Bell correction, tunnelling

Becke and Roussel (BR) functional, 185 Bell, Evans, Polanyi (BEP) principle, 364 Bell correction, tunnelling, 391 Bending energy, in force fields, 11 BLYP function, 188 Boltzman probability function, 374 Bond critical points, electron density analysis, 225... 

An analysis of a tunneling correction from the experimental kn/ko relies upon a calculation of the transition-state structure to obtain the bond-stretch kn/kn, and the tunneling effect. Once the transition-state structure is calculated, the tunneling probability is primarily a function of the imaginary frequency (v ) for the reaction coordinate. The truncated Bell correction is shown in Eq. (10.8). This correction... 

The enzyme, soybean lipoxygenase-1 (SLO), offers an excellent system in which to illustrate the power of Eq. (10.20) in reproducing experimental data. Several characteristics of SLO are impossible to interpret through a Bell-like tunneling correction. The KIE on k at, k at = 81 + 5, is nearly temperature independent,... 

Thus each of the factors (a), (b) and (c) suggests that the tunnelling contribution to /ch//cd will show the same qualitative variation as that from zero-point energy, with a maximum value for a symmetrical transition state. Provided that the correction is not large it will represent only a minor accentuation of the Westheimer effect. A number of calculations have modified Westheimer s model to include behaviour of this type [26, 85, 87]. Typical maximum values of coJ,(H) used have been 700/and 1000/cm for which at 25 C, Bell s tunnel... 

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. 

Bell, R. P. Tunnel effect corrections for parabolic potential barriers,Trans. Far. Soc. 55, 1 (1959). 

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. 

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... 

A further point of interest regarding this problem has been raised by Bell et al. (1971). Calculations based on an electrostatic charge cloud model indicate that the variation in kH/kD is primarily determined by the tunnel correction. Different reactions will have different barrier widths, hence different tunneling probabilities, and, in the context of this hypothesis, different variations of isotope effects. The hypothesis still predicts, however, that for a given system kH/kD will have maximum value for the symmetrical transition state where the probability of tunneling is highest. 

This variation in the isotope effect, due to variation in the isotope sensitivity of vf, has been called the Westheimer symmetry effect, and it will be one of the central ideas of this paper. However, in connection with proton transfers it has been attacked, for when more realistic potential energy surfaces are used (that is pf > 0), a much greater degree of force constant asymmetry is required to get a much reduced isotope effect9-11. Bell has therefore suggested that much of the observed variation in the isotope effect is due to variation in the tunnel correction, QhIQd-... 

Attempts to calculate theoretical values for the isotope effects and their temperature dependence were made using a linear activated complex model and a Sato potential energy surface. Various tunneling corrections were applied but only the Bell model ° predicts the curvature observed in log (fcio/ ii) versus l/T. Similar theoretical isotope effect predictions were found using a non-linear transition state model. 

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