Setting reaction coordination changes

For reaction networks with well-separated constants coordinates of left eigenvectors Z are close to 0 or 1. We can use the left eigenvectors for coordinate change. For the new coordinates z,- = Vc (eigenmodes) the simplest equations hold Zi = 2 Z . The zero-one law for left eigenvectors means that the eigenmodes are (almost) sums of some components z, = some sets of numbers Vj. 

It is often useful to transform from simple Cartesian coordinates to other sets of coordinates when we study collision processes including chemical reactions. In a collision process, it is obvious that the relative positions of the reactants are relevant and not the absolute positions as given by the simple Cartesian coordinates. It is therefore customary to change from simple Cartesian coordinates to a set describing the relative motions of the atoms and the overall motion of the atoms. For the latter motion the center-of-mass motion is usually chosen. In the following we will describe a general method of transformation from Cartesian coordinates to internal coordinates and determine its effect on the expression for the kinetic energy. 

Jaume et al. (1984) studied the contribution of solvent relaxation to the reaction coordinate of the F (H20) + CH3F(H20)SN2 reaction. Potential energy calculations were performed using the ab initio MO method with the 3-21G basis set. The authors found large variation of the solvation parameters along the reaction path and concluded that solvent coordinates are an important part of the reaction coordinate. They showed that the solvent acts not only as a medium for the reaction but also as a rectant. Thus, the solvent does not adjust its position to the changing chemical system but rather takes part in it. 

Tliis new variable e, called the reaction coordinate, characterizes the extent or degree to winch a reaction has taken place. Equations (13,2) and (13.3) define changes in s with respect to clianges in the numbers of moles of tire reacting species. The definition of e itself is completed for each application by setting it equal to zero for the initial state of the system prior to reaction. Thus, integrationof Eq. (13.3) from an initial unreacted state where e = 0 and n, = ,(, to a state reached after an aibitrary amount of reaction gives ... 

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) ... 

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