Generating Symmetry Coordinates Using the Projection Operator Method

4 GENERATING SYMMETRY COORDINATES USING THE PROJECTION OPERATOR METHOD  

To this point in our discussion of molecular vibrations, we have shown how to generate the symmetries of the vibrational modes by reduction of the representation and subtraction of translational and rotational modes. When pictures of the normal modes of vibration are available, we have also been able to demonstrate how to assign the different symmetries to the correct normal modes of vibration. However, we have not yet attempted to show the origins of the normal modes themselves. The reason for this is because a full-blown norma/ mode analysis (NAM) is a 

TABLE 9.10 Application of the projection operator method using the bond stretch as the basis. 

Both of these SALCs ultimately need to be normalized, so the coefficients in front of the brackets do not really matter at this point. In order to normalize a wave function, the sum of the square of the coefficients in front of the wave functions making up the linear combination must be equal to one. The final wave functions must also be orthogonal to each other. It is the coefficients in front of each basis function that are relevant. For SALC I, these coefficients are 2, — I, and 

Vector diagram for the Tg -i basis set of BCIj, showing how orientation of SALC 1 along the y-axis results in SALC 2 lying along the x-axis in order for the two SALCs to be orthogonal. SALC 2 is a linear combination of - t 2 + Ps because Tj has a negative x-component and has a positive x-component. 

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