Application of Floquet theory to the nuclear magnetic resonance spectra of homonuclear two-spin systems in rotating solids

Abstract

The theory for the nuclear magnetic resonance (NMR) spectra of homonuclear two-spin systems under sample spinning has been developed. The propagator describing the time-evolution of the systems, which is driven by the homogeneous Hamiltonian composed of the chemical-shift difference and the flip–flop parts of the dipolar and J couplings, is treated using Floquet theory; the Floquet eigenvalues and eigenvector components determine the resonance frequencies and intensities, respectively. Nondegenerate Rayleigh–Schrödinger perturbation theory is used to solve the Floquet secular equation. Recurrence equations for the perturbation corrections have been derived, allowing us to evaluate efficiently very high-order terms, such as 20th-order terms. Analytical expressions for the resonance frequency obtained in the low-order approximations provide an intuitive understanding of the main spectral features; the second-order equations can describe the conspicuous solid-state effects on the magic-angle spinning (MAS) spectra, such as the unique line shapes and the additional shifts, while the zeroth and first-order approximations yield spectra similar to those in solution-state NMR. The spectra should, however, be calculated up to the perturbation order where line shape function converges, to reproduce precisely the experimental results, especially the change of the additional shifts with the spinning frequency which were obtained in the MAS spectra for 13C doubly-labeled sodium acetate. The theoretical treatment shows that off-magic angle spinning (OMAS) spectra of two-spin systems should exhibit line shapes dependent on the mutual orientations of the dipolar and the chemical-shift tensors of the two nuclei. Also, the nondegenerate perturbation treatment yields criteria for the conditions under which the normal and n=0 rotational resonance occur.