VTST, quantum mechanical effects

Like Eq. (27.2), Eqs. (27.11) and (27.12) are also hybrid quantized expressions in which the bound modes are treated quantum mechanically but the reaction coordinate motion is treated classically. Whereas it is difficult to see how quantum mechanical effects on reaction coordinate motion can be included in VTST, the path forward is straightforward in the adiabatic theory, since the one-dimensional scattering problem can be treated quantum mechanically. Since Eq. (27.12) is equivalent to the expression for the rate constant obtained from microcanonical variational theory [7, 15], the quantum correction factor obtained for the adiabatic theory of reactions can also be used in VTST. 

Tunneling occurs when a configuration, that has an energy lower than an energy barrier, nonetheless surmounts it due to quantum mechanical effects. In such cases, adjustments of the rate constant due to tunneling become necessary to obtain improved accuracy. These corrections in TST and VTST are in the form of a correction coefficient K such that... 

Abstract Some of the successes and several of the inadequacies of transition state theory (TST) as applied to kinetic isotope effects are briefly discussed. Corrections for quantum mechanical tunneling are introduced. The bulk of the chapter, however, deals with the more sophisticated approach known as variational transition state theory (VTST). 

There are two corrections to equation (12) that one might want to make. The first has to do with dynamical factors [19,20] i.e., trajectories leave Ra, crossing the surface 5/3, but then immediately return to Ra. Such a trajectory contributes to the transition probability Wfia, but is not really a reaction. We can correct for this as in variational transition-state theory (VTST) by shifting Sajj along the surface normals. [8,9] The second correction is for some quantum effects. Equation (14) indicates one way to include them. We can simply replace the classical partition functions by their quantum mechanical counterparts. This does not correct for tunneling and interference effects, however. 

Methods like PI-QTST provide an alternative perspective on the quasithermody-namic activation parameters. In methods like this the transition state has quantum effects on reaction coordinate motion built in because the flux through the dividing surface is treated quantum mechanically throughout the whole calculation. Since tuimeling is not treated separately, it shows up as part of the free energy of activation, and one does not obtain a breakdown into overbarrier and tuimel-ing contributions, which is an informative interpretative feature that one gets in VTST/MT. 

In our own applications of VTST to bimolecular reactions we have also used the "quantum mechanical" formulation of section II.A in which internal states are quantized (quasiclassically) and tunneling effects are included (quantum mechanically or semiclassically). We have compared the rate constants calculated by our quantized formulation, with and without tunneling corrections, to accurate quantum mechanical equilibrium rate constants for the same assumed potential energy surfaces for several collinear reactions and one three-dimensional reaction.This work, reviewed elsewhere,shows that ... 

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