The D_2h Point Group

This point group contains the following symmetry operations:

E    the identity operation
C₂   a twofold principal symmetry axis
2 * C₂   two twofold symmetry axes orthogonal to the principal axis
i     inversion through a center of symmetry
σ_h   a horizontal mirror plane intersecting the principal symmetry axis
2 * σ_v   two vertical mirror planes aligned with the principal symmetry axis

A simple example for a D_2h symmetric molecule is ethylene (C₂H₄), here in its HF/6-31G(d) optimized structure:

In this case the symmetry of the system is reflected in the Z-Matrix through the use of identical variable names for hydrogen atoms H5 - H8 and through constraining all atoms to the symmetry plane. This reduces the number of independent structural variables from 12 (for an asymmetric, non-linear molecule containing six centers) to 3 and thus accelerates geometry optimizations.

Molecular orbitals as well as harmonic vibrations (if calculated) are labeled according to their symmetry properties as belonging to one of the eight irreducible representations ( A_g, A_u, B_1g, B_2g, B_3g, B_1u, B_2u, B_3u, ) of the D_2h point group.