Emerging expansion schemes in the NMR field


In quantum mechanics, the Schrödinger equation is a partial differential equation describing the timely modification of quantum systems. The equation was postulated by Schrödinger about a century ago and has stayed at the helm of quantum mechanics being an equivalence of Newton’s second law in classical mechanics. With a broad range of problems arising from the Schrodinger representation, researchers have adopted solutions based on exponential time-dependent perturbation theories. In particular, Floquet-Magnus expansion and Fer expansion have attracted significant research attention.

Herein, Dr. Eugene Stephane Mananga from the City University of New York and New York University, recently explored the two potential expansion schemes developing in the NMR field. The Floquet-Magnus and Fer expansion were specifically used to investigate the spin dynamics process in a spin-locking radiation experiment. The aim was to use the expansion schemes in the calculation of the effective Hamiltonians and propagators to improve the existing understanding of the spin-locking radiation for the benefit of solid-state NMR and magnetic resonance in general. The work is currently published in the journal, Chemical Physical Letters.

The proposed criterion for the Floquet-Magnus expansion is presented through the iterative based approach. Additionally, theoretical analysis was vital in controlling the systems during spin radiation. In the past analysis, it was concluded that Fer expansion was advantageous in calculating correct higher-order as compared to its counterparts. To ascertain this result, the author examined a magic-echo sequence using the Fer expansion and compared the results to the existing literature. Finally, the applications and relationship between the two approaches was established.

Comparing and contrasting the two, the two orders of both expansions were observed to be identical for time-dependent Hamiltonian. At higher orders, however, the discrepancy between the two orders was evident. For instance, the third order of Floquet-Magnus expansion looked more complicated than Fer expansion. The results were in good agreement with the existing resulting regarding the Floquet-Magnus expansion and Fer expansion expansions and more so the simplicity and efficiency of Fer expansion in the calculation of higher-order corrections. However, for solving spin dynamics problems in solids, the Floquet-Magnus expansion was more suitable and less complex to use. Since the two approaches are developed generally for different purposes, it is not guaranteed that they may be equal at a particular instance.

In summary, the study presented the applications and a comparison of the Floquet-Magnus and Fer expansion approaches of the Schrodinger equation in spin-locking radiation of solid-state NMR. The results could be extended to the construction and implementation of numerical integrators. The work is identified by the Advances in Engineering committee as a useful contributor to the general field of spin dynamics. In a statement to Advances in Engineering, Dr. Eugene stated that his theoretical and numerical investigations will particularly be of great interest in advancing not only the solid-state NMR but the general magnetic resonance community at large.

Emerging expansion schemes in the NMR field - Advances in Engineering

About the author

Dr. Eugene S. Mananga is the Director of Grants and STEM Program, a member in the Executive Board and Board of Directors-at-Large of The CUNY ACADEMY FOR HUMANITIES AND SCIENCES. He is a Doctoral Faculty in the Ph.D. Programs in Physics & in Chemistry at the Graduate Center of the City University of New York (CUNY).

He is an Adjunct Professor of Applied Physics at New York University (NYU), and an Assistant Professor of Physics and Nuclear Medicine at BCC of CUNY. In 2019, Professor Mananga was selected by the US Department of Energy (DOE, Office of Science) in the Faculty Program (VFP) at Argonne National Laboratory. He completed his Ph. D in Physics from the Graduate Center of the City University of New York, and holds 6 additional graduate degrees and training from various institutions including Harvard University, Massachusetts General Hospital, and City College of New York.

Eugene did his postdoctoral studies in the National High Magnetic Field Laboratory of USA, Harvard Medical School, and Massachusetts General Hospital. Prior to joining Harvard – MGH, Dr. Mananga was an “Ingenieur de Recherche” in the French Atomic Energy Commission and Alternative Energies (“Commissariat a l’Energie Atomique de France”, CEA-SACLAY, NEUROSPIN). During his tenure with CEA-SACLAY, Dr. Mananga worked in collaboration with Dr. Thibault Charpentier to introduce the Floquet-Magnus expansion in the field of Solid-State Nuclear Magnetic Resonance. This important work contributes to advance the understanding of spin dynamics in nuclear magnetic resonance, quantum physics, and beyond.

Eugene has published more than 60 peer-review scientific articles (mainly as first and corresponding author) including prestigious and major scientific journals such as Physics Reports, Royal Society of Chemistry, the Journal of Chemical Physics, PLOS ONE, the Journal of Physical Chemistry, Chemical Physics, Chemical Physics Letters, Journal of Magnetic Resonance, Solid-State Nuclear Magnetic Resonance, Journal of Nature and Science, etc… and has been serving as editorial board member for more than 40 international scientific journals.

He is currently the Editor-in-Chief of the journal of Molecular Physics, the journal of Drug Design Development Delivery, and recently, he served as the Editor-in-Chief of the Journal of Imaging Science and also served the most prestigious position of “Chief Editor” for the editorial board of ”The Scientific Journal of Molecular Physics”. Dr. Mananga has been an honorable Scientific Adviser and Organizing Committee Member for several major international scientific conferences in the USA and around the world. On April 2018, he was selected as one of the 2018 STEM Faculty Leaders in the USA to participate in the 2018 PKAL STEM LEADERSHIP INSTITUTE II. His scientific contribution in the field of Nuclear Magnetic Resonance was honored during the 70th anniversary (1946 – 2016) of the Russian Academic of Sciences. Dr. Mananga is the Laureate of 2017 Henry Wasser Award in Physics from the CUNY Academy of Humanities and Sciences in recognition of “Outstanding Scholarship and Academic Performance”.

His potential for leadership in SCIENCE is evidenced by his recent recognition as the recipient of the 2018 Distinguished Scientist Award from the American Chemical Society (New York & Westchester Section) for “Contributions and Advanced Studies in the Theory of Spin Dynamics in Solid-State Nuclear magnetic Resonance and Quantum Mechanics”. From 2018 to 2023, Professor Mananga serves as a Faculty Mentor in physics on the National Science Foundation (NSF) S-STEM scholarship $5 million scholarship grant on “Developing a Growth Mindset Model to Build Resiliency in Underrepresented STEM Students”.

Research Interest: Professor Eugene Mananga research interests are in Condensed Matter Physics, Atomic and Molecular Physics, Solid-State Nuclear Magnetic Resonance, Medical Physics, Radiological Sciences and Nuclear Medicine. Currently, he is focused on advancing theory, simulation and methodology in Solid-State Nuclear Magnetic Resonance, Spin Dynamics, Quantum Physics, Imaging Science, Medical Physics and Nuclear Medicine.


Mananga, E. (2019). Application of Floquet-Magnus and Fer expansion approaches during spin-locking radiation in solid-state NMR. Chemical Physics Letters, 730, 153-164.

Go To Chemical Physics Letters

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