Quantum effects on reaction coordinate motion

Calculations of Generalized Transition State Number of States 

The generalized transition state number of states needed for microcano-nical variational theory calculations counts the number of states in the transition state dividing surface at s that are energetically accessible below an energy E. Consistent with approximations used in calculations of the partition functions, we assume that rotations and vibrations are separable to give 

In the previous sections, we quantized the F — 1 degrees of freedom in the dividing surface, but we still treated the reaction coordinate classically. As discussed, such quantum effects, which are usually dominated by tunneling but also include nonclassical reflection, are incorporated by a multiplicative transmission coefficient k(T). In this section, we provide details about methods used to incorporate quantum mechanical effects on reaction coordinate motion through this multiplicative factor. 

The SCT, LCT, and OMT tunneling calculations differ from onedimensional models of tunneling in two key respects  

The wave function decays most slowly if the system tunnels where the effective barrier is lowest however, the distance over which the decay is operative depends on the tunneling path. Therefore, the optimum tunneling paths involve a compromise between path length and effective potential along the path. As a consequence, the optimum tunneling paths occur on the concave side of the minimum energy path i.e., they cut the corner 

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The hybrid or quasidassical approach is very old [1]. As the next step we go beyond the standard treatment, and we discuss using the adiabatic theory to develop a procedure for including quantum effects on reaction coordinate motion. A critical feature of this approach is that it is only necessary to make a partial adiabatic approximation, in two respeds. First, one needs to assume adiabaticity only locally, not globally. Second, even locally, although one uses an adiabatic effective potential, one does not use the adiabatic approximation for all aspects of the dynamics. 

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. 

At this point one can include optimized multidimensional tunneling in each (i = 1,2,..., 7) of the VTST calculations. The tunneling transmission coefficient of stage 2 for ensemble member i is called and is evaluated by treating the primary zone in the ground-state approximation (see the section titled Quantum Effects on Reaction Coordinate Motion ) and the secondary zone in the zero-order canonical mean shape approximation explained in the section titled Reactions in Liquids , to give an improved transmission coefficient that includes tunneling ... 

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. 

Quantum Mechanical Effects on Reaction Coordinate Motion... 

The density of reactive states p( ) defined by Eq. (6) is the quantum mechanical analogue of the transition state theory p ( ) of Eq. (14). Transition state theory with quantum effects on the reaction coordinate motion and recrossing predicts that the CRP will increase in smooth steps of height kt at each energy level of the transition state and that p( ) will be a sum of bell-shaped curves, each centered at an energy E. We have found clear evidence for this prediction in the densities of reactive states p(E) that we have calculated by accurate quantum dynamics. 

The quantum mechanical effect on the motion along the reaction coordinate is included in the kinetics calculations by multiplying the CVT rate constant by a temperature-dependent transmission coefficient c(T) which accoimts for tunneling and non-classical reflexion. Therefore, the final expression for the rate constant is given by ... 

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