Temperature-dependent transmission coefficients

The quantum mechanical effect on the motion along the reaction coordinate is included in the kinetics calculations by multiplying the CVT rate constant by a temperature-dependent transmission coefficient c(T) which accoimts for tunneling and non-classical reflexion. Therefore, the final expression for the rate constant is given by ... 

The temperature dependent transmission coefficient X(S) is related to the absorption factor k by ... 

Note that both k and AG 0 depend on temperature. The transmission coefficient is sometimes called the tunneling transmission coefficient because tunneling is the main quantum effect on the reaction coordinate. 

Figure 2 (left panel) shows the energy profile for a two-level system weakly coupled to the reaction coordinate. Both the ground and excited state surfaces have two minima separated by a high barrier at (Rq) = The right panel of this figure compares the time dependent rate coefficients for quantum (QRB) and classical (CRB) treatments of the reaction coordinate for a moderately low temperature (/3 = 2). At t = 0, the CRB result for the time-dependent transmission coefficient, (t) = where is determined from a... 

With help of the Wentzel-Kramers-Brillouin approximation the field dependent transmission coefficient was calculated for the vacuum potential barrier. Integrating over all energies for a free electron gas at room temperature results in an exponentially increasing I-V curve that reaches 1 nA at 0.7 kV. With a band gap well above 2 eV proteins are good insulators, their electric breakdown is believed to be a conduction mechanism, which occurs at a voltage of about 200 V. The necessary potential for tip-emission can be further decreased to about 100 V using nano-fibers. To create transparent... 

However, often the minimum in Si or Ti which is reached at first is shallow and thermal energy will allow escape into other areas on the Si or Ti surface before return to So occurs (Fig. 3, path e). This is particularly true in the Ti state which has longer lifetimes due to the spin-forbidden nature of both its radiative and non-radiative modes of return to So-The rate of the escape should depend on temperature and is determined in the simplest case by the height and shape of the wall around the minimum, similarly as in ground state reactions (concepts such as activation energy and entropy should be applicable). In cases of intermediate complexity, non-unity transmission coefficients may become important, as discussed above. Finally, in unfavorable cases, vibronic coupling between two or more states has to be considered at all times and simple concepts familiar from ground-state chemistry are not applicable. Pres-... 

In summing up we see that the assumption of a much looser complex might have raised k by a factor of 8 to 20, depending on the temperature, while the uncertainty in the transmission coefficient could introduce another factor of 2. Considering the over-all defects of the theory we may thus estimate the uncertainties in the preexponential factors calculated in this way at about an average factor of 10. The agreement obtained here for Hs, a factor of 2, is thus well within these expected uncertainties. 

Khan [174] studied the electrooxidation of ferrocene at a Pt electrode in polar solvents ranging from methanol to heptan-l-ol. Experimental data concorded well with the calculated results when solvent influence on the pre-exponential coefficient was considered. In calculations v = rb was used. Khan [174] points out that expressed by Eq. (36) exhibits a temperature dependence different from that predicted by the classical expression = k T/h. Another conclusion which may follow from the same paper is that the transmission coefficient for the electrochemical outer-sphere electron-transfer reactions in polar alcoholic solvents may not be equal to unity. 

Bauer et al. [344] have used a similar formalism, but included a transmission coefficient. Petermann [345] purported to show that the temperature dependence of the transmission coefficient for the system H2/Ni 100 decreases from 0.1 to 0.02 between 690 and 830 K. Such temperature dependence is not completely surprising since it relates to the efficiency of energy transfer and Suhl et al. [346] have shown that there is a change in the transmission coefficient near a paramagnetic to ferromagnetic transition. 

Figure F.2 Illustrations of the transmission coefficient T as a function of the energy-dependent variable 9. The heavy line in all panels shows the form when a = T = hf2 = 1. Panel A shows in addition the values for a = 0.5 and a = 1.5. Panel B shows changes with temperature as r = 0.5 and r = 1.5. Panel C gives the form for different barrier heights /i = 1.5 and h = 2.5. Panel D indicates changing barrier widths when a = t = h/2 = 0.25 and a — T — h/2 = 4.

While the quantum instanton approximation cannot describe the recrossing effect, this effect, which is predominantly classical, could be included empirically as a multiplicative factor (a transmission coefficient ) obtained in a classical molecular dynamics simulation. The main challenge for future research is to develop accurate and efficient approximations for the rate constant, its temperature dependence, and the kinetic isotope effect that include the quantum and recrossing effects simultaneously and rigorously. 

Let us dwell now on the dependence of the rate constant on the isotopic composition of the reactant molecules which is usually called kinetic isotope effect. Various types of isotope effects are illustrated in Table 3. Assuming that the transmission coefficient is independent of the isotopic composition, Eq. (11.1) would yield the ratio of the rate constants ki/kg for reactions of molecules with a different isotopic composition. This ratio depends on the symmetry of reactants, their zero-point vibrational energies and effective masses corresponding to motions along the reaction coordinate (for detail see [222, 304]). In the classical limit (E2, EJ < kT), the ratio kj/kg depends on the ratio of effective masses rather than on temperature. In the essentially quantum case (E, EJ kT), the value of kj/kg is influenced mainly by the change in zero-point energies however, the ratio X1/X2 Iso substantially differ from unity [308, 309] as demonstrated for different isotopic variants of reaction H + Hg H2 + H. It is just the difficult calculation of the transmission coefficient that limits the applicability of the transition-state method to the calculation of the isotope effect. 

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