Class, group theory

Atoms have complete spherical synnnetry, and the angidar momentum states can be considered as different synnnetry classes of that spherical symmetry. The nuclear framework of a molecule has a much lower synnnetry. Synnnetry operations for the molecule are transfonnations such as rotations about an axis, reflection in a plane, or inversion tlnough a point at the centre of the molecule, which leave the molecule in an equivalent configuration. Every molecule has one such operation, the identity operation, which just leaves the molecule alone. Many molecules have one or more additional operations. The set of operations for a molecule fonn a mathematical group, and the methods of group theory provide a way to classify electronic and vibrational states according to whatever symmetry does exist. That classification leads to selection rules for transitions between those states. A complete discussion of the methods is beyond the scope of this chapter, but we will consider a few illustrative examples. Additional details will also be found in section A 1.4 on molecular symmetry. 

Before going on to consider applications of group theory in physical problems, it is necessary to discuss several general properties of irreducible representations. First, suppose that a given group is of order g and that the g operations have been collected into k different classes of mutually conjugate operations. It can be shown that the group Q possesses precisely k nonequivalent irreducible representations, T(1), r(2).r(t>, whose dimen-... 

Each set of four numbers ( 1) constitutes an irreducible representation (i.r.) of the symmetry group, on the basis of either a coordinate axis or an axial rotation. According to a well-known theorem of group theory [2.7.4(v)], the number of i.r. s is equal to the number of classes of that group. The four different i.r. s obtained above therefore cover all possibilities for C2V. The theorem thus implies that any representation of the symmetry operators of the group, on whatever basis, can be reduced to one of these four. In summary, the i.r. s of C2 are given by Table 1. 

RENKES Symbolic Computer Programs Applied to Group Theory Table II. Terminal Display of Classes... 

We have found three distinct irreducible representations for the C3v symmetry group two different one-dimensional and one two dimensional representations. Are there any more An important theorem of group theory shows that the number of irreducible representations of a group is equal to the number of classes. Since there are three classes of operation, we have found all the irreducible representations of the C3v point group. There are no more. 

Normally we would choose to do the sum over classes rather than over group elements. Equation (20) is an extremely useful relation, and is used frequently in many practical applications of group theory. 

Benzene Vibrations. With =12 atoms, benzene has 3N- 6 = 30 vibrational modes, but 10 are degenerate (E type) due to symmetry so that only 20 unique vibrational frequencies are predicted. Using group theory these are classed according to the symmetry species of the point group of benzene as... 

The theory requires two assumptions that the dissociable groups may be divided into a small number of classes, each of which may be characterized by a single intrinsic dissociation constant (Kini)i and that the ampholyte may be represented as a sphere over Avhich the net charge is uniformly distributed. If n,- denotes the total number of groups of class i, and r< of these are dissociated at a given pH where the average net charge on the molecule is Z, then... 

The process of finding all the matrices of lowest order (irreducible representations) of a group is usually very tedious. In many cases it is necessary to know only their trace, (40), 10.VIIIN, which in group theory is called the character, denoted by %. The trace of a representation f(<) is denoted by —" (A ), X(i)(A2),etc.,Au A2, A3i. .. being members of a group. The character of every element in a single class is identical. The class is represented by a subscript, Xi(t), etc. If the column gives the class and the row the representation, then ... 

Since AiG g) 1 Abelian group, any element is a class on its own. Consequently, AiG ) has four irredudble representations, denoted as Fo to Fa. The characters of the elements of AiG g) he individual irredudble representations are given in Table 4, which is the character table of the group A(Gig)> For determining the characters X(Aj,Fjk) reference should again be made to the literature on group theory. 

This preference originates from the group theory requirement of maximizing the number of unique classes of symmetry operations associated with a molecule, to be discussed in Section 4.3.3. 

A group is defined as a collection of elements (in the case of symmetry groups, symmetry operations) that are related to each other by a given set of rules. Group theory allows us to apply manipulations such as multiplication, definition of subgroups and classes, etc. 

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