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 D_5h 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 D_5h point group.

last changes: 08.06.2006, HZ
questions & comments to: zipse@cup.uni-muenchen.de