Kramer model

According to Kramers model, for flat barrier tops associated with predominantly small barriers, the transition from the low- to the high-damping regime is expected to occur in low-density fluids. This expectation is home out by an extensively studied model reaction, the photoisomerization of tran.s-stilbene and similar compounds [70, 71] involving a small energy barrier in the first excited singlet state whose decay after photoexcitation is directly related to the rate coefficient of tran.s-c/.s-photoisomerization and can be conveniently measured by ultrafast laser spectroscopic teclmiques. 

This ensures the correct connection between the one-dimensional Kramers model in the regime of large friction and multidimensional imimolecular rate theory in that of low friction, where Kramers model is known to be incorrect as it is restricted to the energy diflfiision limit. For low damping, equation (A3.6.29) reduces to the Lindemann-Flinshelwood expression, while in the case of very large damping, it attains the Smoluchowski limit... 

According to Kramers model, for flat barrier tops associated with predominantly small barriers, the transition from the low- to the high-damping regime is expected to occur in low-density fluids. This expectation is home... 

Cartesian kinetic SDEs with unprojected, geometrically projected, and inertially projected random forces require the same correction forces in certain special cases. Inertial and geometric projections are completely equivalent for models with an equal bead mass m for all beads, for which the mass tensor m v = is proportional to the identity. Unprojected and geometrically projected random forces require identical correction forces in the case of local, isotropic friction with an equal friction coefficient for all beads, as in the Rouse or Kramers model, for which the friction tensor ... 

The standard language used to describe rate phenomena in condensed phases has evolved from Kramers one dimensional model of a particle moving on a one dimensional potential, feeling a random and a related friction force. In Section II, we will review the classical Generalized Langevin Equation (GEE) underlying Kramers model and its application to condensed phase systems. The GLE has an equivalent Hamiltonian representation in terms of a particle which is bilinearly coupled to a harmonic bath. The Hamiltonian representation, also reviewed in Section II is the basis for a quantum representation of rate processes in condensed phases. Eas also been very useful in obtaining solutions to the classical GLE. Variational estimates for the classical reaction rate are described in Section III. 

If were white noise this would be the Kramers model of VIII.7, but we now assume only that has a short tc. Equation (5.15) takes the form... 

Extensions of the Kramers model are considered necessary [92-94, 97-99] although there are refined versions of the original formulation [100, 101]. Such non-Markovian dynamics has been taken into consideration... 

Often, one defines nonexponential relaxations in terms of a time-depen-dent rate coefficient k(t) through p(t) = exp(—k(t)t). For the fractional Kramers model one therefore obtains the rate coefficient k(t) = ln a(—r ta) /t which leads to two limiting cases, the short-time self-simi-... 

Let us discuss a possible application of this fractional Kramers model to protein dynamics. It was noted before [106-109] that the dynamic process in proteins, like the rebinding process of carbon monoxide CO to myoglobin Mb,... 

Two different expressions can be obtained for /< model. From the Kramer model [20], we get ... 

Another approach to solvent fluctuation control of reactions in solution based on the Kramer model (Kramer, 1940 Sumi, 1999 and references therein). According to this model a transition over a double-well potential W(q) occurs as a result of zigzag diffusion. An important parameter of the theory is the relaxation time of the average motion of the medium... 

Inspired by Christiansen s treatment of a chemical reaction as a diffu-sional problem, Kramers studied the model of a particle in Brownian motion in a one-dimensional force field and predicted the existence of three fundamental kinetic regimes, depending on the magnitude of the friction. The basic hypothesis and results of this work will be summarized below, as many of the results most recently obtained using more sophisticated models are still best described by reference to Kramers original model and reduce to Kramers models when the appropriate limits are taken. 

The motion of a particle (mass M) in the Kramers model may be described by the following Langevin equation ... 

Figure 2. The reaction rate in the Kramers model relative to its transition state value. The low friction rate is plotted for (a). 2 (ft), 5 (c), 10 (d), and 20 (e).

In the model studied by Kramers,the particles are assumed to be initially at the well around A and to be lost as they escape above the barrier. Many physical processes, however, are more realistically modeled by a bistable potential (see Fig. 3), namely when two states A and B may be inter-converted. In the original Kramers model no back-crossings from B to A were considered the particles were somehow absorbed as they arrived at B. 

Figure 16. A double-well potential for reaction IV X) along a one-dimensional reaction coordinate X in the Kramers model, and a reactive diffusive trajectory represented by a zigzag line surmounting a reaction barrier from the reactant to the product well.

Usually, T is proportional to the viscosity, rj, of solvents. Therefore, the Kramers model predicts that the rate constant becomes proportional to in viscous solvents. Indeed, in solution reactions, including electron transfer reactions, rate constants are known to decrease with an increase in rj, as mentioned before. However, the observed dependence is not but... 

The Need for Generalization of the Kramers Theory The Generalized Kramers Model Non-Markovian Effects in the One-Dimensional Case The Escape Rate of a Non-Markov Multidimensional Process... 

This chapter reviews the generalizations of the Kramers model that were develojjed during the past few years. The result of this effort, which we may call the generalized Kramers theory, provides a useful framework for the theoretical description of activated rate processes in general and of chemical reaction rates in condensed phases in particular. Some applications of this framework as well as its limitations are also discussed. In the last few years there has also been substantial progress in the study of the quantum mechanical Kramers model, which may prove useful for condensed phase tunneling reactions. This aspect of the problem is not covered by the present review. 

The Kramers model consists of a classical particle of mass m moving on a one-dimensional potential surface V(x) (Fig. 1) under the influence of Markovian random force R(t) and damping y, which are related to each other and to the temperature T by the fluctuation dissipation theorem. 

Although the Kramers model contains much of the essential physics of the activated escajje problem, it cannot be used for quantitative discussion of many realistic activated processes. In particular the model is too oversimplified for the original application intended by Kramers for chemical rate processes. The theory needs to be generalized to correct the following shortcomings of the Kramers model. 

The friction (more generally, molecule-solvent interaction) is taken in the Kramers model to be a constant, independent of the position along the reaction coordinate. As seen, generalization to position-dependent friction is trivial in the Smoluchowski limit. In many systems position-dependent friction should be considered also in the underdamped case. An obvious example is desorption where the dissociating particle ceases to feel the thermal bath as it draws further away from the surface. 

As far as comparison with experimental data is concerned, the fractional Klein-Kramers model under discussion may be suitable for the explanation of dielectric relaxation of dilute solution of polar molecules (such as CHCI3, CH3CI, etc.) in nonpolar glassy solvents (such as decalin at low temperatures see, e.g., Ref. 93). Here, in contrast to the normal diffusion, the model can explain qualitatively the inertia-corrected anomalous (Cole-Cole-like) dielectric relaxation behavior of such solutions at low frequencies. However, one would expect that the model is not applicable at high frequencies (in the far-infrared region), where the librational character of the rotational motion must be taken... 

Hynes and Kramers models for this type of process are compared with the experimental data. 

Dynamical effects in barrier crossing—The Kramers model... 

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