Rectangular and Skewed Coordinates

The relative position of the three atoms A, B, C is described by three coordinates. These can be the distance rg between atoms B and C, the distance between atom A and the center of mass of BC, and the angle y between vectors and rgQ (Fig. 8). For the linear configuration, there are only two coordinates R and rg. In the R — rgQ plane, the potential surface is represented by the contour lines corresponding to the same energy. Transforming to other coordinates 5, 2 which make the kinetic energy equal to the product of the effective mass (p/2) by the sum of the squares of velocity components for motions along the and 2 coordinates, the dynamics of the linear system ABC can be simulated by the motion of the heavy mass point pi over the potential surface U(5i, 52) These coordinates are related to R and rg by a scale transformation 1/a and a  

Here a is defined by the effective mass of the three-atom system g and the reduced mass ggQ of the diatomic complex BC  

The coordinates and 2 e convenient for the description of the relative motion of A and BC they are generally used in VT energy exchange studies. In the interpretation of exchange reactions, these coordinates are only suitable for the start of the reaction at which A and BC approach each other. However, towards the end, the reaction (receding particles AB and C) is more appropriately described by distance r g between atoms A and B and distance R between atom C and the center of mass of AB. The relevant coordinates rji and 1Q2 obtained by scaling are 

Transformations (10.1) and (10.3) permit the description of the kinetic energy of a linear three-atom system in a unified way 

Two rectangular coordinate systems 2 related by rotation over the 

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