C4v

C4v is a point group used in molecular symmetry to describe objects that possess a principal fourfold rotation axis and vertical mirror planes. It contains eight symmetry operations and is common for molecules with a square-based arrangement around the axis.

The symmetry operations of C4v are: the identity E; two rotations about the principal axis by 90 degrees and 270 degrees (C4 and C4^3); a rotation by 180 degrees around the same axis (C2); two vertical mirror planes that contain the axis (sigma_v); and two dihedral mirror planes that also contain the axis but bisect the angles between the primary directions (sigma_d). The eight operations fall into five classes: E; 2C4; C2; 2sigma_v; 2sigma_d.

C4v has five irreducible representations: four one-dimensional representations A1, A2, B1, B2, and one two-dimensional representation E. The character table (summarized) indicates how these irreps transform under the symmetry operations. A1 and A2 are symmetric under most operations; B1 and B2 differ in the behavior under sigma_v and sigma_d; E is two-dimensional and corresponds to degenerate pairings. Basis functions and Cartesian components that transform as these irreps are known: z transforms as A1; (x, y) transform as E; Rz as A2; x^2 − y^2 as B1; xy as B2; and xz, yz as E.

A common example of a molecule with C4v symmetry is XeOF4, which has a square arrangement of ligands around a central axis. C4v is used in spectroscopy and quantum chemistry to determine selection rules and vibrational modes for such systems.