Theory Formulation of Reaction Rates

We imagine a current of systems in an initial quantum state n moving in the nuclear configuration space from the reactants region R towards the col of the potential energy surface V(x,y ) parallel to the reaction coordinate x. Any system can be described by a wave packet which has an average momentum 

We consider first, in a quasiclassical treatment, the systems with momentum in the range between p and flnx of these sys- 

The hypersurface S (x = const) may be placed anywhere in configuration space because of the condition (108,11) of current conservation in reactive x-direction, which yields the relation (16,III), Consequently, the rate expression (22,111) is invariant in respect to the position of the surface. However, to calculate the reaction velocity by this expression, it is necessary to know not only the transition probability also the distribution function 

Introducing (28.Ill) and (29.HI) in (27.HI) and changing the order of summation and y -integration yields the expression 

The derivation of the expression (32.Ill) presumes a continuous or quasi-continuous energy distribution for the motion along the reaction coordinate, given by (17.Ill) therefore, it is valid when this motion corresponds to either a relative translation or a low frequency vibration of reactants. In the later case, however, an alternative derivation of (32.Ill) is possible, which can be generalized to include the situation of a high-frequency vibration along the reaction coordinate where the quantum-statistical formula (18.Ill) for a discontinuous energy distribution applies. 

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