Theory of multiple quantum dynamic NMR spectroscopy

Abstract

A formalism for calculating dynamic multiple quantum NMR line shapes obtained by the time proportional phase increment (TPPI) method from spin-1/2 systems is developed. The formalism is essentially an extension of the related formalism in Liouville space used in the theory of conventional dynamic NMR of strongly interacting spin systems. It is subsequently used to calculate the expected multiple quantum proton NMR line shapes of a number of (hypothetical and real) systems consisting of compounds dissolved in liquid crystalline solvents and undergoing intramolecular rearrangement. These include cyclobutadiene, cyclohexatriene, and cyclooctatetraene undergoing bond shifts and s-trioxane undergoing ring inversion. Since the computations involve diagonalization of high-dimensional matrices extensive use is made of symmetry factorization. It is shown that the resulting line shapes depend on the mechanism and rate of the dynamic processes, and may therefore be used to derive kinetic parameters from multiple quantum experiments. The high-quantum order spectra are particularly useful because for intermediate and large spin systems they are much simpler than the corresponding conventional single quantum spectra. Approximate expressions for the multiple quantum line shapes are also derived for the slow and fast exchange limits. It is found that except for an intermediate dynamic region these equations faithfully reproduce the exact line shapes in the appropriate limits.