Nuclear tunneling effects

DR. EPHRAIM BUHKS (University of Delaware) I would like to ask your opinion about a possible interpretation of proton transfer in terms of nuclear tunneling effects. Might it be possible that as the energy of the vibrational modes becomes very large, the classical rate theory might not work ... 

In the classical activated-complex formalism nuclear tunneling effects are neglected. In addition, the electron transfer is assumed to be adiabatic. These assumptions are relaxed in the semiclassical model. 

From the expressions given for example in Refs. [4,9,29], it can be seen that the nuclear factor, and consequently the electron transfer rate, becomes temperature independent when the temperature is low enough for only the ground level of each oscillator to be populated (nuclear tunneling effect). In the opposite limit where IcgT is greater than all the vibrational quanta hco , the nuclear factor takes an activated form similar to that of Eq. 1 with AG replaced by AU [4,9,29]. The model has been refined to take into account the frequency shifts that may accompany the change of redox state [22]. 

The principal focus of the present article is on the "mass-independent isotope fractional effect" (MIF) found in atmospheric and laboratory produced ozone. When this MIF occurs, a plot of the positive or negative "enrichment" in samples versus that of in those same samples has a slope of approximately unity, rather than its typical value of about 0.52. The 0.52 is the value expected using conventional transition state theory when nuclear tunneling effects are absent. For an isotope Q, 5Q is defined in per mil as 1000 [(Q/0)/(Q/0)std - 1]/ where Q/O is the ratio of Q to in the sample and std refers to its value in some standard sample, standard mean ocean water. An example of a three-isotope plot showing a slope of 0.52 is given in Figure 2.1. 

The theoretical ket values were nearly two orders smaller than those experimentally obtained. This may be attributed to disregarding the internal vibration and/or the nuclear tunneling effect. An estimation of the former effect by Sumi-Marcus theory predicts one-two orders increase in the rate with conserving the relative difference in the rates in different solvents. 

If the electron transfer is nonadiabatic (class I) and nuclear tunneling effects are neglected, then the rate constant for electron transfer within the precursor complex is ... 

The above treatment neglects nuclear-tunneling effects. The nuclear-tunneling factors calculated for a (hypothetical) electron transfer having the inner-sphere parameters of and [Fe(HjO)jp ions (see Table 1 in 12.2.3.3.4) and no solvent re-... 

The condition (6A) is necessary for the definition of the activated complex in Eyring s theory as far as the translation motion along the classical reaction path is concerned. If this condition is not fulfilled, the quantum-mechanical penetration of the potential barrier, i.e ,the nuclear tunnel effect, has to be taken into account. Then, the formula (5A) has to be corrected by an additional factor... 

Recent calculations indicate that for reactions with participation of light atoms, such as H and its isotopes, the nuclear tunnel effect is not negligible /17-19/ and the Wigner tunneling cor-... 

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. 

In any case, nuclear tunneling cannot be avoided. The nuclear tunneling effect depends on the fact that at a low temperature the nuclei do not have enough energy to pass the activation barrier. Instead, tunneling through the barrier by the nuclei becomes relatively more important. 

To deal with the ET rate in such a case, our strategy is to combine the generalized nonadiabatic transition state theory (NA-TST) and the Zhu-Nakamura (ZN) nonadiabatic transition probability.The generalized NA-TST is formulated based on Miller s reactive flux-flux correlation function approach. The ZN theory, on the other hand, is practically free from the drawbacks of the LZ theory mentioned above. Numerical tests have also confirmed that it is essential for accurate evaluation of the thermal rate constant to take into account the multi-dimensional topography of the seam surface and to treat the nonadiabatic electronic transition and nuclear tunneling effects properly. 

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