Introduction of the Floquet-Magnus expansion in solid-state nuclear magnetic resonance spectroscopy

Abstract

In this article, we present an alternative expansion scheme called Floquet-Magnus expansion (FME) used to solve a time-dependent linear differential equation which is a central problem in quantum physics in general and solid-state nuclear magnetic resonance (NMR) in particular. The commonly used methods to treat theoretical problems in solid-state NMR are the average Hamiltonian theory (AHT) and the Floquet theory (FT), which have been successful for designing sophisticated pulse sequences and understanding of different experiments. To the best of our knowledge, this is the first report of the FME scheme in the context of solid state NMR and we compare this approach with other series expansions. We present a modified FME scheme highlighting the importance of the (time-periodic) boundary conditions. This modified scheme greatly simplifies the calculation of higher order terms and shown to be equivalent to the Floquet theory (single or multimode time-dependence) but allows one to derive the effective Hamiltonian in the Hilbert space. Basic applications of the FME scheme are described and compared to previous treatments based on AHT, FT, and static perturbation theory. We discuss also the convergence aspects of the three schemes (AHT, FT, and FME) and present the relevant references.

Theories in Spin Dynamics of Solid-State Nuclear Magnetic Resonance Spectroscopy

“…effective Hamiltonian that will be useful to describe spin dynamics processes in solid-state NMR and understand different synchronized or non-synchronized experiments [34] [52]- [55]. Furthermore, our first applications of FE approach to study interactions in solid-state NMR when irradiated with the magic-echo sequence support this goal [52].…”

“…This unique approach (FME) is an extension of the popular Magnus expansion and average Hamiltonian theory and is useful to shed new lights on AHT and FLT [27] [28]. The aims of the FME is to bridge the AHT to the Floquet Theorem but in a more concise and efficient formalism [34]. Calculations can then be performed in a finite-dimensional Hilbert space instead of an infinite dimensional space within the Floquet theory.…”

“…The Floquet Magnus expansion is a new theoretical tool for describing spin dynamics recently introduced in solid-state NMR and spin physics [34]- [36]. This unique approach (FME) is an extension of the popular Magnus expansion and average Hamiltonian theory and is useful to shed new lights on AHT and FLT [27] [28].…”

“…Methods developed over the past 3.5 years enabled us to obtain simplified calculations for the common form of Hamiltonian in solid-state NMR and multimode Hamiltonian in its generalized Fourier expansion Hamiltonian [34]. Based on these and other unpublished findings, we now believe that the FME provides a quick and efficient means to calculate higher order terms allowing the disentanglement of the stroboscopic observation and …”

“…Solid-state NMR is a powerful method to elucidate molecular structure and dynamics in systems not amenable to characterization by other methodologies and its importance stands in its ability to accurately determine intermolecular distances and molecular torsion angles [34] [49]- [52].…”