Kinetic Energy of a Triatomic System

In order to describe the dynamics of the triatomic system A + BC a convenient coordinate system must be constructed. Six degrees of freedom are required to completely describe the relative motion of the three particles. These may be chosen in many ways. The most familiar set of coordinates is that which we used in describing the ABC potential-energy surface in Section 9.2 the AB and BC distances, the A-B-C angle, and three angular coordinates to specify the orientation of the ABC system in space. Such a coordinate set is shown in Fig. 10.1, where the vectors r B and Fbc provide all the information required to specify the relative motion and the rotational motion of the three particles. While a natural set of coordinates, it is not convenient for calculation. The discussion is greatly simplified if we describe the system in terms of BC relative motion and the relative motion of A with respect to the center of mass of BC these coordinates are also shown in Fig. 10.1. 

Complete analysis of the dynamics of collision in the A + BC system is extremely difficult even if A, B, and C are atoms. To avoid the complications introduced by molecular rotation we limit consideration to col-linear configurations such as those shown in Fig. 10.2 then only A-B 

For the collinear system the total kinetic energy of relative motion is the sum of the vibrational kinetic energy of the BC diatom and the kinetic energy of A relative to the center of mass of BC. Letting and 

In addition to its obvious interpretation as the relative kinetic energy of the three collinear particles, (10.4) may also be interpreted as the kinetic energy of a single particle of mass + mQ)lM moving on a planar surface. 

The relation between x and y and the chemically interesting coordinates Tab and Tbc niay be found from (10.2) and (10.3), 

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