the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 05 Dec 2025
The Origin of Mirror Symmetry in High-Resolution NMR Spectra
Abstract. A correlation between the symmetry of NMR spectra, including higher-order spectra, and the properties of the spin system has been established. It is shown that for a spectrum to be symmetric about the mid-resonance frequency (ν0), two conditions must be satisfied: the resonant frequencies of the spins must be symmetrically positioned about ν0, and the J‑coupling matrix must be symmetric about the secondary diagonal. The results were validated by calculating theoretical spectra for 3-, 4-, 5-, and 6-spin systems.
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CC1: 'Comment on mr-2025-15', Tom Barbara, 12 Dec 2025
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AC1: 'Reply on CC1', Dmitry Cheshkov, 13 Dec 2025
Thank you very much for pointing us to Corio's monograph. We will certainly include a reference to the Symmetry Theorem for AnBn systems described therein in the revised version of our manuscript.
Citation: https://doi.org/10.5194/mr-2025-15-AC1
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AC1: 'Reply on CC1', Dmitry Cheshkov, 13 Dec 2025
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RC1: 'Comment on mr-2025-15', Michael Tayler, 16 Dec 2025
This manuscript reports an interesting symmetry theorem for high-field NMR, which I admit I didn't know of before, but think is useful to describe to community. The theorem is well explained, and although perhaps lacking a rigorous "proof", it is enough to be convincing. Some minor comments for improvement: (1) The authors should use existing terminology for the matrices -- a matrix that is symmetric about it's secondary diagonal is called a persymmetric matrix, which, indeed, has known connections to permutation matrices/operators, see https://en.wikipedia.org/wiki/Persymmetric_matrix , and (2) the theorem implicitly relies on spin eigenstates that are Zeeman product states. One must therefore state that the rule breaks down in low fields, for example, in the molecules containing 1H-19F coupled spins at fields comparable to or weaker than earth's field.
Citation: https://doi.org/10.5194/mr-2025-15-RC1 -
CC2: 'Reply on RC1', Tom Barbara, 17 Dec 2025
I agree that it would be of some interest to find a more "analytical" approach and my guess is that one could tackle it by way of the "method of moments". I vaguely understand that this is related to the trace of multiple commutators, and since it involves the trace, the results should be independent of any basis states one would like to work with. The topic has many layers for sure. My background is from deuterium solids and of course, if chemical shift effects can be ignored, the Pake powder pattern is supposed to be symmetric and all the signal can be phased into "one channel". In that situation, pulse imperfections can come into play and it is not entirely a trivial situation to analyze. I only bring this up to illustrate how interesting it can get.
Citation: https://doi.org/10.5194/mr-2025-15-CC2
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CC2: 'Reply on RC1', Tom Barbara, 17 Dec 2025
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AC2: 'Comment on mr-2025-15', Dmitry Cheshkov, 26 Dec 2025
Dear Colleagues,
Thank you very much for your valuable comments and suggestions. We will address all points and incorporate the necessary corrections and clarifications in the revised version of the manuscript.
Inspired by your feedback, we have spent the past few weeks working on a rigorous proof for the theorem on symmetric NMR spectra introduced in our work. We are pleased to report substantial progress, and you can find our preliminary findings in the attached file.
We plan, however, to keep the focus of the current manuscript on the more illustrative and phenomenological aspects. The complete theoretical results will be developed into a separate, dedicated paper.
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CC3: 'Reply on AC2', Tom Barbara, 27 Dec 2025
Very Impressive! Of course, I did work on the moment approach out of pure curiosity and happy to see your efforts. Congrats.
Citation: https://doi.org/10.5194/mr-2025-15-CC3
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CC3: 'Reply on AC2', Tom Barbara, 27 Dec 2025
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RC2: 'Comment on mr-2025-15', Norbert Mueller, 17 Jan 2026
The authors address a topic that has not been given much attention in the recent literature, the relation between symmetry of NMR-spectra and symmetry of molecules. This may erroneously be believed to be a “trivial” problem, but clearly may range from deceptively simple to highly complex. While the symmetry of first order multiplets is a well-understood fact, the case(s) of symmetry with respect to the entire spectrum proves to be a more intricate problem. The authors provide two criteria: (1) Unsurprisingly, the frequency distribution around the center frequency must be symmetric, (2) the symmetry of the J-coupling matrix with respect to what the authors call “secondary diagonal” is presented as the necessary condition for spectral symmetry.
There is not much current literature pertaining to this topic. The book by Corio, which is mentioned in comments on the MR web-site, would surely help. I tried for a week to get hold of a copy but failed. There is one related paper by Corio (Journal of Chemical Education, Volume 46, Number 6, June 1969 )but it highlights only aspects (e.g. AA’BB’ systems) of what could be found in that book. But it would be worth comparing the results.
Note: I believe Mohamed Sabba (U. of Southampton) recently did some highly advanced work on coupling networks , dealing with simulations of large complicated spin systems. He has some novel approaches on his blog (https://arxiv.org/html/2404.03560v2) and may be the right person to propose improvements to the approaches of the current paper. I am not completely certain but he posted some impressive simulations on what used to be “twitter”. Dr. Sabba’s work does not address exactly the cases the authors consider but may lead to a more elegant general solution. Also the aspect of dynamics needs to be addressed or at least it should be clearly stated that this approach is (or is not) applicable in cases of dynamic averaging of spin system parameters.
Reviewers conclusion: The paper draws attention to an important overlooked question, which maybe relevant also in the context of QIP (quantum information processing) and proposes a solution. But it seems a bit incomplete, an elegant mathematical or algorithmic solution to the ordering of the coupling matrix elements is missing, in my humble opinion. Nevertheless I recommend it for publication after revision, so as to provide a basis for continued deliberation of this topic by this and other research groups.
Addressing the formal criteria required for a review report:
Scientific impact: good
Scientific quality: OK, there is some lack of mathematical rigor, see comments by Michael Tayler and Tom Barbara on the MR-website..
Novelty: OK, but the AA’BB’ system simulations have been around for some time.
Presentation quality: OK, room for improvement, the figure captions could be more informative
The language could be improved by some native speaker’s expertise.
Additional references as mentioned in this review and by the commenters on the MR-website should be added.
So, I recommend a major revision.
Citation: https://doi.org/10.5194/mr-2025-15-RC2
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The appearance of high symmetry in high resolution NMR spectra is always a wonder to behold. They can be so complicated and yet striking visually. I have not studied the exposition offered here and hope to do so in the near future. In the meanwhile, the authors may find it of some interest that there is a similar result offered for AnBn spin systems in the authoritative monograph by Paul Corio: "Structure of High Resolution NMR Spectra" Academic Press, 1966. His theorem arises from the use of the "spin inversion" operator where for each spin the projection quantum number m changes sign. It is something of a tragedy in the field of NMR that it is probably difficult to find this monograph today. It is a real gem and worth the search to find a copy. I know of it from my undergraduate studies in NMR at University of Kentucky, where I was fortunate to have Paul as my undergrad advisor.