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We present the theories used in solid-state nuclear magnetic resonance and the expansion schemes used as numerical integrators for solving the time dependent Schrodinger Equation. We highlight potential future theoretical and numerical directions in solid-state nuclear magnetic resonancesuch as the Chebychev expansion and the transformation of Cayley.

The time-dependent Schrodinger Equation [

The main theories used so far in solid-state nuclear magnetic resonance include the average Hamiltonian theory [

Computing the exponential of a matrix is an important task in quantum mechanics and in solid-state nuclear magnetic resonance in particular where all theories used rely on exponential Hamiltonian operator propagators. The approximation of the matrix exponential is among the oldest and most extensive research topics in numerical mathematics [

Simulation is very important in molecular quantum dynamics especially in the case of chemical exchange or relaxation, where line shapes can be overlapping. The main difficulty encountered in spectrum simulation is the rapid increase of computational requirements with an increasing number of spins. This is due to the exponential scaling of the spin vector space. Specifically, a state of an

Although the theoretical advantages of the Chebyshev method are still not entirely realized for currently feasible computations, the Chebyshev approach has the potential to be extensively used in spin quantum dynamics and emerged as an important approach for numerical simulations of many systems encounters. Our motivation here is to present the Chebyshev approximation as a potential surrogate of the popular expansions in NMR for the task of numerical simulations in spin dynamics paradigm stems from its numerical stability and high accuracy. In addition to the Chebyshev approximation, we also present the Cayley transformation [

This new method provides a useful alternative to the exponential mapping relating the Lie algebra to the Lie group. This fact is particularly important for numerical methods where the evaluation of the exponential matrix is the most computation-intensive part of the algorithm. It is noteworthy that the combinations of two or more of the theories known in NMR will continue to provide a framework for treating time-dependent Hamiltonian in quantum physics and NMR in a more efficient way that can be easily extended to all types of modulations.

Nearly three decades ago, Tal-Ezer and Kosloff introduced the Chebyshev method as a means of solving the time-dependent Schrodinger Equation in the field of molecular dynamics [

where

with well-chosen coefficients

which gives the final approximation

The Chebyshev method has two main advantages: first, it exploits the sparsity of the Liouvillian (Hamiltonian) by expressing the propagator in terms of a sequence of

The second approach we are presenting called Cayley transform, provides a useful alternative to the exponential mapping relating the Lie algebra to the Lie group. This fact is particularly important for numerical methods where the evaluation of the exponential matrix is the most computation-intensive part of the algorithm. Blanes and co-workers shown that the solution of the linear ordinary differential Equation can be written as [

with

Note that the choice of

where

Three matrix-matrix products are required in addition to the three commutators involved in the computation of

We hope this article will encourage the use of Magnus and Fer expansions as numerical integrators as well as the use of Floquet-Magnus expansion as an alternative approach in designing sophisticated pulse sequences and analyzing and understanding of different experiments. We also hope that this letter will contribute to motivating NMR spin dynamics experts to consider other theoretical and numerical perspectives such as Chebyshev or Cayley methods. Although much progress has been made in the theory of NMR compared to other spectroscopic techniques, a lot still needs to be done. So far, the NMR literature does not present theoretical treatment of problems with more than three frequencies analyzed using Floquet theory or Floquet-Magnus expansion approach. With the increase of the level of sophistication of NMR experiments, second and third order terms are of increasing importance, such as in diffusion experiments, or like triple-resonance continuous waves (CW) radio frequency irradiation under MAS. These are some examples of unsolved problems in the theory of NMR. In respect with the developments in the mathematical structure of AHT, FLT, FME, and FE, we expect that the realm of applications of the Floquet Magnus expansion and Fer expansion will also be wide over the years and generate new contributions like the generation of efficient numerical algorithm for geometric integrators [

The intention of writing this comment is to help bring the current and future prospective theoretical aspects of spin dynamics to the attention of the NMR community and lead new interactions between NMR experts and other specialists in related fields. All of these points strongly support the idea that the Floquet-Magnus expansion, the Fer expansion, and the Chebyshev approach can also be the very useful and powerful tools in quantum spin dynamics.

The author acknowledges support from Harvard University/Harvard Medical School, Massachusetts General Hospital, and the National Institute of Health (NIH), under Grants R01-HL110241 and T32 EB013180. The contents of this paper are solely the responsibility of the authors and do not represent the official views of NIH.