Zero-point vibrations tunneling

Based on C-H versus C-D zero point vibrational differences, the authors estimated maximum classical kinetic isotope effects of 17, 53, and 260 for h/ d at -30, -100, and -150°C, respectively. In contrast, ratios of 80,1400, and 13,000 were measured experimentally at those temperatures. Based on the temperature dependence of the atom transfers, the difference in activation energies for H- versus D-abstraction was found to be significantly greater than the theoretical difference of 1.3kcal/mol. These results clearly reflected the smaller tunneling probability of the heavier deuterium atom. 

One of the simplest chemical reactions involving a barrier, H2 + H —> [H—H—H] —> II + H2, has been investigated in some detail in a number of publications. The theoretical description of this hydrogen abstraction sequence turns out to be quite involved for post-Hartree-Fock methods and is anything but a trivial task for density functional theory approaches. Table 13-7 shows results reported by Johnson et al., 1994, and Csonka and Johnson, 1998, for computed classical barrier heights (without consideration of zero-point vibrational corrections or tunneling effects) obtained with various methods. The CCSD(T) result of 9.9 kcal/mol is probably very accurate and serves as a reference (the experimental barrier, which of course includes zero-point energy contributions, amounts to 9.7 kcal/mol). 

Fig. 4.5 Schematic projection of the energetics of a reaction. The diagram shows the Born-Oppenheimer energy surface mapped onto the reaction coordinate. The barrier height AE has its zero at the bottom of the reactant well. One of the 3n — 6 vibrational modes orthogonal to the reaction coordinate is shown in the transition state. H and D zero point vibrational levels are shown schematically in the reactant, product, and transition states. The reaction as diagrammed is slightly endothermic, AE > 0. The semiclassical reaction path follows the dash-dot arrows. Alternatively part of the reaction may proceed by tunneling through the barrier from reactants to products with a certain probability as shown with the gray arrow...

The main significance of the works [8] was in revealing the existence, irrespective of the barrier shape, of the finite low-temperature limit of the rate constant K(0). Even for Eckart barrier V x)= V /ch (2x/d), having an infinite width at = 0, the tunneling probability remains finite due to the existence of zero-point vibrations. 

However, such electron transfer can hardly be called a chemical reaction in the full sense of this term rather it is closer to the class of tunneling phenomena in the physics of solids and electronics. A chemical reaction, by definition, must include the reconstruction of molecules, the spatial rearrangement of atoms, and changes of lengths and angles of valence bonds the definition simply does not apply to small shifts over distances of the order of the zero-point vibration amplitude. 

It is very important to remember that this definition of a PES is based on the assumption that the atomic positions can be exactly specified, which is the ultimate condition for the structure or shape of a molecule. This means adoption of the Born-Oppenheimer (B.O.) approximation, in which the nuclei are viewed as stationary point charges, whereas the electrons are described quantum mechanically [5]. This approximation is justified by the fact that the electrons are much lighter than the nuclei and hence are moving faster. The classical nature of the atomic nuclei is usually a valid approximation, but the zero-point vibrational energy of molecules or the tunneling effect, for example, make it evident that it does not always hold. 

If zero-point vibration amplitudes of the dot are comparable with the Fermi length of the electrons, the shuttling takes place at small bias voltage. This is the case for cold dots. The constructive interference of electron waves in the tunnel gap center effectively charges the dot. In the quantum limit, this charging requires a justification of the tunnel-term concept based on the Schrodinger equation. In next section we address a more rigorous quantum mechanical picture based on the "ab-initio" SET model. 

There are, however, important advantages in focusing on the nuclear arrangements. Nuclei within a molecule are more particlelike than electrons. Whereas the Heisenberg uncertainty relation renders the concept of precise position of an electron within a molecule nearly meaningless, the concept of nuclear position within a molecule is a useful one, even if one must modify the classical idea of position with qualifiers such as zero-point vibration and tunneling. 

The CP method may be considered as a semi-classical dynamics approach where the electrons are treated quantum mechanically while the nuclear motion is treated classically. The latter implies that for example zero point vibrational effects are not included, nor can nuclear tunnelling effects be described this requires fully quantum methods, as described in the next section. 

Equations (52)-(54) show that finite rates of creep, relaxation of Young s modulus, and fracture at T - 0 are conditioned by tunneling of the atoms through the potential barrier and the existence of zero-point vibrations. These phenomena are not taken into account in the classic Arrhenius-type equations. The tunneling contribution in the kinetics of the processes examined depends on the characteristic temperature. 

When the central barrier in a double energy minimum decreases below the lowest zero point vibrational level, the situation is equivalent to a true single minimum case (Forsen et al., 1978). A carbon tunnelling mechanism suggested to explain the dynamic behaviour in 2-norbornyl cations, (Fong, 1974 Dewar and Merz, 1986) has been shown to be unimportant (Yannoni etaL 1985). 

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