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Top of Page Molecular Structures of Organic Compounds - Point Groups
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 (C3) as well as three vertical mirror planes (σv), and thus belongs to the point group named C3v.

Symmetry of Chloroform

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).

Symmetry and Point Groups
Symmetry Elements of a 3D-Object
Top of Page Point Groups
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 Cs). Animated examples for this racemization process can be found on this web site (movies with MPG format or online animations using Jmol):

Racemization of [8]Helicene

*) 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').

Symmetry of Ethane
Symmetry elements of ethane (staggered and ecliptic conformations)
Symmetry of Cyclohexane
Symmetry elements of cyclohexane (chair and boat conformations)
Symmetry of 1,2-Dichlorocyclopropane
Symmetry elements of 1,2-dichlorocyclopropane (cis- and trans-configuration)
Top of Page Crystallographic Point Groups and Stereographic Projections
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 Cn 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   C1Stereographic Projection of Point Group C1 C2Stereographic Projection of Point Group C2 C3Stereographic Projection of Point Group C3 C4Stereographic Projection of Point Group C4 C6Stereographic Projection of Point Group C6
   
triclinic
monoclinic
trigonal
tetragonal
hexagonal
Cnv  
-
C2vStereographic Projection of Point Group C2v C3vStereographic Projection of Point Group C3v C4vStereographic Projection of Point Group C4v C6vStereographic Projection of Point Group C6v
     
orthorhombic
trigonal
tetragonal
hexagonal
Cnh  
(C1h)

Cs
Stereographic Projection of Point Group Cs C2hStereographic Projection of Point Group C2h C3hStereographic Projection of Point Group C3h C4hStereographic Projection of Point Group C4h C6hStereographic Projection of Point Group C6h
   
monoclinic
monoclinic
hexagonal
tetragonal
hexagonal
Dn  
-
D2Stereographic Projection of Point Group D2 D3Stereographic Projection of Point Group D3 D4Stereographic Projection of Point Group D4 D6Stereographic Projection of Point Group D6
     
orthorhombic
trigonal
tetragonal
hexagonal
Dnh  
-
D2hStereographic Projection of Point Group D2h D3hStereographic Projection of Point Group D3h D4hStereographic Projection of Point Group D4h D6hStereographic Projection of Point Group D6h
     
orthorhombic
hexagonal
tetragonal
hexagonal
Dnd  
-
D2dStereographic Projection of Point Group D2d D3dStereographic Projection of Point Group D3d
-
-
     
tetragonal
trigonal
   
Sn  
(S1 = Cs)
(S2)

Ci
Stereographic Projection of Point Group Ci
(S3 = C3h)
S4Stereographic Projection of Point Group S4 S6Stereographic Projection of Point Group 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 σh, and a S2 axis corresponds to an inversion center. In addition, any Sn axis with odd n is identical to the combination of a Cn and σh, and therefore these axis need not to be considered separately.

Stereographic Projection of Point Group S1 Stereographic Projection of Point Group S2 Stereographic Projection of Point Group S3 Stereographic Projection of Point Group S4 Stereographic Projection of Point Group S5 Stereographic Projection of Point Group S6
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 Ih of higher symmetry (icosahedral point groups) which we need to consider (see also 'Three-dimensional Stereographic Representations of Point Groups').

Top of Page Point Groups of Crystal Classes
The various point groups may be categorized by the corresponding crystal classes as shown by the table below:

Crystal System      Unit Cell Parameters        Point Groups
Triclinic abc
αβγ
Triclinic Unit Cell
C1 Ci          
Monoclinic abc
α = β = 90 ≠ γ
Monoclinic Unit Cell
Cs C2 C2h        
Orthorhombic abc
α = β = γ = 90
Orthorhombic Unit Cell
C2v D2 D2h        
Tetragonal a = bc
α = β = γ = 90
Tetragonal Unit Cell
C4 S4 C4h C4v D2d D4 D4h
Trigonal a = b = c
α = β = γ ≠ 90
Trigonal Unit Cell
C3 S6 C3v D3 D3d    
Hexagonal a = bc
α = β = 90; γ = 120
Hexagonal Unit Cell
C3h C6 C6h C6v D3h D6 D6h
Cubic a = b = c
α = β = γ = 90
Cubic Unit Cell
T Th Td O Oh    

Top of Page High-Symmetry Point Groups of Platonic Solids
The high-symmetry point groups in which more than one Cn 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.

 
Tetrahedron
Octahedron
Cube
Dodecahedron
Icosahedron
Graphics
tetrahedron
octahedron
cube
dodecahedron
icosahedron
Faces
4 triangles
8 triangles
6 squares
12 pentagons
20 triangles
Vertices
4
6
8
20
12
Edges
6
12
12
30
30
Point Group     
Td
Oh
Oh
Ih
Ih

For each of the point groups Td, Oh, and Ih there exists sub-groups T, O, and I which contain all Cn symmetry elements, but none of the Sn operations (including inversion and reflection, see also 'Three-dimensional Stereographic Representations of Point groups'). Adding a σh mirror plane or an inversion center to the T group yields Th.

For more information on other research topics, please refer to the complete list of publications and to the gallery of graphics and animations.

© Copyright PD Dr. S. Immel

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