Theoretical studies of barrierless and very low barrier chemical reactions in solution

Abstract

The above review of experimental investigations on the photophysics of stilbene and related compounds and the torsional dynamics of TPM in dyes in condensed phase clearly indicate a strong influence of viscosity and polarity of the solvent and also of the temperature and the wavelengths of excitation and probe pulses on the non-radiative decay dynamics. Although these effects are related to each other, attempts have been made to separate the influence of each individual parameter on the dynamics. Ultrafast spectroscopic techniques with sub-picosecond time resolution have played a significant role in separating the pure static effects from the dynamic ones. However, the above studies also reveal that it is not possible always to separate the intramolecular and intermolecular effects. The experimental studies project the inadequacy of the existing one-dimensional theories in explaining the isomerization dynamics. These studies also point out the limitations of hydrodynamic models for the 'effective reactive friction.' Thus we conclude that further improvement of the existing barrierless models may be helpful in quantitative understanding of the non-radiative decay dynamics. The present thesis is an effort towards such an improvement and generalization. The main predictions of the model are the decreased non-exponentiality of the short-time decay compared to that of the pinhole sink and the weak dependence on the excitation wavelength. The latter follows from the assumption that the relaxation from the upper surfaces to the rim region in S? surface is much faster than the relaxation along the rim. The present model does not rule out a probe-wavelength dependence because of the variation of the ground-state potential energy surface along the reaction coordinate. An essential ingredient of the Mexican hat model is the multidimensional potential surface in which it is embedded. The present study can be extended in several directions, such as in the use of a more realistic sink function which can be obtained from the quantum chemical calculations carried out by Rettig [1]. We hope that because of the simplicity of the Mexican hat model of Rettig [1], it will find use in fitting to the experimental data. In this thesis, a theoretical study of various aspects of barrierless chemical reactions in solution is presented. Many of the important chemical and biological relaxation processes are known to be barrierless. These reactions defy the traditional picture of a chemical reaction as a passage over a high activation barrier. In contrast to the impressive developments and understanding of high-barrier reactions achieved in recent years, our understanding of barrierless reactions is still far from complete. Just as in the case of high-barrier reactions, these barrierless reactions also show rich and complex dynamical behavior. They reveal not only many aspects of the excited-state potential energy surfaces but also offer scope for testing the stochastic theories of chemical reaction dynamics and the relevant concepts. The studies reported in this thesis revealed several new features of barrierless reactions. Firstly, we have shown that even a small barrier can affect the dynamics of the reaction profoundly. It is found that although the Arrhenius exponential activation energy dependence of the rate breaks down for very small barriers, for the present model it is recovered at values which are still surprisingly small. Another interesting feature is the fractional power-law dependence of the rate for very small barriers. Note that this contradicts the claim of Robinson et al. [1] of the validity of Arrhenius behavior even for very small barriers. Secondly, a numerical study of barrierless reactions on a two-dimensional potential surface has been presented. This study has clearly shown that the multidimensionality of the reaction potential energy surface can have a profound effect on the dynamics of a barrierless reaction. In fact, if the frequencies of motion along the two surfaces are comparable, then the concept of reaction coordinate loses much of its meaning as there can be no preferred path for the reaction to occur. Also, the effective friction can have an important role in determining the course of the reaction. Next, we have studied the dynamics of barrierless reaction for the new model, called the Mexican hat, which has been introduced recently by Rettig [2] to explain the experimental results on electronic relaxation of TPM dyes in solution. For the special case of perfect absorption on two sides of the sink, a closed-form analytic expression has been derived. This model predicts less non-exponentiality in the short-time decay than the pinhole sink model. And lastly, a quantum chemical calculation of the intrinsic non-radiative decay rate from the excited-state surface is presented. The calculation is based on the generalized quantum Liouville equation proposed recently by Coalson and Karplus [3] and uses the split-propagator path integral method of Hellsing and Metiu [4] to obtain the density matrix and hence the decay rate. The calculated rates show the well-known exponential dependence on the energy gap between the initial and the final surfaces for small energy gaps; deviations are seen when the energy gap becomes large. The position (or the reaction coordinate) dependence of the rate is also calculated and found to have the expected sharp decrease as x is changed from the minimum of the excited-state potential surface. Although significant progress has been achieved in recent years in our understanding of the dynamics of barrierless reactions, there are still many aspects that have not been explored. We next present a list of theoretical problems which we believe need careful study in future. (i) Study of non-Markovian (or memory) effects All the theoretical studies reported so far are based on a Markovian theory of solvent response to the reactive motion. Such an approximation is clearly inadequate when the reaction time is in the picosecond or in the sub-picosecond domain. Some of the slow solvent modes that contribute to the macroscopic viscosity cannot respond to such fast motions and therefore, we must consider a non-Markovian theory with frequency-dependent friction. The importance of non-Markovian effects is well-known in high-barrier reactions and they have been intensely studied in the last decade or so. However, such a study is lacking for barrierless reactions, although much of the necessary theoretical work has already been done by Okuyama and Oxtoby [5] who have derived a non-Markovian Smoluchowski equation which can be used, with proper sink functions, to study barrierless reactions. This is certainly a worthwhile problem. (ii) Multidimensional reactions An important problem not addressed yet is the involvement of a solvent mode in the reaction. Many photoisomerization reactions involve significant charge separation during the course of the reaction, especially when the twisting motion is around a double bond. In such cases, the solvent polarization modes can get heavily involved in the reaction. An approximate way to include the influence of the solvent polarization modes is to add a second dimension whose motion is controlled by an average potential from these modes. The friction along this second dimension is given by the dielectric friction. A realization of this situation may happen in the isomerization of trans-stilbene and related compounds in pure solvents. The study presented in this thesis can be extended to understand the dynamic solvent effects in such reactions. (iii) Barrierless electron transfer reaction Experimental studies have shown that many fast, excited-state, intramolecular electron transfer reactions proceed without any appreciable activation barrier. The rate of electron transfer appears to be still controlled by solvent polarization fluctuations because the rate scales (often linearly) with the longitudinal polarization relaxation rate, ?L ( = (?m / ??) ?D ) where ?0 and ?? are, respectively, the zero and infinite frequency dielectric constants and ?D is the principal Debye relaxation time). Marcus and coworkers have recently suggested that the dynamic solvent effects on such electron transfer reactions can be understood by modeling the electron transfer reaction as a barrierless process with the solvent polarization as a reaction coordinate. Thus, the model of Sumi and Marcus is virtually identical with the model of Bagchi, Fleming, and Oxtoby. The model of Sumi and Marcus is, however, limited by its total neglect of intermolecular correlations that are present in a dense dipolar liquid and which are known to be important in solvation dynamics and in collective orientational relaxation. The correct approach would be to find the proper "microscopic" time dependence of the reaction coordinate which is like the solvation energy of an ion and then use the proper kinetic description to find the reaction rate. The multidimensionality of the reaction can also be important here, as stressed by Maroncelli et al. [5]. This is certainly an interesting problem for future study. (iv) Calculation of the time-dependent intrinsic rate In many situations, the reaction proceeds in a time-dependent environment. For example, the potential energy surfaces may themselves be functions of time because of solvent relaxation. The generalized quantum Liouville equation described in Chapter 6 should be extended to describe such situations. The above list is by no means exhaustive but it gives a glimpse of many important problems that remain unexplored in the field of barrierless reaction dynamics. We believe that this field will remain an interesting area of research in the coming years for both experimentalists and theoreticians alike.