D4h

D4h is a point group in molecular symmetry (Schönflies notation). It describes objects with a square-planar or tetragonal geometry that possess a principal fourfold rotation axis and a horizontal mirror plane. The group has 16 symmetry operations and is one of the Dnh/D4h families used to classify molecular vibrations, orbitals, and transitions in spectroscopy.

The symmetry elements of D4h include the identity E; two C4 proper rotations around the principal axis, C4 and C4^3; a C2 rotation about the principal axis, C4^2; two C2 axes perpendicular to the principal axis (C2' through the midpoints of opposite sides) and two C2 axes along the diagonals (C2'' through opposite vertices); an inversion center i; two S4 improper rotations around the principal axis; the horizontal mirror plane sigma_h; two vertical mirror planes sigma_v that contain the principal axis and a C2' axis; and two diagonal mirror planes sigma_d that contain the principal axis and bisect the angles between C2' axes.

The principal axis is the z axis; C2' axes lie along x and y; C2'' axes lie along the diagonals of the square. The sigma_h plane coincides with the molecular plane for many square-planar species; sigma_v and sigma_d are vertical planes relative to that plane.

The irreducible representations of D4h consist of ten one- and two-dimensional species: A1g, A2g, B1g, B2g, and Eg (gerade), and Au, A2u, B1u, B2u, Eu (ungerade). Eg and Eu are two-dimensional, giving rise to degenerate pairs.

Common examples of molecules with D4h symmetry include square-planar transition-metal complexes such as PtCl4^2−, PdCl4^2−, and Ni(CN)4^2−. The group is a subgroup of Oh and is widely used to analyze vibrational modes, electronic structure, and selection rules in spectroscopy.