Orthogonal coordinates, role

Direction cosines, which can be used to define the direction of a vector in an orthogonal coordinate system, play an essential role in accomplishing coordinate transformations. As illustrated in Fig. A. 1, there is a vector V oriented in a (z,r,9) coordinate system. Because our concern here is only the direction of the vector, the physical dimensions are sufficiently small so that the curvature in the 9 coordinate is not seen (i.e., the coordinate system... 

G. R. Fleming Yes, the role of orthogonal coordinates could be significant. I believe that Prof. J. Jean (Washington University, St. Louis) is beginning to address this issue via Redfield theory. 

If the M(CO)2 angle is acute, the interior vertical dir orbital (dyz in the coordinate system of Fig. 18) will be stabilized and occupied. A cis alkyne would then be in the plane bisecting the M(CO)2 unit so that 17] would see the filled dyz orbital while ir would encounter the vacant dxz orbital. Alternatively, an obtuse M(CO)2 angle would reverse the roles of dyz and dxz and lead to a predicted alkyne orientation orthogonal to the M(CO)2 bisector plane (Fig. 18). Note that these arguments are valid only for d4 monomers and completely irrelevant for d6 dicarbonyl alkyne derivatives such as W(CO)2(ZC=CZ)2(dppe) (30). 

As we shall see in later sections, orthogonal matrices play an important role in defining the coordinate transformations that are used in characterizing the symmetry properties of molecules. 

The introduction of the foregoing discussion—unnecessary for the development of the point groups—is intended to emphasize the possible role of sets of symmetry operators as coordinates that define the symmetry space of the object under consideration. Thus, certain properties of an object with known symmetry may be expressable in terms of basis vectors that span the space of the symmetry group elements of the object. We will return to this approach in later chapters. The important idea is that the group elements themselves form a space of dimension equal to the [lumber of elements, and as such it should be possible to find orthogonal basis vectors for this space and then to use these basis vectors to study physical consequences of the symmetry of the system. This is the root of the utility of group theory in physics and chemistry. 

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