Linking Results for Symmetry-Inequivalent Sets of Atoms

7 Linking Results for Symmetry-Inequivalent Sets of Atoms 

These do not involve or because these basis vectors are not symmetry related to b. Another two functions can be obtained by projection of, say, b, to give 

To get the overall picture of the vibrations we need to link together the modes for the two sets of H atoms. This has to be done so that only modes of like symmetry are combined. This ensures that in, say, an Aj mode for the molecule as a whole the movement of each subset of atoms must conform to the same irreducible representation. 

However, the subsets may move in phase with one another or out of phase with one another. So, adding superscripts to indicate the subsets, the two pairs of basis functions give four possible results  

Note that from a basis of four bond vectors we have again arrived at four vibrational modes. The addition of two normalized functions in Equation (6.54) gives new functions that are no longer of unit magnimde. However, it is straightforward to use the procedure of Section 6.5 to renormalize, i.e. scale by a factor which restores the magnitude to one. 

---