Character table for D3h point group, linear; 253 quadratic rotations x + y? R) (xz)
Character table for D3h point group, linear; 253 quadratic rotations x + y? R) (xz)
The Correct Answer and Explanation is:
Character Table for D3h Point Group
The D3h point group is common in molecules with a trigonal planar shape, such as BF3 or NO3−. It contains symmetry elements including:
- E (identity)
- 2C3 (rotation by ±120°)
- 3C′2 (rotation about axes in the molecular plane)
- σh (horizontal mirror plane)
- 2S3 (improper rotations)
- 3σv (vertical mirror planes)
Here is the character table for the D3h point group:
| D3h | E | 2C3 | 3C′2 | σh | 2S3 | 3σv | Basis Functions |
|---|---|---|---|---|---|---|---|
| A′1 | 1 | 1 | 1 | 1 | 1 | 1 | z |
| A′2 | 1 | 1 | –1 | 1 | 1 | –1 | |
| E′ | 2 | –1 | 0 | 2 | –1 | 0 | (x, y), (x²–y², xy) |
| A″1 | 1 | 1 | 1 | –1 | –1 | –1 | |
| A″2 | 1 | 1 | –1 | –1 | –1 | 1 | Rz |
| E″ | 2 | –1 | 0 | –2 | 1 | 0 | (Rx, Ry), (xz, yz) |
Explanation
The D3h point group is of particular importance in chemistry for describing the symmetry of trigonal planar molecules. The character table above organizes the irreducible representations (irreps) of this group and associates them with symmetry-adapted basis functions like coordinates and orbital shapes.
The question asks about the symmetry of linear, quadratic, and rotational functions such as x + y, Rz, and xz.
- Linear functions (x, y, z):
- x, y transform as the E′ irrep. This is a 2D degenerate representation.
- z transforms as A′1, since it remains unchanged under the main symmetry operations.
- Rotational functions (Rx, Ry, Rz):
- Rx and Ry belong to the E″ representation.
- Rz, rotation about the principal axis, transforms as A″2.
- Quadratic functions:
- x² – y² and xy are in the E′ irrep.
- xz and yz fall under E″, as they mix in-plane and out-of-plane axes.
As for x + y, it is a linear combination of x and y and still belongs to the E′ representation. The individual components don’t affect the irreducible representation, only the symmetry behavior matters.
Thus:
- x + y → E′
- Rz → A″2
- xz → E″
Understanding these assignments is essential in vibrational spectroscopy, molecular orbital theory, and predicting activity in IR or Raman spectra. By matching a molecular vibration or orbital with these symmetry types, chemists can predict which transitions are allowed and interpret experimental data more effectively.
