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Unformatted text preview: 31Jan2011 Chemistry 21b Spectroscopy Lecture # 12 Group Theory in Spectroscopy Molecular symmetry is the unifying thread throughout spectroscopy and molecular structure theory. It makes it possible to classify states, and, more importantly, to determine selection rules without having to do any sophisticated calculations. The application of symmetry arguments to atoms and molecules has its origin in group theory developed by mathematicians in the 19th century, and it is for this reason that the subject is often presented in a rigorous mathematical formulation. However, it is possible to progress quite a long way in understanding molecular symmetry without a detailed mathematical knowledge of the theory of groups, and only a simple introduction to the subject will be outlined below. Besides that in Atkins & Friedman and Harris & Bertolucci, other treatments of molecular symmetry and group theory are given by J.I. Steinfeld, Molecules and Radiation (1974), Ch. 6, F. A. Cotton, Chemical Applications of Group Theory (1963), and M. Tinkham, Group Theory and Quantum Mechanics (1964). The latter book contains a more mathematical discussion for those interested in probing the details of this subject. The great utility of group theory lies in its abstractness. Provided any set of elements A,B,C,D,... obeys the following four conditions, they form a group: (1) Closure . If A and B are any two members of the group, then their product A B must also be a member of the group. (2) Associativity . The rule of combination must be such that the associative law holds. That is, if A , B , and C are any three elements of the group, then ( A B ) C = A ( B C ). (3) Identity . The group must contain a single element I such that for any element A of the group, A I = I A = A . I is called the identity element . (4) Inverse . Each element A of the group must have an inverse A- 1 that is also a member of the group. By the term inverse we mean that A A- 1 = A- 1 A = I , where I is the identity element. Each molecule has a number of so-called symmetry elements, which together comprise the point group to which the molecule belongs (Point groups are so named because of the fact that the symmetry operations in the groups leave at least one point in space unchanged. Space groups leave lines, planes, or polyhedra unchanged, and so are very useful in the crystallographic study of solids.). As Harris & Bertolucci describe, for molecules the symmetry operations (in addition to the identity operation) that must be considered include rotation about an axis, reflection about a plane, inversion through a point, or a combination of these operations....
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This note was uploaded on 01/03/2012 for the course CH 21b taught by Professor List during the Fall '10 term at Caltech.
- Fall '10