refers to the mathematical entity called
"group"
and deals with properties that can be deduced for these
mathematical entities. Group theory has found use in chemistry, because
symmetry operations of molecules,
crystals, ..., form groups.
Group theory is used by theoreticians, crystallographers, and
spectroscopists for a wide range of applications:
to unambiguously name symmetries,
to derive and apply spectroscopical selection rules,
to speed up quantum chemical computations, and so on.
Point groups and space groups
are the groups used in chemistry. In general,
point groups, line groups, plane groups, and space groups are distinguished,
depending on which geometrical element remains in place under all symmetry
operations of the group.
In case of an isolated molecule, this geometrical element
is the centre of mass, and one speaks of point groups.
If a line remains in place, one speaks of line groups or frieze groups.
If the connecting element is a plane, one speaks of plane groups
or wall paper groups.
In case of a three-dimensional, periodic arrangement of
molecules or ions in a crystal, the common geometrical element is
the three-dimensional space, and one speaks of space groups.
Based on the expertise of the author, this
tutorial is focussed on point groups. It is not meant to replace a
textbook or a lecture series; instead, it should be used
as an augmentation of such sources, in the presentation as well as
the learning process.
Notation.
The various point groups.
Dr. Michael Ramek.
