Solvent friction, Kramers

From these potential energy curves, the reaction rate can be calculated with the aid of Kramers theory. In the limit of a high solvent friction y, the rate is given by Kramers [1940] and Zusman [1980]... 

In Kramers theory that is based on the Langevin equation with a constant time-independent friction constant, it is found that the rate constant may be written as a product of the result from conventional transition-state theory and a transmission factor. This factor depends on the ratio of the solvent friction (proportional to the solvent viscosity) and the curvature of the potential surface at the transition state. In the high friction limit the transmission factor goes toward zero, and in the low friction limit the transmission factor goes toward one. 

If equilibrium solvation is the only cause of the solvent effect then the Mu reaction should also be a factor 35 faster in aqueous solution compared to the gas phase. This was not observed, the increase of its rate constant in water for addition to benzene amounts to only a factor of 3-5 (Figure 8), and it is not limited by diffusion. The difference was ascribed to a dynamic solvent effect and taken as evidence of Kramers solvent friction which increases with frequency and is thus obviously far more important for the reaction of Mu, the lighter isotope [33]. 

Recently, a QUAPI procedure was developed suitable for evaluating the full flux correlation function in the case of a one-dimensional quantum system coupled to a dissipative harmonic bath and applied to obtain accurate quantum mechanical reaction rates for a symmetric double well potential coupled to a generic environment. These calculations confirmed the ability of analytical approximations to provide a nearly quantitative picture of such processes in the activated regime, where the reaction rate displays a Kramers turnover as a function of solvent friction and quantum corrections are small or moderate, They also emphasized the significance of dynamical effects not captured in quantum transition state models, in particular under small dissipation conditions where imaginary time calculations can overestimate or even underestimate the reaction rate. These behaviors are summarized in Figure 7. 

The first major modification of TST was made by Kramers [12], who considered in more explicit detail the dynamics of barrier crossing and investigated a particular role of the solvent On one hand, the solvent fluctuations occasionally bring particles to the barrier top, but fluctuations in a solvent are indelibly connected with another property of the solvent it causes friction. Kramers s idea is best worded by the abstract to his famous paper ... 

The first term on the right-hand side represents the force due to the parabolic barrier, the second term is the frictional force with C the solvent friction experienced during the motion, and the final term F (t) is a random force which causes the fluctuations in the position and velocity. The mass m must be interpreted as the reduced mass related to the reactive mode in nuclear motion. This equation is equivalent to the Fokker-Planck equation used by Kramers, but it is much easier to solve. The solution of this equation is straightforward, and from the solution the current correlation function in the integral (Eq. 9.2) over the barrier can be calculated with the given boundary conditions. Some standard mathematics, well, actually quite a bit, leads to the result aheady derived by Kramers for the rate ... 

In the Smoluchowski limit, one usually assumes that the Stokes-Einstein relation (Dq//r7)a = C holds, which fonns the basis of taking the solvent viscosity as a measure for the zero-frequency friction coefficient appearing in Kramers expressions. Here C is a constant whose exact value depends on the type of boundary conditions used in deriving Stokes law. It follows that the diffiision coefficient ratio is given by ... 

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