Tunnel effects

The origin of the mnnel effect can be found in the wave-particle duahty proposed by de Broglie 

In quantum mechanics, the wave function that describes the movement along a direction X of a system with a mass m and energy E is 

This equation is rather precise, even for energies close to the barrier maximum, A V, and it is exact for potential barriers with the shape of an inverted parabola. When E is much smaller than AV, a case commonly found in molecular systems, the above expression can be simplified to 

Finally, a potential barrier that closely approaches the classical reaction path of an atom-transfer reaction and that also has an analytical solution for the transmission probability, is the Eckart barrier [14] 

In a real system we do not have one particle with a given eneigy colliding with ihe barrier, but many molecules with a Boltzmann distribution of energies at the temperature T. Thus, for real systems, we have to integrate the ttansmission probability G E) over all the energies 

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Truong T N 1997 Thermal rates of hydrogen exchange of methane with zeolite a direct ab initio dynamics study on the importance of quantum tunneling effects J. Rhys. Chem. B 101 2750... 

Bell, R.P., 1980, The Tunnel Effect in Chemistry (Chapman and Hall, London). 

It appears that the bending vibrations, which are ignored in the above discussion, may act to increase as can the tunneling effect, so it is possible for the... 

The main approximation of such one-dimensional corrections is that the tunnelling is assumed to occur along the MER This may be a reasonable assumption for reactions having either early or late (close to either reactant or product) transition states. For reactions where both bond breaking and formation are significant at the TS (as is usually the case), the dominant tunnel effect is comer cutting (Figure 16.5), i.e. the favoured... 

Lifnbach et al. [92JA9657 97BBPG889] made an exhaustive study of proton transfer in solid pyrazoles. For instance, the activation barriers, isotope and tunneling effects of the dimer 67, the trimer 68, and the tetramer 69 were determined. Catemers, like pyrazole itself, do not show dynamic behavior. 

Many computational studies in heterocyclic chemistry deal with proton transfer reactions between different tautomeric structures. Activation energies of these reactions obtained from quantum chemical calculations need further corrections, since tunneling effects may lower the effective barriers considerably. These effects can either be estimated by simple models or computed more precisely via the determination of the transmission coefficients within the framework of variational transition state calculations [92CPC235, 93JA2408]. 

The percolation theory [5, 20-23] is the most adequate for the description of an abstract model of the CPCM. As the majority of polymers are typical insulators, the probability of transfer of current carriers between two conductive points isolated from each other by an interlayer of the polymer decreases exponentially with the growth of gap lg (the tunnel effect) and is other than zero only for lg < 100 A. For this reason, the transfer of current through macroscopic (compared to the sample size) distances can be effected via the contacting-particles chains. Calculation of the probability of the formation of such chains is the subject of the percolation theory. It should be noted that the concept of contact is not just for the particles in direct contact with each other but, apparently, implies convergence of the particles to distances at which the probability of transfer of current carriers between them becomes other than zero. 

Theoretically, the problem has been attacked by various approaches and on different levels. Simple derivations are connected with the theory of extrathermodynamic relationships and consider a single and simple mechanism of interaction to be a sufficient condition (2, 120). Alternative simple derivations depend on a plurality of mechanisms (4, 121, 122) or a complex mechanism of so called cooperative processes (113), or a particular form of temperature dependence (123). Fundamental studies in the framework of statistical mechanics have been done by Riietschi (96), Ritchie and Sager (124), and Thorn (125). Theories of more limited range of application have been advanced for heterogeneous catalysis (4, 5, 46-48, 122) and for solution enthalpies and entropies (126). However, most theories are concerned with reactions in the condensed phase (6, 127) and assume the controlling factors to be solvent effects (13, 21, 56, 109, 116, 128-130), hydrogen bonding (131), steric (13, 116, 132) and electrostatic (37, 133) effects, and the tunnel effect (4,... 

Among the theories of limited applicability, those of heterogeneous catalysis processes have been most developed (4, 5, 48). They are based on the assumption of many active sites with different activity, the distribution of which may be either random (23) or thermodynamic (27, 28, 48). Multiple adsorption (46, 47) and tunnel effects (4, 46) also are considered. It seems, however, that there is in principle no specific feature of isokinetic behavior in heterogeneous catalysis. It is true only that the phenomenon has been discovered in this category and that it can be followed easily because of large possible changes of temperature. 

Figure 22 shows an application of the present method to the H3 reaction system and the thermal rate constant is calculated. The final result with tunneling effects included agree well with the quantum mechanical transition state theory calculations, although the latter is not shown here. 

More recent theoretical work has raised questions about these conclusions, how-ever. Particularly extensive calculational treatment of the rearrangement of 54 to vinyl chloride by several research groups failed to duplicate the predictions of an atypical kinetic isotope effect. These later studies indicate that tunneling effects should indeed be greater for H-shift than for the heavier D rearrangement. Consequently, the k /ko ratio should actually decrease at higher temperatures. The discrepancy in predicted results was eventually traced to an error in the earlier calculations. Nevertheless, it... 

Figure 3.5 Graphical representation of the quantum mechanical tunnelling effect between tip and sample. The probability P of a particle with kinetic energy E tunnelling through a potential barrier cf> is shown as a function of sample-tip separation z.

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). 

Hie possibility that a particle with energy Jess than the barrier height can penetrate is a quantum-mechanical phenomenon known as the tunnel effect. A number of examples are known in physics and chemistry. The problem illustrated here with a rectangular barrier was used by Eyring to estimate the rates of chemical reactions. ft forms the basis of what is known as the absolute reaction-rate theory. Another, more recent example is the inversion of the ammonia molecule, which was exploited in the ammonia maser - the fbiemnner of the laser (see Section 9.4,1). 

Many molecules have more than one well-defined structure - or even none. If there is more than one equilibrium structure the passage from one to another can take place because of the tunnel effect , although it may be impossible from a purely classical point of view. The best known example is certainly the ammonia molecule, NH3. 

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... 

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