D3hsymmetry

D3h is a point group in molecular symmetry describing molecules with a trigonal planar arrangement that possess a horizontal mirror plane. It has 12 symmetry operations and is part of the dihedral family D3h.

The symmetry operations include the identity E; two C3 rotations about a principal axis; three C2' rotations about axes in the molecular plane; a horizontal mirror plane sigma_h; two improper rotations S3; and three vertical mirror planes sigma_v that contain the C3 axis. The C3 axis is perpendicular to the molecular plane; the C2' axes lie in the plane and pass through the atoms. The presence of sigma_h indicates the molecular plane is a mirror plane; the three sigma_v planes bisect the angles between adjacent atoms. The S3 operations combine a C3 rotation with reflection in sigma_h.

The irreducible representations of D3h are A'1, A'2, E', A''1, A''2, and E''. The primes denote symmetry with respect to the sigma_h plane, and the double primes indicate antisymmetry with respect to it. The E' and E'' representations are doubly degenerate. Common basis functions show how these transform: z (perpendicular to the plane) transforms as A''2, while x and y (in the plane) transform as E'. These transformation properties underpin the selection rules for electronic and vibrational transitions in D3h systems.

Molecules exhibiting D3h symmetry include BF3, SO3, and NO3-. This symmetry framework helps predict degeneracies, spectroscopic activity, and the splitting patterns of energy levels in trigonal systems.