Representations - Group Theory Applied to Chemistry - page 63

upper index, Ω , which stands for the irrep, and the lower index, i . The latter in-
dex is sometimes called the subrepresentation and also contains a specific piece of
information. It determines the component of the irrep, i.e., it refers to a particular
column in the irreducible representation matrix that describes the transformation of
this component:

j | Φ j D ji (R)

R | Φ i =

(4.47)

This symbol of course makes sense only if we do not limit ourselves to the charac-
ters, but also determine the representation matrices for all nondegenerate irreps of
the group. These will depend on the choice of a particular canonical basis set. Tabu-
lar material containing suitable sets of irrep matrices is rather sparse. Some standard
choices are provided in Appendix C . Now we construct the projector P based on the
available matrices:

dim (Ω )
|

P Ω

G D Ω

kl (R) R kl =

(4.48)

G |

R ∈

| Φ i

Let us apply this projector to the SALC . This requires the combination of Eqs.

( 4.47 ) and ( 4.48 ) and exploits the full GOT potential:

dim (Ω )
|

kl (R)D ji (R) Φ j

P Ω D Ω

Φ i =

kl |

G |

R ∈ G j

δ Ω ,Ω δ kj δ li Φ j =

j

= Φ Ω k δ Ω ,Ω δ li (4.49)

The action of the projector entails a twofold selection, both at the level of the repre-
sentation and of the subrepresentation, indicated by the two Kronecker deltas. First,
it compares the irreps of the operator and of the SALC. If they do not match, then
the SALC is simply destroyed. Second, the δ li selection rule comes into play—
under the “protection,” as it were, of the first Kronecker delta, which assures that
the second selection rule will matter only when we are already inside the same ir-
rep. This second rule compares index l of the projection operator with index i of
the target, and annihilates the target unless they are the same; it therefore selectively
picks out the SALC that transforms exactly as the l th component of the Ω irrep.
Third, instead of delivering as result this particular component, the projector also
has a built-in ability to act as a ladder operator and turn the l th component so ob-
tained into a k th component. If we do not want this ladder aspect, we simply use the
diagonal projection operator with k l . Let us illustrate this for the Q Ex hydrogen

bending mode in Fig. 4.2 . If we want to obtain from this the Q Ey component, we
should use a projection operator that recognizes the x and replaces it by y ; hence, it
must belong to the E irrep, its row index should be 2, and its column index 1. This

=