Elements of group theory and applications

Examination of the geometries of isolated molecules shows that there are four types of symmetry elements reflection planes, axes of rotation, inversion centres, and improper axes of rotation. A symmetry operation moves a molecule from an initial configuration to another, equivalent configuration, either hy leaving the position of the atoms unchanged, or by exchanging equivalent atoms. The ideas of symmetry element and symmetry operation are of course very closely linked, since a symmetry operation is defined with respect to a given element of symmetry, and, conversely, the presence of a symmetry element is established hy the presence of one or more symmetry operations that are associated with that element. 

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For a given application of group theory it is furthermore necessary to both the set of elements X,Y,Z... and the law of combination ( multiplication or other). The following examples should clarify this question. It is to be noted that the order of the group is defined as the number of its members. 

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. 

Chapters 8 and 9 are devoted to a discussion of applications of the VSEPR and LCP models, the analysis of electron density distributions to the understanding of the bonding and geometry of molecules of the main group elements, and on the relationship of these models and theories to orbital models. Chapter 8 deals with molecules of the elements of period 2 and Chapter 9 with the molecules of the main group elements of period 3 and beyond. 

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. 

Since the interaction of two crystallographic symmetry elements results in a third crystallographic symmetry element, and the total number of them is finite, valid combinations of symmetry elements can be assembled into finite groups. As a result, mathematical theory of groups is fully applicable to crystallographic symmetry groups. 

Volumes 3-6 describe developments in the coordination chemistry of the metallic elements since 1982 (x, p, and /-block metals, transition metals of Groups 3-6 7-8 9-12). These volumes correspond to Volumes 3, 4, and 5 in CCC. A review of technetium coordination chemistry was unavailable when CCC was published, and a complete account of its development from the earliest discoveries to present-day applications is incorporated in the new work. In these volumes space limitations restrict the material that can be presented. The information that appears has been selected to give a near comprehensive coverage of new discoveries, new interpretations of experiment and theory, and applications, where relevant. 

The book by E. P Wigner, Group Theory and its Application to the Quantum Mechanics of Atomic Spectra (Academic Press, New York, 1959), ind the text by B. R. Judd, Second Quantization and Atomic Spectroscopy (The Johns Hopkins Press, Baltimore, 1967), provide the necessary material. The latter monograph eJso discusses the relations between the reduced field operator matrix elements and the coefficients of fractional parentage. 

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