Quantum Effects on the Reaction Coordinate

Up to this point we have incorporated quantum mechanics in the f — 1 bound degrees of freedom (where E is the total number of bound and unbound 

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Note that both k and AG 0 depend on temperature. The transmission coefficient is sometimes called the tunneling transmission coefficient because tunneling is the main quantum effect on the reaction coordinate. 

In the present chapter, we have described a formalism in which overbarrier contributions to chemical reaction rates are calculated by variational transition state theory, and quantum effects on the reaction coordinate, especially multidimensional tunneling, have been included by a multidimensional transmission coefScient. The advantage of this procedure is that it is general, practical, and well validated. 

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. 

To be sure all experimental methods need to be complemented by theoretical techniques. The calculational techniques started with ab-initio and quantum calculational methods, such as MOP AC and GAMESS. These methods focus on the solution for the wave functions of the system being modeled. Those computations enabled calculations to be done in a sequence of frozen configurations of the catalyst and the gas phase molecules approaching the surface. The calculations produced thermodynamic energetic and entropic effects as the reaction coordinate changed, bring a reactant closer to the... 

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. 

Eyring adopted an ad hoc procednre that has been successful in including quantum effects in TST. First, the classical partition functions are replaced by their quantum analogues. Second, the classical rate is mnltiplied by a transmission coefficient that takes into account the qnantnm effects along the reaction coordinate. The quantum partition functions assume that the transition state can be treated as a stable system. The separate quantisation of the reaction coordinate is based on the vibrational adiabaticity assumption, that is, all the other vibrational modes of the reactive system very rapidly adjust to the reaction coordinate and maintain the continuity and smoothness of the PES. 

In contrast to the subsystem representation, the adiabatic basis depends on the environmental coordinates. As such, one obtains a physically intuitive description in terms of classical trajectories along Born-Oppenheimer surfaces. A variety of systems have been studied using QCL dynamics in this basis. These include the reaction rate and the kinetic isotope effect of proton transfer in a polar condensed phase solvent and a cluster [29-33], vibrational energy relaxation of a hydrogen bonded complex in a polar liquid [34], photodissociation of F2 [35], dynamical analysis of vibrational frequency shifts in a Xe fluid [36], and the spin-boson model [37,38], which is of particular importance as exact quantum results are available for comparison. 

The description of the PES requires quantum chemical calculations of the energy as well as the first and the second derivatives of the energy with respect to the nuclear coordinates at several points on the reaction path. These calculations can be effectively performed by using the recently developed methods (82-100,104-116) and programs (224-226) for systems consisting of five to six atoms. 

The tunnel correction is an artificial device to correct an artificial treatment, in which the real vibrations of the transition state are treated quantum mechanically, but the passage along the reaction coordinate over the barrier is treated classically. This correction has the property that Qh > Qd > 1 it thus can only increase the isotope effect. It is dependent on several factors12 including especially i>, and the QhIQd can only be much greater than unity if i>fH >. It can be shown11,13 ... 

On the other hand, the proton potential of the 5-bromo compound is exactly symmetrical with reference to the reaction coordinate of the tautomerization. Consequently, the proton transfer can proceed through the tunnelling mechanism. This is the reason why the paraelectric behaviour is maintained even at 4 K. The suppression of the antiferroelectric phase transition may be derived from a quantum tunnelling effect. Such paraelectric behaviour can be regarded as quantum paraelectricity , which is a notion to designate the phenomenon that (anti)ferroelectric phase transitions are suppressed even at cryogenic temperatures due to some quantum-mechanical stabilization, proton tunnelling in this case. 

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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