The D5h Point Group
This point group contains the following symmetry operations:
E the identity operation
C5 a fivefold principal symmetry axis
5 * C2 five twofold symmetry axes orthogonal to the principal axis
σh a horizontal mirror plane intersecting the principal symmetry axis
5 * σv five vertical mirror planes aligned with the principal symmetry axis
S5 a fivefold improper rotation axis
A simple example for a D5h symmetric molecule is the cyclopentadienyl anion (C5H5-), 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 carbon atoms C3 - C7 and for hydrogen atoms H8 - H12, through fixing the bond angles between adjacent centers to 360/5 = 72 degrees, and through constraining all atoms to the horizontal symmetry plane. This reduces the number of independent structural variables from 24 (for an asymmetric, non-linear molecule containing ten centers) to 2 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 ( A1', A2', E1', E2', A1'', A2'', E1'', E2'', ) of the D5h point group.