A simple formulation of dynamic magic-angle spinning NMR derived from relaxation and Floquet theories

A simple method proposed in an insightful paper by A. J. Vega [J. Magn. Reson. 65 (1985) 252–267] is applied for calculating the effects of chemical exchange on magic-angle spinning (MAS) NMR spectra in the case of a two-site rotational jump motion. This approach which only requires two basic expressions of $T_2$ for the limiting cases of fast and slow exchange is compared with exact numerical calculations for arbitrary rates of motion and spinning frequencies. This comparison justifies the application of relaxation theory (RT) to calculate fast-exchange lineshapes but the slow-exchange $T_2$ time constant originally derived by A. Schmidt and S. Vega [J. Chem. Phys. 87 (1987) 6895–6907] using Floquet-perturbation theory (FPT) fails to account for the differences in the spinning sideband linewidths. In this paper, the complete FPT (cFPT) expression of the MAS spectrum is shown to account for all details of differential sideband broadening observed in the slow-exchange regime. Moreover, the RT and cFPT solutions give insight into the effects of molecular dynamics on the MAS spectra and decrease dramatically the computation time. The calculation procedure using the RT and cFPT formulas yield lineshape simulations that are in very good agreement with exact numerical results except in the intermediate-exchange regime when the sideband linewidths become comparable with or larger than the MAS rate. This is a minor drawback in practice as fast relaxation then makes quantitative measurements difficult.