The time-dependent Schrodinger Equation [1] serves as a benchmark for various theories used in nuclear magnetic resonance (NMR). This time-dependent linear differential Equation (Schrodinger) is the basic framework unifying the quantum mechanical, mathematical physics, quantum chemistry, atomic and molecular physics, quantum electrodynamics, quantum computing or control theory etc., and statistical descriptions of matter [2] - [6] . This framework is the unique one permitting a consistent treatment of the spin dynamics in quantum physics in general and solid-state NMR in particular. Such spin dynamics are central in the description of the quantum measurement process leading to the NMR signal and also in designing sophisticated pulse sequences and understanding of different experiments [7] -[16] .

The main theories used so far in solid-state nuclear magnetic resonance include the average Hamiltonian theory [17] and the Floquet theory [18] -[20] , and the developing theories are the Fer expansion [21] or the FloquetMagnus expansion [22] . These theories provide solutions to the time-dependent Schrodinger Equation which is a central problem in quantum physics in general and solid-state NMR in particular. The expansion schemes of these theories including the Magnus [23] and the Fer expansions can be used as numerical integrators for solving the time dependent Schrodinger Equation. However, there arepotential future theoretical and numerical directions for the calculation of the time propagation and evolution operators use Chebychev expansion [24] -[27] and Cayley method [27] [28] that need further attention in the NMR community.

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 [27] [29] . This is due to the fact that many physical, biological, and economic processes are governed by mathematical models described by systems of linear, constant coefficient ordinary differential Equations. Although many efficient algorithms have been developed, so far, the problem is still not having being solved in general. For instance, the evaluation of, where is a (real or complex) matrix, represents the major factor in the overall computational cost required by this class of algorithms and is probably one of the most problematic aspects. For instance, in the reference [29] , twenty different numerical algorithms for computing the exponential of a matrix are carefully reviewed. Approaches such as scaling and squaring with Pade approximation, Chebyshev approximation, Krylov space methods, or splitting methods, have been used to approach the exponential of a matrix problem [27] .

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 spin system is represented by a vector with dimensions. Therefore, it is essential to establish the most efficient numerical methods for dealing with the very large matrices that result from more challenging simulations. Simulation of spin system dynamics requires the numerical solution of the Liouville von Neumann Equation, or equivalently the numerical exponential of a Liouville matrix [30] .

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 [31] .

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 [37] -[39] .

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.