Introduction

The key to relating Group theory is to be able to identify the 'Point'. The possible symmetry operations connected through a molecule are computed through the symmetry components possessed via that molecule. Consequently the 1^st step in applying group theory to molecular properties is to recognize the entire set of symmetry elements possessed via the molecule. This needs the individual to visually identify the elements of symmetry in a three-dimensional object.

Definition of Symmetry

The H₂O molecule, for instance, has the symmetry elements E, C₂, v₁ and v. It should be noted that there is no h for H₂O since the planes present are parallel to the principal axis C₂. The sets of symmetry operations connected through these collections of symmetry elements are termed to as point groups. The adjective "point" is utilized since the symmetry operations related through all the symmetry elements that are dealt via here leave a point of or in the molecule fixed in space (Babarinde, 2008).  

Each collection of symmetry procedures constitutes a point group, and each point group is specified a symbol. The symbol usually written in bold face kind is depends on the principal elements of symmetry of that point group. Group theory is one of the most influential mathematical tools utilized in quantum chemistry and Spectroscopy. It permits the user to forecast, interpret, rationalize, and frequently simplify compound theory and data.

At its heart is the fact that the set of operations related through the symmetry components of a molecule constitute a mathematical set said Group. This permits the application of the mathematical theorems connected by these groups to the symmetry operations.

Conditions for the Formation of Symmetry Point Group

1. Any consequence of two or more operations must generate the similar effect as application of one operation inside the group, that is, the group multiplication table must be closed for instance, H₂O, which has E, C₂ and 2v's. The group multiplication table attained is presented below. The table is closed, that is, the consequence of 2 operations produces an operation in the group.

Fig: Conditions for the Formation of Symmetry Point Group

2. Must have an identity ( )

3. All components must have an inverse, that is, for a specified operation ( ) there must exist an operation ( ) such that  

Classification of the Symmetry of Molecules

Certain symmetry operations can be present simultaneously, while others can't.

There are certain combinations of symmetry operations that can happen together.

 Symmetry groups enclose elements and there mathematical operations. For instance, one of the symmetry elements of H₂O is a C₂-axis. The equivalent operation is rotation of the molecule via 180° about an axis.

Table: Properties of Symmetry Point Groups

Point group determination chart

The flow chart bellow provides the steps to the determination of point groups for dissimilar molecules.

Fig: Condensed Flow Chart for the Determination of Point Groups in Molecules

Identification of Symmetry Elements in a Molecule

Some molecules have many symmetry components, several of which aren't immediately obvious as in X_eF₄.

Hence the complete list of symmetry element is

E, C₄, C₂, S₄, i, 2C₂', 2C₂", σ_n, 2σ, 2σ' = D_4h

Examples:  List the symmetry elements of the following molecules:

(i)  BCl₃ (ii) NH₃ (iv) BF₃

Symmetry in Crystal Structures

The X-ray analysis of crystals has revealed that they are made up of easy chapter consisting of atoms, ions or molecules. Each crystal is just a repetition of the basic unit. Each crystal unit is said the unit cell, or space lattice. A crystal is a regular 3-dimensional arrangement of the atoms, ions or molecules of which it is built up. The three-dimensional design of a crystal is composed of identical units forming a three- dimensional network of cells, which separate the space into equal volumes through no space excluded.  This is only probable when the points are arranged in these a way that each of the points has precisely the similar environments as any similar point  in  the  system. Such points through identical surroundings are termed to as lattice points whilst the three- dimensional arrays of such points are said space lattices.

All crystals fall into any of the seven crystal systems; cubic, monoclinic, triclinic, orthorhombic, tetragonal, rhombohedral, and hexagonal. Usually, 3 imaginary lines are drawn between the centres of opposite faces to symbolize the 3 axes of symmetry. The system to that a crystal belongs depends on the relative lengths of the axes of symmetry and the angle at which the axes meet in the unit cell. Due to the  regularity in  the  distribution of  the points in a space  lattice  its necessary geometry can be determined through three lattice vectors, said lattice constants (a, b, c); and  the inter-axial angles, and. Such lattice parameters describe the unit cell or the smallest repeating unit of lattice. A. in X-ray crystallography, the lattice constants are repeat distances along axes x, y, and z, correspondingly. They are termed to as axial ratios while the inter-axial angles, and are said axial angles. Bravais illustrated that there are only 14 different ways in that lattice points can be arranged in a regular 3-dimensional pattern.

Table: The 14 Bravais Lattices of the Seven Crystal Systems

Table: Symmetry Point Groups of 3-Dimensional Solids

Implication of Symmetry on Properties of Molecules

There are instant consequences of symmetry on properties of molecules. Such are specified below.

1. Electric dipole moment: Only molecules belonging to the groups C_n, C_nv and Cs might have electric dipole moment. Out of the 3 point groups, C_n, and C_nv must have their electric dipole moment lying along the rotation axis. A molecule having a C_n axis can't have a dipole moment perdicular to its axis but it might exist parallel to the axis. The similar rule applies to C_nv.

Optical activity: The plane of a polarized light can only be rotated via a molecule if it can't be superimposed on its mirror image. This implies that symmetry operation that should be looked for in order to find out the optical activity is the improper axis of rotation, S_n. If S_n is present, the object can be superimposed on its mirror image, therefore can't be optically active. On the other hand, absence of S_n implies that superposition is impossible, and optical activity may occur. It should be noted that any offensive rotation axes that might be implied through the group must included. All the groups in C_nh have Sn concealed in them since they enclose C_n and σh. In common, any point group having the centre of inversion, i, as a symmetry component as well possesses Sn, where the minimum value of n is 2. It should be understood that an inversion can be regarded as a C₂ (180° rotation) followed via a σ_h. Consequently, any molecule through centre of inversion can't be optically active. However, it isn't every one molecule through a centre of inversion that is optically active. For instance, a molecule that has symmetry point group S₄ but has no centre of inversion illustrates optical activity (Atkins, 1982; Barrow, 1982).

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