Skewed angle coordinate system

D.1.2 Mass-weighted skewed angle coordinate systems... 

Sometimes a mass-weighted skewed angle coordinate system rather than a rectangular system is used to plot the potential energy surface and the trajectories for a simple triatomic reaction like... 

We note that the coordinate X directly reflects the distance between atoms one and two, whereas the coordinate X2 reflects a combination of both distances. Therefore, a knowledge of the two coordinates does not directly tell us what the distances are between the involved atoms. Also, for the potential energy function in a collinear collision, the natural variables will be the distances between atoms A and B and atoms B and C. These variables appear as the components along a new set of coordinate axes, if instead of a rectangular coordinate system we use a mass-weighted skewed angle coordinate system. 

Fig. D.1.1 Sketch of an ordinary Cartesian coordinate system and the mass-weighted skewed angle coordinate system.

If we consider the potential energy as a function of the Jacobi coordinates X and X2 and draw the energy contours in the X1-X2 plane, then the entrance and exit valleys will asymptotically be at an angle to one another and in the mass-weighted skewed angle coordinate system parallel to its axes. So the idea with this coordinate system is that it allows us to directly determine the atomic distances as they develop in time and that it shows us the asymptotic directions of the entrance and exit channels. 

A thermodynamic example may be illustrative. Consider Maxwell s model of the Gibbs USV surface for water (Fig. 1.1), as depicted schematically in Fig. 9.1. In this model, the physical (U, S, V) coordinates are associated with mutually perpendicular axes, and three chosen points on this surface form a triangle whose edges, angles, and area are as shown in Fig. 9.1a. However, the model might have been constructed (with equal thermodynamic justification) in a skewed Mcworthogonal axis system (Fig. 9.1b) in which the... 

In this way we obtain the centre-of-mass motion separation (the first term). The next two terms represent the kinetic energy operators for the independent pairs AB and BC, while the last one is the mixed term Tabc, whose presence is understandable atom B participates in two motions, those associated with Rab and Rgc- We may eventually get rid of Tabc introducing a skew coordinate system with the Rab and Rgc axes (the coordinates are determined by projections parallel to the axes). After a little derivation, we obtain the following condition for the angle 6 between the two axes, which assures the mixed term ... 

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,... 

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. ... 

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