Dynamic solvent effects, Kramers theory

In this chapter we consider dynamical solvent effects on the rate constant for chemical reactions in solution. Solvent dynamics may enhance or impede molecular motion. The effect is described by stochastic dynamics, where the influence of the solvent on the reaction dynamics is included by considering the motion along the reaction coordinate as (one-dimensional) Brownian motion. The results are as follows. 

In our discussion of the transition-state theory with static solvent effects, it was noticed that it is a mean field description where the effects of dynamical fluctuations in the solvent molecule positions and velocities were excluded. 

Most modern investigations of the effects of a solvent on the rate constant, where dynamical fluctuations are included, are based on a classical paper by Kramers from 1940 [1], His theory is based on the transition-state theory approach where we have identified the reaction coordinate as the normal mode of the activated complex that has an imaginary frequency. In ordinary transition-state theory, we assume that the motion in that coordinate is like a free translational motion with no recrossings. This 

It should be emphasized that these dynamical effects can lead to significant corrections to conventional transition-state theory where recrossings are neglected. However, 

Kramers idea was to give a more realistic description of the dynamics in the reaction coordinate by including dynamical effects of the solvent. Instead of giving a deterministic description, which is only possible in a large-scale molecular dynamics simulation, he proposed to give a stochastic description of the motion similar to that of the Brownian motion of a heavy particle in a solvent. From the normal coordinate analysis of the activated complex, a reduced mass pi has been associated with the motion in the reaction coordinate, so the proposal is to describe the motion in that coordinate as that of a Brownian particle of mass g in the solvent. 

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We have seen that dynamical solvent effects in the friction can lead to a breakdown of TST. As stressed above, this is also a breakdown in the equilibrium solvation assumption for the transition state and configurations in its neighborhood. In fact, the standard TST view is a special one-dimensional equilibrium perspective, i.e. a mean potential curve for the reacting species is visualized and no friction of any sort is considered. The solvent influence can be felt solely via this potential, hence it is assumed that for each configuration of the reacting species, the solvent is equilibrated. On the contrary, the discussion above about Kramers and Grote-Hynes theories has documented the importance of nonequilibrium solvation effects in a frictional language. 

Dynamic medium effects in solution kinetics were first recognized by Kramers [41], He treated the problem on the basis of the Langevin equation [42] according to which the velocity of the reactants along the reaction coordinate and the friction of the surrounding medium play a role. Details of Kramers theory are not given here but an introduction to this subject can be found elsewhere [G3], The parameters involved in quantitatively assessing the dynamic solvent effect are the frequency associated with the shape of the barrier of the transition state and a friction parameter which is related to solvent viscosity. 

Key words Kramers theory - Reaction dynamics -Solvent effects - Stochastic processes - Escape over a potential barrier... 

Chapters 9-11 deal with elementary reactions in condensed phases. Chapter 9 is on the energetics of solvation and, for bimolecular reactions, the important interplay between diffusion and chemical reaction. Chapter 10 is on the calculation of reaction rates according to transition-state theory, including static solvent effects that are taken into account via the so-called potential-of-mean force. Finally, in Chapter 11, we describe how dynamical effects of the solvent may influence the rate constant, starting with Kramers theory and continuing with the more recent Grote-Hynes theory for... 

Clearly, a general theory able to naturally include other solvent modes in order to simulate a dissipative solute dynamics is still lacking. Our aim is not so ambitious, and we believe that an effective working theory, based on a self-consistent set of hypotheses of microscopic nature is still far off. Nevertheless, a mesoscopic approach in which one is not limited to the one-body model, can be very fruitful in providing a fairly accurate description of the experimental data, provided that a clever choice of the reduced set of coordinates is made, and careful analytical and computational treatments of the improved model are attained. In this paper, it is our purpose to consider a description of rotational relaxation in the formal context of a many-body Fokker-Planck-Kramers equation (MFPKE). We shall devote Section I to the analysis of the formal properties of multivariate FPK operators, with particular emphasis on systematic procedures to eliminate the non-essential parts of the collective modes in order to obtain manageable models. Detailed computation of correlation functions is reserved for Section II. A preliminary account of our approach has recently been presented in two Letters which address the specific questions of (1) the Hubbard-Einstein relation in a mesoscopic context [39] and (2) bifurcations in the rotational relaxation of viscous liquids [40]. 

The authors proceed to calculate the reaction rates by the flux correlation method. They find that the molecular dynamics results are well described by the Grote-Hynes theory [221] of activated reactions in solutions, which is based on the generalized Langevin equation, but that the simpler Kramers model [222] is inadequate and overestimates the solvent effect. Quite expectedly, the observed deviations from transition state theory increase with increasing values of T. 

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