Punktgruppen

Punktgruppen, or point groups, are mathematical groups formed by the symmetry operations that leave at least one point fixed in three-dimensional space. They describe the rotational and reflectional symmetries of finite objects such as molecules and nanoparticles, as well as the non-translational symmetry of crystals when translations are disregarded. The symmetry operations include the identity, proper rotations about axes, mirror reflections in planes, inversion through a center, and improper rotations (a rotation followed by reflection).

In practice, point groups classify the overall shape and how it behaves under symmetry. They form finite groups for most molecules and nanoscale objects. Space groups extend point groups by incorporating translational symmetry, which is relevant for the full crystal lattice. The conventional notations combine Schoenflies notation (for point groups) and, in crystallography, Hermann–Mauguin notation. There are 32 crystallographic point groups, which are compatible with a lattice and common in solid-state contexts.

Character tables provide a compact summary of a point group’s irreducible representations, their characters for each symmetry operation, and the selection rules for transitions. They underpin the prediction of vibrational modes, infrared and Raman activity, and degeneracies in spectra, as well as the symmetry classification of molecular orbitals and transitions.

Common examples include: C2v for water (H2O), C3v for ammonia (NH3), D6h for benzene (C6H6), and Td for methane (CH4). Oh describes highly symmetric systems like octahedral or cubic environments, while D∞h and C∞v describe idealized linear diatomic or linear molecules.

Punktgruppen thus provide a framework to analyze and predict physical properties based on geometric symmetry.