Point Groups:

• Point Groups
• Crystallographic Point Groups and Stereographic Projections
• Point Groups of Crystal Classes
• High-Symmetry Point Groups of Platonic Solids

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 (C₃) 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 Hermann-Mauguin 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 trans-1,2-dichlorocyclopropane 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 three-dimensional geometry and its (very often, but not necessarily) low-energy conformation. Transition states are often said to have a certain symmetry, which may imply details on the simultaneity of the bond-breaking and/or bond-forming processes.

For example, the racemization of the chiral [8]helicene (left- or right-handed helix with point group C₂, 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 three-dimensional connectivity of atoms in molecules and stereoisomers (e.g. cis- and trans-1,2-dichlorocyclopropane), whereas the conformation describes the exact 3D-arrangment of all atoms in molecules (e.g. the rotameric forms of ethane or chair/boat-conformations 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 five-fold axis of rotation C₅, but a five-, seven, or eight-fold rotational symmetry of crystals is incompatible with their three-dimensional translational lattice. The only rotational symmetries allowed for crystals are 1-, 2-, 3-, 4-, and 6-fold axes of rotation. However individual molecules may have rotation axis C_n with any integral value of n > 0 (including n = 5, 7, 8, ...).

General*		Non-cubic Crystallographic Point Groups and Stereographic Projections*
		n = 1		n = 2		n = 3		n = 4		n = 6
Cn		C1		C2		C3		C4		C6
		triclinic		monoclinic		trigonal		tetragonal		hexagonal
Cnv		-		C2v		C3v		C4v		C6v
				orthorhombic		trigonal		tetragonal		hexagonal
Cnh		(C1h) Cs		C2h		C3h		C4h		C6h
		monoclinic		monoclinic		hexagonal		tetragonal		hexagonal
Dn		-		D2		D3		D4		D6
				orthorhombic		trigonal		tetragonal		hexagonal
Dnh		-		D2h		D3h		D4h		D6h
				orthorhombic		hexagonal		tetragonal		hexagonal
Dnd		-		D2d		D3d		-		-
				tetragonal		trigonal
Sn		(S1 = Cs)		(S2) Ci		(S3 = C3h)		S4		S6
				triclinic				tetragonal		trigonal
		Cubic Crystallographic Point Groups
T		T		Th		Td
O		O		Oh
[*] Chiral point groups are marked by bold-face point group symbols. All other point groups are achiral. The stereographic projections are two-dimensional illustrations of the set of symmetry operations of an object (i.e. a molecular geometry). Objects are placed inside a sphere around their geometrical center, the equator plane of that sphere being perpendicular to the highest-order axis of rotation (the principal axis). All symmetry elements intersect in the center of the sphere, and are projected into the equator plane: A horizontal mirror plane σh coincides with the equator plane and is indicated by a bold outline of this equator. The n-fold principal axis of rotation is perpendicular to the equator plane and indicated by a regular n-sided polygon in the center. For rotation axes Cn this polygon is drawn solid, whereas for rotary reflection axes Sn open polygons are drawn. Rotation axes perpendicular to the principal axis are indicated by dashed lines in the equator plane including the appropriate polygons at both ends. If this axis is contained in a vertical mirror plane σv these lines are drawn solid. For a random point which does not lie on any symmetry element, all symmetry equivalent positions are generated and projected into the equator plane. Points above the equator plane are indicated by crosses, point below that plane are shown as open circles. The pattern of symmetry related points (crosses and circles) is characteristic for each point group and the corresponding combination of symmetry elements.

From the above stereographic projections it can be seen that a rotary-reflection axis S₁ is equivalent to a horizontal mirror plane σ_h, and a S₂ 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.

S1 = σh	S2 = i	S3 = C3 + σh	S4	S5 = C5 + σh	S6 = C3 + i

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 'Three-dimensional Stereographic Representations of Point Groups').