Symmetry and Elements of Group Theory

In this chapter, we first discuss the concept of symmetry and the identification of the point group of any given molecule. Then we present the rudiments of group theory, focusing mainly on the character tables of point groups and their use. 

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A group is a collection of elements that are interrelated according to certain rules. We need not specify what the elements are or attribute any physical significance to them in order to discuss the group which they constitute. In this book, of course, we shall be concerned with the groups formed by the sets of symmetry operations that may be carried out on molecules or crystals, but the basic definitions and theorems of group theory are far more general. 

All molecules possess the identity element of symmetry, for which the symbol is / (some authors use E, but this may cause confusion with the E symmetry species see Section 4.3.2). The symmetry operation / consists of doing nothing to the molecule, so that it may seem too trivial to be of importance but it is a necessary element required by the mles of group theory. Since the C operation is a rotation by 2n radians, Ci = I and the symbol is not used. 

A group is a set of abstract elements (members) that has specific mathematical properties. In general it is not necessary to specify the nature of the members of the group or the way in which they are related. However, in the applications of group theory of interest to physicists and chemists, the key word is symmetry. 

Formally the thermal theory can be established, via TFD, within c algebra (I. Ojima, 1981 A.E. Santana et.al., 1999) and symmetry groups (A.E. Santana et.al., 1999), opening a broad spectrum of possibilities for the study of thermal effects. For instance, the kinec-tic theory has been formulated for the first time as a representation theory of Lie symmetries (A.E. Santana et.al., 2000) and elements of... 

The four properties are the defining properties of a group A group is a set of elements of whatever kind for which we can define the product of a pair of elements and which has the properties (i)-(iv) above. Hence it follows that the set of all the symmetry operations belonging to a molecule forms a group, the symmetry group of the molecule, and that the methods of group theory can be used to discuss the symmetry of a molecule. 

This illustrates a type of symmetry only seen in crystals and other extended arrays. That is, the symmetry operation combines both elements of point symmetry (as seen in molecules) and translation (which generates arrays). Here you can see that the repeat of this operation yields a vertical translation of one unit. The two-and three-dimensional space groups are realizations of the more general topic of group theory, which has been one of the tremendous scientific achievements in the last two centuries in the field of pure mathematics. 

These matrix elements can be reduced with the aid of group theory. To show the symmetry species of the vibrational mode and electronic state explicitly, we express the normal coordinate and electronic states as mry), and nT y ), respectively. Equation (47) is rewritten as... 

Schoenflies symbols are widely used to describe molecular symmetry, the symmetry of atomic orbitals, and in chemical group theory. The terminology of the important symmetry operators and symmetry elements used in this notation are given in Table A3.1. 

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