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Particularly we will consider the following point groups which molecules can belong to.
- 1. The groups , , and .
- A molecule belongs to the group if it has no
elements other than identity . Example: DNA. A molecule belongs to the group , if it
consist of two operations: the identity and the inversion . Example: meso-tartaric
acid. A molecule belongs to the group ,
if it consists of two elements: identity and a mirror plane . Example: .
- 2. The group .
- A molecule belongs to the group if it has a n-fold axis.
Example: molecule belongs to the group as it has the elements and .
- 3. The group .
- A molecule belongs to the group if in addition to the
identity and a axis, it has vertical mirror planes . Examples:
molecule belongs to the group as it has the symmetry elements , , and two
vertical mirror planes which are called and . The molecule
belongs to the group as it has the symmetry elements , , and three
planes. All heteroatomic diatomic molecules and belong to the group
because all rotations around the internuclear axis and all reflections
across the axis are symmetry operations.
- 4. The group .
- A molecule belongs to the group if in addition to the
identity and a axis, it has a horizontal mirror plane . Example:
butadiene , which belongs to the group, while molecule belongs to
the group. Note, that presence of and operations imply the presence
of a center of inversion. Thus, the group consists of a axis, a horizontal
mirror plane , and the inversion .
- 5. The group .
- A molecule belongs to the group if it has a n-fold
principal axis and two-fold axes perpendicular to . is of cause
equivalent with and the molecules of this symmetry group are usually classified as
.
- 6. The group .
- A molecule belongs to the group if in addition to the
operations it possess dihedral mirror planes . Example: The twisted,
allene belongs to group while the staggered confirmation of ethane belongs
to group.
- 7. The group .
- A molecule belongs to the group if in addition to the
operations it possess a horizontal mirror plane . As a
consequence, in the presence of these symmetry elements the molecule has also necessarily
vertical planes of symmetry at angles 360/2 to one another.
Examples: has the
elements , , , and and thus belongs to the group.
has the elements , , , and and thus belongs to the
group. All homonuclear diatomic molecules, such as , , and others belong to the
group. Another examples are ethene (),
(), ().
- 8. The group .
- A molecule belongs to the group if it possess one
axis. Example: tetraphenylmethane which belongs to the group . Note, that the
group is the same as , so such molecules have been classified before as .
- 9. The cubic groups.
- There are many important molecules with more than one principal
axes, for instance, and . Most of them belong to the cubic groups, particularly
to tetrahedral groups , , and , or to the octahedral groups
and . If the object has the rotational symmetry of the tetrahedron, or octahedron,
but has no their planes of reflection, then it belongs to the simpler groups , or .
The group is based on , but also has a center of inversion.
- 10. The full rotational group .
- This group consists of infinite number of
rotational axes with all possible values of . It is the symmetry of a sphere. All atoms
belong to this symmetry group.
Next: Group Multiplication Table
Up: The Symmetry Classification of
Previous: Definition of the Group
Contents
Markus Hiereth
2005-02-09
Auf diesem Webangebot gilt die Datenschutzerklärung der TU Braunschweig mit Ausnahme der Abschnitte VI, VII und VIII.