Group theory,

Group theory, although a topic of pure mathematics has had such a powerful influence on the physical sciences that many features thereof have been adapted especially to emphasize scientific rather than mathematical aspects. For the same reason it has become traditional for physical scientists to present group theory as a consummate mixture of science and maths. The same bias will be evident in this chapter despite a conscious effort to present the predominantly physical aspects at their more appropriate places in the text. 

Even mathematicians often find it hard to present group theory in totally abstract terms. 

There are several texts that deal with group theory for non-mathematicians 

A mathematical group, by definition, consists of a set of distinct elements G — E, A, B,C, D. endowed with a law of composition (such as multiplication, addition, or some other operation), such that the following properties are satisfied  

This property is known as the closure property of the group. The group is said to be closed under the given law of composition. 

Examples where group theory can be applied successfully are molecular orbital theory, molecular spectroscopy, multipole expansions, and ligand field theory. In general, group theoretical results can be used as classification scheme and/or to simplify numerical calculations. 

In Sec. 1.5, the symmetry and the point group allocation of a given molecule were discussed. To understand the symmetry and selection rules of normal vibrations in polyatomic molecules, however, a knowledge of group theory is required. The minimum amount of group theory needed for this purpose is given here. 

Consider a pyramidal XY3 molecule (Fig. 1.18) for which the symmetry operations /,, Cf, Qi, Q2, and a3 are applicable. Here, and denote rotation through 

It is seen that a group consisting of the mathematical elements (symmetry operations) /, A, B, C, D, and E satisfies the following conditions  

These are necessary and sufficient conditions for a set of elements to form a group. It is evident that operations I,A,B, C, D, and E form a group in this sense. It should be noted 

The six elements can be classified into three types of operation the identity operation I, the rotations C and C3, and the reflections ai, Q2, and a3. Each of these sets of operations is said to form a class. More precisely, two operations, P and Q, which satisfy the relation X PX=PorQ, where X is any operation of the group and X is its reciprocal, are said to belong to the same class. It can easily be shown that C3 and C, for example, satisfy the relation. Thus the six elements of the point group C31, are usually abbreviated as I, 2C3, and 3a . 

The same result can be obtained more easily for a field 5 in the z direction, since the corresponding Pauli matrix 

FIGURE 9.2 Yin-yang symbol and its color inverse. The original could then be retrieved by a 180° rotation. 

Such principles can be formalized as gauge field theories, which provide the basic structure of the standard model for electromagnetic, weak and strong interactions. 

The strong nuclear force is insensitive to the distinction between neutrons and protons. These can be treated as alternative states of a single particle called a nucleon, differing in isotopic spin or isospin. It is found, for example, that the nuclei and He have similar energy-level spectra. Isospin is, however, only an approximate symmetry. It is broken by electromagnetic interactions since protons have electric charge, while neutrons do not. Broken symmetry is a central theme in fundamental physics. An open question is how our universe evolved to break the symmetry between matter and antimatter, so that it is now dominated by matter. 

FIGURE 9.3 Symmetry operations on an equilateral triangle, shown with the aid of shaded areas. 

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Also used in group theory to denote particular energy slates, e.g. with s = 2. 

Mulliken symbols The designators, arising from group theory, of the electronic states of an ion in a crystal field. A and B are singly degenerate, E doubly degenerate, T triply degenerate states. Thus a D state of a free ion shows E and Tj states in an octahedral field. 

In applications of group theory we often obtain a reducible representation, and we then need to reduce it to its irreducible components. The way that a given representation of a group is reduced to its irreducible components depends only on the characters of the matrices in the representation and on the characters of the matrices in the irreducible representations of the group. Suppose that the reducible representation is F and that the group involved... 

Cotton F A 1990 Chemical Applications of Group Theory 3rd edn (New York Wiley)... 

Tinkham M 1964 Group Theory and Quantum Mechanics (New York MoGraw-Hill)... 

Kleinian V, Gordon R J, Park H and Zare R N 1998 Companion to Angular Momentum (New York Wiley) Tinkliam M 1964 Group Theory and Quantum Mechanics York McGraw-Hill)... 

Fisher M 1983 Scaling, universality and renormalization group theory Critical Phenomena (Lecture Notes in Physics vol 186) (Berlin Springer)... 

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. 

We now turn to electronic selection rules for syimnetrical nonlinear molecules. The procedure here is to examme the structure of a molecule to detennine what synnnetry operations exist which will leave the molecular framework in an equivalent configuration. Then one looks at the various possible point groups to see what group would consist of those particular operations. The character table for that group will then pennit one to classify electronic states by symmetry and to work out the selection rules. Character tables for all relevant groups can be found in many books on spectroscopy or group theory. Ftere we will only pick one very sunple point group called 2 and look at some simple examples to illustrate the method. 

Freed K F 1987 Renormalization Group Theory of Macromolecules (New York Wiley-Interscience)... 

E. P. Wigner, Group Theory and Its Applications to the Quantum Mechanics of Atomic Spectra, Academic Press, New York, 1959. 

The stereochemistry of reactions has to be handled in any detailed modeling of chemical reactions. Section 2.7 showed how permutation group theory can be used to represent the stereochemistry of molecular structures. We will now extend this approach to handle the stereochemistry of reactions also [31]. 

Let us first repeat the essential features of handling the stereochemistry of molecular structures by permutation group theory ... 

Figure 3-23. The treatment ofthe stereochemistry ofa further S,y2 reaction by permutation group theory.

The stereochemistry of reactions can also be treated by permutation group theory for reactions that involve the transformation of an sp carbon atom center into an sp carbon atom center, as in additions to C=C bonds, in elimination reactions, or in eIcctrocycHc reactions such as the one shown in Figure 3-21. Details have been published 3l]. 

The stereochemistry of reactions can be treated by means of permutation group theory. 

Chemical Applications of Group Theory, F. A. Cotton, Interscience, New York, N. Y. (1963)- Cotton. 

It is recommended that the reader become familiar with the point-group symmetry tools developed in Appendix E before proceeding with this section. In particular, it is important to know how to label atomic orbitals as well as the various hybrids that can be formed from them according to the irreducible representations of the molecule s point group and how to construct symmetry adapted combinations of atomic, hybrid, and molecular orbitals using projection operator methods. If additional material on group theory is needed. Cotton s book on this subject is very good and provides many excellent chemical applications. 

We have found three distinet irredueible 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 irredueible representations of a group is equal to the number of elasses. Sinee there are three elasses of operation, we have found all the irredueible representations of the C3V point group. There are no more. 

G. Davidson, Group Theory for Chemists MacMillan, Hampshire (1991). 

J. R. Ferraro, J. S. Ziomek, Introductory Group Theory Plenum, New York (1975). 

D. M. Bishop, Group Theory and Chemistry Clarendon, Oxford (1973). 

M. Tinkham, Group Theory and Quantum Mechanics McGraw-Hill, New York (1964). R. McWeeny, Symmetry An Introduction to Group Theory and its Applications Pergamon, New York (1963). 

Other techniques that work well on small computers are based on the molecules topology or indices from graph theory. These fields of mathematics classify and quantify systems of interconnected points, which correspond well to atoms and bonds between them. Indices can be defined to quantify whether the system is linear or has many cyclic groups or cross links. Properties can be empirically fitted to these indices. Topological and group theory indices are also combined with group additivity techniques or used as QSPR descriptors. 

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. 

There will be many occasions when we shall need to multiply symmetry species or, in the language of group theory, to obtain their direct product. For example, if H2O is vibrationally excited simultaneously with one quantum each of Vj and V3, the symmetry species of the wave function for this vibrational combination state is... 

Davidson, G. (1991) Group Theory for Chemists, Macmillan, London. 

Vincent, A. (2000) Molecular Symmetry and Group Theory, 2nd edn., John Wiley, Chichester. 

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