit [94]. Here the theory of radiationless transitions [95–97] was used to calculate the ET rate constant. The approach employed essentially the model of vibrational oscillators of a solid. Afterward a general quantum mechanical expression for the ET rate constant in terms of dielectric spectral density of solvent and intramolecular vibrational modes was derived [98]. Further theoretical studies uncovered importance of highfrequency intramolecular vibrations in ET by Dogonadze and Kuznetsov [99], Efrima and Bixon [100], Jortner and co-workers [101,102] and Ulstrup [103]. It was shown that reorganization of intramolecular high-frequency modes can strongly increase the ET rate in the inverted region due to reduction of the activation energy. Development of the stochastic model [87] opened the way beyond the Golden rule limit. This model predicted dynamic solvent effect that restricted the maximum ET rate by the reciprocal solvent relaxation time. In the solvent-controlled regime a saturation of the ET rate on the electronic coupling was predicted. Although initial experimental studies [104–106] supported this conclusion, there are still relatively few systems in which solvent control has been observed [107,108]. Despite this, ET rate constants exceeding the solvent control limit, 1/ L , by two and more orders of magnitude have been reported [109–111]. These results required development of new models that could explain the contradiction. One possible explanation was suggested in papers [112–115]. The stochastic model was modified to account for reorganization of ultrafast solvation and


intramolecular high-frequency (but still classical) modes. The modification led to widening the Zusman ı-sinks. This model does not demonstrate saturation of the rate constant with increasing electronic coupling and, thus, increases considerably the upper ET rate limit, but even so it strongly underestimates ET rate in the Marcus inverted region [116–118]. One more important advantage of the model is its prediction of nonexponential kinetics [112–115,119–126], observed in experiments [110]. Another important modification of the stochastic model was involving reorganization of intramolecular high-frequency vibrational modes [47,48,127,128]. These modes were recognized to play crucial role in speeding up the ET rate in the Marcus inverted region [3,43,129]. Later a hybrid model, including a high-frequency quantum vibrational mode, a low-frequency classical vibrational mode, and a slow solvent polarization mode, has combined the ideas of Refs. [43,112–115] with those developed to describe ET occurring in nonequilibrium regime [130–135]. In the hybrid model, the ET rate constant depends on the solvation coordinate X(t) as follows [44]

kET =

2 e−S Sn (X) G0→n 2 2 −1/2 V (4Erlf kB T ) exp − el n! 4Erlf kB T

(3)

n

Here Vel is the electronic coupling energy, Erlf is the reorganization energy of the low-frequency vibrational modes, which are treated classically, S is the Huang–Rhys factor equal to S = Erhf /(), where Erhf and are the reorganization energy and the average frequency of high-frequency vibrational modes coupled to the ET, G0→n (X) = GET + n + (1 − 2X(t))Ers

(4)

where GET is the ET free energy, Ers is the solvent reorganization energy. The hybrid model was able to describe large rate constants observed in the inverted region [44]. It was also successfully used to explain the absence of the Marcus normal region in excited donor–acceptor complexes [40]. The hybrid model is based on the time-dependent Golden Rule that considerably limits the region of its applicability. In particular, it is valid only when the total probability of non-thermal transition through all the sinks, which the wave packet has gone by during its relaxation, is small. The generalized stochastic model [51] involves all the elements of the hybrid model, but its applicability is broader, since the generalized stochastic model can describe non-equilibrium (hot) electronic transitions with large probabilities [135]. 2.2. Stochastic model of ultrafast photoinduced CT in multi-mode polar solvents Now we represent a general model of photoinduced ET reaction in a donor–acceptor system placed in a viscous polar medium. First we introduce a set of quantum states |s (s = 1, . . ., S) that correspond to different locations of the transferred electron within the molecular system. In the case of intermolecular ET, these states are usually associated with the reacting particles (electron donor(s) and acceptor(s)). For intramolecular ET, the |s states are generally attributed to different centers within a single supramolecule. The CT reaction of the type |s |s commonly involves rearrangement of polar solvent around the reacting particles. This rearrangement is quantitatively described by the solvent reorganization energy Erm . For real molecular systems in real solvents, Erm values are difficult to calculate from the first principles, since they are determined not only by the solvent polarization properties, but the spatial distributions of charges in the reactant and product states as well. Some quantitative approaches to the problem are however known (see Refs. [1,18,136–147]). In our analysis, presented below in this report, the Erm values are generally considered

53

as unknown quantities. These values, however, in some cases can be determined indirectly as a result of fitting to the experimental data. We consider here primarily photoinduced ET processes, where electronic and vibrational excited states of the reactants and products play a significant role. To take these states into account, we assume hereafter that electron-vibrational excitation of the molecular system does not in general produce significant redistribution of the electronic density. This assumption however relates to the locally excited electronic states only, but does not apply to DACs, where excitation of the CT band results in transfer of an electron between the reactant molecules. In the case of DACs, ground and excited states of the molecular system are considered as the reactant and product states, corresponding to different s. Local excitation of a molecular system thus does not cause reorganization of the surrounding solvent. In what follows we assign a quantum number k (k = 0, 1, . . .) to each locally excited electronic substate of the |s state. The electronic state of the system is thus described by a pair of integers s = (s, k). Vibrational excitation of the molecular system is described by the vector of vibrational quantum = {n1 , n2 , . . ., nM }. These numbers correspond to the numbers n high-frequency intramolecular vibrational modes with frequencies ˛ (˛ = 1, . . ., M). The free energy surfaces (FESs) for the electronic states | s = |s, k are usually written in terms of reaction coordinates Qi (i = 1, . . ., N), which are generally associated with solvent relaxation modes. We consider here the general case of multi-mode polar solvent with N relaxation times i (i = 1, 2, . . ., N), that is, a solvent with relaxation function [148–151] N

X(t) =

xi e−t/i

(5)

i=1

Here xi is the weight of the i-th component of the relaxation function. Within the linear response approximation [18] the diabatic FESs are harmonic in the Qi coordinates. They can be written in the form ) (n

1 (s) 2 (Qi − Q˜ i ) + n˛ ˛ + G s 2

=

U s

N

M

i=1

˛=1

(6)

) (s) (n Here Q˜ i are the coordinates of the U s FES minimum. It follows from Eq. (6) that arrangement of the minima positions for the reactant (s = 1) and product (s = 2) FES in the Qi coordinate space relates to the solvent reorganization energy Erm as

1 ˜ (1) (2) 2 (Qi − Q˜ i ) 2 N

Erm =

(7)

i=1

The quantity G s in Eq. (6) is the free energy of the | s electronic state. The free energy gap between the | s and | s electronic states is then calculated as G s s = G s − G s (Fig. 3). Temporal evolution of the photochemical system is described by the set of differential equations for the probability distribution ) (n functions s (Q , t) [87,72,73,152,153]

∂ s(n) ∂t −

) (n = Lˆ s s −

2 2 (n ,n ) ) ) ) ) (n (n (n (n V s, s F × ı(U s − U s )( s − s ) s = / s

1 (k,s)

ic

1

) (n

s ın ,0 +

(k+1,s)

ic

n

(0)

k+1,s ın ,n ∗ −

(n ) s (˛,n˛ )

˛

v

∗

+

(n ˛ ) s

(˛,n˛ +1)

˛

v

(8)

54


(0)

(0)

Fig. 3. The free energy surfaces of the ground U1,0 (blue) and the locally excited U1,1 (0) U2,0

(red) electronic states of ET reactants, and the ground electronic state (green) of ET product. Vibrational sublevels of the electronic states are not shown.

where s = (s, k), s = (s , k ), s = / s. First term in the righthand side ) (n of Eq. (8) describes diffusion in the parabolic potential U s (Q ). The Smoluchowski operator Lˆ s is written in the form Lˆ s =

N 1 i=1

2

∂ ( s) ∂ 1 + (Qi − Q˜ i ) + kB T ∂Qi ∂Qi2

i

(9)

| s , n The electronic-vibrational transitions of the type | s, n between the states with different localization of the electron are described by the second term in Eq. (8). We invoke here the Zusman approximation [87] about the ı-localization of these electronic | s , n transitions is transitions. The intrinsic rate of the | s, n determined by the electronic coupling energy V s, s (s = / s ) and the Franck–Condon factor F

,n ) (n

=

e

−S˛

n˛ !n˛ !

˛

n˛ +n˛ −2l 2 L (−1)n˛ −l ( S˛ ) l=0

l!(n˛ − l)!(n˛ − l)!

(10)

In this equation L = min(n˛ , n˛ ), S˛ = Erv˛ / ˛ is the Huang–Rhys electron-vibrational coupling parameter, Erv˛ is the reorganization energy of the ˛-th quantum intramolecular vibrational mode. The remaining terms in Eq. (8) describe internal conversion |s, (k,s) k + 1 |s, k with the time constant ic and relaxation of highfrequency vibrational modes n˛ → n˛ − 1 with the rate 1/v(n˛˛ ) , (1)

where v(n˛˛ ) = v˛ /n˛ [154]. Initial conditions for Eq. (8) depend on details of the photoexcited state preparation and are specified separately for each case considered below. Together with the basic equations they determine mathematical formulation of the problem. Equations of motion were solved numerically using the recrossing algorithms of the Brownian simulation method [48,51]. The time-dependent populations of all electronic states of interest were calculated with the equation P s (t) =

) (n s (Q , t) dQ

(11)

n

This multi-channel point-transition stochastic model describe kinetics of ultrafast CT reactions in which non-equilibrium is created by the pumping pulse and by separate stages of the reaction itself.

Fig. 4. Multi-channel non-equilibrium CR in DACs resulting in excitation of intramolecular high-frequency vibrations. Red vertical arrows show vibrational relaxation, blue and black arrows — solvent relaxation.

3. Non-equilibrium effects in charge recombination of the excited donor–acceptor complexes Ultrafast kinetics of CR in excited DACs show peculiarities which can be attributed to non-equilibrium of solvent and intramolecular nuclear degrees of freedom. Among them is the spectral effect, that is the dependence of the CR rate on the excitation pulse carrier frequency [51,53]. Physical origin of this effect is simple: with different excitation wavelengths, the wave-packets produced by photoexcitation are formed at different positions on the excited state free energy surface, so it takes different time for these packets to reach the CR reaction zone. This time delay results in different effective reaction rate constants. Another manifestation of the nuclear non-equilibrium is nonexponentiality of the CR kinetics, that reflects the wave-packet motion to the FES minimum [51]. Absence of the MNR can also be interpreted as a manifestation of non-equilibrium created by the pumping pulse [35,53,40,52]. In this section we discuss these effects and their reflection in CR kinetics of DACs in polar liquids. The CR processes in DACs are typically described in terms of two electronic states pictured in Fig. 4. It is assumed that the system is initially in the ground neutral state | s = 1, k = 0 =|Gr with nuclear coordinates distributed according to the Boltzmann law. Short laser pulse with duration e (much shorter than relaxation times of the solvent i (i = 1, . . ., N)) transports the system to the charge separated state | s = 2, k = 0 =|CS. If carrier frequency of the pump pulse is close to the red edge of the CT band then only the ground state of intramolecular high-frequency vibrational modes is populated. If however laser pulse frequency corresponds to the CT band maximum, the high-frequency vibrational modes can be excited. The intramolecular vibrational redistribution often proceeds much faster than CR, therefore it is often supposed to proceed = 0). from the vibrational ground state (n 3.1. The spectral effect Studies of ultrafast CT in donor–acceptor molecular systems indicate an important role of nuclear non-equilibrium in such processes [36,40,44,47,135]. Direct evidence of non-equilibrium nature of CR in DACs is a dependence of the effective CR rate on the excitation pulse carrier frequency. Both experimental and theoretical studies have shown the spectral effect to be relatively small but reliably observable [41,51,53,74]. Experimental measurements were performed for DACs consisting of hexamethylbenzene


Fig. 5. Cuts in the FESs of the ground and the excited states of a DAC. The dashed parabolas represent vibrational excited states of the ground electronic state and CS state. The initial excited state distribution produced by a laser pump is pictured by red and blue colors corresponding to the red and blue edges of the CT band.

(HMB), pentamethylbenzene (PMB), and isodurene (IDU) as electron donors and TCNE as electron acceptor in series of polar solvents [53]. Numerical results related to the spectral effect simulations in these DACs are reported in Ref. [51]. We outline here the results of fitting experimental data [53] to the theory. In simulations, basic ) (n equations (8) for the probability distribution functions Gr (Q , t) , t) in this case were completed with initial conditions and CS (Q [51]: , t = 0) = Z −1 CS (Q

S n

˛

˛

e−S˛

n˛ ! n˛

exp

( n)

−

(0)

2

(UCS − UGr − ωe ) e2 22

(0)

−

UGr

kB T

(12)

˛

The pump pulse was assumed to have a Gaussian form

E(t) = E0 exp

iωe t −

t2 e2

(13)

with duration e to be short enough so that medium is considered to be frozen during excitation. Equation (12) shows the initial distri , t = 0), to be affected both by excitation pulse carrier bution, CS (Q frequency, ωe , and by duration of the pumping pulse, e . This result is illustrated in Fig. 5 where two initial distributions, calculated for different excitation wavelengths 620 nm (red line) and 480 nm (blue line), are shown. Since CR in excited DACs is a reverse process to the CT band excitation, most energetic parameters of the DAC can be obtained from its resonance Raman and stationary absorption spectra. An exception is the electronic coupling, Vel , which cannot be extracted directly from experimental data. This parameter is considered here as the only variable parameter in the fitting. We define the spectral effect here as =

620 − 480 eff eff 480 eff

(14)

where the superscript indicates the excitation wavelength in nm. The effective CR time is determined by the equation

eff =

∞

PCS (t) dt

(15)

0

The kinetics of CT in the IDU/TCNE complex were rather well described while there is some overestimation of the spectral effect. For the PMB/TCNE and HMB/TCNE complexes the solvent effect is correctly reproduced but not the spectral effect. A reason of this

55

Fig. 6. The free energy gap dependence of the CR rate constant, kCR in s−1 , in the excited DACs: 1, Per-TCNE; 2, Py-TCNE; 3, Naph-TCNE; 4, Py-TCNQ; 5, Naph-TCNQ; 6, Per-PMDA; 7, Py-PMDA; 8, Chr-PMDA; 9, Naph-PMDA; 10, Per-PA; 11, An-PA; 12, Py-PA (the experimental data are taken from [35]); 13, TMB-TCNE; 14, HMB-TCNE; 15, PMB-TCNE; 16, IDU-TCNE; 17, TeMB-PDMA; 18, TrMB-PDMA; 20, VER-PDMA; 19, DMB-PDMA; 21, ANI-PDMA (from [40,53]). Solid curves (lines a, b and c) are the results of numerical simulations. The thermal rate constant, kth , is shown as a dashed curve. Vibrational spectrum involving 5 high-frequency modes is employed [160]. Parameters used: Vel = 0.0065 eV, Erm = 0.6 eV (line a); Vel = 0.02 eV (line b), Vel = 0.095 eV (line c), Erm = 0.5 eV, Erv = 0.51 eV.

discrepancy can be connected with ignoring the excitation of intramolecular high-frequency vibrational modes in the model. Recent experimental and theoretical investigations of ultrafast photoinduced CT and CR kinetics [69,156–158] have shown that the effect of intramolecular high-frequency vibrational mode excitation can considerably exceed the spectral effect associated with solvent non-equilibrium [51]. It should be noted that non-equilibrium model provides good correlation between the experimental and calculated dependencies of the CR rate on solvent viscosity. The model also predicts nonexponential decay of the excited state population that can be described by the equation: f (t) = exp

s t −

(16)

Simulations predict an increase of the nonexponentiality factor, s, with decreasing the excitation wavelength and its value larger than unity, s > 1, in full accord with experimental data [51]. 3.2. Free energy gap law in charge recombination kinetics of excited DACs Another important manifestation of solvent non-equilibrium in excited DACs is the lack of the MNR [35,40,53]. Simulations of FEG law for ultrafast charge recombination in DACs confirmed absence of the MNR [52]. Results of simulation for the experimentally accessible region of CR free energies, −2.5 eV kth , is fulfilled. In strongly exothermic region the effective CR rate is close to the thermal one. Deviation of the effective CR rate constant from its thermal value is caused by nonequilibrium nature of the CR. To demonstrate the extent of non-exponentiality of the CR kinetics, time dependencies of the excited-state population decay are shown in Fig. 7. Generally, CR here proceeds in two stages: as fast initial non-equilibrium decay the rate kCR , and slower thermal decays with the rate kth (kCR kth ). This two-stage regime manifests itself most clearly if solvent reorganization energy considerably exceeds reorganization energy of high-frequency vibrational modes (solid lines). Otherwise CR kinetics approaches closely to a single-exponential regime (dashed lines in Fig. 7). Exponential decay of the excited state population is not however a direct evidence of equilibrium CR regime. For example, simulations show that CR in the TMB/TCNE complex (GCR =−0.4 eV, symbol 13 in Fig. 6) proceeds with a single rate constant. Moreover, kCR in this case nearly coincides with kth , but nonetheless CR proceeds due to hot electronic transitions in the course of solvent and intramolecular relaxation [40]. The multi-channel stochastic model predicts nearly linear dependence of the logarithm of CR rate constant on the CR free energy only for sufficiently large electronic couplings that fits well to the series of the complexes studied in Refs. [40,53] (solid line c). Although for DACs with smaller electronic couplings the dependence deviates from the linear one but it still deviates stronger from that predicted for thermal reactions. This argues in favor of the proposed mechanism related to the non-equilibrium solvent state created by the pumping pulse.

Fig. 7. Time dependence of the excited state population kinetics in DACs. The parameters are: GCR =−0.4 eV, Erm = 0.8 eV, Erv = 0.2 eV (solid lines); GCR =−0.56 eV, Erm = 0.5 eV, Erv = 0.51 eV (dashed lines). The numbers near the lines indicate the value of electronic coupling, Vel , in eV.

3.3. Dynamic solvent effect The dynamic solvent effect (DSE) [162] in excited DACs implies the dependence of the CR rate constant on dynamic properties of solvent. This effect is most known in thermal reactions. For example, the rate of thermal ET in Debye solvents is inversely proportional to the longitudinal relaxation time, L . In solvents with two (and more) relaxation times, 1 , 2 , the first one is associated with fast inertial relaxation of the medium, whereas the second with slow diffusion, 1 < 2 . It is well known that 1 is almost independent of the medium viscosity, while 2 is proportional to the viscosity. Therefore DSE should be largely attributed to the influence of diffusion relaxation component, 2 . DSE can be strong in the area of sufficiently large electronic couplings. For thermal ET reactions, this conclusion was confirmed in a series of experiments [104,105,108,163]. It was also shown by numerical simulations [164,165] that reorganization of highfrequency vibrational modes influences DSE only slightly in the Marcus normal region, but suppresses it effectively in the inverted and activationless regions. This picture is just the opposite to ultrafast CR reaction in excited DACs. Here DSE is weak in the normal region, and is strong enough in the activationless and inverted regions. In Fig. 8 the CR rate dependencies on GCR in solvents with 2 varying in the range from 0.5 ps to 5 ps are displayed. The mechanism of this result is the following. ET coupling to the high-frequency mode opens up new CR channels (Fig. 9). In the area of strong exergonicity, −GCR > Erm + Erv , these new sinks are located in vicinity of UGr minimum. In this case the CR rate is largely controlled by diffusive delivery of the wave packet to the CR zone and thus depends strongly on 2 . On he other hand, for weakly exergonic CR (in the normal region) the wave packet appears immediately in zone of CR sinks. As a result DSE here is weak or completely absent. 3.4. Resume The majority of experimentally observed manifestations of nuclear non-equilibrium in CR kinetics in the excited DACs (the spectral effect, the absence of MNR, the non-exponential decay, the dynamic solvent effect), can be explained and quantitatively reproduced within the two electronic level model taking into account


57

Fig. 10. Cuts of FESs in DACs with two charge transfer bands, CT1 and CT2 (vertical red and blue arrows).

Fig. 8. The free energy dependence of the CR rate constant, kCR in s−1 . Parameters used: Erm = 0.5 eV, Erv = 0.51 eV; Vel = 0.07 eV (line a), Vel = 0.12 eV (line b), 2 = 0.5 ps (solid lines a and b), 2 = 4.9 ps (dashed lines a and b).

the reorganization of intramolecular high-frequency vibrational modes. 4. Charge recombination in excited donor–acceptor complexes with two charge transfer absorption bands In this section we consider CR kinetics in a large group of DACs showing two CT absorption bands. In what follows we will refer to this bands as CT1 and CT2. In Ref. [53] the CR kinetics in DACs consisting of TMB as electron donor and TCNE as electron acceptor was reported. In VaCN solution the exponential kinetics were observed while in ACN the CR was fast at short times and much slower at longer times. The TMB-TCNE complex also shows two CT absorption bands [53,166] corresponding to transitions into different excited states of the DAC. These transitions are characterized by different sets of ET energy parameters and most likely lead to complexes with different spatial geometries [166].

resembles the CR process in DACs with a single absorption band described in the previous section. More complex situation arises when the DAC is excited by a 480 nm pulse. Such a pulse forms wave packets in both | e2 and | e1 excited states due to overlapping of the CT2 and CT1 bands in this spectral region. Difference in evolution of these wave packets is caused only by difference in their initial positions. The wave packet on the Ue2 term has however more evolution paths. First, direct CR (transition from | e2 to the ground state | Gr) is possible. It was, however, shown in Ref. [53] that such a transition would occur much more slowly than the transition observed experimentally because of a large energy gap. Second, there is a radiationless transition from | e2 to | e1. This transition is close to the vertical one if its time is shorter than solvent relaxation time. It is further assumed that the wave packet created at the Ue2 term experiences instantaneous vertical transition to Ue1 . This approximation reduces calculations of the CR dynamics in the three-level model to calculations in terms of the two-level model described in [48,51] with the initial conditions [167]: , t = 0) = Z −1 ej (Q

Sjn e−Sj n

4.1. Model To clarify the nature of strong non-exponential CR kinetics in the TMB/TCNE complex the CR kinetics was simulated numerically. For such complexes the model should minimally involve three electronic states. Corresponding FESs are shown in Fig. 10. Excitation of the DAC by a 620 nm pulse causes population of the Ue1 term (red wave packet, excited state |e1). In this case CR

exp

n!

⎧ 2 ⎨ (U (n ) − U (0) − ωe ) e2 Gr ej −

⎩

22

−

(0) UGr

kB T

⎫ ⎬ ⎭

,

j = 1, 2.

(18)

This equation explicitly connects initial position of the wave packet on the excited state FES with the carrier frequency of the excitation pulse, ωe .

Fig. 9. Location of the CR reaction zone in the Marcus normal (panel A) and inverted (panel B) regions. Initial wave packet is pictured by a blue line.

58


Fig. 11. Stationary optical absorption spectrum of TMB/TCNE in ACN (thin line) and its approximation by a sum of two asymmetric Gaussians (19) (bold line).

4.2. Determination of electronic transition parameters from the absorption spectra To separate CT bands the TMB/TCNE absorption spectrum and to determine contribution of each transition, the spectrum (Fig. 11) was approximated by a sum of two asymmetric Gaussian functions. Each band was approximated by the following expression Aj (ωe ) = Cj

Sjn e−Sj n

n!

exp

−

(j)

(Gej − Erm − n − ωe ) (j)

4Erm kB T

2

,

j = 1, 2

(19)

Fig. 12. Excited state kinetics of TMB/TCNE in ACN: experimental data [53] (solid lines) and simulations (dashed lines). Pumping pulse wavelengths are 480 nm (blue) and 620 nm (red).

the coupling element. Large couplings however lead in ultrafast CR, which ends already at the non-equilibrium stage. DACs with two absorption bands have however different optimal geometries corresponding to maximum transitions to the first and the second CT states. As a result, the geometry of a DAC that underwent transition to the second excited state and radiationless transition to the first excited state is not optimal for subsequent CR. Smaller coupling parameter results in incomplete CR at the hot stage and, as a consequence, thermal CR stage becomes observable. 4.4. Resume

The best fit was obtained with the following parameters [54]: (1) (2) Erm = Erm = 1.02 eV, Ge1 =−0.39 eV, Ge2 =−1.48 eV, S1 = 3.34, S2 = 3.22, = 0.17 eV. Fairly good agreement between the theory and experimental data was obtained almost over the whole region of CT bands.

Kinetics of two-stage CR observed in experiments with the TMB/TCNE complex in ACN reflect the presence of two recombination steps, the non-equilibrium (hot) and the equilibrium (thermal) reaction stages [54]. This observation [53] directly evidences in favor of non-equilibrium mechanism of CR in excited DACs.

4.3. Fitting the CR kinetics in excited donor–acceptor complexes

5. Effect of intramolecular high-frequency vibrational mode on ultrafast photoinduced CT and CR kinetics

Time-dependent populations of the first excited state of the TMB/TCNE complex in ACN induced by excitation pulses at different wavelengths are shown in Fig. 12 by solid (experimental data) and dashed (simulations) lines. The model therefore quantitatively reproduces experimental data over the whole time interval presented in the figure. Note that at times t > 0.5 ps, a plateau appears, which can be treated as thermal CR stage in the framework of our approach. The height of the plateau is determined by the part of DACs avoided recombination at hot stage. The thermal stage of CR has a much lower effective rate, so that TMB/TCNE complex demonstrates the biphasic CR behavior. In slower solvents (valeronitrile and octanonitrile) the efficiency of non-equilibrium CR increases, so the thermal stage became unobservable, which is in complete agreement with the experimental data [53]. This is an expected behavior, because probability of non-thermal transitions at each intersection increases in slower solvents [135]. The reason of observation of two-stepped CR kinetics only for complexes with two CT absorption bands is likely as follows. DACs exist in several quasi-equilibrium configurations due to their lability. Complexes with one CT band mostly absorb radiation in configurations with large electronic couplings to the ionic state, since the probability of absorption is proportional to the square of

Excitation of intramolecular high-frequency modes by a short pumping pulse can alter ultrafast photoinduced CT kinetics in molecular systems with relatively slow intramolecular vibrational relaxation. Manifestations of vibrational excited states in CT kinetics were observed in various donor–acceptor pairs [65,66,168–178]. Recently systematic experimental investigations of the effect of high-frequency vibrational mode on the CT kinetics in complexes consisting of zinc(II) mesotetrasulfonatophenylporphyrin (ZnTPPS4− ) and magnesium(II) meso-tetra sulfonatophenylporphyrin (MgTPPS4− ) as the electron donor and a series of viologens as electron acceptors were reported [69]. The Q-band absorption spectra of these compounds show a well seen vibrational structure. This structure reflects excitation of a vibrational mode of the porphyrin ring with the frequency 0.17 eV and rather long relaxation time to compete with photoinduced charge separation. Such a spectrum provides a possibility to excite separately the first vibrational excited state n = 1 of electronic locally excited state (S11 ) or the ground vibrational state n = 0 (S10 ). These experiments clearly confirmed that excitation of highfrequency vibrational mode affects both the forward and backward (into the ground state) ET rate [69]. Simulations of the effect also were carried out and reported in Refs. [156–158,179].


59

(iii) there are quantitative discrepancies between the experimental and numerical data. One of the reasons of these discrepancies can be connected with rather strong non-exponentiality of the CT kinetics. The magnitude of the effective rate constant thus depends on its definition. 5.2. The mechanism of the effect

Fig. 13. Scheme of electronic states involved in photoinduced ET with a few vibrational sublevels.

0.8 0.6

(6)

0.4 (3)

χf

0.2 0

(5)

(2)

(4)

-0.2 -0.4

(1)

-0.6 -1.2

-1

-0.8

-0.6

-0.4

To appreciate physical mechanisms of the above results let us consider ET in the low exergonic area, where inequality −GCS < CS is met. Fig. 13 shows that in this area the charge separated Erm state is predominantly formed in the ground vibrational state due to smaller activation energy. The rate of charge separation from the first excited vibrational state S11 is larger than that from the ground state S10 due to the following reasons: (i) the Franck–Condon factor is larger for the vibrational excited state, F10 /F00 = SCS since SCS > 1; (ii) activation barrier for charge separation from the S10 state is also larger. This leads to the positive effect. Moreover, f raises with decreasing −GCS because the difference between the heights of the activation barriers increases. When charge separation rate becomes smaller than vibrational relaxation rate, 1/ v , the effect starts to decrease with decreasing −GCS . For GCS approaching 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 Fig. 14 only the beginning of this trend is seen. CS + E , charge sepIn vicinity of ET rate maximum, −GCS = Erm rv aration proceeds mainly through the sinks placed at the bottom of (n) ∗ the free energy surface ULES hence the ratio between kFET and kFET is determined by the ratio of corresponding Franck–Condon factors [156,158] ∗ kFET

ΔGCS, eV

kFET

Fig. 14. Effect of vibrational mode excitation f as a function of the free energy gap GCS . Experimental data are pictured with +. Numbers referring different comCS = 0.65 eV, plexes are the same as in Ref. [69]. The parameters are: Erv = 0.2 eV, Erm Ve = 0.025 eV, Vg = 0.030 eV, v = 1.0 ps.

∼

F1m∗ F0m∗ −1

=

(m∗ − S)2 m∗

(21)

where m∗ =

CS + −GCS − Erm

(22) (1)

is the number of the sink nearest to the minimum of ULES . The area of the reaction exergonicity where the negative effect is expected is determined by the equation

5.1. Effect of intramolecular high-frequency vibrational mode excitation on the CT kinetics To get insight into the mechanism of the effect we consider now Fig. 13, where the free energy surfaces of the donor–acceptor system are shown. Two reasons for the effect can be distinguished here: the Franck–Condon factors and the height of the activation barrier separating reactants and products in vibrational excited states. To characterize quantitatively the influence of intramolecular high-frequency vibrational mode excitation on the CT kinetics the following parameter was introduced [158]

f =

∗ kFET

− kFET

kFET

(20)

∗ where kFET and kFET are the effective rate constants for CS from the first excited vibrational state n = 1 (S11 ) and from the ground vibrational state n = 0 (S10 ), correspondingly. GCS -dependence of f [158] is presented in Fig. 14. Experimental data are borrowed from Ref. [69], simulation parameters are taken from the fitting to the FEG law for CT from the ground vibrational state. The effect of vibrational mode excitation is therefore calculated without adjusting parameters. Simulation data presented in Fig. 14 show: (i) the effect, f , can be both positive and negative in accord with experimental data; (ii) magnitudes of experimental and simulated effects are comparable;

∗ kFET

kFET

=

(m∗ − S)2 100 fs [156]. 5.4. Resume The main conclusion of the simulations is that the excitation of high-frequency vibrational mode in Zn–porphyrin compounds increases the photoinduced ET rate in the both regions of strong and weak exergonicity and decreases at moderate exergonicity provided Erv > [156–158]. Despite the facts that both theory and experiment show a rather large effect of high-frequency vibrational mode excitation its observation is a rare fact. We can point out two reasons of the effect suppression. Very fast redistribution/relaxation of vibrational energy in electronic excited state. However decreasing the temperature can considerably enlarge the effect since the vibrational relaxation is expected to slow down with the temperature lowering [178]. The second reason is caused by large number of the vibrational degrees of freedom in molecules. The Franck–Condon active modes at the stage of the photoexcitation can be inactive at the stage of the charge transfer. In this case strong suppression of the vibrational mode excitation effect is expected. This can strongly limit the circle of polyatomic molecules in which the vibrational spectral effect can be observed. The way out of this situation could be associated with usage of a synchronous excitation of the electronic and vibrational transitions. This type of experiment could also figure out which vibrational modes are active at the stage of the charge transfer. Moreover, the synchronous IR pumping can provide a rather effective tool of photoinduced CT rate control.

6. Ultrafast kinetics of S1 , S2 , and charge separated state populations in electron transfer quenching of the S2 state of directly linked Zn–porphyrin–imide dyads In a series of studies of ultrafast fluorescence quenching of the second excited state the kinetics of both relative populations of the S2 and S1 state population kinetics were measured with high time resolution [31,55–63,65,66,69–71,180]. Experimental data on CS state population kinetics also were recently reported for similar supramolecular systems consisting of Zn(II)–porphyrin–naphthaleneimide and Zn(II)–porphyrin–amino naphthalene diimide [67,68]. These investigations have revealed that the CS state population exhibits two pronounced maxima: one on 0.2–0.4 ps and the second on the time scale from tens to hundred picoseconds [67,68]. This behavior strongly evidences in favor of the following chain of successive transitions [67] CR

S2 → CS−→S1 → CS → S0

(25)

where the CR to the S1 state may proceed only in the nonequilibrium mode when the free energy of the S1 state is larger than that of CS state. Such a two humped kinetic curve of the charge separated state population is a peculiar feature of the ET quenching of the S2 state. The sequence of the CT processes created by the second excited state ET quenching has been quantitatively described in terms of the multi-channel stochastic model. The model [48,49,51,52,87] allowed quantitative reproducing the kinetics of the population of the S2 and S1 states for a series of directly linked Zn–porphyrin–imide dyads in THF solution [72] reported in Ref. [59] as well as the CS state population kinetics [73,153,181]. In particular, the results of these simulations reproduce the two-humped experimental kinetic curve of the CS state population.

6.1. Similarities and differences between ET quenching of the S1 and S2 states We outline now the main difference between the kinetics of transformations launched by ET quenching of S2 and S1 states of the donor–acceptor molecular system. In both cases photoinduced processes starting with CS may be followed by CR, but CR is expected to proceed at qualitatively different regimes depending on the value of the CR free energy gap [182,183]. To comprehend this difference we consider electronic FESs pictured in Fig. 2. The CS can be visualized as an appearance of a wave packet in the vicinity of the crossing points of the US2 or US1 and UCS terms. When the CR driving force is large, −GCR > Er , here Er is the total reorganization energy, the CR proceeds after completion of the relaxation. This implies that CR occurs as a thermal reaction. In deep Marcus inverted region the CR is slow due to high activation barrier and the CS and CR are well separated in time. This term configuration is typical for photoinduced CT from the first excited state due to large energy gap between S1 and the ground state S0 (see Fig. 2A). Only a few exceptions exist. The donor–acceptor pair consisting of perylene and TCNE is the most known example [34,182,184]. However, the CS in this pair most likely proceeds through the population of excited states of the radical ions so that the CR has different and more complicated mechanism [184]. The CS state created by CT from the second excited state can be followed by CR into both the first locally excited S1 and the ground S0 states. Because of relatively small free energy gap between the second and the first singlet locally excited states, the CR into the first excited state proceeds in the MNR and moreover the free energy of the CR can be even positive (see Fig. 2B). This quantitative difference leads to a dramatic change of the kinetics and the mechanism of photochemical processes occurring from the second and the first excited states. Point is that in the case of small CR driving force, −GCR < Er , the CR is expected to mainly proceed at the stage of the solvent and vibrational relaxation that is visualized as a wave packet motion to the FES minimum of the CS state (see Fig. 2B) [36,44,47–49,135,183]. In this regime the nuclear relaxation and the CR are not separable due to their overlapping in time. The mechanism of such a non-equilibrium CR is very similar to that in the excited DACs which is discussed in the previous sections. This is the reason why one can expect that the study of the kinetics of the CT from the second excited state sheds light on the mechanism of CR in the excited DACs. 6.2. Fitting to ultrafast kinetics of S1 and S2 state populations Detailed experimental investigations on the kinetics of the populations of the second and first excited states in a series of supramolecular systems consisting of the Zn–porphyrin directly linked to an electron acceptor (a series of imide compounds) were reported in Refs. [31,57–59]. In what follows we use the abbreviations for these supramolecular systems suggested in these papers. The multi-channel stochastic model was used to fit the kinetics of the population of the S2 and S1 states for a series of directly linked Zn–porphyrin–imide dyads in THF solution [72]. In simulations a part of the parameters were accepted as invariable. These are: the dynamic parameters of the solvent (THF solution) x1 = 0.443, x2 = 0.557, 1 = 0.226 ps, 2 = 1.520 ps [185], the free energy gap between two excited states G12 =−0.68 eV [58], the reorganization energies of the solvent Erm = 0.5 eV and intramolecular high-frequency vibrational mode Erv = 0.4 eV with the frequency = 0.17 eV and the relaxation time v = 0.030 ps. Only the CS free energy, GCS , and the electronic couplings, VCS = VCR , were the variable parameters. Although the equality VCS = VCR was accepted as the first approximation to reduce the number of variable parameters, it appeared to be a good approach.


61

In the fitting the positive sign of GCR was obtained for all dyes. The sign of GCR is a factor controlling the kinetics of the S1 population at moderate times. If a dye had a negative value of GCR , the initial rise of the S1 population conditioned by non-equilibrium transitions during vibrational relaxation of CS state would be followed by a further rise of S1 state population due to thermal CR. However the experimental results show rather a decay of the S1 population in 1–3 ps window with the rate constants varying from 1 to 0.01 ps−1 . This evidences strongly in favor of GCS and GCR values obtained in the fitting. 6.3. Role of reorganization of high-frequency intramolecular vibrational modes

Fig. 15. Population dynamics of S2 (left panel) and S1 (right panel) states. The data are plotted with light gray lines (experimental results [59]) and black lines (simulation results). Values of GCS and Vel are listed in Table 2. Other parameters are identical for CS and CR: Erm = 0.5 eV, Erv = 0.4 eV, = 0.17 eV, v = 0.03 ps, e = 0.110 ps. Table 2 The parameters of CS and CR of ZP-I series, in eV. Dyad

GCS a

GCR a

GCS

GCR

VCS = VCR

ZP-NI ZP-PI ZP-Cl4 PH ZP-Cl2 PH ZP-ClPH ZP-PH ZP-MePH

−1.42 −1.15 −0.87 −0.68 −0.63 −0.57 −0.56

+0.74 +0.47 +0.19 +0.00 −0.05 −0.11 −0.12

−1.37 −1.09 −0.96 −0.74 −0.73 −0.75 −0.77

+0.69 +0.41 +0.28 +0.06 +0.05 +0.07 +0.09

0.023 0.030 0.035 0.033 0.032 0.026 0.022

a

The data are taken from Ref. [59].

6.4. Kinetics of the charge separated state population

To compare the experimental and theoretical data the instrumental response time should be accounted for. The time dependent fluorescence intensity measured in the experiments, A(t), was considered to be a convolution of the population with the instrumental response function A(t) = (e2 )

−1/2

t

P(t − ) exp −∞

For description of the kinetics of the CS state population the model [72] was modified in three aspects [73,153]. Firstly, the recombination into the ground state was included into consideration. Secondly, the internal conversion S2 → S1 was explicitly described. Thirdly, the intramolecular reorganization was described in terms of several vibrational high-frequency modes because the resonance Raman spectra of DACs show that about ten vibrational modes associate with CT transition [50,51,160,161,186,187]. Moreover, the models with single mode and those with real spectrum of the quantum vibrational modes predict essentially different kinetics for the thermal [164] and the non-equilibrium reactions [51,52]. Unfortunately, the spectra (frequencies and Huang–Rhys factors) are known only for a few molecular systems [50,51,160,161,186,187]. However, kinetics of charge transfer is not sensitive to the choice of the vibrational mode spectrum provided that the number of active modes is equal to or larger 5 [159]. Although vibrational modes active in the Soret band excitation are known from the vibronic spectra but there is no information how many and which vibrational modes are active at ET stage. If the number of active modes is larger or equal to 5 then it is possible to use the parameters of a CT complex [159]. In all investigated charge transfer processes this condition was met. This is the reason why we accept this hypothesis for the dyads including Zn porphyrin.

−

2 e2

d

(26)

where e = 110 fs [58] is the instrumental response time. With the set of parameters stated above a satisfactory fitting to the experimental data on population kinetics of both excited states was obtained. The results of the simulations are presented in Fig. 15 and the parameters obtained in the fitting are listed in Table 2. One may see that the fitting is in a rather good agreement with the experimental data and there is no necessity to vary VCS and VCR independently. In the dyes ZP-NI, ZP-PI, and ZP-Cl4 PH a fast rise of S1 population is followed by rather fast decay. This decay to produce again the CS state essentially proceeds as the thermal ET and its high rate is conditioned by the proximity of the CS to the barrierless region. The S1 populations of the dyes ZP-Cl2 PH, ZP-ClPH, ZP-PH, and ZP-MePH also decay but much slower due to a higher activation barrier between the S1 and CS states. The timescales of these slow decays are a few tens of ps.

The model elaborated was applied for fitting to experimental kinetics observed for Zn(II)–porphyrin covalently linked to naphthaleneimide dyads in dimethylformamide (DMF) solution and reported in Ref. [67]. The fitting was aimed to reproduce the available experimental data obtained for Zn(II) porphyrin covalently linked to naphthaleneimide in the DMF solution [67]: (i) the population of the CS state at three key points in time: PCS (tmax ) = 0.16 at the time of the first maximum, PCS (t = 3 ps) = 0.07, corresponding to the minimum between two maxima, and PCS (t = 100 ps) = 0.19 in the vicinity of the second maximum, (ii) the time scale of the S2 state decay, CS2 = 0.4 ps, the time scale of the charge separation from the first excited state CS1 = 170 ps, and the CR to the ground state CR0 = 38 ps. The invariable parameters were borrowed from independent measurements and estimations: GS0S2 =−2.9 eV, GS1S2 =−0.8 eV, GCS =−1.025 eV, the dynamic parameters of the DMF solvent are [185] x1 = 0.508, x2 = 0.453, x3 = 0.039, 1 = 0.217 ps, 2 = 1.70 ps, 3 = 29.1 ps, the time scale of the internal conversion (1) −1 IC = kIC = 2.0 ps, the time constant of vibrational relaxation v = 50 fs. The variable parameters are the reorganization energies Erm , Erv , and the electronic couplings, VCS , VCS , and VCR0 . The best fit pictured in Fig. 16 was obtained with the parameters [73]: Erm = 0.91 eV, Erv = 0.36 eV, VCS = 0.023 eV, VCR1 = 0.048 eV, and VCR0 = 0.00325 eV. These parameters are quit reasonable except for the magnitude of Erm that seems to be a little bit too large for

62


Fig. 16. Kinetics of the CS state population of Zn(II)–porphyrin covalently linked to naphthaleneimide in DMF solution. The left half of the figure shows data for the first 3 ps, and the right half shows data for 3–200 ps. The values of the parameters are listed in the text.

intramolecular ET. However, all attempts to fit the experimental data with smaller Erm are failed. This fitting allows reproducing the magnitudes of the key experimental parameters: PCS (tmax ) = 0.16, PCS (t ≈ 1 ps) = 0.07, and PCS (t = 100 ps) = 0.19. The S2 state decay is nearly exponential and its time scale is equal to CS2 = 0.24 ps. This value differs essentially from experimental 0.4 ps. However, it cannot be increased because its magnitude for given medium relaxation time scale is rigidly determined by the height of the first maximum and the height of the plateau in the area approximately from 1 ps to 3 ps (see Fig. 16). In other words the values PCS (tmax ) = 0.16 and PCS (t ≈ 1 ps) = 0.07 can be obtained with different sets of the model parameters but CS2 is invariably equal to 0.24 ps. The discrepancy may be caused by either the roughness of the model or the experimental errors. The fact is that the decay of measured optical signal at a given frequency is treated as a result of the state population decay. However, the relaxation of the medium and the intramolecular reorganization can considerably affect the decay of the signal on such a short time scale. One more reason connected with reorganization of an intramolecular relaxation mode is discussed in the next section. 6.5. Influence of the dyad geometry on the quantum yield of ultrafast charge separation The experiments and theory point out that ultrafast photoinduced intramolecular ET in the D–A dyads upon the Soret-band excitation of the donor compound is strongly affected by the two unwanted processes: internal conversion to the first excited state, |S1 , and non-equilibrium CR during solvent relaxation within the first few picoseconds. To achieve high quantum yield of the charge separated state, YCS , at least two conditions should be fulfilled [188]: (i) forward ET from |S2 to |CS1 should be faster than internal conversion, and (ii) non-equilibrium recombination of D+ -A− pairs should be inefficient. To simulate the influence of the CT distance and space sizes of the dyad compounds the electronic coupling dependence on effective radii of the donor and acceptor R1 , R2 and CT distance R12 was supposed to follow the model equation (c)

R −R −R 12 1 2

Vk = Vk exp −

L

(27)

(c)

Here k = CS or CR, Vk is the maximal value of electronic coupling reached at the contact distance, R12 = R1 + R2 , and L is the electron tunneling length. The dyad geometry also influences on the magnitude of the solvent reorganization energy. The Marcus expression relating the reorganization energy E12 of the medium to the donor and acceptor effective radii R1 and R2 is used [18] E12 =

cP e 2 2

1

R1

+

1 2 − R2 R12

(28)

where e is the electric charge transferred, cP = 1/ε∞ − 1/ε0 , ε∞ and ε0 are the optical and stationary dielectric susceptibilities, and R12 is the center-to-center distance between the spherical cavities modeling donor and acceptor. Although Eq. (28) was obtained in the limit of large center-to-center distance, Rij , it gives a good approximation for any distances while the spheres do not overlap [140]. This condition was fulfilled in all simulations. The quantum yield of ultrafast charge separation is defined as an integral population of thermalized charge-transfer states of the molecular dyad at some moment of time tth when the initial ultrafast (non-equilibrium) phase of the reaction is over, while the following slow (thermal) phase does not start yet [189] (see Fig. 16, where such a quasi-plateau are well seen in the region t > 1 ps). As a measure for tth one can use the characteristic time of solvent relaxation L . It was shown in Ref. [189] that 5 L can be used as a good estimate for tth . YCS is thus YCS = PCS (5L )

(29)

The stochastic multi-channel model includes a lot of parameters. Since the dyads containing zinc–porphyrin (ZnP) as an electron donor and different imides as electron acceptors most often are exploited their characteristics are taken as typical. −1 Namely, internal conversion time scale |S2 → |S1 D = kIC = 2 ps [59], the free energy gap between the second and first excited states, −GS2S1 = 1 eV, the contact values of electronic couplings ˚ the are V(c) = 0.05 eV, the electron tunneling lengths are L = 1.5 A, (1) intramolecular vibrational relaxation time is v = 300 fs, the frequency of the intramolecular quantum vibrational mode is taken = 0.1 eV. In simulations the parameters of ACN solvent are used: the dielectric constants ε∞ = 1.806 and ε0 = 36.64 and the solvent relaxation time L = 0.5 ps. Since both the CS rate constant and the non-equilibrium CR probability increase with the rise of the electronic coupling one can suggest a non-monotonous dependence of the charge separation yield, YCS , on the distance, R12 , between the donor and acceptor compounds. This conclusion is confirmed by numerical results shown in Fig. 17. The calculated R12 -dependencies of YCS have pronounced maxima at R12 noticeably larger than the contact radius. The results presented in Fig. 17 show one more trend, namely, strong dependence of YCS on the free energy gap, −GCS . Since the maximum value of the CS rate constant is reached when the free energy barrier for the CS reaction is low the equation should be fulfilled 2

# GCS =

(E12 + Erv + GCS ) kB T 4E12

(30)

to get maximum value of YCS . The simulations performed clearly show that the quantum yield of ultrafast charge separation in zinc–porphyrin/imide systems in acetonitrile is not expected to be higher than ∼50% [188]. This limitation cannot be overcome due to two reason: (i) the values of the electronic couplings for CS from the second excited state and CR into the first excited state are close each other [72] and (ii) fast internal conversion, |S2 → |S1 , in zinc–porphyrin occurring with the time constant, D = 2 ps. These reasons suggest two possibility to enlarge the yield, YCS . First it is to design donor–acceptor dyads in which the values of the electronic couplings for CS from the second excited state is much larger than that for CR into the first excited state [73]. The second is to exploit a compounds with larger second excited state lifetime [188]. This is a real way to circumvent the problem because there are photochemical systems with strong |S0 → |S2 absorption band and slower rates of |S2 → |S1 internal conversion. For example, in xanthione-containing systems the lifetime of the second excited state, D , can be as large as 100 ps and even more [190]. In such systems fast non-equilibrium CR is


63

Fig. 17. Quantum yield of thermalized charge separated states in the D–A dyads after the Soret-band excitation as a function of the donor–acceptor distance for a few values of GCS and R1 , R2 (indicated in the figure). The parameters are: the free energy gap between the second and first excited states GS2S1 = GS2 − GS1 = 1 eV.

avoided much easily. Results reported in Ref. [188] support this conclusion. In the dyads with larger second excites state lifetime, D = 100 ps, the quantum yield of charge separated state can be close to unity. The results reviewed in this section show that special attention should be paid to the geometry of designed dyads to reach high quantum yield, YCS . Indeed, for a given set of the parameters a rather narrow area of CT distances exists where an efficient charge separation can occur. Since the position of this area depends on the values of the other parameters a subtle adjustment of all parameters is needed to design donor–acceptor dyads with high efficient charge separation. 6.6. Resume Exploration of the mechanism of CS from the second excited state has shown that a distinguishing feature of CT is effective non-equilibrium CR to the first excited state. The multi-channel stochastic model was applied for the fitting to the experimental kinetics of S1 , S2 , and charge separated state populations observed for Zn(II)–porphyrin covalently linked to imides. The key parameters of the two-humped kinetic curve have been quantitatively reproduced. Three time scales of the S2 state decay, CS2 , charge separation from the first excited state, CS1 , and CR into the ground state, CR0 , are also have been calculated and they are in a good accord with the experimental data. So, one can conclude that the multi-channel stochastic model can quantitatively describe the kinetics of the ET from the second excited state including effective non-equilibrium reverse ET to the first excited state. The main results are summarized as follows. The CS state population kinetics display two maxima for a wide region of the parameters. So, the two-humped kinetic curve in Zn–porphyrin derivatives seems to be rather universal provided that the relaxed CS state lies below the S1 state and that the nonequilibrium recombination occurs to the S1 state following CS from the S2 state. The fraction of molecules in the CS state that has escaped the non-equilibrium recombination varies in the interval from 0.01 to 0.5 in the region of reasonable parameters. The models with single and many active high-frequency vibrational modes predict quantitatively different kinetics of the CS state population. This

indicates that the model with a real spectrum of the high-frequency modes should be exploited to get a quantitative description of the experimental data. The experiment clearly demonstrates an effective CR into the first locally excited state that can proceed only in nonequilibrium regime because the thermal process requires large activation energy (see Fig. 2B). The non-equilibrium CR strongly lowers the charge separated state population. For example for Zn(II)–porphyrin covalently linked to naphthaleneimide in the DMF solution it is only 0.07. The particles survived in the charge separated state are responsible for a quasi-plateau which is seen in Fig. 16 on the time interval t > 1 ps. The theory predicts an analogous quasi-plateau also for CR in excited DACs which, however, has been only once directly observed in a complex with two CT absorption band [53]. Accounting for similarities of the mechanisms of two considered processes and larger values of electronic coupling in DACs, we may suppose that in excited DACs the non-equilibrium CR effectiveness is close to unity and the DACs survived in the excited state are not seen due to their small number. 7. Fitting to the two hump kinetic curve of the charge separated state population in rigid dyads In this section microscopic mechanism of photoinduced ET from the second excited electronic state in Zn–porphyrin–amino naphthalene diimide (Zn–TPP–ANDI) dyad in toluene solution is discussed. The system was recently experimentally studied and the results were reported in Ref. [68]. The dyad is very similar to that considered in the previous section, but there is one important difference in the nature of the solvent. The toluene is weakly polar so that a considerably smaller solvent reorganization energy is expected. 7.1. Modeling the charge separation and recombination kinetics in rigid dyads Fitting to the experimental two-humped kinetic curve of the CS state population was performed in the framework of the multichannel stochastic point-transition model. A part of the model parameters is well known for the dyad Zn–TPP–ANDI. They are:

64


Fig. 18. Kinetics of the CS state population of porphyrin–amino naphthalene diimide dyad in toluene solution: no slow intramolecular reorganization. The black line is the experimental curve [68], the red line is the result of the best fit.

the free energy changes accompanying the transitions considered, GS0S2 =−2.9 eV, GS1S2 =−0.8 eV, GCS =−1.2 eV, and the internal conversion rate constant, kIC = 0.5 ps−1 [67,68]. Relaxation parameters of toluene are: x1 = 0.59, x2 = 0.19, x3 = 0.22, 1 = 0.08 ps, 2 = 0.65 ps, 3 = 3.0 ps [191]. It is also supposed that reorganization of intramolecular low frequency vibrational modes at the stage of charge separation and recombination is negligible. The relaxation times for the quantum vibrational modes are set the same for all (1) modes and are equal to v = 100 fs. So, the variable parameters in the fitting are: the reorganization energies of the low frequency classical modes, Erm , and the intramolecular high-frequency vibrational modes, Erv , the electronic couplings, VCS , VCR1 , and VCR0 . Supposing that the transient at 650 nm (see Fig. 5B in Ref. [68]) is fully caused by the radical cation absorption and, hence, is proportional to the CS state population, fitting of the simulated population to transient absorption at 650 nm was performed [181]. The results of the fitting to the experimental two-humped kinetic curve is pictured in Fig. 18 with red line (the experimental data are presented with the black line). The best fit parameters are: Erm = 0.93 eV, Erv = 0.42 eV, VCS = 0.18 eV, VCR1 = 0.09 eV, and VCR0 = 0.003 eV. These parameters are rather typical for intramolecular ET except for the magnitudes of VCR0 and Erm . The magnitude of VCR0 is much smaller than VCS and VCR1 . On the contrary, the magnitude of Erm is considerably larger than that expected for the intramolecular ET in non-polar solvents [192,193]. Nevertheless, all the attempts to fit the experimental data with smaller Erm were unsuccessful. Moreover, the width of the peak in the femtosecond domain obtained in the fitting is considerably narrower than that in the experiment. This problem is discussed in next section. 7.2. What factors control the width of the peak in the femtosecond time interval? To get deeper insight into the origin of inconsistencies in previous fitting we notice that the kinetic curve with two maxima can be obtained only if the time of the charge separation from the second excited state, 1/kCS , is shorter than the lifetime, relax , of the non-equilibrium CS states from which CR is possible. One can see that relax is equal to the time that a particle created on the CS state curve needs to lower under the state S1 where the hot CR terminates. Considering that the particles can leave the area of the non-equilibrium recombination at the stage of the relaxation of the fast mode of the solvent, this time interval can be estimated as relax ≤ 1 = 0.08 ps. So, the width of the peak in the femtosecond domain, t, should be t 1/kCS + 1 < 21 . This value is much less than that obtained in the experiment and cannot be decreased because it is totally determined by the dynamical properties of the solvent which are borrowed from the independent experiments.

Fig. 19. Kinetics of the CS state population of porphyrin–amino naphthalene diimide dyad in toluene solution: accounting for slow intramolecular reorganization. The black line is the experimental curve [68], the red dashed line is the best fit.

In the discussion above the instrument response function was not considered. The fact is that accounting for this function does not improve the quality of the fitting because convolution of the obtained here kinetic curve with the instrument response function results in strong suppression of the maximum in the femtosecond domain so that it disappears at all [181]. Accounting for the S2 absorption that can also contribute to the absorption at 650 nm can improve the fitting. However, a good fitting is achieved only when the S2 absorption dominates that excludes this possibility of explanation of the discrepancy with the experiment data [181]. This inconsistency points out that the reorganization energy, Erm , may be related not only to the solvent but to reorganization of slow intramolecular vibrational mode too. Such a mode is associated with the conformational changes of a molecule and is characterized by overdamped motion of a large amplitude. This kind of the reorganization can be described in terms of the stochastic model as an additional relaxation mode in the relaxation function Eq. (5) with a large weight and relatively long relaxation time. Indeed, the fitting to the CS state population in Zn–TPP–ANDI dyad in the toluene solution allowed to get some information on the reorganization energy and the relaxation time of slow intramolecular vibrational mode. To simulate the low frequency overdamped intramolecular vibrational mode, the fourth term with a weight, x4 , and a relaxation time constant, 4 , was added in the relaxation function, X(t), [181,194]. The weights of all modes are met the condition 4

xi = 1

(31)

1

The result of the best fit is shown in Fig. 19. It indicates that the slow intramolecular mode should have the relaxation time constant about 4 = 6.8 ps and large weight, x4 = 0.7 [194]. Dynamics of such modes also depend on the solvent viscosity [195,90]. Another conclusion is that the width of the peak in the femtosecond domain is very sensitive to the dynamic properties of the solvent and intramolecular relaxation modes. Quantum-chemical simulations support the hypothesis that a slow intramolecular mode with large amplitude are active in ET. They point out that the slow intramolecular mode can be associated with rotations around C C bonds in the ANDI moiety [194].


Fig. 20. The wavelength-selective photoinduced ET in the zinc–porphyrin/imide compounds. |S1 and |S2 are the first and second locally excited electronic states –D+ –AR and AL –D+ –A− , of ZnP, |CSL and |CSR are the charge separated states A− L R respectively.

7.3. Resume Modeling the ultrafast population kinetics of the CS state inevitably leads to much smaller width of the first peak than that obtained in the experiment. The finding has a simple and reliable physical interpretation. Since for a given height of the peak its width is determined by the relaxation time of the solvent and the relaxation characteristics of the solvent are taken from an independent experiment, the discrepancy between the results of simulations and the experiment can be overcome only if one supposes that the charge separation is associated with the reorganization of additional (not connected directly with the solvent) slow nuclear modes, for example, slow intramolecular rotation or bending of large amplitude. In solutions such motions occur with high friction and, hence, present a kind of the Brownian motion. 8. Intramolecular charge separation from the second excited state: suppression of hot charge recombination by electron transfer to the secondary acceptor Recent experimental studies, conducted by the group of Prof. L. Hammarstroem have shown [67] that molecular triads of the type acceptorL –donor–acceptorR (AL –D–AR ) having two locally excited electronic states of the donor could function as a molecular optical switch offering the possibility of selective ET from the donor to the “right” or “left” acceptor depending on the wavelength of excitation. Suitable molecular compounds, containing zinc–porphyrin (ZnP) as electron donor and different imides as acceptors, have been synthesized [67]. In these compounds the first locally excited electronic state of ZnP, |S1 , is quenched by ET to the AL , while the second excited state, |S2 , is quenched by ET to the AR on the opposite side of the porphyrin ring. Direct measurements however revealed rather low efficiency of charge separation after the Soret-band excitation of these devices. Only 10–20% of ion pairs formed in the |CSR state (i.e. by photoinduced ET from |S2 ) were detected to avoid recombination within the first few picoseconds. As it was suggested in [67], the reason of low efficiency of charge separation in the right-hand branch of the imideL –ZnP–imideR compounds is nonequilibrium back ET from the charge separated state |CSR to the first locally excited state of the donor |S1 (see the scheme in Fig. 20).

65

To increase the efficiency of charge separation in such molecular devices, the problem of hot recombination of charges in the right-hand branch of the imideL –ZnP–imideR compound has to be solved. Recent numerical simulations have shown the quantum yield of hot CR in these systems to be large for any reasonable values of model parameters [73,153]. This conclusion is consistent with experimental data that universally demonstrate highly effective CR to the |S1 state that follows quenching of the |S2 fluorescence by either intermolecular [61] or intramolecular photoinduced ET in the ZnP–acceptor systems [59,62,63,67]. Efficient method of charge separation with high quantum yields is however known in nature. It is implemented in photosynthetic reaction centers of bacteria and plants where primary photoseparation of charges proceeds as a sequential multi-step ET from the excited donor to more and more distant acceptors [196]. Utilizing several acceptors instead of a single one helps photosynthetic centers to stabilize the charge separated state and to minimize unnecessary excitation loss due to recombination. This mechanism of CR suppression is relevant to the ZnP-based molecular switches too [68]. Making use a compound with two sequentially linked acceptors on the right-hand side of the porphyrin could result in screening of back ET to the |S1 state. One should however mention here essential difference in physical mechanisms of CR suppression in photosynthetic reaction centers and in ZnP-based compounds. In natural photosynthesis, primary separation of charges during a few consequent initial steps compete with CR to the donor ground state too, but the rate of recombination is relatively low. The reason is large exergonicity of CR that shifts the reaction to the Marcus inverted region. Therefore CR in photosynthetic centers proceeds as activated (thermal) reaction after the completion of solvent and intramolecular vibrational relaxation. Entirely different situation is realized in ZnP derivatives where CR is low-exergonic and proceeds in the normal region as ultrafast non-equilibrium electron transfer [72]. Kinetics and quantum yield of photoinduced CS from the second excited state in ZnP–imide dyads were studied earlier in Chapter 7. In this chapter we explore the influence of the secondary acceptor on charge separation in the donor–acceptor1 –acceptor2 molecular system (for short D–A1 –A2 ) supposing that CS here proceeds from the second locally excited state of the donor. Therefore we exclude the whole left branch of the switch from consideration, assuming it has no significant effect on photochemical processes in the right branch of the device. Note that essential influence of A2 on photochemistry in D–A1 –A2 system can be expected only if charge shift to the secondary acceptor, A1 to A2 , is ultrafast (see Fig. 21). Only under this condition competition of A1 → A2 electronic transitions with hot CR to the |S1 state could be successful and the quantum yield of charge separation from |S2 could be high.

8.1. The model and the mechanism of ultrafast CR suppression In general, description of ultrafast two-step ET from the donor to the primary and then to the secondary acceptor requires one new element, accounting for correlation between two reaction coordinates connected with the two stages. Reorganization of solvent at different ET steps can however be taken into account properly by introducing additional reaction coordinate(s). In particular, electronic transitions in a three-center molecular system in a single-mode polar solvent can be described using two Marcustype reaction coordinates (see [132,134,197,198,200] for details). We restrict our model here exactly to this situation and introduce two classical coordinates Q1 and Q2 associated with solvent polarization around the reactants. Free energy surfaces for the electronic

66


Fig. 21. Photochemical processes in D–A1 –A2 triad following the Soret band excitation of the donor molecule. CS, CR and CSh are the initial charge separation (D** A1 A2 → D+ A− A ), charge recombination to the first excited state (D+ A− A → D* A1 A2 ) and 1 2 1 2 A → D+ A1 A− ). charge shift to the secondary acceptor (D+ A− 1 2 2

states of interest (|S2 , |CS1 , |CS2 and |S1 ) in these coordinates are written as US2 =

(Q1 −

2E12 )

2

+

2

(n)

UCS1 =

(m)

UCS2 =

Q12 2

Q22

+

2

(Q1 −

Q22

(32)

2

+ nv + GCS1

2E23 cos )

2

2

+

(Q2 −

(33)

2E23 sin ) 2

+ mv + GCS2

(l)

US1 =

(Q1 −

2E12 )

2

2

+

2

(34)

Q22 2

E12 + E23 − E13

2

E12 E23

.

and UCS1 = UCS2 (green) cross-lines, where hot charge recombination and hot charge shift proceeds, respectively. Non-equilibrium electronic transitions play a dominant role here, since only a small part of particles reaches the bottom of the UCS1 term. It seems clear from this picture that outcome of competition between ultrafast charge recombination and charge shift in this photochemical system is determined by mutual arrangement of UCS1 = US1 and UCS1 = UCS2 cross-lines on the (Q1 , Q2 ) plane. Intrinsic rates of hot electronic transitions along these cross-lines are also important. Hereafter we proceed to numerical calculations of the quantum yield YCS of ultrafast photochemical separation of charges from the second excited state in molecular triads with two sequentially linked electron-acceptor compounds. 8.2. Influence of D–A1 –A2 system parameters on the hot CS quantum yield

+ lv + GS1

(35)

where E12 and E23 are the solvent reorganization energies for ET from the donor to the primary acceptor, and from the primary to the secondary acceptor, respectively. is the angle between the directions of |S2 → |CS1 and |CS1 → |CS2 transitions in the (Q1 , Q2 ) space. This quantity can be treated as a measure of correlation of solvent response to the consequent ET steps. It is not however independent parameter and determined by solvent reorganization energies according to the equation [197] cos =

Fig. 22. Free energy surfaces for the electronic states of the D–A1 –A2 molecular system (circles) on the (Q1 , Q2 ) plane. Pair intersections between these FESs are shown by colored straight lines.

(36)

Consider now a scenario of photoinduced ET in D–A1 –A2 system in context of the free energy surfaces (32) and electronic transitions between them. Contour plots of these FESs are pictured in Fig. 22 as circles, pair intersections are shown as straight solid (0–0 intersections) and dashed (0–n intersections) lines. |S2 state is produced by a short laser pulse, so that initial population of the second locally excited state can be visualized as an equilibrium wave packet in vicinity of the US2 minimum. Primary separation of charges in the triad proceeds as fast low-barrier thermal reaction (n) along the US2 = UCS1 cross-lines (red in Fig. 22). Electronic transitions from D to A1 form non-equilibrium wave packets on the UCS1 surface (one of them is pictured as a yellow spot) that start to move to the FES bottom, reflecting relaxation of solvent polarization around the reactants. These packets then pass UCS1 = US1 (blue)

Now we investigate how different dynamic and energetic parameters of the D–A1 –A2 system can influence the efficiency of ultrafast CS from the second excited state. Key parameters among them are solvent and intramolecular reorganization energies (Eij and Erv ), free energies of electronic transitions (Gk ) and electronic coupling energies (Vk ). Our goal here is to find the regions of model parameters where YCS is high. First of all we emphasize that intramolecular high-frequency vibrational mode(s) usually play a determining role in nonequilibrium ET, since larger Erv opens up more effective channels for hot electronic transitions to vibrationally excited sublevels of the product state. For the molecular system considered, this conclusion is however applicable both to hot charge recombination and charge shift. One therefore can expect non-monotonous dependence of YCS on intramolecular reorganization energy Erv due to competition between these reaction channels. The above expectation is supported by simulations results presented in Fig. 23, where YCS () dependencies are pictured for a few values of Erv varying from 0 to 0.35 eV. These results clearly show, that hot CS yield depends sharply on electron-vibrational coupling and reach maximum ∼0.6 at Erv = 0.15 eV. Consider now the influence of solvent reorganization on the efficiency of the CS processes in the D–A1 –A2 system. Fig. 24 shows calculated YCS () curves for some values of E12 ranging from 0.5 to 1.2 eV (indicated in the figure caption). In these calculations E23 was changed proportionally to E12 , keeping the ratio E23 /E12 = 0.5. These results also reveal non-monotonous dependence of hot CS


67

Fig. 25. Spatial parameters of a triad. Definitions of the radii, Ri , interparticle distances, Rij , and the bending angle, ϕ, are presented. Table 3 max in molecular dyads D–A Maximal values of calculated charge separation yield YCS at different GCS (see Fig. 17). R1 , R2 and R12 values indicate the corresponding geometric parameters of the donor and acceptor. Fig. 23. Quantum yield of hot charge separation YCS as a function of for a few values of Erv (indicated in figure). Simulation parameters: E12 = 0.9 eV, E23 = 0.4 eV, E13 = 0.8 eV, GCS1 = GCS2 = GS1 =−0.8 eV, VCS = VCSh = VCR = 0.05 eV.

GCS , eV

R1 and R2 , A˚

R12 , A˚

max YCS

−1.4 −1.2 −1.0 −0.8

4 5 6 7

10.4 12.5 14.7 17.0

0.54 0.50 0.47 0.40

ultrafast charge separation in this system depends crucially on ET energetics, especially at the stage of charge shift from A1 to A2 . We invoke here the well-known Marcus expression relating the reorganization energy Eij of the continuous polar medium to the donor and acceptor effective radii Ri and Rj [18] Eij =

cP e 2 2

1 1 2 + − Ri Rj Rij

(37)

This equation is obtained in the limit of large center-to-center distances, but it gives a good approximation for any distances while the spheres do not overlap [140]. Further we adopt the values ε∞ = 1.806 and ε0 = 36.64 corresponding to ACN as a solvent. The electronic coupling energies Vk (k = CS, CR and CSh), on the other hand, depend on Ri and Rij according to the following model equation Fig. 24. The same as in Fig. 23 for a few values of solvent reorganization values E12 and E23 . E12 and E23 are changed proportionally, keeping relation E23 = E12 /2 in all simulations. Parameters: E12 = 1.2 (1), 1.1 (2), 1.0 (3), 0.9 (4), 0.8 (5), 0.7 (6), 0.6 (7), 0.5 (8) eV, v = 0.1 eV, Erv = 0.15. Other parameters are the same as in Fig. 23 caption.

yield on E12 and E23 with a maximum value YCS ≈ 0.63 at E12 = 0.7 eV (E23 = 0.35 eV) and ≈ 50◦ . Another important energetic characteristic of ET reaction in a polar solvent is known to be the free energy change GET . Calculated YCS () dependencies (not shown here) for a few values of GCS2 however demonstrate only a shift of position of YCS maximum. The maximal value of the hot CS yield does not therefore depend on GCS2 to any considerable extent. These results provide a rather simple way to maximize YCS by adjusting the free energy of the |CS2 state. 8.3. Role of spatial geometry of the triad Now we focus on the effects of molecular structure of the donor–acceptor system, namely the roles of the donor and acceptors effective radii, the ET distances, and the bending angle of the triad. The mechanism of influence of these parameters on YCS is rather transparent: geometry of reactants affect solvent reorganization energies Eij and coupling energies Vk corresponding to the different ET processes. Our previous results show that efficiency of

(c)

Vk = Vk

exp

−

Rij − Ri − Rj Lk

(38)

Here i and j indices correspond to the initial and the final states of (c) ET, Vk is the maximal value of electronic coupling reached at the contact radius, Rij = Ri + Rj , and Lk is the electron tunneling length (Fig. 25). To search out optimal molecular geometries of the triad yielding high efficiencies of ultrafast charge separation, we employ the following strategy: first we search for optimal spatial configurations in the single-acceptor system (i.e. in molecular dyad D–A) and then add the secondary acceptor to the model in order to increase the quantum yield of charge separation further. The results of numerical simulations in molecular dyads D–A are presented earlier in Fig. 17. Optimal geometries of the donor–acceptor system correspond to the maxima of the YCS (r) dependence. Positions of these maxima depend on energetic parameters of the CS and CR reactions, and are summarized in Table 3. To explore the influence of the triad geometry on the CS yield max we take optimal spaand to find upper obtainable values of YCS tial configurations of the donor and primary acceptor from Table 3 (parameters R1 , R2 , R12 for different GCS ) and will vary R3 (the effective radius of the secondary acceptor) and R23 (the distance between the primary and secondary acceptors) in a wide range. Results of these simulations are presented in Fig. 26 as a series of

68


Fig. 26. Contour plot of the quantum yield YCS in molecular triads as a function of the secondary acceptor radius, R3 , and the surface-to-surface distance between two acceptors S23 = R23 − R2 − R3 . The values of R1 , R2 , R12 and GCS are taken from Table 3 and indicated in the panels. GCSh = 0, the rest parameters are the same as in Fig. 17.

contour plots YCS (R3 , S23 ), where S23 = R23 − R2 − R3 is a surface-tosurface distance between A1 and A2 . These results demonstrate a prominent rise of the CS yield in ˚ arranged molecular triad in the area of rather large A2 (R3 ∼ 8–9 A) ˚ Maximal increase of YCS due to A2 howclosely to A1 (S23 1 A). ever depends on GCS , amounting YCS ≈ 0.08 at GCS =−0.8 eV to YCS ≈ 0.12 at GCS =−1.4 eV. This suggests more effective CS in molecular triads with compact distribution of electronic densities in |S2 and |CS1 states, that is smaller R1 and R2 and, accordingly, larger E12 (1 eV and more). On the other hand, the energy of solvent reorganization at the second ET step, E23 , should be less than 0.5 eV, which requires large R3 and close spatial locations of A1 and A2 . 8.4. Resume Numerical results presented in this section show that secondary acceptor can considerably increase the yield of ultrafast charge separation in donor–acceptor1 -acceptor2 triads but this is possible only if CS-controlling parameters are precisely adjusted. To maximize this quantum yield the following conditions should be fulfilled: (1) radius of the secondary acceptors should be rather ˚ (2) the secondary acceptor should be arranged large (R3 ∼ 8–9 A); closely to the primary acceptor. Adjustment of certain energetic parameters of the triad can also favor charge separation efficiency. For example, (1) the reorganization of the high-frequency vibrational mode at the CS and CSh stages should be considerable (Erv > 0.2 eV); (2) the free energy gap of charge shift should be moderate −0.3 eV