Application of Floquet-Magnus expansion and Fer expansion to investigate the chemical shift anisotropy in solid-state NMR when irradiated with the triple oscillating field technique

The Floquet-Magnus and Fer expansion schemes were introduced in solid-state nuclear magnetic resonance (NMR) in 2011 and 2006, respectively. Key features of the Floquet magnus expansion are its ability to account for the calculations developed in a finite-dimensional Hilbert space instead of an infinite-dimensional space within the Floquet theory as well as its use of its distinguishable function from other theories such as average Hamiltonian theory, Floquet theory, and Fer expansion, which facilitates the evaluation of the spin behavior in between the stroboscopic observation points. This article focuses on revisiting the Floquet-Magnus and Fer expansion approaches and applying both methods to calculate the effective Hamiltonians and propagators, which control the spin system evolution during the Triple Oscillating Field Technique radiation experiment (TOFU). The TOFU pulse sequence is an important technique that was shown to avoid the dipolar truncation problem and form a new basis for accurate distance measurement by solid-state NMR. We take advantage of the interaction frequencies and the time modulation arising from the TOFU pulse sequence allowing selective recoupling of specific terms in the Hamiltonian that fulfill determined specific conditions. The work presented unifies and generalizes existing results of the Floquet-Magnus and Fer expansions and delivers illustrations of novel springs that boost previous applications that are based on the classical information. We believe that the generality of this work points to potential applications in problems related to theoretical developments of spectroscopy as well as interdisciplinary research areas whenever they include the spin dynamics concepts.