Group theory began as a branch of pure mathematics. Mathematicians, through their study of the way numbers behave and how they may be treated, often discover theorems and relationships that may appear to have no immediate application to the physical world. On the other hand, physical scientists--chemists and physicists--attempt to understand how objects in the physical world behave, for instance how chemical molecules interact with each other in chemical reactions. They seek ways of visualizing these reactions and methods of predicting and calculating whether or how fast a reaction may be expected to occur. Since the days of Gallileo (1564-1642) and Isaac Newton (1642-1727), scientists have found that mathematics provides useful methods for scientists to accomplish this task. Group theory is one of a number of branches of mathematics that have proven useful to chemists and physicists in their work.
In our everyday use of the term, we commonly employ the word "group" to mean a collection of things which have something in common: a group of children, a musical group, a group of ducks. In a mathematical group, however, these "things" which have something in common are called elements, and there are rules that define what a set of elements must have in common in order to be regarded as a mathematical group. In their pioneering work with groups, Evariste Galois (1811-1832), Camile Jordan (1838-1921), and Felix Klein (1849-1925) developed the rules for mathematical groups and the methods for working with them. The result is the branch of mathematics known as group theory.
There are a number of groups that are important in chemistry. The chemical elements are grouped into sets in the periodic table (such as halides and alkali metals) based on the chemical behavior they have in common. Functional groups (such as alcohols, acids, and amines) are identified in organic chemistry. The commonalties shared by these groups are not of a mathematical nature. These groups do not follow the rules for a mathematical group, and group theory is, therefore, of no help in understanding them.
Many chemical molecules are symmetrical. Their structure is such that the molecule may be moved in certain ways, and the molecule in its new position appears to be identical to that before the movement occurred. Let us suppose that a molecule is made up of one atom of element A and three atoms of element B (B', B'', B'''). Let us further stipulate that the atoms of the molecule are all located in the same plane. This molecule is shown as Figure 1. Now rotate the molecule 120° about an axis which is perpendicular to the molecular plane and which passes through the central A atom. The configuration of the molecule after this rotation is shown in Figure 2. But if all of the B atoms are chemically and physically the same, configurations shown in Figures 1 and 2 are equivalent as far as their chemical and physical properties are concerned. If we now rotate the molecule by another 120° about the same axis, the molecule shown in Figure 3 will appear. Again we see that the configuration shown in Figure 3 is equivalent to Figures 1 and 2. Rotation of Figure 3 by 120° results in Figure 4, which is clearly not only equivalent to Figures 1,2 and 3, but is identical to Figure 1. The AB3 molecule is thus said to possess a three-fold axis of symmetry.
The planar AB3 molecule also possesses other symmetry elements. For instance, if you interchange the positions of atoms B' and B'' leaving atoms A and B''' in their original places, the structure shown in Figure 5 is obtained. Again chemically and physically the molecule seen in Figure 5 is equivalent to the molecule in Figure 1. The molecule is said to possess mirror image symmetry relative to a hypothetical mirror perpendicular to the plane of the molecule and passing through atoms A and B'''.
The collection of all of the symmetry elements that a molecule possesses forms a group. Interestingly, this group is found to follow the rules required for a mathematical group. The methods of group theory should, therefore, be applicable in the study of the behavior of symmetrical molecules such as AB3.
The application of group theory to chemical molecules is called chemical group theory. It has been particularly useful in providing simpler methods for constructing and visualizing hybrid molecular orbitals and in simplifying the quantum mechanical calculations in molecular orbital theory. Group theory has also provided the basis for the development of ligand field theory and in understanding and predicting the vibrations of symmetrical molecules. The usefulness of group theory in these applications results from the fact that, as we have seen, some configurations of symmetrical molecules, as they move about, are equivalent. Because of this, restrictions are placed on the solutions of quantum mechanical equations that apply to the orbital structure and behavior of these molecules. As a result, solutions of these equations are greatly simplified.
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