Reaction coordinates order parameters

Order Parameters, Reaction Coordinates, and Extended Ensembles... 

There have been a number of investigations of the formulation of the problem of electron transfer accompanied by atom transfer particularly with regard to the simultaneous movement of the proton (which, in view of its small mass, may in fact be an atypical case). A possible model for such processes would assume a conservation of bond order along the reaction coordinates (Johnston, 1960). It is of interest that the results of such calculations are similar to those for electron transfer for weak coupling, although the interpretation of the process and parameters (such as a) are different. 

It is also possible to build the order parameter directly into the potential energy function. This would enable one to determine the free energy difference between a given potential and a reference potential. In a sense, this is complementary to the previous discussion of reaction coordinates rather than further subdividing configuration space by various order parameters, we are considering multiple configuration... 

The order parameter can be defined in two different ways. It can be either a function of atomic coordinates or just a parameter in the Hamiltonian. Examples of both types of order parameters are given in Sect. 2.8.1 in Chap. 2 and illustrated in Fig. 2.5. This distinction is theoretically important. In the first case, the order parameter is, in effect, a generalized coordinate, the evolution of which can be described by Newton s equations of motion. For example, in an association reaction between two molecules, we may choose as order parameter the distance between the two molecules. Ideally, we often would like to consider a reaction coordinate which measures the progress of a reaction. However, in many cases this coordinate is difficult to define, usually because it cannot be defined analytically and its numerical calculation is time consuming. This reaction coordinate is therefore often approximated by simpler order parameters. 

Free energy with several order parameters. The previous formulas can be extended to the case of multiple reaction coordinates i,..., p. For example, the derivative of the free energy becomes a gradient and... 

Stationary points on a PES are points where dE/dq = 0 for all q, where q is a geometric parameter. The stationary points of chemical interest are minima (d2Eldq]q-i > 0 for all q) and transition states or first-order saddle points d2E/dqiqj < 0 for one q, along the reaction coordinate (intrinsic reaction coordinate, IRC), and > 0 for all other q. Chemistry is the study of PES stationary points and the pathways connecting them. 

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