Solvent Polarization Relaxation Research Articles

Polarization Caging in Diffusion-Controlled Electron Transfer Reactions in Solution

In some bimolecular diffusion-controlled electron transfer (ET) reactions such as ion recombination (IR), both solvent polarization relaxation and the mutual diffusion of the reacting ion pair may determine the rate and even the yield of the reaction. However, a full treatment with these two reaction coordinates is a challenging task and has been left mostly unsolved. In this work, we address this problem by developing a dynamic theory by combining the ideas from ET reaction literature and barrierless chemical reactions. Two-dimensional coupled Smoluchowski equations are employed to compute the time evolution of joint probability distribution for the reactant (P((1))(X,R,t)) and the product (P((2))(X,R,t)), where X, as is usual in ET reactions, describes the solvent polarization coordinate and R is the distance between the reacting ion pair. The reaction is described by a reaction line (sink) which is a function of X and R obtained by imposing a condition of equal energy on the initial and final states of a reacting ion pair. The resulting two-dimensional coupled equations of motion have been solved numerically using an alternate direction implicit (ADI) scheme (Peaceman and Rachford, J. Soc. Ind. Appl. Math. 1955, 3, 28). The results reveal interesting interplay between polarization relaxation and translational dynamics. The following new results have been obtained. (i) For solvents with slow longitudinal polarization relaxation, the escape probability decreases drastically as the polarization relaxation time increases. We attribute this to caging by polarization of the surrounding solvent. As expected, for the solvents having fast polarization relaxation, the escape probability is independent of the polarization relaxation time. (ii) In the slow relaxation limit, there is a significant dependence of escape probability and average rate on the initial solvent polarization, again displaying the effects of polarization caging. Escape probability increases, and the average rate decreases on increasing the initial polarization. Again, in the fast polarization relaxation limit, there is no effect of initial polarization on the escape probability and the average rate of IR. (iii) For normal and barrierless regions the dependence of escape probability and the rate of IR on initial polarization is stronger than in the inverted region. (iv) Because of the involvement of dynamics along R coordinate, the asymmetrical parabolic (that is, non-Marcus) energy gap dependence of the rate is observed.

Acceleration of the Recombination of Photoexcited Donor—Acceptor Complexes with a High-Frequency Vibrational Mode

The recombination dynamics of ion pairs formed by the photoexcitation of donor—acceptor complexes in a polar solvent was investigated using the stochastic approach. The reorganization of high-frequency intramolecular vibrational modes plays an important role in these reactions. It was shown for the perylene—tetracyanoethylene complex in acetonitrile solution that the overwhelming majority of ion pairs recombined at the hot stage, during solvent polarization relaxation.

Theory of ion solvation dynamics in mixed dipolar solvents

Time dependent density functional theory in its “extended linear” or “surrogate” form is used to investigate the dynamics of selective ion solvation in binary dipolar solvents. It is shown that simple analytical approximations that trap the basic physics of the solvation process can be obtained. In particular, it is found that the relaxation of the solvent number densities about a charged solute is governed by two distinct modes clearly associated with electrostriction and redistribution processes. This is consistent with the physical picture suggested by molecular dynamics (MD) simulations. The solvent polarization relaxation is also dominated by two modes associated with the two rotational diffusion constants of the binary solvent. In addition to the analytical approximations, full numerical solutions of the extended linear theory are obtained and the dependence of the relaxation on solvent density and solute charge is discussed. Detailed comparisons of the theory with MD simulations for a closely related model indicate that the theory is qualitatively correct, but quantitatively poor generally predicting relaxation rates which are too fast. This is due mainly to the neglect of inertial or non-Markovian effects in the theoretical approach.

Non-exponentiality in electron transfer kinetics: Static versus dynamic disorder models

Non-exponential electron transfer kinetics in complex systems are often analyzed in terms of a quenched, static disorder model. In this work we present an alternative analysis in terms of a simple dynamic disorder model where the solvent is characterized by highly non-exponential dynamics. We consider both low and high barrier reactions. For the former, the main result is a simple analytical expression for the survival probability of the reactant. In this case, electron transfer, in the long time, is controlled by the solvent polarization relaxation—in agreement with the analyses of Rips and Jortner and of Nadler and Marcus. The short time dynamics is also non-exponential, but for different reasons. The high barrier reactions, on the other hand, show an interesting dynamic dependence on the electronic coupling element,V el.

Effects of solvent polarization relaxation on nonadiabatic outersphere electron transfer reactions in ultrafast dipolar solvents

Since the important work of Efrima and Bixon [J. Chem. Phys. 70, 3531 (1979)], it is believed that solvent polarization relaxation is usually too slow (compared to the rate of electron transfer) or the amplitude of energy fluctuation too large to have any noticeable effect on the dynamics of the nonadiabatic (NA) electron transfer reactions. On the other hand, recent studies have demonstrated that solvent polarization relaxation in several common dipolar liquids can proceed at a rate much faster than that anticipated in the earlier studies. This calls for a re-examination of the role of solvent dynamics on NA electron transfer reactions in these ultrafast solvents. In this paper, the results of such studies are presented for NA reactions in water and acetonitrile. It is found that because of ultrafast solvation, many NA reactions may lie in the dynamic region where the solvent effects are just beginning to be important. The present study further reveals the following new results. (i) In the case of high barrier reactions in solvents such as acetonitrile, the polarization relaxation in the reactant well can contribute significantly to the total rate of the nonadiabatic electron transfer reaction. (ii) In water, on the other hand, the reactive friction is still sufficiently high to make energy diffusion in the reactant well efficient and so, the solvent effects are predicted to be negligible. This is in accord with the earlier theoretical suggestions and is demonstrated here clearly for real systems. (iii) We find an interesting limiting situation where the long time rate can be significantly larger than the rate given by the Marcus expression.

Effects of the Solvent Dynamics and Vibrational Motions in Electron Transfer

Abstract Recent theoretical and experimental progress concerning electron transfer (ET) in solution is reviewed by focusing on the mechanism of ET, which occurs much faster than solvation dynamics. Theories of ET in solution are briefly reviewed with particular emphasis placed on the relation to solvent dynamics. Experimental methods to investigate solvent polarization relaxation are described. Ultrafast intramolecular ET, which is found in back ET from the photo-induced charge-transfer state to the ground state, is described concerning highly polar betaines and mixed-valence compounds. Ultrafast intermolecular ET has been observed for the systems of various dyes in electron-donating solvents. A non-exponential process with a significant temperature dependence was observed in aniline. A faster ET with a single exponential decay as fast as 1013 s−1 was observed with no temperature dependence in a system of oxazine 1 in N,N-dimethylaniline. The ET rate constants of excited coumarins in electron-donating solvents drastically depend on the substituent groups of the coumarin. Relatively small Stokes shifts in steady-state fluorescence spectra of ultrafast reacting molecules in solution are evidence of a “chemical timing” effect; namely, the reaction occurs in a non-equilibrium configuration of the solvent. These experimental observations are explained in terms of the extended Sumi–Marcus theory, in which the effect of solvation dynamics and low- and high-frequency vibrational modes are taken into account.

Qualitative kinetics of electron transfer reactions

We revise the conventional theory of electron transfer reactions and give a qualitative picture of the turnover from non-adiabatic to adiabatic regime consistent with the general principle that equilibrium properties can depend on no kinetic parameters. It is shown that if the solvent polarization fluctuations are fast enough then the reaction is non-adiabatic and the solvent follows the instantaneous charge distribution. No kinetic characteristics of the solvent enters the expression for the rate constant which in this case is determined by the total reorganization energy of the system. If a slow component of polarization fluctuations can be identified then a high frequency dielectric permittivity of the solvent can be introduced self-consistently. In the normal region the reaction is adiabatic and the charge distribution follows the solvent polarization relaxation in the potential which has the form of the free energy of the two level system with no interaction between the states at fixed value of the slow reaction coordinate. In this regime the reaction rate is determined by the reorganization energy of the slow degrees of freedom. It is also shown that in the inverted region the reaction can never be adiabatic.

Solvation structure and dynamics of the Fe2+–Fe3+ ion pair in water

Computer simulations are performed to study the structure of the coordination shells of Fe+2 and Fe+3 ions fixed at a very close reactive separation. The simulations show that it is possible for the two octahedral aquo complexes, i.e., Fe(H2O)62+ and Fe(H2O)63+, to come as close as 5 Å without disrupting their coordination shells. The reorientational dynamics within the hydration shells of these ions at this separation is examined by studying the time correlation functions (TCFs) of the unit vectors on the water molecules and along the iron–oxygen vector. The quantities related to the solvent polarization relaxation during a change in the charge, the dipole moment, and the quadrupole moment located at the solute ions are examined by studying the corresponding TCFs in the system. The TCFs exhibited a bimodal response, with a very fast initial relaxation due to inertial motions of the solvent, followed by a long tail corresponding to a diffusive component. The polarization fluctuations are also estimated via the cavity field time correlation function (CFTCF), which is useful in the theory of electron transfer processes. The memory kernel or the time dependent friction (TDF) of the solvent is also estimated from the force–force time correlation function. The cross correlations between the Coulombic and the non-Coulombic components of the forces at the ions contribute significantly to the TDF.

Effects of molecular size in solvation dynamics

The effects of molecular size on the dynamics of polar solvation are studied by using a microscopic theory which includes the translational relaxation modes of the solvent consistently. It is shown that while in the absence of the translational contribution the solvation rate increases with the size of the solute (in agreement with the conclusions of the nonequilibrium MSA theory), a complete reversal of the solute size dependence occurs when translational modes make a significant contribution to the solvent polarization relaxation

Polarization relaxation, dielectric dispersion, and solvation dynamics in dense dipolar liquid

A unified treatment of polarization relaxation, dielectric dispersion and solvation dynamics in a dense, dipolar liquid is presented. It is shown that the information of solvent polarization relaxation that is obtained by macroscopic dielectric dispersion experiments is not sufficient to understand dynamics of solvation of a newly created ion or dipole. In solvation, a significant contribution comes from intermediate wave vector processes which depend critically on the short range (nearest-neighbor) spatial and orientational order that are present in a dense, dipolar liquid. An analytic expression is obtained for the time dependent solvation energy that depends, in addition to the translational and rotational diffusion coefficients of the liquid, on the ratio of solute–solvent molecular sizes and on the microscopic structure of the polar liquid. Mean spherical approximation (MSA) theory is used to obtain numerical results for polarization relaxation, for wave vector and frequency dependent dielectric function and for time dependent solvation energy. We find that in the absence of translational contribution, the solvation of an ion is, in general, nonexponential. In this case, the short time decay is dominated by the longitudinal relaxation time but the long time decay is dominated by much slower large wave vector processes involving nearest-neighbor molecules. The presence of a significant translational contribution drastically alters the decay behavior. Now, the long-time behavior is given by the longitudinal relaxation time constant and the short time dynamics is controlled by the large wave vector processes. Thus, although the continuum model itself is conceptually wrong, a continuum model like result is recovered in the presence of a sizeable translational contribution. The continuum model result is also recovered in the limit of large solute to solvent size ratio. In the opposite limit of small solute size, the decay is markedly nonexponential (if the translational contribution is not very large) and a complete breakdown of the continuum model takes place. The significance of these results is discussed.

Microscopic expression for time-dependent solvation energy of ions and dipoles in dense polar liquids

Microscopic expressions for the time dependence of solvation energies of newly created ions and dipoles in a dense dipolar liquid are presented. The dynamics of solvation of an ion differ considerably from that of a dipole, especially that the long wavelength (k = 0) component of solvent response is totally absent for dipoles. The translational modes of the solvent molecules lead to a breakdown of Onsager's conjecture on the distance dependence of solvent polarization relaxation.

The role of translational diffusion in the polarization relaxation in dense polar liquids

The role of translational diffusion in the dynamics of polar solvation is investigated by using a Smoluchowski-Vlasov equation. It is shown that translational diffusion plays an important role in solvent polarization relaxation at intermediate wave numbers ( k≈2π/σ, σ is the solvent molecular diameter) and that the microscopic structure of the polar liquid must be considered in order to understand the role of translational diffusion. The translational contribution to polarization relaxation depends on the relative sizes of the solute and the solvent molecules, on the relative values of translational and rotational diffusion coefficients and also on the order of the multipole that gives the solute charge distribution.

Dynamical polar solvent effects on solution reactions: A simple continuum model

The influence of polar solvent dynamics in solution reactions is investigated for a simple model. In this model, a charge is subject to a chemical free energy barrier, the successful crossing of which constitutes reaction. The retarding influence of the time dependent solvent polarization is described at the continuum Debye relaxation level. The reaction rate constant k is determined as a function of the barrier curvature, the charge-solvent interaction strength, and the solvent polarization relaxation time. The reduction of k with increasingly slower solvent relaxation is found to depend sensitively on the magnitude of charge-solvent interactions. When the latter are weak, the behavior of k differs qualitatively from a standard Kramers prediction. The analogous reaction problem for a dipole is discussed briefly. Directions for a more realistic treatment of polar solvent effects on reaction rates are described.