Skew coordinate system

There is an important relationship between vectors and skew-symmetric tensors. Suppose A and B are two vectors in a three-dimensional rectangular coordinate system whose components are connected by... 

The skew-symmetric part S 4 is equivalent to a vector (x t)/2 with components (/. t),/2 = (/.jtk — /.ktj)/2, involving correlations between a libration and a perpendicular translation. The components of S 4 can be reduced to zero, and S made symmetric, by a change of origin. It can be shown that the origin shift that symmetrizes S also minimizes the trace of T. In terms of the coordinate system based on the principal axes of L, the required origin shifts p, are... 

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. 

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 ... 

Instanton trajectories (solid) with several different energies calculated by the normal form theory are shown in the skewed coordinate system. The dotted curve shows the IRC. 

The axes and 7)2 form a skewed coordinate system which is also sometimes used for the description of the motion of the system (Eig. 9). 

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,... 

With the help of this example we can also see the general relations between potential mountains and activation heats firstly we see immediately that the agreement between height of the saddle and activation heat also exists truly in the general case, since the fact that a pictorial ball put on the height of the saddle rolls into the v llley at the smallest push, remains intact also for the skewed coordinate system. 

The presented method allows to avoid application of any interface element, the shape of the contact zone is consistent with the used finite element. The method of setting the local coordinate systems is compatible with the one used for shells consisted of plane triangles (linear tetrahedral case) and quadratic triangles for the second order tetrahedrzds. The way of introducing the method for 3D case is compatible with previously developed algorithm for 2D case which has been succesfuliy applied to the solution of bulk material flow problems. The most important feature of the method is the use of skew and conical... 

Such effects as the efficient conversion of the exoergicity into product vibration on an attractive surface can readily be visualized in the skewed coordinate system. On an attractive surface the ball is rolling downhill as Q decreases and hence it enters the bend along the reaction path with a high speed. The ball will then fail to make its exit along the products valley and instead it will climb the shoulder of the potential (the bobsled effect) and thereby convert much of the exoergicity to product vibration. 

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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