Quantum-tunneling

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  

We also know that the one imaginary frequency associated with the transition state has the form rvfm. These quantities can be combined to form a crossover temperature, Tc, defined by 

A final complication with the version of transition-state theory we have used is that it is based on a classical description of the system s energy. But as we discussed in Section 5.4, the minimum energy of a configuration of atoms should more correctly be defined using the classical minimum energy plus a zero-point energy correction. It is not too difficult to incorporate this idea into transition-state theory. The net result is that Eq. (6.15) should be modified 

A zp is the zero-point energy corrected activation energy defined in Eq. (6.22). At intermediate temperatures, Eq. (6.24) smoothly connects these two limiting cases. In many examples that do not involve H atoms, the difference between the classical and zero-point corrected results is small enough to be unimportant. 

An interesting outcome from Eq. (6.25) is that the prefactor in this Arrhenius expression is independent of the system s vibrational frequencies. In Section 6.1 we noted that this prefactor could be estimated in terms of typical vibrational frequencies as being 1012-1013 s-1. Equation (6.25) supports this estimate As T varies from 100-500 K, kBT/h changes from 2 x 1012 to 

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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... 

Pugliano N and Saykally R J 1992 Measurement of quantum tunnelling between ohiral isomers of the oyolio water trimer Science 257 1936-40... 

The distortion caused by the field allows an electron to pass from the molecule to the tip if the applied potential is positive or from the tip to the molecule if the potential is negative. This is called field ionization (FI), and the electron transfer occurs through quantum tunneling. Little or no vibrational excitation occurs, and the ionization is described as mild or soft. 

Quantum tunnelling in chemical reactions can be visualised in terms of a reaction coordinate diagram (Figure 2.4). As we have seen, classical transitions are achieved by thermal activation - nuclear (i.e. atomic position) displacement along the R curve distorts the geometry so that the... 

While this generally has only small effects on the static magnetic properties, it may play a relevant role in providing a quantitative agreement with experimental resonance data, and in explaining quantum tunnelling relaxation of the magnetization at low temperature [51]. 

Gatteschi, D. and Sessoli, R. (2003) Quantum tunneling of magnetization and related phenomena in molecular materials. Angew. Chem. Int. Ed., 42,... 

O., Coronado, E., Cardona-Serra, S., Marti-Gastaldo, C., Clemente-Juan, J.M., Sese, J., Drung, D. and Schurig, T. (2010) Spin-lattice relaxation via quantum tunneling in an Er3+-polyoxometalate... 

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