Tunneling zero point energy

The square of the wavefunction is finite beyond the classical turrfing points of the motion, and this is referred to as quantum-mechanical tunnelling. There is a further point worth noticing about the quantum-mechanical solutions. The harmonic oscillator is not allowed to have zero energy. The smallest allowed value of vibrational energy is h/2jt). k /fj. 0 + j) and this is called the zero point energy. Even at a temperature of OK, molecules have this residual energy. 

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). 

Wong K-Y, Gao J (2008) Systematic approach for computing zero-point energy, quantum partition function, and tunneling effect based on Kleinert s variational perturbation theory. J Chem Theory Comput 4(9) 1409-1422... 

Thnnelling has sometimes been regarded as a mysterious phenomenon by chemists. It is worth stressing, therefore, that tunnelling has the same firm foundation in quantum mechanics as zero-point energy, which is the most important component of a KIE both these phenomena are a consequence of Heisenberg s uncertainty principle. 

Because of their dependence on mass, KIEs have been used in two ways to detect tunnelling. One is that primary deuterium KIEs are larger than predicted on the basis of zero-point energy alone when tunnelling makes a significant contribution to the KIE. For example, primary deuterium KIEs larger than 25 have been reported (Lewis and Funderburk, 1967 Wilson et al., 1973) for proton transfer reactions where tunnelling is important. 

Qualitatively the results are explained in the following way. Although the transferring deuterium atom does not introduce a primary isotope effect due to zero-point energy differences into ko/ko, there is less tunnelling when deuterium is transferred than when hydrogen is transferred. Therefore, the tunnel correction to the secondary /c°//cd is small relative to that for k /k. Thus, the experimental results are in agreement with the results of the model calculations. 

Finally, it is important to realize that the application of several criteria is advisable if reliable estimates of tunnelling are to be obtained. Just one criterion, e.g. the magnitude of the secondary KIE, may be misleading since a large KIE may be the result of a small tunnelling contribution and a large zero-point energy contribution. 

Fig. 10.1 Zero point energy diagrams, (a) An H or D atom attacking an H2 molecule. The TST isotope effect is negative (inverse, kn > kn) because there is no zero point isotope effect in the ground state, and tunneling is ignored in the TST approximation, (b) An H atom attacking either an H2 or D2 molecule. The isotope effect calculated in the TST approximation is positive (normal, kH > kn) because the zero point isotope effect in the ground state is larger than that in the transition state.

Non-unit kinetic isotope effects such as the rate-constant ratio kn/k-Q also derive from isotopic zero-point energy differences in the reactant state and in the transition state. A second manifestation of the Uncertainty Principle may also contribute to kinetic isotope eff ects, namely isotopic differences in the probability of quantum tunneling through the energy barrier between the reactant state and the product state. 

Fig. 2 Schematic representation of the so-called semiclassical treatment of kinetic isotope effects for hydrogen transfer. All vibrational motions of the reactant state are quantized and all vibrational motions of the transition state except for the reaction coordinate are quantized the reaction coordinate is taken as classical. In the simplest version, only the zero-point levels are considered as occupied and the isotope effect and temperature dependence shown at the bottom are expected. Because the quantization of all stable degrees of freedom is taken into account (thus the zero-point energies and the isotope effects) but the reaction-coordinate degree of freedom for the transition state is considered as classical (thus omitting tunneling), the model is ealled semielassieal.

All approaches for the description of nonadiabatic dynamics discussed so far have used the simple quasi-classical approximation (16) to describe the dynamics of the nuclear degrees of freedom. As a consequence, these methods are in general not able to account for processes or observables for which quantum effects of the nuclear degrees of freedom are important. Such processes include nuclear tunneling, interference effects in wave-packet dynamics, and the conservation of zero-point energy. In contrast to quasi-classical approximations, semiclassical methods take into account the phase exp iSi/h) of a classical trajectory and are therefore capable—at least in principle—of describing quantum effects. 

Fortunately, it is relatively simple to estimate from harmonic transition-state theory whether quantum tunneling is important or not. Applying multidimensional transition-state theory, Eq. (6.15), requires finding the vibrational frequencies of the system of interest at energy minimum A (v, V2,. . . , vN) and transition state (vj,. v, , ). Using these frequencies, we can define the zero-point energy corrected activation energy ... 

Several theoretical studies have addressed the problem of the relative stabilities of vinylidene and acetylene, one of the most recent concluding that the classical barrier to Eq. (1) is 4 kcal zero-point energy effects lower this to 2.2 kcal (4). Tunneling through this barrier is extremely rapid the calculated lifetime of vinylidene is ca. 10 " sec, which agrees with a value of ca. 10-10 sec deduced from trapping experiments (5). 

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