The C_s Point Group

This point group contains only two symmetry operations:

E    the identity operation
σ   a mirror plane

A simple example for a C_s symmetric molecule is methanol (CH₃OH) in its staggered conformation, here in its HF/6-31G(d) optimized structure:

In this case the symmetry of the system is reflected in the Z-Matrix only through the use of identical variable names for hydrogen atoms H5 and H6 and through constraining atoms 1 - 4 to the symmetry plane. This reduces the number of independent structural variables from 12 (for an asymmetric, non-linear molecule containing six centers) to 8 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 two irreducible representations (A' and A'') of the C_s point group. Those properties belonging to irreducible representation A' are symmetric to both the identity operation E as well as reflection through the mirror plane. Those properties belonging to irreducible representation A'' are symmetric with respect to the identity operation E but antisymmetric with respect to reflection through the mirror plane.