Improper Axis Of Rotation

In group theory, geometry, representation theory and molecular symmetry, a symmetry operation is a transformation of an object that leaves an object looking the same after it has been carried out. For example, as transformations of an object in space, rotations, reflections and inversions are all symmetry operations. Such symmetry operations are performed with respect to symmetry elements (for example, 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.
# Physical properties must be invariant with respect to symmetry operations.
# 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 ''E'' or ''I''. 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

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 σ (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 z-axis), it is designated as a vertical mirror plane, which is indicated by a subscript "v" (σ_v).

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 "h" (σ_h).

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 "d" (σ_d).

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

Inversion operation

In an inversion through a centre of symmetry, i (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. x,y,z to -x,-y,-z). The symbol used to represent inversion center is i. When inversion operation is carried out in n number of times, it is denoted by iⁿ, where iⁿ =E when n is even and iⁿ=E when n is odd. AB₆, planar AB₄ planar and trans AB₂C₂ and ethylene are some of examples of molecules that have inversion center. Examples of molecules without inversion center C₅H₅⁻ and tetrahedral AB₄.

Proper rotation operations or n-fold rotation

This is simple rotation about an axis. These are denoted by ''C_n^m'' and ''C_n'' is a rotation through of 360°/''n'', performed ''m'' times. The superscript ''m'' is omitted if it is equal to one. Here the molecule can be rotated into equivalent positions around an axis. C₁ is a rotation through 360°, where n=1. It is equivalent to Identity (E) operation. Another example is H₂O molecule. It has C₂ axis of rotation (n=2). C₂ is a rotation through 180° and the axis passes through Oxygen atom. If the water molecule is rotated by 180°, no detectable difference before and after the C₂ operation is observed. Order n of an axis can be regarded as a number of times the least rotation which give an equivalent configuration must be repeated to give a configuration not just equivalent to the original structure but also identical to it. Example of this is C₃ proper rotation which rotate by 2π/3, 2* 2π/3, 3*2π/3. the symbol C₃ represent the first rotation around the C₃ axis which is 2π/3, C₃² represent the second rotation which is by 2* 2π/3 while the C₃³ represent rotation by 3* 2π/3. the C₃³ is the identical configuration because it gives the original structure, and it is called identity element(E). Therefore, C₃ as an order of three and is often refer to as threefold axis.{{Cite book, last=Cotton, first=Albert, title=Chemical Applications of Group Theory, publisher=Wiley-Interscience , year=1990, location=United States , pages=23

Improper rotation operations

An improper rotation involves two operation steps: a proper rotation follow by reflection through a plane perpendicular to the rotation axis. The improper rotation is represented by the symbol ''S_n'' where n is the order. Since the improper rotation is the combination of a proper rotation and a reflection, ''S_n'' will always exist whenever ''C_n'' and a perpendicular plane exist separately. ''S₁'' is usually denoted as σ, a reflection operation about a mirror plane. ''S₂'' is usually denoted as ''i'', an inversion operation about an inversion center. When ''n'' is an even number ''S_nⁿ'' = ''E'', but when ''n'' is odd ''S_n²ⁿ'' = ''E''.

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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 ''z'' axis of the molecule to contain the principal rotation axis.

Examples

Dichloromethane, CH₂Cl₂. There is a ''C₂'' 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 ''C₂'' axis, the ''xz'' plane as containing CH₂ and the ''yz'' plane as containing CCl₂. A ''C₂'' rotation operation permutes the two hydrogen atoms and the two chlorine atoms. Reflection in the ''yz'' plane permutes the hydrogen atoms while reflection in the ''xz'' plane permutes the chlorine atoms. The four symmetry operations ''E'', ''C₂'', σ(''xz'') and σ(''yz'') form the point group C_2v. 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, CH₄. In addition to the proper rotations of order 2 and 3 there are three mutually perpendicular ''S₄'' axes which pass half-way between the C-H bonds and six mirror planes. Note that ''S₄²'' = ''C₂''.

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

Physical chemistry
Symmetry