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:

#P HF/6-31G(d) opt=(Z-Matrix,tight) test1 HF/6-31G(d) opt methanol 0 1 H1 C2 1 r2 O3 2 r3 1 a3 H4 3 r4 2 a4 1 180.0 H5 2 r5 3 a5 1 d5 H6 2 r5 3 a5 1 -d5 r2=1.08105957 r3=1.39956997 r4=0.94629421 r5=1.08744287 a3=107.170012 a4=109.447010 a5=112.035787 d5=118.773010

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.