In the activated complex, motion along the reaction coordinate can be separated from the other degrees of freedom (exact within the framework of a normal-mode description) and treated classically as a free translation. [Pg.141]

The reaction coordinate is found by a normal-mode analysis at the saddle point and is therefore separable from the other degrees of freedom in the activated complex the motion in this coordinate is treated as that of a free particle. Then, according to Eq. (A. 13), at sufficiently high temperatures, we have [Pg.142]

This zeroth-order approach, however, neglects any quantum mechanical aspect of the reaction coordinate motion (the Fth degree of freedom). If one assumes that the reaction coordinate is separable for the F — 1 degrees of freedom on the dividing surface, then the Heaviside function in Eq. (19) is replaced by a one-dimensional tunneling probability. [Pg.393]

Following up on the discussion in Section 15.2 about the nature of the activated complex, the TS structure should be recognized as a species that is a minimum in 3N - 7 degrees of freedom - the missing degree of freedom is the reaction coordinate. Thus, we may readily define the electronic, translational, and rotational components of the partition function associated with the TS structure in the usual way. For the vibrational component, we will separate [Pg.525]

The second point is that the new phase-space representation permits the definition of a true dividing surface in phase space which truly separates the reactant and product sides of a reaction. Traditional transition state theory of chemical reactions, based simply on coordinate-space definitions of the degrees of freedom, required an empirical correction factor, the transmission [Pg.21]

For bimolecular reactions (i.e. where the reactant is two separate molecules) and contribute a constant —4 RT. The translational and rotational enttopy changes are substantially negative, —30 to —50 e.u., due to the fact that there are six translational and six rotational modes in the reactants but only three of each at the TS. The six remaining degrees of freedom are transformed into the reaction coordinate and five new vibrations at the TS. These additional vibrations usually make a few kcal/mol [Pg.304]

The first requirement is the definition of a low-dimensional space of reaction coordinates that still captures the essential dynamics of the processes we consider. Motions in the perpendicular null space should have irrelevant detail and equilibrate fast, preferably on a time scale that is separated from the time scale of the essential motions. Motions in the two spaces are separated much like is done in the Born-Oppenheimer approximation. The average influence of the fast motions on the essential degrees of freedom must be taken into account this concerns (i) correlations with positions expressed in a potential of mean force, (ii) correlations with velocities expressed in frictional terms, and iit) an uncorrelated remainder that can be modeled by stochastic terms. Of course, this scheme is the general idea behind the well-known Langevin and Brownian dynamics. [Pg.20]

This re-derivation of the one-dimensional TST result emphasizes the effective character of the potential used in one-dimensional treatments of barrier crossing problems. The one-dimensional model, Eq. (14.11), will yield the correct TST result provided that the potential V (x) is taken as the effective potential of the reaction coordinate, that is, the potential of mean force along this coordinate where all other degrees of freedom are in thennal equilibrium at any given position of this coordinate." It should be stressed, however, that this choice of the one-dimensional effective potential assumes that such a coordinate can be identified and that a point along this coordinate can be identified as the transition point that separates reactants from products. [Pg.494]

Moreover, in this linear-response (weak-coupling) limit any reservoir may be thought of as an infinite number of oscillators qj with an appropriately chosen spectral density, each coupled linearly in qj to the particle coordinates. The coordinates qj may not have a direct physical sense they may be just unobservable variables whose role is to provide the correct response properties of the reservoir. In a chemical reaction the role of a particle is played by the reaction complex, which itself includes many degrees of freedom. Therefore the separation of reservoir and particle does not suffice to make the problem manageable, and a subsequent reduction of the internal degrees of freedom in the reaction complex is required. The possible ways to arrive at such a reduction are summarized in table 1. [Pg.7]

Statistical theories treat the decomposition of the reaction complex of ion-molecule interactions in an analogous manner to that employed for unimolecular decomposition reactions.466 One approach is that taken by the quasiequilibrium theory (QET).467 Its basic assumptions are (1) the rate of dissociation of the ion is slow relative to the rate of redistribution of energy among the internal degrees of freedom, both electronic and vibrational, of the ion and (2) each dissociation process may be described as a motion along a reaction coordinate separable from all other internal [Pg.199]

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