the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Simulation of NMR spectra at zero- and ultra-low field from A to Z – a tribute to Prof. Konstantin L’vovich Ivanov
- 1Univ Lyon, CNRS, ENS Lyon, UCBL, Université de Lyon, CRMN UMR 5280, 69100 Villeurbanne, France
- 2Laboratoire des biomolécules, LBM, Département de chimie, École normale supérieure, PSL University, Sorbonne Université, CNRS, 75005 Paris, France
- 1Univ Lyon, CNRS, ENS Lyon, UCBL, Université de Lyon, CRMN UMR 5280, 69100 Villeurbanne, France
- 2Laboratoire des biomolécules, LBM, Département de chimie, École normale supérieure, PSL University, Sorbonne Université, CNRS, 75005 Paris, France
Abstract. Simulating NMR experiments may appear mysterious and even daunting for those who are new to the field. Yet, broken down into pieces, the process may turn out to be easier than expected. Quite to the opposite, it is in fact a powerful and playful means to get insights into the spin dynamics of NMR experiments. In this Tutorial Paper, we show step by step how some NMR experiments can be simulated, assuming as little prior knowledge from the reader as possible. We focus on the case of NMR at zero- and ultra-low field, an emerging modality of NMR in which the spin dynamics is dominated by spin-spin interactions rather than spin-field interactions, as is usually the case of conventional high-field NMR. We first show how to simulate spectra numerically. In a second step, we detail an approach to construct an eigenbasis for systems of spin-½ nuclei at zero-field. We then use it to interpret the numerical simulations. In this attempt to make NMR simulation approachable, the authors wish to pay a tribute to Prof. Konstantin L’vovich Ivanov, a great scientist and pedagogue who passed away on March 5th 2021.
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Quentin Stern and Kirill Sheberstov
Status: open (until 09 Dec 2022)
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RC1: 'Comment on mr-2022-18', Anonymous Referee #1, 07 Nov 2022
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The discussion by Stern and Sheberstov introduces to readers some basic principles by which to simulate NMR spectra of simplistic molecules during free evolution of their nuclear magnetization in a zero or ultralow magnetic field (ZULF). Specifically, the compounds must contain two sets of magnetically equivalent spins-1/2, each with a different gyromagnetic ratio, for example a carbon-13 and hydrogen-1.
Technically speaking, the work is correct, but readers should note that the content is not so original, and in my opinion a narrow viewpoint on the topic of near-zero-field NMR. Butler et al. wrote a key paper back in 2013 describing the theory of zero-field NMR in not only AXn systems as presented in this work, but more complex spin systems AmXn and AXmBn as well. Many of the results were derived analytically without simulations, for example, using perturbation theory.
Simulations of zero-and-ultralow-field NMR spectra in AXn, AmXn and AXmBn are also presented in detail in several PhD theses from the early 2010s: see for example Dr Thomas Theis (2012, UC Berkeley, https://escholarship.org/uc/item/01d528kh ), Dr John Blanchard (2014, UC Berkeley, https://escholarship.org/uc/item/2mp738zn ) and Dr Tobias Sjolander (2017, UC Berkeley, https://escholarship.org/uc/item/2kj4v04n ). Readers are encouraged to consult these original sources.
I encourage the authors to revise the paper by including more original content. For example, simulating the zero-field NMR spectra of compound that has not yet been studied experimentally, or magnetic fields bordering the zero-field condition where the perturbation theory starts to break down. Alternatively, by going beyond summarizing the main results of Butler-2013 and reviewing other works where zero-field NMR spectra are calculated. I believe this would be very useful to readers interested in simulation, not only to do justice to the works listed above. Perhaps a quick way is to use a table: compound or spin system, simulation approach (e.g. exact, perturbation theory), software, literature reference .
Additional comments:
- Authors and Readers should be aware of review article on zero-and-ultralow-field NMR, plus applications, published by Jiang M, Peng X et al. in 2021. This deserves to be mentioned: https://doi.org/10.1016/j.fmre.2020.12.007
- Some of the equations can provide valuable physical insight into how a ZULF NMR experiment works, but it is not explained. Let us take Equation 57 as an example. The total magnetization operator Mz = gI Iz + gS Sz is applied to an eigenstate of the zero field, that is |F, mF> in the case of AXn. The result is immediately quoted in terms of Clebsch-Gordan coefficients. There is a slightly different way to write the result where the operator is given as Mz = gI (Iz + Sz) + (gS - gI) Sz. Here the first term on the right-hand side of the (=) sign leads to the eigenvalue mF gI, while the second term leads to a sum over other operators, and overall proportional to (gS - gI) times the C-G coefficient. The authors may want to mention that this second term leads to ZULF signals where the amplitude of peaks scale with (gS - gI).
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AC1: 'Reply on RC1', Quentin Stern, 14 Nov 2022
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The referee is right in saying that the content of our paper is “not so original”. It is in fact not original at all. Our aim in writing this paper was not to present original theory but rather to present known theory in a pedagogical way. We chose to focus on the simplest systems to make the paper easiest to follow for the newcomer. Yet, to make the paper more appealing to the expert, we will add a section showing the transition from ZULF to high-field for a CH3 group.
We agree with the referee that highly valuable and pedagogical material can be found in PhD dissertations. One may therefore argue that papers that are only meant to be pedagogical are not useful. However, PhD dissertations are usually not as easily accessible. The starting Master student or PhD student who wants to discover the field will not come across a PhD dissertation easily by searching online. It’s usually by word of mouth that people are advised to read PhD dissertations. That’s why we believe that a paper presenting the basic concepts in detail is useful and can serve as an entry point to newcomers. That being said, we will add references to the PhD dissertations of T. Theis, J. Blanchard, and T. Slojander. Not only this will be useful for the reader but, as pointed out by the referee, it will be a due acknowledgment of the work done by these authors. We will also mention clearly that Butler et al’s 2013 paper presents analytical solutions to similar cases (and more).
We would like to point out to the referee that we did not present numerical simulations only to understand the appearance of ZULF spectra. The primary aim was to show how to numerically simulate spectra. Again, whether such a paper is in itself useful is up for debate.
We thank the referee for the additional comments. We will take them into account in the next version of the paper (once we have received feedbacks from other referees)
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CC1: 'Reply on RC1', Tom Barbara, 15 Nov 2022
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Perhaps just as important as originality, perhaps even more so, is clairity of exposition. It may be the case that those thesis publications are not as clear as they could be. And as far as originality goes, I am sure that everything anyone needed to know could be found in Paul Corio's very complete monograph "Structure of High Resolution NMR specta" Academic Press 1966.
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CC2: 'Reply on CC1', Tom Barbara, 16 Nov 2022
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Late last night I recalled another aspect of the NMR spectra simulation problem. This method is not covered in Paul Corio's book and though old, it does appear to supply an "original" perspective not usually encountered in the NMR field. This is the method of Primakoff and Holstein and the close cousin invented by Schwinger, where bosonic states are used to map angular momentum states. A nice, relatively recent effort at expostion is offered by J.A. Gaymfi and this can be found on the physics arxiv:
arXiv:1907.07122v1The author works out the use of these for zero field Hamiltonians. Beware that it is "very mathematical".
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CC2: 'Reply on CC1', Tom Barbara, 16 Nov 2022
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RC2: 'Reply on RC1', Anonymous Referee #1, 17 Nov 2022
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In response to the Author and Community comments so far:
I think that this paper could eventually end up being a good source of reference for ZULF NMR as the authors intend, but for now it lacks in both clarity and detail.
I still find the objective unclear. The abstract promises “detail, including the tricks that are usually omitted from research papers and assuming as little prior knowledge from the reader as possible”. However, the paper is not written clearly, several details are omitted, neither appear any tricks, and there are several inaccuracies (see the end) that can confuse readers regardless of their level of familiarity with theory of NMR.
Here is what I would expect the paper to contain for those entering the ZULF NMR field, or those with little experience of simulations, in this order:
- An overview of common situations where ZULF NMR simulations may be useful, maybe because analytical expressions are not available in: (a) complex spin systems, (b) free evolution near the edge of the ZULF regime, (c) evolution under pulses, e.g. dc pulses, composite pulses, (d) evolution during field sweeps, level crossings, such as those used in ZULF parahydrogen-induced polarization protocols, (e) evolution during dynamical decoupling sequences or the ZULF-TOCSY type of spin locking experiments done by the Ivanov group, (f) relaxation, (g) errors in pulse sequences or spin Hamiltonian parameters, (h) simulation of geometric effects such as sample shape, detector position, or inhomogeneous fields. The authors do not have to cover all of these in detail, but it would help to specify what they are, and give literature examples for each.
- An overview of the simulation “choices” that can be made by beginners: (a) eigenvector/eigenvalue analysis in either Hilbert of Liouville space, so that for time intervals where the Hamiltonian is constant you simply take an exponential of the eigenfrequency, rather than the whole matrix, (b) use of other symmetry-adapted bases, e.g. Zeeman vs. total angular momentum, and treat smaller subspaces one by one, (c) brute-force propagation of the Liouville von-Neumann or Schrodinger equation, say in the Zeeman basis. In the manuscript, it appears that a combination of (a) and (c) has been used, but it is not completely clear when and how.
- A few illustrative examples from the list (a)-(g) above. Novelty is not required if these examples reproduce the results of previous work with appropriate citation. Such as “In work (author et al., 20xx) this was shown. We will now show the reader how this can be simulated by …” and then proceed to do so, step by step, without leaving any gaps or diverting the reader to sources elsewhere. I will concede that the examples given by the authors are acceptable for a tutorial paper, but they do lack the complete start-to-finish detail that would allow a graduate student with a basic knowledge of quantum mechanics to sit down at a computer and work through them without getting stuck.
In my opinion, the authors are assuming a lot more of the reader than they realize, so their work can only be followed by someone who already has an expert level of experience in spin dynamics simulations for NMR, and who is probably capable of looking up the PhD theses already mentioned in the previous review comment.
Below are some specific or technical comments that I recommend the authors address:
- Line 21: “Use frequency dispersion” instead of “chemical shift dispersion”
- Line 24: The reference “(Thayer and Pines, 1987)” does not seem an appropriate citation with the 1.2 GHz NMR magnet?
- Line 29: “Mumetal”, rather than “μ-metal”
- Line 31: “does no longer influence the outcome of the experiment” is highly ambiguous. Consider replacing with “where the Larmor period is much longer than the coherence time, so that the field can be completely neglected”.
- Line 47: “at ZULF, there is no such intuition as the vector model”. Is that really true? The AX spin system can often partitioned into an isolated 2LS where cyclically commuting operators can be represented as a vector model picture, see for example https://doi.org/10.1063/PT.3.1948 or https://doi.org/10.1021/acs.jpca.6b04017 supporting information.
- Line 56 and 77: “using a theoretical framework coming from atomic physics”. I don’t think this is ok. You apply the rules of addition of angular momentum. This is not specific to atomic physics – it is just as widely used in molecular (e.g. rotational) spectroscopy and NMR.
- Line 69: “we assume the reader is familiar with general concepts of NMR but not necessarily with simulation”. This is not a clear sentence.
- Line 82: Spinach already has a large set of examples for zero-field NMR, e.g., http://spindynamics.org/wiki/index.php?title=Zerofield.m and http://spindynamics.org/wiki/index.php?title=Zulf_abrupt.m (sudden field drop), and probably SpinDynamica does too. These could be mentioned more directly, even though the above links are not permanent ones.
- Line 269: “is sometimes referred to as the sandwich formula” needs a reference. The only time I have read about “sandwich formulae” in the context of NMR is the rotation sandwich formulae in Levitt’s book “Spin Dynamics: basics of nuclear magnetic resonance”. These are specific to cyclically commuting triads of operators, i.e. fictitious spin-half representation of a two-level subspace
- Line 272: the “propagation operator” is a propagator
- Starting line 341: For the sake of clarity, let us not call signal or time a vector. Perhaps “list” or “array” of time points or time-amplitude pairs
- Line 505- 507: The phenomenon described is not nutation. Better to refer to the excitation curves as “Rabi curves”
- Style comment: please cite equations as “…, see Equation (x)” rather than “see (x)”.
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AC1: 'Reply on RC1', Quentin Stern, 14 Nov 2022
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Quentin Stern and Kirill Sheberstov
Model code and software
Fourier transform and simulations of spectra Quentin Stern and Kirill Sheberstov https://doi.org/10.5281/zenodo.7271319
Quentin Stern and Kirill Sheberstov
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