Character Tables
For each point group, a character table summarizes information on its symmetry operations and on its irreducible representations. As there are always equal numbers of irreducible representations and classes of symmetry operations, the tables are square.
The table itself consists of characters that represent how a particular irreducible representation transforms when a particular symmetry operation is applied. Any symmetry operation in a molecule's point group acting on the molecule itself will leave it unchanged. But, for acting on a general entity, such as a vector or an orbital, this need not be the case. The vector could change sign or direction, and the orbital could change type. For simple point groups, the values are either 1 or −1: 1 means that the sign or phase (of the vector or orbital) is unchanged by the symmetry operation (symmetric) and −1 denotes a sign change (asymmetric).
The representations are labeled according to a set of conventions:
- A, when rotation around the principal axis is symmetrical
- B, when rotation around the principal axis is asymmetrical
- E and T are doubly and triply degenerate representations, respectively
- when the point group has an inversion center, the subscript g (German: gerade or even) signals no change in sign, and the subscript u (ungerade or uneven) a change in sign, with respect to inversion.
- with point groups C∞v and D∞h the symbols are borrowed from angular momentum description: Σ, Π, Δ.
The tables also capture information about how the Cartesian basis vectors, rotations about them, and quadratic functions of them transform by the symmetry operations of the group, by noting which irreducible representation transforms in the same way. These indications are conventionally on the righthand side of the tables. This information is useful because chemically important orbitals (in particular p and d orbitals) have the same symmetries as these entities.
The character table for the C2v symmetry point group is given below:
| C2v | E | C2 | σv(xz) | σv'(yz) | ||
|---|---|---|---|---|---|---|
| A1 | 1 | 1 | 1 | 1 | z | x2, y2, z2 |
| A2 | 1 | 1 | −1 | −1 | Rz | xy |
| B1 | 1 | −1 | 1 | −1 | x, Ry | xz |
| B2 | 1 | −1 | −1 | 1 | y, Rx | yz |
Consider the example of water (H2O), which has the C2v symmetry described above. The 2px orbital of oxygen is oriented perpendicular to the plane of the molecule and switches sign with a C2 and a σv'(yz) operation, but remains unchanged with the other two operations (obviously, the character for the identity operation is always +1). This orbital's character set is thus {1, −1, 1, −1}, corresponding to the B1 irreducible representation. Likewise, the 2pz orbital is seen to have the symmetry of the A1 irreducible representation, 2py B2, and the 3dxy orbital A2. These assignments and others are noted in the rightmost two columns of the table.
Read more about this topic: Molecular Symmetry
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