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This short review article presents theories used in solid-state nuclear magnetic resonance spectroscopy. Main theories used in NMR include the average Hamiltonian theory, the Floquet theory and the developing theories are the Fer expansion or the Floquet-Magnus expansion. These approaches provide solutions to the time-dependent Schrodinger equation which is a central problem in quantum physics in general and solid-state nuclear magnetic resonance in particular. Methods of these expansion schemes used as numerical integrators for solving the time dependent Schrodinger equation are presented. The action of their propagator operators is also presented. We highlight potential future theoretical and numerical directions such as the time propagation calculated by Chebychev expansion of the time evolution operators and an interesting transformation called the Cayley method.

The Schrodinger equation is the fundamental equation of physics for describing quantum mechanical behavior. In classical physics, the Schrodinger equation predicts the future behavior of a dynamic system and plays an important role of Newton’s laws and conservation of energy [

This short review presents some applications of major theories used in NMR spectroscopy such as the average Hamiltonian theory (AHT) and the Floquet theory (FLT), as well as the developing approaches including the Fer expansion (FE) and the Floquet-Magnus expansion (FME) [

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 [

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 [

effective Hamiltonian that will be useful to describe spin dynamics processes in solid-state NMR and understand different synchronized or non-synchronized experiments [

element in a given Lie algebra group, both approaches have the required structure and evolve in the desired group (Lie group). In addition, this is also true for their truncation to any order. We are thus poised to perform more work to ascertain the feasibility of Fer expansion in handling cases involving non-periodic and non-cyclic cases, and to use the expansion schemes of the Magnus (AHT) and the Fer expansions as numerical integrators for solving the time dependent Schrodinger equation which remains the central problem in quantum physics. Theoretical approaches in NMR are challenging, but the potential payoff is substantial, and could ultimately lead not only to a more accurate and efficient spin dynamics simulation, but also to the development of sophisticated RF pulse sequences, and understanding new experiments. Since the first demonstration of nuclear magnetic resonance in condensed matter in 1946 [

The overall goals of this review article is to support theories in NMR in order to continue to a) apply the average Hamiltonian theory to problems including (but not limited to): a class of symmetrical radio-frequency pulse sequences in the NMR of rotating solids, the symmetry principles in the design of NMR multiple-pulse sequences, the composite pulses, and the problems still unsolved such as the AHT for 3 spins [

Since its first application in NMR in 1968 by Evans, Haeberlen and Waugh, the average Hamiltonian theory has evolved as a powerful technique of analysis in the development of high resolution NMR spectroscopy [

in terms of exponentials of combinations of the coefficient matrix

for

provides

The FLT introduced to the NMR community in the early 1980’s simultaneously by Vegaand Maricqis another illuminating and powerful approach that offers a way to describe the time evolution of the spin system at all times and is able to handle multiple incommensurate frequencies [

The notation

Subsequently, the Floquet density operator and the Hamiltonian operator are represented by

The Floquet Hamiltonian

operators defined both in the spin

modulation) with the off-diagonality represented by the indices

the Floquet transition amplitudes was evaluated to:

coupled I = 1/2 spin pairs by evaluating two uncoupled homonuclear spins under magic angle sample spinning conditions

Analysis and numerical implementation of Magnus expansions is not a trivial task. Therefore, an alternative to the Magnus expansion which is called the Fer expansion can be useful for solving the time-dependent Schrodinger differential equation. This approach was formulated more than half a century ago by Fer and wasrecently introduced to the NMR community by Madhu and Kurur [

tials. The expansion is generated by the recursive scheme,

The Fer expansion involves a series of nested commutators resulting in

The Floquet Magnus expansion is a new theoretical tool for describing spin dynamics recently introduced in solid-state NMR and spin physics [

FLT that could be very useful in simplifying calculations and providing a more intuitive understanding of spin dynamics processes. The approach of FME is essentially distinguished from other theories with its famous function

the option of a more general representation of the FME with

formal derivation of higher order terms. In the above equations, the ^{th} order term of the argument of the operator that introduces the frame such that the spin system operator is varying under the time independent Hamiltonian

For the sake of simplicity, we considered the Hamiltonian:

mon form of Hamiltonian in solid-state NMR.

second rank m-order spherical tensor describing the spin system as defined by

perturbation theory (SPT) in terms of the irreducible tensor operators gives the diagonal Hamiltonian,

interaction frame where the Hamiltonian becomes time-dependent

is in agreement with the static perturbation theory and Van Vleck transformations. This is not the case of the Magnus expansion. This agreement can be easily explained by the connection that exists between the SPT and

FME propagators written as

Considering the generalized Fourier expansion of the Hamiltonian (

larly, calculation of second order terms is straightforward [

For example, applying the first contribution terms of FME to the dipolar Hamiltonian when irradiated with the BABA (

reference [

the degree of recoupling magnetic dipolar between nuclear spins which is useful for preparing and detecting double quantum coherence [

teraction of BABA and C7 pulse sequences. The size of

which indicates the degree of efficiency of the scheme. In reference [

serve that all curves are strictly monotonous. This tells us that, the strength of the DQ terms increase continously with time and no decoupling conditions occur in the BABA (with delta-pulse) and C7 pulse sequences.

Now, let us Consider, BABA pulse sequence with finite pulse width where the relation

only DQ terms in the function

bers

to BABA with delta-pulse width shows that the magnitude of the DQ terms of BABA with finite pulses is small

compared to the magnitude of BABA with

magnitude of the double quantum terms decreases, as expected. When

maximum corresponding to the delta-pulse sequence. However, when

a full decoupling is possible, which occurs at

Application of the first contribution terms of the Floquet-Magnus expansion to the chemical shift anisotropy when irradiated with the BABA pulse sequence lead to an important condition for the CSA to be averaged out in each rotor period

of the argument of the propagator operator in FME approach was evaluated to

A numerical analysis for a simple case consisting of one spin system with

neous values of the function

molecule and depend on the orientation of the molecule and on the CSA tensor elements. This complex function

can also be ploted versus the dimensionless number

ferent orientation of the molecule.

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

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 [

cated Chebyshev expansion of

advantages: first, it exploits the sparsity of the Liouvillian (Hamiltonian) by expressing the propagator in terms of a sequence of

The 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 [

of Equation (1) can be written as

such that if

various expansion coefficients. Blanes and co-workers obtained the time-symmetric methods of order 4 and 6, based on the above Cayley transform where the efficiency of Cayley based methods can be built directly from Magnus based integrators [

In this publication, we have thoroughly reviewed the abiding applications of average Hamiltonian theory, Floquet theory, and Floquet-Magnus expansion from very different perpectives in spin quantum physics of nuclear magnetic resonance. We also have presented some potential theories in NMR such as Fer expansion, Chebychev approximation, and possibly Cayley method. The combinations of two or more of the theories therein described will provide a framework for treating time-dependent Hamiltonian in quantum physics and NMR in a way that can be easily extended to both synchronized and several non-synchronized modulations. We hope this publication will encourage the use of Floquet-Magnus and Fer expansions as numerical integrators as well as the use of Floquet-Magnus expansion as alternative approach in designing sophisticated pulse sequences and analyzing and understanding of different experiments. We also hope that this review will contribute to motivate spin dynamics experts in NMR to consider other perspectives and approaches beyond the scope of the current popular or used theories in the field of nuclear magnetic resonance. They are also many remarkable applications of the theory of NMR that we do not discuss in this review such as quantum information processing and computing. For example, the nuclear magnetic resonance quantum calculations of the Jones polynomial are interesting theoretical problems to tackle as well as theoretical treatment of problems with more than three frequencies analyzed using Floquet theory or Floquet-Magnus expansion approaches. 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 wide over the years. With new application in the field of NMR, we also expect the FME to generate new contributions like the generation of efficient numerical algorithm for geometric integrators.

The intention of writing this overview of theories and applications in nuclear magnetic resonance spectroscopy is to help bring the current and future prospective theoretical aspects of spin dynamics in NMR to the attention of the NMR community and lead new interactions between NMR experts and other specialists in mathematics, physics, chemistry, physical chemistry, and chemical physics. All these points strongly support the idea that the Floquet-Magnus expansion, the Fer expansion, the Chebyshev approach, and possibly the Cayley method can also be the very useful and powerful tools in quantum spin dynamics.

E. S. Mananga appreciates the moral supports of Profs. Joseph Malinsky, Andrew Akinmoladun and Akhil Lal, Mr. Hamad Khan and Mr. Alfred Romito.