C3v

C3v is a point group used in molecular symmetry to describe objects that have a principal threefold rotation axis and three vertical mirror planes. The group contains six symmetry operations: the identity E, two C3 rotations (C3 and C3^2) about the principal axis, and three σv reflections in planes containing the axis. The order of the group is 6.

The character table for C3v includes three irreducible representations: A1, A2, and E. The two one-dimensional representations are A1 and A2, and there is one two-dimensional representation E. The classes are E, 2C3, and 3σv, with the following characters:

A1: 1, 1, 1

A2: 1, 1, -1

E: 2, -1, 0

In many C3v molecules, the principal axis is taken as the z-axis. The z coordinate transforms as A1, while the x and y coordinates transform as the two components of the E representation.

Common molecules exhibiting C3v symmetry include ammonia (NH3), phosphine (PH3), and methyl chloride (CH3Cl) in their idealized geometries. These molecules typically feature a pyramidal arrangement with three identical substituents around a central atom.

Irreducible representations of C3v are used to classify electronic states and molecular vibrations, and to establish selection rules for transitions. Vibrational modes decompose into A1, A2, and E species; this classification informs infrared and, in some cases, Raman activity and the degeneracy of modes.