Graph theoretical characterization of NMR groups, nonrigid nuclear spin species and the construction of symmetry adapted NMR spin functions

Abstract

NMR group defined by Woodman is shown to be the generalized wreath product group of the NMR graph which is constructed with protons as vertices and edges representing nuclear couplings. The NMR graph can be expressed as a generalized composition of its quotient graph and various types. It is shown that the NMR group is the automorphism group of the NMR graph that preserves the coupling matrix defined here. Using the representation theory of generalized wreath product groups developed in an earlier paper of the present author, the character tables of NMR groups can be obtained. The character tables of certain NMR groups are presented. The quotient graph of the NMR graph is shown to be the composite particle graph. The automorphism group of the quotient NMR graph is the symmetry group of the composite particles, preserving the NMR coupling constants. An iterative algorithm is formulated for obtaining the automorphism group of the composite particle tree. A combinatorial approach is expounded for identifying the nuclear spin species of nonrigid and rigid molecules. The coalescence diagrams describing the effect of nonrigidity on nuclear spin species are introduced and obtained to exemplify the utility of the proposed method. It is shown that using the automorphism group of the quotient graph, the symmetry adapted composite particle spin functions can be constructed. We illustrate this method with 2,3-dimethylbutane. The nuclear spin Hamiltonian matrix of 2,3-dimethylbutane of order 214×214 is reduced to a matrix of order 64×64 by the composite particle method. This matrix can be blocked in the symmetry adapted basis set constructed through the automorphism group of the quotient graph into 40 matrices of order 1×1 and 12 matrices of order 2×2.