Skewed and scaled coordinates

Fig. 9.31 PE contours scaled and skewed coordinates for three different mass combinations. L light atom H heavy atom.

The skew angle is the angle between the r g and rgQ axes in the scaled and skewed coordinate system that diagonalizes the kinetic energy. ... 

These equations can be visualised too as a rolling ball with the corresponding representation of as a function of b and c. To this end a skewed coordinate system with different scales has to be introduced, in which the equations of motion of a ball are obtained as follows. FVom Fig. 16 the angle between the skewed fr- and c-coordinates is -f and the ratio between the scales of the two coordinates is /. Then, when q eind V get the interpretation following from the figure and the ball has mass m,... 

Insight about the dynamics of a unimolecular reaction can be obtained by examining the reaction s potential energy contour map. Usually this is at best only a qualitative analysis. However, it can be made quantitative for a linear triatomic ABC molecule by using skewed and scaled coordinates (Glasstone et al., 1941 Levine and Bernstein, 1987). The significance of these coordinates becomes readily apparent by considering the internal coordinate classical Hamiltonian for the linear ABC molecule that is. 

Figure B5.1 The two (equivalent) pairs of Cartesian coordinates, / bc and Ft, or, / ab and R, that allow the kinetic energy to be written as a sum of squares [adapted from F. T. Smith, J. Chem. Phys. 31, 1352 (1959)]. The reactants and products valleys are along the new and old physical bond distances, / ab and Rbc, and so appear to be at a skew angle / with respect to one another. The mass-scaled coordinates Qare defined in the text.

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