Water molecule cartesian displacement vectors

The details of the F-G matrix procedure are best explained by working through a simple example, such as the water molecule. This belongs to the point group C2l.. The nine Cartesian displacement vectors, three on each atom, give rise to the representation... 

First, an appropriate basis set has to be found. Considering that a molecule has 37V degrees of motional freedom, a system of 37V so-called Cartesian displacement vectors is a convenient choice. A set of such vectors is shown in Figure 5-5 for the water molecule. A separate Cartesian coordinate system is attached to each atom of the molecule, with the atoms at the origin. The orientation of the axes is the same in each system. Any displacement of the atoms can be expressed by a vector, and in turn this vector can be expressed as the vector sum of the Cartesian displacement vectors. 

Continuing with the water molecule as an example, the basis of the Cartesian displacement vectors will consist of nine vectors... 

Figure 5-5. Cartesian displacement vectors as basis for representation of the water molecule.

Continuing with the water molecule as an example, the basis of the Cartesian displacement vectors will consist of nine vectors (see Figure 5-5). Operation E brings all of them into themselves, and the character is 9. Operation Cj changes the position of the two hydrogen atoms, so only the three coordinates of the oxygen atom have to be considered. The corresponding block of the matrix representation is... 

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