A survey on application of representation theory to molecular vibration
Representations Theory is used extensively in many of the physical sciences as every physical system has a symmetry group G. Various differential equations determine the vibration of a molecule, and the symmetry group of the molecule acts on the space of solutions of these equations. In this paper we use CH{sub 4} (methane) molecule, which has four hydrogen atoms at the corners of a regular tetrahedron, and a carbon atom at the center of the tetrahedron. The four hydrogen atoms in CH{sub 4} are permuted by the action of the symmetry group and this action fixes the carbon atom. At each of the 5 vertices, we assign three unit vectors, called the standard basis vectors in directions of the three edges which are joined to the vertex. The symmetry group G of the molecules permutes the 15 standard basis vectors, so we may regard Q{sup 15} as a GG By expressing Q{sup 15} as a direct sum of irreducible GG-modules, the problem of finding the normal modes of vibration is reduced to that of computing the eigenvectors of some small matrices.