Kramers theory, modified

The second approach starts from the modified Langevin equation Eq. (37) and uses the equivalence of the Kramers theory to the multi-dimensional TST. It has been established by numerical comparison that there is agreement between the two approaches. 

Fig. 14.9 The isomerization rate on Zraw.v-stilbene displayed as a function of the inverse self-diffusion coefficient of the solvent at T = 298 K. The measurements are done at different pressures in supercritical and hquid alkane solvents Ethane (circles), propane (triangles), and -butane (squares). The solid lines represent fit to the Markovian Kramers theory that use solvent modified barrier height (A n) and barrier frequency ( >b ) From Jorg Schroeder, Ber. Bunsenges. Phys. Chem. 95, 233 (1991).

Kramers [67], Northrup and Hynes [103], and also Grote and Hynes [467] have considered the less extreme case of reaction in the liquid phase once the reactants are in collision where such energy diffusion is not rate-limiting. Let us suppose we could evaluate the (transition state) rate coefficient for the reaction in the gas phase. The conventional transition state theory needs to be modified to include the effect of the solvent motion on the motion of the reactants as they approach the top of the activation barrier. Kramers [67] used a simple model of the... 

Here the first two terms just give ma = Force as mass times the second time derivative of the friction equal to the F as the negative derivative of potential, y is the memory friction, and F(t) is the random force. Thus the complex dynamics of all degrees of freedom other than the reaction coordinate are included in a statistical treatment, and the reaction coordinate plus environment are modeled as a modified one-dimensional system. What allows realistic simulation of complex systems is that the statistics of the environment can in fact be calculated from a formal prescription. This prescription is given by the Fluctuation-Dissipation theorem, which yields the relation between the friction and the random force. In particular, this theory shows how to calculate the memory friction from a relatively short-time classical simulation of the reaction coordinate. The Quantum Kramers approach. 

Both the embedded atom method (EAM) and effective medium theory (EMT) have been used with some success to model the structure and composition of metal systems. In EAM, the potential is not linear in the number of neighbors and depends on the electron density. The parameters for both theories can be deduced from experiments or theory 1 . These systems, for the most part, have been limited to looking only at the properties of only the metal atoms without adsorbates since there were few metal-adsorbate potentials. Recent simulations by van Beurden and Kramer indicate that with some effort modified embedded atom (MEAM) potentials can be established for adsorbates in order to simulate the dynamics of the species on metal substrates. 

We have presented several approaches to calculate the rate constants of electron transfer occurring in solvent from the weak to strong electronic couplings. In the fast solvent relaxation limit, the approach based on the nonadiabatic transition state theory can be adopted. It is related to the Marcus formula by a prefactor and referred as a modified Marcus formula. When the solvent dynamics begin to play a role, the quantum Kramers-like theory is applicable. For the case where the intramoleeular vibrational motions are much faster than the solvent motion, the extended Sumi-Mareus theory is a better ehoice. As the coherent motion of eleetron is ineorporated, such as in the organic semiconductors, the time-dependent wavepaeket diffusion approach is proposed. Several applications show that the proposed approaches, together with electronic structure calculations for the faetors eontrolling eleetron transfer, can be used to theoretically predict electron transfer rates correctly. 

One of the central issues in studying reaction dynamics isf how you describe the flux over the reaction barrier, whether from simple collision theory, or transition state theory, to modified generalized Langevin theory, Kramer s theory...or even to quantum mechanical tunnelling - This is a selective not exhaustive list of possibilities (7,8). When the timescale of the photo chemical event (be it photodissociation, isomerization, charge transfer)... 

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