The D_5h 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
σ_h a horizontal mirror plane intersecting the principal symmetry axis
5 * σ_v 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:

#P RHF/6-31G(d) opt=(Z-Matrix,tight) RHF/6-31G(d) opt min cyclopentadienyl anion D5h -1 1 X1 X2 1 1.0 C3 2 r3 1 90.0 C4 2 r3 1 90.0 3 +72.0 C5 2 r3 1 90.0 4 +72.0 C6 2 r3 1 90.0 5 +72.0 C7 2 r3 1 90.0 6 +72.0 H8 2 r8 1 90.0 7 +72.0 H9 2 r8 1 90.0 8 +72.0 H10 2 r8 1 90.0 9 +72.0 H11 2 r8 1 90.0 10 +72.0 H12 2 r8 1 90.0 11 +72.0 r3=1.19285213 r8=2.27287212

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.