Density matrix theory for calculating magnetization transfer and dynamic lineshape effects

NMR is a powerful tool for the study of chemical rate processes and the mechanisms by which they occur. Dynamic processes give rise to characteristic changes in transverse and longitudinal relaxation, which manifest themselves in the NMR lineshape and in magnetization transfer phenomena, respectively. Analysis of these effects permits the extraction of kinetic data. The mathematical theory central to this analysis is illustrated using the formalism of the density matrix for a two-spin system undergoing mutual intramolecular exchange in isotropic solution. The spin Hamiltonian and exchange operators are introduced, and the quantum mechanical equation of motion of the density matrix is presented, emphasizing those density matrix elements that correspond classically to longitudinal and transverse magnetization. The necessary Liouville and exchange superoperators are explicitly constructed from the corresponding Hamiltonian and exchange operators using tensor algebra. The equation of motion is then solved for the populations and single-quantum coherences to illustrate the origin of magnetization transfer and dynamic lineshape effects. The extension of the theory to arbitrarily large exchanging spin systems is addressed and illustrated for two mutually exchanging spins scalar coupled to a third spin. Finally, the two-spin lineshape calculation is fully worked out using a steady-state method that reduces the problem to the solution of a set of simultaneous algebraic equations.