Tunneling masses

Quantum-chemical calculations of PES for carbonic acid dimers [Meier et al. 1982] have shown that at fixed heavy-atom coordinates the barrier is higher than 30kcal/mol, and distance between O atoms is 2.61-2.71 A. Stretching skeleton vibrations reduce this distance in the transition state to 2.45-2.35 A, when the barrier height becomes less than 3 kcal/mol. Meier et al. [1982] have stressed that the transfer is possible only due to the skeleton deformation, which shortens the distances for the hydrogen atom tunneling from 0.6-0.7 A to 0.3 A. The effective tunneling mass exceeds 2mn-... 

As stated by inequality (2.81) (see also section 4.2 and fig. 30), when the tunneling mass grows, the tunneling regime tends to be adiabatic, and the extremal trajectory approaches the MEP. The transition can be thought of as a one-dimensional tunneling in the vibrationally adiabatic barrier (1.10), and an estimate of and can be obtained on substitution of the parameters of this barrier in the one-dimensional formulae (2.6) and (2.7). The rate constant falls into the interval available for measurements if, as the mass m is increased, the barrier parameters are decreased so that the quantity d(Vom/mn) remains approximately invariant. 

The novelty of this picture is that we have established rather generally a multiparticle character of the tunneling events. This is counterinmitive because, naively, the larger the number of particles involved in a tunneling event, the larger the tunneling mass is, and the harder the mnneling becomes. This is... 

The temperature dependence of this rate constant was measured by Al-Soufi et al. [1991], and is shown in Figure 6.17. It exhibits a low-temperature limit of rate constant kc = 8x 105 s 1 and a crossover temperature 7 C = 80K. In accordance with the discussion in Section 2.5, the crossover temperature is approximately the same for hydrogen and deuterium transfer, showing that the low-temperature limit appears when the low-frequency vibrations, whose masses are independent of tunneling mass, become quantal at Tisotope effect increases with decreasing temperature in the Arrhenius region by about two orders of magnitude and approaches a constant value kH/kD = 1.5 x 103 at T[Pg.174]

In contrast, if the transfer is non-degenerate, a situation may occur as illustrated in Fig. 6.7. At the transition state there is remaining ZPE in the antisymmetric stretch. This will lead to a decrease in the difference between the effective barriers for H and for D, as has been proposed by Westheimer [45]. This decrease has also been called the Westheimer-efifect [46]. Tunneling pathways may no longer involve only changes in q, but also a substantial heavy atom motion. This means that the effective tunneling masses will be increased by an additional mass Am as illustrated schematically. 

With = 1 and mP = 2, the low-temperature rate constant is then determined mainly by for a given value of E. The low-temperature and temperature independent kinetic isotope effect k /k is, therefore, determined by which is obtained experimentally at high temperatures. In other words, kg /kg and the high-temperature kinetic isotope effects cannot be varied independently of each other, which is not in agreement with experimental data. This effect can be associated with heavy atom tunneling during the H-transfer. The tunneling mass is increased and the low-temperature H/D isotope effect decreased. 

The heavy atom contribution reduces generally the value of kg /kg. For example, if during H-tunneling in an OHO-hydrogen bond over 2a = 0.5 A both oxygen atoms are displaced, each by 2a = 0.05 A, it follows that Am= 0.32, and the total tunneling mass is 1.32 instead of 1. 

The tunneling mass of a given isotopic reaction is written as... 

If on the other hand AFJ is much smaller than Ipv, the eigenstates 1), 12) essentially coincide with the handed states L), /2). In that case the absolute value of the difference E2 — 1] between the eigenstates is approximately 2 V v - We can estimate the size of the tunneling splitting for typical chiral molecules for instance within the Wentzel-Kramers-Brillouin approximation. If we assume a quartic double well potential with a barrier height of 200 kJ mol a barrier Avidth of 200 pm and a tunneling mass... 

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