One-dimensional barrier

Fig. 3. One-dimensional barrier along the coordinate of an exoergic reaction. Qi(E), Q i(E), QiiE), Q liE) are the turning points, coo and CO initial well and upside-down barrier frequencies, Vo the barrier height, — AE the reaction heat. Classically accessible regions are 1, 3, tunneling region 2.

Fig. 5. Arrhenius plot of k T) for one-dimensional barrier with to Iota = 1. 0.5, 0.25 for the curves 1-3, respectively 2nVa/hota = 10, prefactor is taken constant.

The zero mode, which is associated with the longitudinal fluctuations, is now included in (4.10), and when wt>0, the determinants in (4.11) do not suffer from the zero-mode problem. The value klD is simply the rate of tunneling (3.67) in the dynamical one-dimensional barrier V%s) along the instanton trajectory. As for Bt, it incorporates the effect of transverse vibrations around the instanton trajectory. To calculate (4.10), one may employ the apparatus of Chapter 3 designed for one-dimensional tunneling. In particular, now it is possible to make use of (3.69) together with (3.66), which gives... 

Figure 5.1 with both the sudden and adiabatic approximations. For the purposes of demonstration, the adiabatic barrier height has been taken to be half the one-dimensional barrier V = V0/2, so that b =, C = Cl. One sees that the sudden approximation is realized only for fairly low vibrational frequencies, while the adiabatic approximation becomes excellent for fl s 2. 

In connection with the aforementioned example, it is useful to note a simple method for estimating the height of a one-dimensional barrier when the values of A and the barrier width are known [Gomez et al., 1967]. For two displaced identical parabolic terms, the tunneling splitting... 

One can form an idea of the magnitude of the corrections from the one dimensional case where, for several potentials, one has exact solutions of the Schrddinger equation which make a power series development unnecessary. Bell t has calculated the penetration of Eckart s one dimensional barrier (Eig. 5a) as function of the temperature. His results for the reaction rate, the classical values, for the same conditions... 

Isotopic variations of H + H2, obtained by replacing hydrogens by deuteriums, were considered (Wu et al., 1973a) within a different computational scheme (Johnson, 1972) and, in connection with threshold behaviours and resonances for varying isotopic combinations. No significant reaction probabilities were found for total energies below the static barrier potential V s). One-dimensional barriers provided reasonable probabilities only... 

Cuccaro et al. (96) interpreted the time delays in Table 2 as resonances and assigned a value of 0 for the third quantum number v without explanation. We identify these resonances as quantized transition states. The analysis presented above of scattering by one-dimensional barriers, with the conclusion that the v = 0 pole is the most important because it is closest to the real energy axis, supplies a justification for the assignment of the third quantum number. 

These functions have been used indeed in Chapter II, Sec.4.1 to calculate the transition probability for a one-dimensional barrier. There, it has been pointed out that the quasiclassical formula (92.11) gives almost the same results as the more accurate expression (99 II) based on the exact wave functions of the harmonic oscillator. This justifies the use of the definition (106.II) of the transition probability, which results from equations (105.11), also in the case of a restricted motion on one or both sides of the barrier if we relate the momentum p. or P2 to the corresponding equilibrium position in the initial or final state of the system, respectively. 

MOTT and WATTS-TOBIN /164/ make use of an adiabatic one-dimensional barrier model for the electro-deposition of metals, which implies that the ion motion is so slow that the quantum states of the solvent and the metal are unchanged. Then, the reaction coordinate is a straight line normal to the metal surface. This adiabatic limitation is, however, not necessary for the derivation of Tafel equation which directly follows from the above general consideration. 

G. A. Arteca and P. G. Mezey, ]. Comput. Chem., 9,728 (1988). Validity of the Hammond Postulate and Constraints on General One-Dimensional Barriers. 

For the present purposes the Bell truncated-parabola treatment will be adopted when a specific barrier is needed. The conclusion should not be drawn that the barrier is a truncated parabola with such and such dimensions E and a, but rather that the data resemble those calculated for a hypothetical one-dimensional barrier of these dimensions. The statement in this form recognizes the artificiality and inadequacy of the model, yet allows a level of quantitative treatment... 

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