Stepwise tunneling model

The effect of the concentration of A particles, i.e. traps for electrons, on the kinetics of tunneling reactions has been studied [4, 21]. These investigations were made for reactions 8 and 9 in Table 1. For these reactions a change in the concentration of A particles has been found to produce no effect on the process rate. In a number of experiments the concentration of A particles would change by a factor of over 3. Such a change in A particle concentration would result in a decrease in the distance from the particles A , A, and B to the nearest A particle by a factor of 31/3. At the initial concentration of A particles of 0.05 M and at reasonable values of ae = 1-2 A, the probability of stepwise tunneling over A particles would increase by more than 102-104 times. However, as has been pointed out, in reality no increase has been observed in the rate of the processes with increasing concentration of A particles. 

the total sum of the experimental data obtained in studying low-temperature electron transfer reactions as well as the results of theoretical analysis indicate that these reactions proceed via direct tunneling rather than via thermal diffusion or stepwise tunneling. 

Ershov, A.K. Pikaev, P.Ya. Glazunov and V.I. Spitsyn, Dokl. Akad. Nauk SSSR, 149 

Khairutdinov and K.I. Zamaraev, Proc. 4th Tihany Symp. Radiat. Chem., Akademiai Kiado, Budapest, Hungary, 1976, Vol. 1, p. 941. 

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This means that at low temperature where P is large the HD reaction is ca. twice as fast as the DD reaction. Equation (6.31) has been used in connection with the Bell-Limbach tunneling model to describe the stepwise double proton transfer in porphyrins, azophenine, and oxalamidines, as will be discussed in Section 6.3. Smedarchina et al. [16] used the same equations for their quantum-mechanical treatment of the porphyrin tautomerism. 

The derivation of expressions for the multiple kinetic isotope effects of the triple hydrogen transfer case is analogous to the HH-transfer but more tedious. Therefore, the reader is referrred to refs. [25] and [26]. The main results are included in Table 6.2. As in the case of the HH-transfer, the kinetic isotope effects derived for the stepwise transfers are valid in the presence of turmeling and are independent of the tunneling model used. By contrast, the kinetic isotope effects of the single barrier reaction are affected by tunneling. 

Figure 6.41 Visualization of a modified Bell tunneling model for degenerate, stepwise double proton transfers involving an intermediate. A minimum energy , is required for proton tunneling, which can take place only in the hatched regions. Ej barrier energy, (a) and (d) E , is given by the energy of the...

Source Apportionment Models for the Cyclohexane-Soluble Fraction of Respirable Suspended Particulate Matter. Stepwise multiple regression analysis was used to determine the coefficients of the source tracers for the models proposed for CYC in equations (7)-(9). These models are shown in Table IV. As expected from the factor analyses, the coefficient for V, accounting for the greatest proportion of the variance of CYC, was fitted first into the equation. Equation (14) was the simplest and the F value was slightly higher than for equations (15) and (16). In addition, as will be discussed later in this paper, the coefficient for PB was in reasonable agreement with the ratio of CYC /PB for samples collected in the Allegheny Tunnel. 

Nonlinear behavior is also observed in the wide-range (0.1-2.5 GPa) pressure dependence of the ESPT rate of DCN2 in alcohols [44[. At low pressure, the proto-lytic photodissociation rate slightly increases, reaching the maximum value. With further pressure increase this rate decreases below the initial value at atmospheric pressure (Fig. 13.11). To explain the unique nonexponential dependence of ESPT rate constants on pressure, as well as temperature, Huppert et al. have developed an approximate stepwise two coordinate proton-transfer model that bridges the high-temperature nonadiabatic proton tunneling limit with the rate constant... 

Pathways [2, 50] is a semiempirical model which it is designed to reproduce electron transfer rates between cofactors in proteins [50, 151 ]. In essence. Pathways includes the contributions to the electronic tunneling from a stepwise path covering aU non-bonded interactions, as well as the bonded ones at the nearest neighbor level. Namely ... 

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