Representation Theory and Physical SystemsFinny KuruvillaDecember 16, 1998Classifying Molecular StructureAdding new elements to character tablesTensoring and IntegrationSome examples. Is it possible that some of the following integrals will be nonzero?The Overlap integral and Molecular OrbitalsSpectroscopy and Selection RulesRepresentation Theory and Physical SystemsFinny KuruvillaMath 126December 16, 1998A Brief History of Representation Theory in Physics and ChemistryRepresentation theory lies at the core of several modern disciplines of science such as particle physics, molecular orbital theory, and quantum dynamics. It is hard to overstate the implications of group representations in these disciplines. The application of group and representation theory to the physical sciences loosely began in the 19th century as crystallographers were devising means to interpret diffraction patterns in order to infer chemical structure. Their efforts utilized concepts from group theory without access to the formalized results that mathematicians would soon develop. The first well-defined example ofrepresentation theory in physical systems was Hermann Weyl’s monumental work entitled Gruppentheorie und Quantenmechanik published in 1928, formally bringing group representations to quantum mechanics toyield very elegant results.After Weyl’s groundbreaking work, heavy applications of group and representation theory were found in chemistry and spectroscopy during the 1920s and 1930s. Nuclear and particle physics found a central place for group representations in the 1930s and 1940s as did high-energy physics in the 1960s. A very beautiful application of group representations can be found in the work of Murray Gell-Mann who used the representations of SU(3) to predict the existence of the quark in 1961. (Its experimental existence would be found three years later at the Brookhaven National Laboratory.)Classifying Molecular StructureThe most obvious way to utilize group theory in physical chemistry is simply as a molecular descriptor. Variations in molecular structure cause immense differences in reactivity and spectroscopy and thus it is important to be able to easily describe the conformation of any given molecule. These structures often have several symmetry elements. Group theory can easily satisfy the need to describe molecular structure since the symmetries of any object naturally form a group. There are two types of molecular groupsymmetries. A point group is a group of symmetries of an object where the elements of the group are restricted to rotations and reflections (elements which hold a point fixed). A space group is a group of object symmetries where the elements of the group include translations, glide reflections, rotations, and reflections. It is clear that no finite molecule can have a translational symmetry so point groups are sufficient to classify ordinary molecules. (Crystals, however, which are treated as infinite repetitions of a particular unit cell can have translations or glide reflections in their symmetry.) Classifying a point group can be doneby enumerating all the symmetry elements of the molecule although chemists have devised flowcharts to rapidly assign a point group to a given molecule (figure 1).Adding new elements to character tablesWhen one thinks of group representations, while it is perhaps most common to think of the vector space as being ℂn, a representation can be over any vector space, including a vector space of functions. Examination of a typical character table from a physical chemistry text (figure 2) shows two additional columns of linear and quadratic functions where a given function corresponds to a particular irreducible representation.Assigning these additional functions to the character table is perhaps best illustrated by example. Let us consider the C3v character table. Fix a coordinate system and let the group elements act on that system, writing a matrix for each conjugacy class of the group. As can be seen in the C3v case (figure 2), thematrices may sometimes be cast is block diagonal form and thus reduced further. With the irreducible formsshown, it is easily seen that the x and y coordinates are associated with the irreducible representation Eand the z coordinate the trivial representation, A1. Hence the assignment of (x, y) and z to their corresponding irreducible representations in the original character table. By direct analogy, the other linear and quadratic functions may be used as a basis for some representation and thus “associated” with a particular irreducible representation. Associating functions with character tables will become useful for evaluating integrals, as will soon be shown.Tensoring and IntegrationLike any vector space, functional vector spaces can be tensored. These new spaces contain all products of the elements from the original spaces. It is a basic fact about representation theory that when vector spaces are tensored, characters multiply. Before implementing tensors in evaluating integrals, an important proposition is required. Proposition. Consider the following integral over some domain D with point group symmetry G.The value of the above integral, y, is zero unless f contains a component that acts a basis of the trivial representation of G.Sketch of proof. In order to be nonzero, the integrand must be invariant under the elements of G. Now suppose that f has no component that can act as a basis of the trivial representation. Then G has some element u under which f is variant. Hence y must be zero.By a simple extension to the above two ideas we obtain the following corollary:Corollary. Consider the following integral over some domain D with point group G:If the tensor product of the n functions does not contain the trivial representation, then the integral must evaluate to zero.Some examples. Is it possible that some of the following integrals will be nonzero?1. f = xy over an equilateral triangle centered at the origin. This domain has point group C3v and from the character table (figure 2), xy may act as a basis for the irreducible representation E. Since E does not contain the trivial representation A1, this integral must always be zero.2. f = z2 over a pentagon. Referring to the point group C5v, z2 is a basis for A1, the trivial representation and thus may be nonzero.3. f = (xy)z over a tetrahedron. The tetrahedron has point group Td (see figure 3 for its character table). Tensoring our two