Generalized reaction coordinate

The LVC model further allows one to introduce coordinate transformations by which a set of relevant effective, or collective modes are extracted that act as generalized reaction coordinates for the dynamics. As shown in Refs. [54, 55,72], neg = nei(nei + l)/2 such coordinates can be defined for an electronic nei-state system, in such a way that the short time dynamics is completely described in terms of these effective coordinates. Thus, three effective modes are introduced for an electronic two-level system, six effective modes for a three-level system etc., for an arbitrary number of phonon modes that couple to the electronic subsystem according to the LVC Hamiltonian Eq. (7). In order to capture the dynamics on longer time scales, chains of such effective modes can be introduced [50,51,73]. These transformations, which are briefly summarized below, will be shown to yield a unique perspective on the excited-state dynamics of the extended systems under study. 

Our description thus far of DOS-based methods has centered on calculation of the density of states. A particularly fruitful extension of such methods involves the calculation of a potential of mean force ( ), or PMF, associated with a specified generalized reaction coordinate, (r) [28,29]. A PMF measures the free energy change as a function of (r) (where r represents a set of Cartesian coordinates). This potential is related to the probability density of finding the system at a specific value of the reaction coordinate (r) ... 

The values of Eg (st), t = 1, - , n provides a first estimate of the free energy along the LFEP. This potential is used as a bias to perform a onedimensional umbrella sampling along the generalized reaction coordinate defined by the LFEP [5,7]. The forces on the atoms of position r due to the umbrella potential are... 

To express the adiabatic energy surface of the solute-solvent system, it is useful to define a generalized reaction coordinate as the energy gap between the diabatic reactant and product EVB states ... 

Fig. 1. Effective potentials for atomic motion during detonation versus generalized reaction coordinate. Different curves correspond to different electronic states vibrational levels for initial and final configurations are also shown.

Truncating the potential of mean force as a parabolic barrier allows diagonalization of the Hamiltonian, as discussed in the previous section. Motion is separable along the generalized reaction coordinate p. TST will therefore be exact (in the parabolic barrier limit) if one chooses the dividing surface / = p — p. Inserting this choice into the TST expression for the rate (31) leads to the well-known KGH expression for the rate in the spatial diffusion limit ... 

In the following the coordinate p will denote the generalized reaction coordinate (the analog of the coordinate / in the previous section) and the coordinate ct the collective bath mode. 

FIGURE 37.8 Changes in the potential energy (kJ/mol) along the generalized reaction coordinate for HF interaction with SiOH group according to quantum chemical calculations [58,61]. 

A second approach to this problem of multiple asymptotia is to introduce specialized coordinates which change smoothly from one arrangement to the others in such a way as to enable one to analyze the wavefunction in the various arrangement limits. This leads to the definition of generalized "reaction coordinates". (10-16) The leading examples of these are the natural collision coordinates first introduced by Marcus (12) and the hyperspherical coordinates pioneered by Delves (10). These have led to important advances in reactive scattering theory but are discussed by others active in their study so we do not pursue them further here. 

This type of elimination is known as an ElcbR (elimination, unimolecular, conjugate base, reversible) reaction, and a generalized reaction coordinate diagram is shown in Figure 10.14. Such reactions exhibit C s-H/D exchange and a 1° hydrogen kinetic isotope effect (ku/ko) of 1.0. An example of an ElcbR is shown in Figure 10.15. ... 

Figure 4.9 Quantum well and potential energy surfaces relevant for charge transfer. Arrows indicate fluctuations about the generalized reaction coordinate leadingto changes in the energy levels of the quantum wells, which corresponds to oscillations along the potential energy surface.

Figure 1 Occupation numbers of natural orbitals in thermally allowed cyclization of butadiene to cyclobutene in dependence on the value of the generalized reaction coordinate cp.

The parameters 0 and cp play in this relation the role of the generalized reaction coordinates, the systematic change of which allows to describe the structure of all transient species lying "inside" the corresponding More O Ferrall diagram. For the analogy with the More O Ferral diagrams becomes even more transparent it is convenient to substitute the primary spherical coordinates 0 and cp by the ordinary More O Ferrall coordinates Qi, Q2 These transformations are described by the set of equations (86). 

Notably, their discussions about non-adiabaticities have been for a reaction coordinate assumed to be the separation between the centers-of-mass of the two reactants. The use of a more general reaction coordinate, as in some VTST methods, may account for some of the non-adiabaticities. Indeed, reaction coordinate optimizations in the variable reaction coordinate (VRC)-TST approach have often found a reduction by a factor of two or more relative to that obtained for the separation between the centers-of-mass. However, these reductions are generally for short range separations where the long-range potential expansions are inappropriate. 

Fig. 2.1. Diagram illustrating the dependence of the system (substrate-product) energy, E, on the generalized reaction coordinate, R.

A more sophisticated approach has been developed by Eyring in the theory of an activated complex [10,11]. The generalized reaction coordinate. 

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