Symmetry-adapted linear combinations

Abstract

In Chapter 6, it was shown how to determine which irreducible representations were present in any reducible representation. For example, a set of bond vectors for a square-planar (D4h) molecule reduced to A1g + B1g + Eu. What is the physical significance of this result? It means that, for this case, the four basis vectors can be used to form four linear combinations (that is, sums and/or differences) such that one combination has A1g symmetry, one B1g symmetry and two (together) Eu symmetry. Each of these corresponds to one vibrational mode of the molecule (see Chapter 8). Similar results are found for any set of n basis vectors or functions — it will be possible to form n symmetry-adapted linear combination (SALCs) of them, corresponding to the appropriate irreducible representations of the molecular point group.