dodecahedron showing the symmetry elements for the Ih group

Molecular Symmetry Examples

Víctor Luaña

Universidad de Oviedo, Departamento de Química Física y Analítica, c/ Julián Clavería 8, 33006-Oviedo, Spain.


This page contains images of the symmetry elements for a collection of molecules spanning all kinds of point groups.

All the models and plots in this page have been created using the new symmetry module of tessel. The module uses a modification of te Velde's algorithm to detect automatically all the symmetry operations of an arbitrarily oriented molecule (see the details in Luaña and Martín Pendás 2005). The rotation axes and mirror planes are then represented according to predefined conventions that can be changed and tuned for each particular case. The resulting molecular model can be rendered with the help of a raytracing code (POVRay) or written in a format suitable to interactive manipulation on screen.

Next section provides a short introduction to molecular symmetry and the particular conventions followed here. It is followed by a succint description of the forms in which tessel produces the molecular models and instructions to view them. The last and main section of the page discusses the symmetry of a representative collection of molecules.

Symmetry elements

All the properties of the molecule are kept invariant under the application of a symmetry operation. In the quantum mechanical jargon, the symmetry operators commute with the molecular hamiltonian, so there is a complete set of symmetry adapted stationary wave functions. By talking about point group symmetry we are restricted to a set of symmetry operations that act on the cartesian coordinates of the molecular constituents in such a way that a single point in the space is always kept fixed: the center of mass of the molecule. Several types of such symmetry operations can be distinguished:

  • the identity, $\hat{E}$, transforms every point of the space into itself;
  • $\hat{C}_n^1$, a rotation of 360/n degrees around a n-order axis. A $\hat{C}_n^m$ rotation is obtained by repeating m times the $\hat{C}_n^1$ operation. The highest order rotation axis is called the main axis of the molecule.
  • the inversion, $\hat{i}$, moves every point in a straight line through the inversion center to the opposite side of the molecule. The inversion center must coincide with the center of mass.
  • $\hat{\sigma}$ is a reflection of the space through a mirror plane that contains the center of mass. The operation is denominated $\hat{\sigma}_h$ (horizontal reflection) if the mirror plane is perpendicular to the main rotation axis, and $\hat{\sigma}_v$ (vertical reflection) if the plane contains the main axis. Diehedral planes, $\hat{\sigma}_d$, are special cases of vertical mirror planes that bisect the angle between two binary axes perpendicular to the main axis.
  • the improper rotation $\hat{S}_n^1$ is formed by two succesive transformations: a rotation of 360/n degrees around an axis followed by the reflection through the perpendicular plane that contains the center of mass. By repeating m times this operation we get the general $\hat{S}_n^m$ improper rotation. $\hat{C}_n^m$ are also called proper rotations to distinguish them from the $\hat{S}_n^m$ operations. Inversion and reflection are particular cases of improper rotations.
The product of two operations, $\hat{A} \hat{B}$, is defined as the result of applying the second operation on the molecule, $\hat{B}$, and then applying the first operation on the result: $\hat{A} \hat{B} \textrm{molecule} = \hat{A} (\hat{B} \textrm{molecule})$. The product is always associative, $\hat{A} (\hat{B} \hat{C}) = (\hat{A} \hat{B}) \hat{C}$, but it is not commutative as a general rule.

The complete set of symmetry operations of a molecule is a mathematical group. This is a consequence of the following properties:

  • the set is closed, in other words, the product of any two symmetry operations of the molecule is another symmetry operation also found on the molecule.
  • the identity, which is always present in the symmetry group, is the neutral element of the product. Therefore, every symmetry operation is kept unchanged when multiplied by the identity.
  • every symmetry operation has an inverse that also belongs to the group. The product of an operation and its inverse is the identity, so whatever an operation does to a molecule the inverse operation does the opposite.

The above description and the names given to the symmetry elements correspond to the Schoenflies notation, perdominantly used by chemists, molecular spectroscopists and methematicians. Crystallographers and solid state scientists in general prefer the use of the Herman-Mauguin or international notation, in which the improper rotations are rotation-inversion rather than rotation-reflection operations. Both ways are equivalent in the end, but not identical.

Images and model formats

The tessel code is able to determine all the symmetry operations of an arbitrarily oriented molecule from the only knowledge of the cartesian coordinates of the atoms in the molecule. The code uses this information to produce a graphical description of the molecule and its symmetry elements. Users enter this information by means of the file:

  • molecule.tess: Input to the tessel code. Everything is produced from here.
Tessel output is writen in any of the following formats:
  • molecule.pov: Molecular model in the format of the POVRay raytracing code.
  • molecule.png: Molecular image in PNG (Portable Network Graphic) format.
  • molecule.wrl: Molecular model in VRML (Virtual Reality Modeling Language) format.
  • Molecular model in the OFF format recognized by geomview.
From these files, molecule.tess is the most useful one if you want to do any change on the plot, molecule.wrl or can be used to see interactively the molecule on your screen, and molecule.pov produces the most beautiful output.

VRML viewers can be installed as plugins of web navigators. The NIST maintains a page with up to date information on VRML viewers for most platforms.

Low symmetry groups

trans 1,2-difluoro 1,2-dichloro ethane

Three groups can be included in this category:

  • $\mathcal{C}_1$, the group of the molecules that have no symmetry element and the only symmetry operation is the identity;
  • $\mathcal{C}_s$, the group of the molecules that only have a mirror plane;
  • $\mathcal{C}_i$, the group of the molecules that only have an inversion center.

Molecule Symmetry .tess .png .pov .wrl .off
FNO $\mathcal{C}_s$ Get See Get See See
C2H2F2Cl2 $\mathcal{C}_i$ Get See Get See See

Cn and related groups


Four different group kinds are included here:

  • $\mathcal{C}_n$ is the basic rotation group, formed by n successive rotations of 360/n degrees around a rotation axis of order n. In other words, the n operations that form the group are $\hat{C}_n^m$ with m=1..n.
  • $\mathcal{C}_{nv}$ is formed by adding n $\hat{\sigma}_v$ mirror planes to the basic rotation group.
  • $\mathcal{C}_{nh}$ is formed by adding a single mirror plane perpendicular to the rotation axis, i.e. a $\hat{\sigma}_h$ plane, to the basic rotation group.
  • $\mathcal{S}_{2n}$ is similar to the $\mathcal{C}_n$ group, but all the operations correspond now to successive applications of $\hat{S}_n^1$ (a rotation of 180/n degrees around an axis followed by the reflection on the plane perpendicular to the axis).

H_2O_2 molecule H_2O molecule NH_3 molecule B(OH)_3 molecule Si_2H_2 molecule in a C2h configuration C_8H_4F_4 molecule

Molecule Symmetry .tess .png .pov .wrl .off
H2O2 $\mathcal{C}_2$ Get See Get See See
H2O $\mathcal{C}_{2v}$ Get See Get See See
NH3 $\mathcal{C}_{3v}$ Get See Get See See
B(OH)3 $\mathcal{C}_{3h}$ Get See Get See See
Si2O2 $\mathcal{C}_{2h}$(*) Get See Get See See
C8H4F4 $\mathcal{S}_{4}$ Get See Get See See

(*) disilyne ground state shows, in fact, a nonclassical bridge structure, Si(H2)Si, of $\mathcal{C}_{2v}$ symmetry (Bogey et al, 1991).

Dihedral groups

C(NO2)3- ion

All dihedral groups contain a main proper rotation axis of order n plus n binary axes perpendicular to the main one. We may have:

  • $\mathcal{D}_n$ is the bare dihedral group, formed by the n rotations around the main axis and n 180 degrees rotations around the perpendicular binary axes.
  • $\mathcal{D}_{nh}$ is obtained by adding a $\sigma_h$ plane to the symmetry elements of the $\mathcal{D}_n$ group.
  • $\mathcal{D}_{nd}$ is the result of adding n mirror planes containing the main axis to the symmetry elements of the $\mathcal{D}_n$ group. Those vertical mirror planes are usually called $\sigma_d$ rather than $\sigma_v$ as a way to indicate that they appear in between the binary axes.

C(NO2)3- ion benzene molecule C3H4 (allene) molecule C8H16 (chair configuration) molecule

Molecule Symmetry .tess .png .pov .wrl .off
C(NO2)3- $\mathcal{D}_3$ Get See Get See See
benzene $\mathcal{D}_{6h}$ Get See Get See See
C3H4 (allene) $\mathcal{D}_{2d}$ Get See Get See See
C8H16 (chair configuration) $\mathcal{D}_{4d}$ Get See Get See See

Cubic and icosahedral groups

Decorated cube with O symmetry

The groups included in this last category contain more than one proper axis of order 3 or larger. Icosahedral groups have a characteristic set of six different C5 proper axes:

  • $\mathcal{I}$, the bare icosahedral group. It can be distinguished because it lacks inversion.
  • $\mathcal{I}_{h}$, the full icosahedral group. It contains the inversion and it is formed by 120 different symmetry operations.
There are many examples of $\mathcal{I}_{h}$ symmetry, starting from the regular icosahedron and dodecahedron. Not so simple it is to find an example of the $\mathcal{I}$ group that could be clearly identified. We can add, however, a decoration to the $\mathcal{I}_{h}$ solids to lower their symmetry.

Decorated icosahedron of I symmetry Regular icosahedron Regular dodecahedron C60 (buckminsterfullerene)

Molecule Symmetry .tess .png .pov .wrl .off
I decorated icosahedron $\mathcal{I}$ Get See Get See See
Icosahedron $\mathcal{I}_h$ Get See Get See See
Dodecahedron $\mathcal{I}_h$ Get See Get See See
C60 (buckminsterfullerene) $\mathcal{I}_h$ Get See Get See See

Cubic groups, on the other hand, have a characteristic set of three different C4 proper axes:

  • $\mathcal{O}$, the bare octahedral group. It can be distinguished because it lacks inversion.
  • $\mathcal{O}_{h}$, the full octahedral group. The regular octahedron and the cube (also called hexahedron) are the two platonic solids that belong to this group.
Tetrahedral groups, finally, are subsets of the cubic groups. The C4 axes of the cubic groups are lost but four different C3 axes remain:
  • $\mathcal{T}$, the bare tetrahedral group is the result of non coincident ternary and binary axes. The group contains four C3 and 3 C2 axes, that give rise to 12 symmetry operations.
  • $\mathcal{T}_{h}$ is the result of adding an inversion center to the basic tetrahedral group. As a result, the 4C3 are also 4S6, and the group contains also three $\sigma_{h}$ mirror planes.
  • $\mathcal{T}_{d}$ results from adding six $\sigma_{d}$ mirror planes to the bare tetrahedral group. As a result, the 3C2 proper axes are also 3S4 improper ones. The group has no inversion center, however, being therefore easy to distinguish from the $\mathcal{T}_{h}$ group. $\mathcal{T}_{d}$ is the full tetrahedral group, to which the regular tetrahedron belongs.

As the tetrahedral groups are subsets of $\mathcal{O}_{h}$ we can start with a cube or an octahedron (both being of $\mathcal{O}_{h}$ symmetry) and decorate their faces to lower the symmetry to any of the cubic groups. The cube decorations shown below are based on the images by Ashcroft and Mermin.

Decorated cube of T symmetry Decorated cube of Th symmetry Decorated cube of Td symmetry Decorated cube of O symmetry Decorated cube of Oh symmetry

Molecule Symmetry .tess .png .pov .wrl .off
T decorated cube $\mathcal{T}$ Get See Get See See
Th decorated cube $\mathcal{T}_h$ Get See Get See See
Td decorated cube $\mathcal{T}_d$ Get See Get See See
O decorated cube $\mathcal{O}$ Get See Get See See
Oh decorated cube $\mathcal{O}_h$ Get See Get See See

More examples

Lennard-Jones 135 cluster

See also

  1. The Lennard-Jones 2-150, Lennard-Jones 310-561, and Lennard-Jones 562-1000 cluster pages.


The images contained in this page have been created and are copyrighted © by V. Luaña (2005). Permission is hereby granted for their use and reproduction for any kind of educational purpose, provided that their origin is properly attributed.


  1. F. A. Cotton, Chemical Applications of Group Theory 3rd ed. (Wiley, New York, 1990) ISBN 0471510947.
  2. J. S. Lomont, Applications of Finite Groups (Dover, New York, 1993) ISBN 0486673766.
  3. M. Hamermesh, Group Theory and Its Applications to Physical Problems (Dover, New York, 1990) ISBN 0486661814.
  4. N. W. Ashcroft and D. N. Mermin, Solid State Physics (Saunders, Philadelphia, 1976).
  5. G. te Velde, Numerical integration and other methodological aspects of bandstructure calculations, PhD Dissertation (Vrije Universiteit Amsterdam, 1990). Available online.
  6. V. Luaña and A. Martín Pendás, Automatic determination of the symmetry elements and point-group symmetry of arbitrary molecules (to be published).
  7. M. Bogey, H. Bolvin, C. Demuynck, and J. L. Destombes, Phys. Rev. Lett. 66 (1991) 413.

Useful links:
Educational pages:

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Page dates: 2005-03-10 (created), 2005-07-21 (last modified).
Web page creation: Víctor Luaña