TUD Organische Chemie  Immel  Tutorials  Symmetry  Point Groups  View or Print (this frame only) 
Point Groups:
The classification of molecules (better: molecular geometries) is done by collecting all their inherent symmetry properties, and putting together those with identical symmetry elements in a certain point group. The names of these point groups are determined by the symmetry elements they posses. For example, chloroform (show below) features a threefold rotational axis (C_{3}) as well as three vertical mirror planes (σ_{v}), and thus belongs to the point group named C_{3v}.
A general flowchart which can be used to determine the point group of arbitrary molecules will be presented later in this section (classification of point groups and point group separation procedure). In this context, we use the Schoenflies system for labeling the point groups of individual molecules (the HermannMauguin system or International system is more frequently used in the discussion of crystal symmetry).



It is important to note that the point groups to which a molecule belongs ultimately depends on its exact molecular geometry and its specific configuration and conformation.^{*)}
Without knowledge of the exact configuration or conformation, the determination of its point group is impossible.
For example the staggered and ecliptic conformations of ethane, or the chair and boat conformations of cyclohexane have different symmetry properties.
Similarly, the stereoisomers cis and trans1,2dichlorocyclopropane have different configurations and therefore different symmetry properties belonging to different point groups. On the other hand, if a molecule is said to belong to a certain point group, we may gain important knowledge on its threedimensional geometry and its (very often, but not necessarily) lowenergy conformation. Transition states are often said to have a certain symmetry, which may imply details on the simultaneity of the bondbreaking and/or bondforming processes. For example, the racemization of the chiral [8]helicene (left or righthanded helix with point group C_{2}, respectively) proceeds through an achiral transition state featuring mirror symmetry (point group C_{s}). Animated examples for this racemization process can be found on this web site (movies with MPG format or online animations using Jmol): ^{*)} The term configuration describes the threedimensional connectivity of atoms in molecules and stereoisomers (e.g. cis and trans1,2dichlorocyclopropane), whereas the conformation describes the exact 3Darrangment of all atoms in molecules (e.g. the rotameric forms of ethane or chair/boatconformations of cyclohexane).
Below a table of all so called 32 crystallographic point groups is provided, together with a list of their essential symmetry elements. In this symmetry section, there is also a flowchart scheme provided (see 'Point Group Separation Procedure'), which allow the determination of the corresponding point group for any arbitrary molecular geometry (see also 'Symmetry Elements of Point Groups' and 'Hierarchy of Point Groups  Symmetry Correlations').



The table below only lists the 32 crystallographic point groups which are in accord with the translational and rotational symmetry of crystals. Please note, that
although a molecule (in its specific conformation) may of course feature a fivefold axis of rotation C_{5}, but
a five, seven, or eightfold rotational symmetry of crystals is
incompatible with their threedimensional translational lattice. The only rotational symmetries allowed for crystals are 1, 2, 3, 4, and 6fold axes of rotation.
However individual molecules may have rotation axis C_{n} with any integral value of n > 0 (including n = 5, 7, 8, ...).
From the above stereographic projections it can be seen that a rotaryreflection axis S_{1} is equivalent to a horizontal mirror plane σ_{h}, and a S_{2} axis corresponds to an inversion center. In addition, any S_{n} axis with odd n is identical to the combination of a C_{n} and σ_{h}, and therefore these axis need not to be considered separately.
Furthermore, for molecules there are additional point groups such as I or I_{h} of higher symmetry (icosahedral point groups) which we need to consider (see also 'Threedimensional Stereographic Representations of Point Groups').


The various point groups may be categorized by the corresponding crystal classes as shown by the table below:


The highsymmetry point groups in which more than one C_{n} axis with n ≥ 3 is present are best visualized by the five regular polyhedra
(Platonic solids) as shown below. In these objects, all faces, vertices, and edges are symmetry related and thus equivalent. The octahedron and the cube
are close related to each other as they contain the same symmetry elements, but in different orientations. The same applies to the dodecahedron and icosahedron.
For each of the point groups T_{d}, O_{h}, and I_{h} there exists subgroups T, O, and I which contain all C_{n} symmetry elements, but none of the S_{n} operations (including inversion and reflection, see also 'Threedimensional Stereographic Representations of Point groups'). Adding a σ_{h} mirror plane or an inversion center to the T group yields T_{h}.
