Tunnelling Corrections

Avery accurate method to calculate tunnelling correction along a unidimensional reaction pafli involves the semi-classical approximation [4]. The transmission coefficient and the transmission probabilities are 

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Miller W H 1979 Tunneling corrections to unimolecular rate constants, with applications to formaldehyde J. Am. Chem. See. 101 6810-14... 

Except for reactions with low barriers (i.e. <10 kcal/mol at 7 = 300 K), or at high temperatures, the quantity /k T is large, and the last series can be neglected. The tunnel correction is then given completely in terms of the magnitude of the imaginary frequency. For small values of the first term may be Taylor expanded to give... 

It is possible to derive tunnel corrections for functional forms of the energy barrier other than an inverted parabola, but these cannot be expressed in analytical form. Since any barrier can be approximated by a parabola near the TS, and since tunnelling is most important for energies just below the top, they tend to give results in qualitative agreement with the Bell foimula. 

Theoretical calculations published concurrently appeared to support these suggestions. Inclusion of tunneling corrections for the calculated rate constants was found to lower the apparent and A values, consistent with the experimentally... 

The rate of hydrogen transfer can be calculated using the direct dynamics approach of Truhlar and co-workers which combines canonical variational transition state theory (CVT) [82, 83] with semi-classical multidimensional tunnelling corrections [84], The rate constant is calculated using [83] ... 

Reaction Rates, Chemical, Collision Theory of (Widom). Reaction Rates, Chemical, Large Tunnelling Corrections 5 353... 

Quantum Theory of the DNA Molecule (Lowdin) Tunnelling Corrections, Large, in Chemical Reaction Rates 8 177... 

Data taken from Saunders (1985). Semiclassical (no tunnelling correction used). 

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. 

A recently proposed semiclassical model, in which an electronic transmission coefficient and a nuclear tunneling factor are introduced as corrections to the classical activated-complex expression, is described. The nuclear tunneling corrections are shown to be important only at low temperatures or when the electron transfer is very exothermic. By contrast, corrections for nonadiabaticity may be significant for most outer-sphere reactions of metal complexes. The rate constants for the Fe(H20)6 +-Fe(H20)6 +> Ru(NH3)62+-Ru(NH3)63+ and Ru(bpy)32+-Ru(bpy)33+ electron exchange reactions predicted by the semiclassical model are in very good agreement with the observed values. The implications of the model for optically-induced electron transfer in mixed-valence systems are noted. 

A second widely used approximation uses the more smoothly shaped Eckart barrier (Fig. 6.1), which for a symmetric barrier may be expressed as V = V sech2(x) = V [2/(ex + e x)]2 where x = jts/a with s a variable dimension proportional to the displacement along MEP, and a a characteristic length. Like the Bell barrier the Eckart potential is amenable to exact solution. The solutions are similar and tunnel corrections can be substantial. In both the Bell and Eckart cases one is implicitly assuming separability of the reaction coordinate (MEP) from all other modes over the total extent of the barrier, and this assumption will carry through to more sophisticated approaches. 

The integrals extend from — so and +So lying at the edges of the barrier. For the truncated parabola the solution reduces to Equation 6.8. Note that the tunneling corrections introduced in Sections 6.3.1 and 6.3.1.1 can be used either with TST or with VTST. 

Results of a PEPICO study of the dissociation dynamics of 2-bromobutane ions have been analysed with tunnelling-corrected RRKM statistical theory using vibrational frequencies obtained from ab initio MO calculations. It has been concluded that the slow rate of loss of HBr, to form the but-2-ene ion, occurs via a concerted mechanism in which tunnelling is a feature of the proton transfer. 

In the theoretical model used the overall rate constant k is represented as the product of the no-tunnelling rate constant k° by the tunnelling correction P, ... 

The tunnelling correction P is the transmission probability through the potential barrier averaged over all possible crossing points and potential energies . An asymmetrical banier of the Eckart type l is assumed in the present model. 

FIGURE 24. Proposed transition stmcture for epoxidation of an aryl-substituted styrene (4-vinyl-biphenyl, left) and the transition stmcture for epoxidation of 1,3-butadiene (right, calculated at the QC1SD/6-31G level). The theoretical isotope effects were calculated at the MP2/6-31G level using the Bell tunneling correction. For a discussion see Reference 19b. Bond lengths are given in A... 

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