Symmetry operation

In mathematics, a symmetry operation is a geometric transformation of an object that leaves the object looking the same after it has been carried out. For example, a Template:Frac turn rotation of a regular triangle about its center, a reflection of a square across its diagonal, a translation of the Euclidean plane, or a point reflection of a sphere through its center are all symmetry operations. Each symmetry operation is performed with respect to some symmetry element (a point, line or plane).

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.

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

In the context of molecular symmetry, quantum 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.

Molecules

Identity Operation

The identity operation corresponds to doing nothing to the object. Because every molecule is indistinguishable from itself if nothing is done to it, every object possesses at least the identity operation. The identity operation is denoted by Template:Mvar or Template:Mvar. In the identity operation, no change can be observed for the molecule. Even the most asymmetric molecule possesses the identity operation. The need for such an identity operation arises from the mathematical requirements of group theory.

Reflection through mirror planes

Reflection operation

The reflection operation is carried out with respect to symmetry elements known as planes of symmetry or mirror planes. Each such plane is denoted as Template:Math (sigma). Its orientation relative to the principal axis of the molecule is indicated by a subscript. The plane must pass through the molecule and cannot be completely outside it.

• If the plane of symmetry contains the principal axis of the molecule (i.e., the molecular Template:Mvar-axis), it is designated as a vertical mirror plane, which is indicated by a subscript Template:Mvar (Template:Math).
• If the plane of symmetry is perpendicular to the principal axis, it is designated as a horizontal mirror plane, which is indicated by a subscript Template:Mvar (Template:Math).
• If the plane of symmetry bisects the angle between two 2-fold axes perpendicular to the principal axis, it is designated as a dihedral mirror plane, which is indicated by a subscript Template:Mvar (Template:Math).

Through the reflection of each mirror plane, the molecule must be able to produce an identical image of itself.

Inversion operation

Inversion operation is shown here with a sulfur hexafluoride molecule. All of the fluorine atoms change their position to opposite side with respect to the sulfur center

In an inversion through a centre of symmetry, Template:Mvar (the element), we imagine taking each point in a molecule and then moving it out the same distance on the other side. In summary, the inversion operation projects each atom through the centre of inversion and out to the same distance on the opposite side. The inversion center is a point in space that lies in the geometric center of the molecule. As a result, all the cartesian coordinates of the atoms are inverted (i.e. Template:Mvar to Template:Mvar). The symbol used to represent inversion center is Template:Mvar. When the inversion operation is carried out Template:Mvar times, it is denoted by Template:Mvar, where $i^n=E$ when Template:Mvar is even and $i^n=-E$ when Template:Mvar is odd.

Examples of molecules that have an inversion center include certain molecules with octahedral geometry (general formula Template:Chem2), square planar geometry (general formula Template:Chem2), and ethylene (Template:Chem2). Examples of molecules without inversion centers are cyclopentadienide (Template:Chem2) and molecules with trigonal pyramidal geometry (general formula Template:Chem2).

Proper rotation operations or n-fold rotation

A proper rotation refers to simple rotation about an axis. Such operations are denoted by Template:Tmath where Template:Mvar is a rotation of Template:Tmath or Template:Tmath performed Template:Mvar times. The superscript Template:Mvar is omitted if it is equal to one. Template:Math is a rotation through 360°, where Template:Math. It is equivalent to the Identity (Template:Mvar) operation. Template:Math is a rotation of 180°, as Template:Tmath Template:Math is a rotation of 120°, as Template:Tmath and so on.

Here the molecule can be rotated into equivalent positions around an axis. An example of a molecule with Template:Math symmetry is the water (Template:Chem2) molecule. If the Template:Chem2 molecule is rotated by 180° about an axis passing through the oxygen atom, no detectable difference before and after the Template:Math operation is observed.

Order Template:Mvar of an axis can be regarded as a number of times that, for the least rotation which gives an equivalent configuration, that rotation must be repeated to give a configuration identical to the original structure (i.e. a 360° or 2Template:Pi rotation). An example of this is Template:Math proper rotation, which rotates by Template:Tmath Template:Math represents the first rotation around the Template:Math axis by Template:Tmath Template:Tmath is the rotation by Template:Tmath while Template:Tmath is the rotation by Template:Tmath Template:Tmath is the identical configuration because it gives the original structure, and it is called an identity element (Template:Mvar). Therefore, Template:Math is an order of three, and is often referred to as a threefold axis.

Improper rotation operations

An improper rotation involves two operation steps: a proper rotation followed by reflection through a plane perpendicular to the rotation axis. The improper rotation is represented by the symbol Template:Mvar where Template:Mvar is the order. Since the improper rotation is the combination of a proper rotation and a reflection, Template:Mvar will always exist whenever Template:Mvar and a perpendicular plane exist separately. Template:Math is usually denoted as Template:Math, a reflection operation about a mirror plane. Template:Math is usually denoted as Template:Mvar, an inversion operation about an inversion center. When Template:Mvar is an even number $S_n^n=E,$ but when Template:Mvar is odd $S_n^{2n}=E.$

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

Examples

Dichloromethane, Template:Chem2. There is a Template:Math 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 Template:Math axis, the Template:Mvar plane as containing Template:Chem2 and the Template:Mvar plane as containing Template:Chem2. A Template:Math rotation operation permutes the two hydrogen atoms and the two chlorine atoms. Reflection in the Template:Mvar plane permutes the hydrogen atoms while reflection in the Template:Mvar plane permutes the chlorine atoms. The four symmetry operations Template:Mvar, Template:Math, Template:Math and Template:Math form the point group Template:Math. Note that if any two operations are carried out in succession the result is the same as if a single operation of the group had been performed.

Methane, Template:Chem2. In addition to the proper rotations of order 2 and 3 there are three mutually perpendicular Template:Math axes which pass half-way between the C-H bonds and six mirror planes. Note that

$S_4^2=C_2.$

Crystals

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.

See also

Molecular symmetry

Crystal structure

Crystallographic restriction theorem

References

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