C2h

C2h is a molecular symmetry point group in the Schoenflies notation. It describes molecules that possess a C2 rotation axis and a horizontal mirror plane, together with an inversion center, forming a four-element group consisting of E, C2, i, and sigma_h. The group is abelian and isomorphic to the Klein four-group.

The symmetry operations of C2h are E (identity), C2 (a 180-degree rotation about the principal axis), i (inversion through the center), and sigma_h (reflection in the horizontal plane perpendicular to the C2 axis). The order of the group is 4, and the elements commute with each other.

The irreducible representations of C2h are A_g, B_g, A_u, and B_u. Their characters, for the operations E, C2, i, and sigma_h, are:

A_g: 1, 1, 1, 1

B_g: 1, -1, 1, -1

A_u: 1, 1, -1, -1

B_u: 1, -1, -1, 1

All irreducible representations are one-dimensional. The g versus u designation indicates symmetry with respect to inversion: g is gerade (inversion-symmetric) and u is ungerade (inversion-antisymmetric).

In spectroscopy, centrosymmetric groups like C2h follow the mutual exclusion principle: gerade representations are typically Raman-active, while ungerade representations are infrared-active, subject to specific selection rules. C2h is used to classify electronic, vibrational, and rotational states and to assign molecular orbitals and transitions in appropriate planar, centrosymmetric molecules.

Related point groups include C2, C2v, Ci, and D2h.