Symmetry Operation

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## Molecules

### Identity Operation

### Reflection through mirror planes

### Inversion operation

### Proper rotation operations

### Improper rotation operations

### Examples

## Crystals

## See also

## References

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

Symmetry Operation

In the context of molecular symmetry, a **symmetry operation** is a permutation of atoms such that the molecule or crystal is transformed into a state indistinguishable from the starting state.
Two basic facts follow from this definition, which emphasizes its usefulness.

- Physical properties must be invariant with respect to symmetry operations.
- Symmetry operations can be collected together in groups which are isomorphic to permutation groups.

Wavefunctions need not be invariant, because the operation can multiply them by a phase or mix states within a degenerate representation, without affecting any physical property.

*C _{1}*, rotation by 360°, is called the Identity operation and is denoted by

A molecule that consists of a plane of symmetry possesses a mirror plane. When this plane of symmetry is parallel to the principal axis of the molecule (molecular z-axis), it is considered as a vertical plane (?_{v}). If the plane of symmetry is perpendicular to the principal axis, then it is denoted as a horizontal mirror plane (?_{h}). A dihedral mirror plane (?_{d}) is the third type of plane of symmetry which bisects the angle between two 2-fold axes perpendicular to the principal axis.
Through the reflection of each mirror plane, the molecule must be able to produce an identical image of itself.

The inversion center is a point in space that lies in the geometric center of the molecule. During an inversion operation, all the atoms are moved through the center of the molecule in the opposite direction. As a result, all the cartesian coordinates of the atoms are inverted (i.e. x,y,z to -x,-y,-z).

These are denoted by *C _{n}^{m}* and are rotations of 360°/

*C _{n}^{n}*,

These are denoted by *S _{n}^{m}* and are rotations of 360°/

Rotation axes, mirror planes and inversion centres are symmetry elements, not operations. The rotation axis of the highest order is known as the principal rotation axis. It is conventional to set the Cartesian *z* axis of the molecule to contain the principal rotation axis.

Dichloromethane, CH_{2}Cl_{2}. There is a *C _{2}* rotation axis which passes through the carbon atom and the midpoints between the two hydrogen atoms and the two chlorine atoms. Define the z axis as co-linear with the

Methane, CH_{4}. In addition to the proper rotations of order 2 and 3 there are three mutually perpendicular *S _{4}* axes which pass half-way between the C-H bonds and six mirror planes. Note that

In crystals screw rotations and/or glide reflections are additionally possible. These are rotations or reflections together with partial translation. These operations may change based on the dimensions of the crystal lattice.

The Bravais lattices may be considered as representing **translational** symmetry operations. Combinations of operations of the crystallographic point groups with the addition symmetry operations produce the 230 crystallographic space groups.

Crystallographic restriction theorem

F. A. Cotton *Chemical applications of group theory*, Wiley, 1962, 1971

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

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