The C_2h Point Group

This point group contains four symmetry operations:

E     the identity operation
C₂   a twofold symmetry axis
i      a center of inversion
σ_h   a horizontal mirror plane

A simple example for a C_2h symmetric molecule is trans -1,2-dichloroethylene, 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 the C-Cl and C-H bond distances and the Cl-C-C and H-C-C bond angles. All atoms are also constrained to the horizontal mirror plane of the C_2h point group. This reduces the number of independent structural variables from 12 (for an asymmetric, non-linear molecule containing 6 centers) to 5.

Molecular orbitals as well as harmonic vibrations (if calculated) are labeled according to their symmetry properties as belonging to one of the four irreducible representations (A_g, B_g, A_u, and B_u) of the C_2h point group.