Bostock, Kathryn Sarah; (1997) Group reduction and its application to chemical problems. Doctoral thesis (Ph.D), UCL (University College London).

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Abstract

The aim of the present thesis is to consider some aspects of the solution of problems in dimensions greater than three. In conventional treatments of problems in chemistry attention is focused on geometrical aspects i.e. the nuclear configurations of molecules and crystals. The symmetry of a system may then be described in terms of the corresponding three dimensional point and space groups. More recently, the geometry of objects in dimensions greater than three has been required in order to understand certain phenomena. This leads both to unfamiliar geometrical notions and to groups which are not encountered in the more familiar problems. An attempt has been made to lead to a treatment of these complicated higher dimensional groups in a relatively simple way. The treatment begins with the consideration of groups in the abstract starting with those of low order and proceeding to more complicated cases. In this way both familiar and unfamiliar groups are given equal emphasis. The groups are examined in terms of operators acting on functions making a connection with the formalisms of quantum mechanics. The treatment leads to the derivation of familiar irreducible representations (and corresponding character tables) for the majority of groups. It is possible to transform the representations to generate groups of orthogonal matrices and this makes a direct comparison with higher dimensional geometries possible. In this way examples of higher dimensional point groups are derived algebraically and their geometrical significance may be examined at a later stage. A common application of group theory is in the classification of quantum states in terms of irreducible representations. In particular, it is customary to classify vibrational states in terms of the underlying point or space group. A complication arises for overtones of degenerate harmonic vibrations which are classifiable in terms of a much higher order summetry group. The summetry adaptation of the corresponding functions to the underlying point group symmetry which is restored by addition of anharmonic terms is solved using the methods developed here.

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