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

Presentation

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, , transforms every point of the space into itself;
 , a rotation of 360/n degrees around
a norder axis. A rotation is obtained
by repeating m times the operation.
The highest order rotation axis is called the main axis of the molecule.
 the inversion, , 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.
 is a reflection of the space through a mirror plane
that contains the center of mass. The operation is denominated
(horizontal reflection) if the mirror plane is perpendicular to the main rotation axis,
and (vertical reflection) if the plane contains
the main axis. Diehedral planes, , are special cases of
vertical mirror planes that bisect the angle between two binary axes perpendicular to the main axis.
 the improper rotation 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
improper rotation.
are also called proper rotations to distinguish them
from the operations.
Inversion and reflection are particular cases of improper rotations.
The product of two operations, , is defined as the result
of applying the second operation on the molecule, , and then applying
the first operation on the result:
.
The product is always associative,
, 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 HermanMauguin or international notation, in which the improper rotations are rotationinversion
rather than rotationreflection 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.
 molecule.off: 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 molecule.off 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

Three groups can be included in this category:
 ,
the group of the molecules that have no symmetry element and the only symmetry operation is the identity;
 ,
the group of the molecules that only have a mirror plane;
 ,
the group of the molecules that only have an inversion center.



Cn and related groups

Four different group kinds are included here:
 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
with m=1..n.
 is formed by adding n
mirror planes to the basic rotation group.
 is formed by adding a single mirror plane perpendicular to
the rotation axis, i.e. a plane, to the basic rotation group.
 is similar to the
group, but all the operations correspond now to successive applications of
(a rotation of 180/n degrees around an axis followed by the reflection on the plane perpendicular to the axis).
(*) disilyne ground state shows, in fact, a nonclassical bridge structure, Si(H_{2})Si, of
symmetry (Bogey et al, 1991).



Dihedral groups

All dihedral groups contain a main proper rotation axis of order n plus n binary axes perpendicular to the
main one. We may have:
 is the bare dihedral group, formed by the n rotations around the main
axis and n 180 degrees rotations around the perpendicular binary axes.
 is obtained by adding a
plane to the symmetry elements of the group.
 is the result of adding n mirror planes containing the main
axis to the symmetry elements of the group. Those vertical mirror planes are
usually called rather than
as a way to indicate that they appear in between the binary axes.



Cubic and icosahedral groups

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 C_{5} proper axes:
 , the bare icosahedral group. It can be distinguished because it lacks inversion.
 , the full icosahedral group. It contains the inversion and it is formed
by 120 different symmetry operations.
There are many examples of symmetry, starting from the regular
icosahedron and dodecahedron. Not so simple it is to find an
example of the group that could be clearly identified. We can add, however,
a decoration to the solids to lower their symmetry.
Cubic groups, on the other hand, have a characteristic set of three different C_{4} proper axes:
 , the bare octahedral group. It can be distinguished because it lacks inversion.
 , 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 C_{4} axes of the cubic groups are lost but four different C_{3} axes remain:
 , the bare tetrahedral group is the result of non coincident ternary and binary
axes. The group contains four C_{3} and 3 C_{2} axes, that give rise to 12 symmetry operations.
 is the result of adding an inversion center to the basic tetrahedral
group. As a result, the 4C_{3} are also 4S_{6}, and the group contains also three
mirror planes.
 results from adding six
mirror planes to the bare tetrahedral group. As a result, the 3C_{2} proper axes are also 3S_{4} improper ones.
The group has no inversion center, however, being therefore easy to distinguish from the
group. is the full
tetrahedral group, to which the regular tetrahedron belongs.
As the tetrahedral groups are subsets of we can start with
a cube or an octahedron (both being of 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.



More examples

See also
 The
LennardJones 2150,
LennardJones 310561,
and LennardJones 5621000
cluster pages.

Copyright

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.



References

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



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