Quantum mechanical VTST

Of course, one is not really interested in classical mechanical calculations. Thus in normal practice the partition functions used in TST, as discussed in Chapter 4, are evaluated using quantum partition functions for harmonic frequencies (extension to anharmonicity is straightforward). On the other hand rotations and translations are handled classically both in TST and in VTST, which is a standard approximation except at very low temperatures. Later, by introducing canonical partition functions one can direct the discussion towards canonical variational transition state theory (CVTST) where the statistical mechanics involves ensembles defined in terms of temperature and volume. There is also a form of variational transition state theory based on microcanonical ensembles referred to by the symbol p,. Discussion of VTST based on microcanonical ensembles pVTST is beyond the scope of the discussion here. It is only mentioned that in pVTST the dividing surface is 

One must recognize that TST is much simpler conceptually than VTST. Thus, there is one transition state in TST and that is located at the maximum energy on the MEP (the saddle point). In VTST the dividing surface is temperature dependent since the partition functions and consequently the free energy of activation are temperature dependent. 

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The quantum mechanical VTST formalism used for the rate-constant calculations presented here is described in detail elsewhere. Here we present a brief review of the theory. 

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). 

In both TST and VTST, quantum mechanical tunneling is introduced into the rate constant expression as a correction factor usually referred to as k. A short discussion of k which is used largely with TST is presented in Section 6.3.1. Tunneling has been explored much more thoroughly in connection with VTST and this work will be discussed later. 

Tunneling in VTST is handled just like tunneling in TST by multiplying the rate constant by k. The initial tunneling problem in the kinetics was the gas phase reaction H -(- H2 = H2 + H, as well as its isotopic variants with H replaced by D and/or T. For the collinear reaction, the quantum mechanical problem involves the two coordinates x and y introduced in the preceding section. The quantum kinetic energy operator (for a particle with mass fi) is just... 

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. 

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. 

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. 

Key words Transition-state theory (TST) - Variational TST (VTST) - Fundamental assumption of TST -Quantum mechanical TST... 

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... 

Accurate quantum mechanical calculations on the D -f H2 reaction allow one to test the quantitative predictive ability of variational transition state theory with multidimensional tunneling contributions. Such VTST calculations agree with accurate quantum dynamics with an average eiTor of only... 

Another important statistical approach to this same problem is the statistical adiabatic channel model (SACM) of Quack and Troe, - which adiabatically correlates the eigenstates of the orthogonal modes along the reaction coordinate, thereby generating rovibrational adiabatic channels. The adiabatic approximation reduces the multidimensional dynamical problem to essentially a one-dimensional barrier-crossing problem. The catch, of course, is that it is extremely difficult to compute the requisite adiabatic channels, though no more difficult than a rigorous quantum mechanical implementation of VTST would be. An authoritative account of adiabatic channel methods is to be found in Statistical Adiabatic Channel Models. 

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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