Computational aspects of motional symmetry in nuclear magnetic resonance spectroscopy

Abstract

The most fundamental approach to calculate the effects of molecular motion on nuclear magnetic resonance spectra involves the density operator and the stochastic Liouville–von Neumann equation. In order to obtain a solution to this equation it is useful to expand the density operator into a set of infinitesimal group generators representing the different alignments and coherences of the nuclear spin system. In this representation the stochastic Liouville–von Neumann equation may be rewritten in the form of a linear homogeneous system of coupled first-order differential equations among the alignments and coherences. The dimension of this system is usually very large and the presence of molecular motion makes the equations highly stiff. This implies that the numerical solution is very difficult and can only be obtained by semi-implicit or implicit integration methods that are sufficiently accurate and stable. The computational efficiency may be improved significantly for any integration method by reducing the dimension of the system. In this paper we introduce a group theoretical methodology to reduce the stochastic Liouville–von Neumann equation for any system exhibiting sufficient motional symmetry. The approach is based on the concept of motional graphs and may be applied to any system independently of the choice of group generators. The results include a set of rules to determine the reducibility of the stochastic Liouville–von Neumann equation. The formalism is exemplified by application to different molecular systems with particular emphasis on demonstrating the results of reducibility.