Let us use *displacement vectors* in the , , and directions
on each atom as our basis functions, and then we will form SALC's from
these to see what the vibrations *should* look like (without the need
for computations). We do this using the technique of *projection
operators*.

Our job is somewhat easier for this example because each atom is
symmetry-equivalent to all the others. Hence, it will suffice to apply
projection operators to only the , , and displacements on *one* of the atoms. Let's label the displacements as , , or
, where is the number of the atom displaced. See Figure
3 for the displacements of atom 1. First, we must determine
what each symmetry operator does to each of our basis vectors. Referring
to Fig. 3, we can construct Table 5, which
provides the results of each symmetry operation on , , and .

To apply a projection operator, we dot each of the rows in Table 5 with the rows of the character table -- there will be a separate projection operator for each irrep. Because the totally symmetric irrep contains all 1's in its row, the rows in the Table are already the result of applying to , , and . Hence, , or (we aren't usually too concerned about normalization). Referring to Figure 2, this is the 126.71 cm normal mode! Similarly, . This is the 1849.60 cm normal mode. Finally, (all the displacements cancel). There is no third mode.

We can apply projection operators to , , and for each irrep
to build up a complete basis of SALC's. As another example,

Consulting Fig. 2, we see that is the 131.73 cm normal mode.

Finally, let's try the irrep:

Now note that there is not a normal mode of vibration! Why is this? If we examine the result of , we see that it displaces