B Appendix Mass-weighted coordinate systems

Consider the collinear reactive collision A - - BC AB - - C. C(Rab,. bc) is then a function of only two bond distances, Rab = Rb Ra and Rbc = Rc Rb, and these are the natnral variables for thinking about the potential. The issue is that the kinetic energy (in the center-of-mass system) is not a simple function of these two coordinates. It is a simple function if, instead, we use the old bond distance, R-qc, and the distance, R, of the attacking atom A to the center of mass of BC, R = Rab (wc/(wb + mc))RBc- Explicitly, using a dot to denote the derivative with respect to time and denoting the reduced masses by /x, 

In terms of either pair of Cartesian coordinates the motion is simple because each coordinate is only coupled to the other by the force and not by the kinetic energy. This is really aU there is to it except that different masses will lead to different motions because of the mass factors. We can eliminate these by scaling the coordinates by the (square roots) of the reduced masses /u.. To do so introduce two new coordinates Qi and Q2 by the transformation 

Here a, b, and cos depend on the masses only, M=mA + rtiB +me is the total mass and the /rs are the reduced masses in the reactants and products channels  

The point of the coordinate transformation from the bond distances to die Qs is that the kinetic energy for the collision (in the c.m. system) has the form 

Equation (B.5.4) has the interpretation of the kinetic energy of a point particle (often referred to as the system point) of unit mass, whose position is specified by the two Cartesian coordinates Q and Q2. 

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