Symmetry and Group Theory

 What is symmetry? 

In simple language we can say that an object has symmetry, if it has some special characteristics, such as pleasing designs, while we look at it. As an example, when we see the telephone posts or electric lamp posts, we say that there is symmetry because they are arranged in a straight line at equal distance. Similarly, when we look at the gates of houses, they will appear symmetric because of their designs. Naturally, our eyes will compare the design on one half of the gate with that of the other half and if they find some characteristic feature such as mirror image or other, then we feel there is symmetry. A suspension bridge, a butterfly, the rose petal etc. are some examples to show the pleasing designs and hence, they are symmetric. 

• Symmetry elements

Of course, we realize the symmetry in the objects when we look at them. However, we must express them scientifically. This could be done with the help of symmetry elements. What are symmetry elements? These are nothing but some physical entities such as line, plane, point etc. The next question is, “Are they present in objects or molecules?” The answer is “no”. These are imaginary.

•  Symmetry operations

These are some mechanical operations, such as, rotation, reflection, inversion etc., performed about the symmetry elements so that indistinguishable structures are produced.

Axis of symmetry or Proper rotational axis of symmetry (Cn).

It is an imaginary line passing through an object or a molecule about which when the object or molecule is rotated by a certain angle, an indistinguishable structure is produced. 

EXAMPLES

  C4 axis of rotation

 The two structures cannot be distinguished, if the letters are removed.

C3 axis of rotation

 The angle between two spheres is equal to  120^o. Hence rotation by 120^ogives an indistinguishable structure.

C2 axis of rotation

When the above V-shaped molecule is rotated by 180^o about the axis passing through the central sphere, (a) and (b) spheres are interchanged. The two structures cannot be distinguished, if the letters are removed.

 The angle between the blue and red spheres is 180^o. Rotaion about the vertical axis by 180^o gives an indistinguishable structure, once the letters are removed.

C6 axis of rotation

 

 60^o rotation about the axis perpendicular to the paper gives an indistinguishable structure, once the letters from the spheres are removed

Order of axis and plane of symmetry

Order of axis

This imaginary line, i.e., the axis of symmetry is represented as Cn, where n is known as the order of the axis. This tells how many times we have to rotate the object to reach the initial structure, i.e., one full rotation.

Or, it tells the angle (360^o/n) by which we have to rotate the molecule to  get the indistinguishable structure.

C4 axis of symmetry, 360/4 = 90^o; that is, 90^orotation will give an in distinguishable structure.

C3 axis of symmetry, 360/3 = 120^o; that is, 120^orotation will give an indistinguishable structure.

C2 axis of symmetry, 360/2 = 180^o; that is, 180^orotation will give an indistinguishable structure.

C6 axis of symmetry, 360/6 = 60^o; that is, 60^orotation will give an indistinguishable structure.

 Principal axis of symmetry

That axis for whhich the n value is maximum is called the principle axis of symmetry.

  Example:

 C4 is the principal axis because n=4 is the maximum number

 

Plane of symmetry (σ)

It is an imaginary plane cutting the molecule or object into two halves which are mirror images.

 Vertical mirror plane (σ_v)

This is the mirror plane parallel to the principal axis of symmetry.

 Horizontal mirror plane (σ_h)

When the mirror plane is perpendicular to the principal axis, it is called horizontal plane of symmetry.

In the first case (plane triangle), the reflection could not be distinguisdhed from the original and the mirror plane is called a horizontal mirror plane, σ_h plane.

 In the other case, (V-shaped), the reflection is inverted and we are able to distinguish this from the original one. Hence, it is not a σ_h plane.

Centre of symmetry, Identity Element, and Improper rotation axis

 Center of symmetry (i)

If we can move in a straight line from every atom or point in a molecule or object through a single point at the center to an identical atom or point on the other side of the center, then the molecule or object is said to possess a center of symmetry

Identity Element (E)

This is nothing but rotating the molecule by 360^o. The original molecule is obtained. The corresponding operation can be called as “doing nothing” operation.

This is important from mathematical considerations.

Improper rotational axis of symmetry or Rotation reflection axis of symmetry (S_n).

Rotation by a particular angle followed by reflection in a plane perpendicular to the rotational axis leads to an indistinguishable structure.

Example: S4 axis: rotation by 360/4 = 90^ofollowed by reflection in a plane perpendicular to C4 axis gives an indistinguishable structure.

Example: SiF₄

•  References 

1.    “Inorganic Chemistry: Principles of Structure and Reactivity”, James E.Huheey, Ellen A.Keiter, Richard L.Keiter, Okhil K.Medhi,  Pearson

Education, Delhi, 2006

2.    ‘Chemical Applications of Group Theory”, 2/e, F.Albert Cotton, Wiley

Eastern, New Delhi, 1986 

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