Using Projection Operators to Construct SALCs

The most important and frequent use for projection operators is to determine the proper way to combine atomic wave functions on individual atoms in a molecule into MOs that correspond to the molecular symmetry. As pointed out in Chapter 5, it is essential that valid MOs form bases for irreducible representations of the molecular point group, we encounter the problem of writing SALCs when we deal with molecules having sets of symmetry-equiv- 

Clearly, for a one-dimensional representation the character and the full matrix are the same thing. Hence, the incomplete projection operator is complete in these cases, and will provide the appropriate SALC unambiguously and automatically. Let us illustrate by asking what SALCs can be formed by the Is orbitals of the four hydrogen atoms in ethylene. 

The three preliminary steps before the SALCs can actually be constructed are, in this and all other cases, the following  

We use the four H Is orbitals as the basis for a representation we chose the coordinates and label the a functions as shown below. It is necessary to 

We can now write a matrix that expresses this transformation  

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