Symmetry elements and symmetry operations

Operations which leave an object looking the same are called symmetry operations . This term is confined to operations where there is definitely no difference in the appearance of a molecule before and after performing the operation.

If there is a point which is not at all affected by the operation, we speak of point symmetry. In contrast, if translation is taken into consideration too, we speak of space symmetry. Dealing with molecules, point symmetry is of relevance, whereas translation and space symmetry plays a role in three-dimensional arrangements of particles, i.e. in crystals.

To characterize symmetry in a concise way, two sets of symmetry symbols have been introduced. The Schoenflies symbols and the Hermann-Mauguin symbols, or, international symbols. The Schoenflies symbols are older and preferentially used for molecular symmetry and in spectroscopy. In the discussion of crystal symmetry, they are outdated. Additionally, there are visual symbols.

A classification of molecules is done with reference to five symmetry operations. A symmetry operation is characterized by a point, a straight line or a plane as symmetry element. Thus, any symmetry element is connected with one ore more symmetry operations that yield an image identical to the original molecule.

  1. Identity E is a symmetry operation without effect, e.g. a rotation with an angle of 360°. Though such an operation seems useless, it is of importance in group theory as any group needs to have one neutral element, i.e. identity. Furthermore, this symmetry operation is the base for the classification of asymmetric molecules like CHBrClF which have no other symmetry.
  2. An n-fold rotation denotes a rotation through an angle of 360°/n, thereby yielding an image indistinguishable from the original. The n-fold axis Cn is the respective symmetry element.
  3. In inversion, any point of an object is taken, moved through a centre of inversion (i.e. the symmetry element i) and placed in equal distance beyond this centre.
  4. Reflection moves any point of the original orthogonally to a point beyond some reflection plane or mirror plane. This plane is the respective symmetry element σ.
  5. A rotary reflection around some rotary reflection axis Sn (i.e. symmetry element) combines an rotation through 360°/n followed by reflection on a horizontal plane.


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