Vibrationally adiabatic path

The classical reaction path of an atom transfer reaction 

together with a procedure to calculate tunnelling corrections, transforms classical ISM into semi-classical ISM. This appendix presents the background material necessary for such a treatment. 

The zero-point energy of the reactants is directly calculated from the Morse curve of the BC bond 

The normal modes of vibration of the linear tri-atomic transition state A- B C are the symmetric stretching, the anti-symmetric stretching and two degenerate bending modes. For a stable tri-atomic molecule, the symmetric and anti-symmetric stretchings are calculated from Wilson s equation 

We note that this equation has the correct limits if all the masses are identical, tiien = mi i if A and C are much heavier than B, then = m H. The reduced mass n is tiiat of the anti-symmetric stretch of a tri-atomic molecule with symmetry, as expected for a linear transition state. As will be seen below, the tunnelling correction employs eq. (IV.6) for calculating the reduced mass. 

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The energy of the maximum point along the vibrationally adiabatic path, eq. (IV.2), minus the zero-point energy of the reactants is the vibrationally adiabatic barrier. 

Hydrogen-atom or proton-transfer transfers between heavy atoms often lead to vibrationally adiabatic paths with two maxima. Formally, the resolution of this problem should be made in the fi amework of the canonically unified theory [5], In practice, given the approximate nature of our treatment, it suffices to use TST with AFj equal to the value for the maximum point. For these cases, the tunnelling correction of the particles wifli energies between that of the highest of the two maxima and the minimum between tiiese is calculated for the highest barrier only. 

The vibrationally adiabatic path of ISM and the LS potential are combined to reflect the fact that a hydrogen-bonded complex brings the structure of the reactants closer to that of the transition state, as shown in Mechanism (V.I). A hydrogen bond can be regarded as an incipient proton transfer, and the bond order at the precursor complex is no longer n=0, but the bond order of the B- -H bond in that complex, n. .. g. Similarly, for the products, the bond order is not n = 1 but the bond order of the H A bond in the successor complex, (1- h - a)- Thus, for a proton transfer in condensed media, the reaction coordinate n is only defined in the interval [%...b. (1 %...a)]- The precursor and successor complexes are included in the classical reaction path of ISM with a simple transformation of the reaction coordinate [3]... 

The LS-lSM vibrationally adiabatic path is then calculated adding the zero-point energies along the reaction path, as indicated in eq. (1V.2), but including also Zab-... 

Figure 5-3. Active site and calculated PES properties for the reactions studied, with the transferring hydrogen labelled as Hp (a) hydride transfer in LADH, (b) proton transfer in MADH and (c) hydrogen atom transfer in SLO-1. (i) potential energy, (ii) vibrationally adiabatic potential energy, (iii) RTE at 300K and (iv) total reaction path curvature. Reproduced with permission from reference [81]. Copyright Elsevier 2002...

The MEP for inversion corresponds to 6 = 0 and is characterized by the barrier height VWhen C/2V, > 1, apart from this MEP, there is a path that includes two segments described by Eq. (8.42) and a second-order saddle point. The barrier along this path is greater than V, and equal to U,(l + 2V0/C). The transverse frequency along the straight-line MEP for inversion has a minimum at the saddle point q = 0, 0 = 0 consequently, the vibrationally adiabatic barrier is lower than the static one. 

When 2Qpv/cob> 1, we are in the slow-flip , or adiabatic , or small curvature limit, where the Q vibration adiabatically follows the 5 coordinate and transfer takes place along the MEP path (i.e. at the saddle point). 

We first consider the case where the reaction probabilities are computed for the adiabatic model with the reaction-path curvature neglected, the so-called vibrationally adiabatic zero-curvature approximation [36]. We approximate the quantum mechanical ground-state probabilities P (E) for the one-dimensional scattering problem by a uniform semiclassical expression [48], which for E < is given by... 

In practice, we approximate the exact transmission coefficient by a mean-field-type of approximation that is we replace the ratio of averages by the ratio for an average or effective potential. For gas-phase reactions with small reaction-path curvature, this effective potential would just be the vibrationally adiabatic ground-state potential. In the liquid phase and enzymes we generalize this with the canonical mean-shape approximation. In any event, though, the transmission coefficient should not be thought of as a perturbation. The method used here may be thought of as an approximate full-dimensional quantum treatment of the reaction rate. 

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