The transmission factor

One fundamental assumption in classical transition state theory is that of no recrossing over the transition state. The advantage, of the RP theory is therefore that since it provides a hamiltonian it can be used in dynamical calculations thereby incorporating the effect of recrossing. However, it is also possible to use the hamiltonian for an estimate of the transmission factor, i.e. the correction to transition state theory from recrossing of the trajectories. An additional correction factor comes from quantum tunneling (see below). Considering the reaction rate constant it may be expressed as 

We notice that for no curvature, Ylk kpQk = h, we would obtain the transition state result, i.e. 

We notice that ti E) 1. Miller [14] also derived a transmission factor for the canonical case. The result was 

We see that the critical energy for which the TST is valid depends on the curvature. If there is no curvature, Ec approaches infinity and the transition state theory is expected to give adequate results - at least in the region where tunneling is not important. 

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These and other studies of the relative substituent effects of X and CH X in nitration were considered in terms of the transmission factor a of the methylene group. To avoid complications from conjugative interactions, attention was focussed mainly on substitution at the meta-position, and ct was defined in terms of partial rate factors by the equation ... 

Next we turn to the magnitudes of the p constants. Evidently if p = 0, there is no substituent effect on reactivity. Moreover because p = -I-1.000 by definition for the aqueous ionization of benzoic acids, we have a scale calibration of sorts. Wiberg gives examples of p as a measure of the extent of charge development in the transition state. McLennan" has pointed out that p values must first be adjusted for the transmission factor before they can be taken as measures of charge devel-... 

Table 2-15 [15] tabulates the transmission factors of the various equations. Most of these are established as correction factors to the correlation of various test data. 

Here, as before, the transmission factor F expresses the fraction of the trajectories which continue to the product state after arriving at X (see Fig. 

FIGURE 2.2. A schematic description of the evaluation of the transmission factor F. The figure describes three trajectories that reach the transition state region (in reality we will need many more trajectories for meaningful statistics). Two of our trajectories continue to the product region XP, while one trajectory crosses the line where X = X (the dashed line) but then bounces back to the reactants region XR. Thus, the transmission factor for this case is 2/3. 

In order to maintain the number of ions arriving at the ion trap, it is necessary to multiply the number above with the transmission factor TF(m), which will be dependent on mass, in order to take into account the permeability of the separation system for atomic number m (analogous to this, there is the detection factor for the SEMP it, however, is often already contained in TF). The transmission factor (also ion-optical transmission) is thus the quotient of the ions measured and the ions produced. 

In the most general case a plurality of gases will make a greater or lesser contribution to the Ion flow for all the masses. The share of a gas g In each case for the atomic number m will be expressed by the fragment factor Ffi g. In order to simplify calculation, the fragment factor g will also contain the transmission factor TF and the detection factor DF. Then the Ion current to mass m, as a function of the overall Ion currents of all the gases Involved, In matrix notation. Is ... 

Variational upper bounds to the quantum rate have been found. The trouble is that they are not very good. Typically, in the deep tunneling regime, where the transmission factor T 1 the best upper bound derived to date goes as VT T. 

Note that the region where solvent is least well equilibrated to the solute is expected to be in the vicinity of the activated complex, since it has so short a lifetime. Since non-equilibrium solvation is less favorable than equilibrium solvation, the non-equilibrium free energy of the activated complex is higher than the equilibrium free energy, and the non-equilibrium lag in solvent response thus slows the reaction. This effect is sometimes referred to as solvent friction and can be accounted for by inclusion in the transmission factor a. 

The transmission factor is related to the transition probability (P0) at the intersection of two potential energy surfaces, as given by the Landau-Zener theory.24... 

The quantities in the square brackets are just the ones known from equ. (4.19) with equ. (4.16). Due to the fixed value of pass it can be seen that for the evaluation of relative intensities the dispersion correction can be omitted. However, the transmission factor Tret(Ekin, pass) which describes the change of transmission caused by the retardation becomes very important in this case, see Fig. 4.16. It has to be determined experimentally, and in ideal cases it can be estimated on the basis of Liouville s theorem for optical systems (see Section 10.3.2). In the example shown in Fig. 4.16 the essential action of the retardation field is to change the brightness B in one dimension. (One has a one-dimensional problem because the lens produces focusing of the line source in one dimension only (for details see [GSa75]).) Following equ. (10.47) one gets (subscripts ( and r denote quantities before and after retardation)... 

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. 

Here, L is the radiance (power/solid angle area, W/cm sr) of the light source, G stands for the optical conductance (solid angle area, sr cm ), and r represents the transmission factor of the system. 

The radiant flux

thermal radiation source through a spectrometer is calculated by multiplying the spectral radiance by the spectral optical conductance, the square of the bandwidth of the spectrometer, and the transmission factor of the entire system (Eq, 3.1-9). Fig. 3.3-1 shows the Planck function according to Eq. 3.3-3. The absorption properties of non-black body radiators can be described by the Bouguer-Lambert-Beer law ... 

Since they are used in communication systems the transmission properties of optical fibers are usually listed in dB values. These are related to the transmission factor r by the following equations ... 

The transmission factor of the entire instrument (including interferometer and Rayleigh filter) is estimated to be t = 0.1. Finally, the radiant power of the Raman line equals ... 

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