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REVIEW 4 major objections 5 minor 50 references

A quaternion-based optimal-control framework that optimizes single-spin effective fields—with chemical-shift anisotropy included in the frame transformation—yields dipolar-recoupling pulse elements that are substantially less sensitive to o

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2026-08-02 03:21 UTC pith:X3D3G5SZ

load-bearing objection A solid methods paper on CSA-robust heteronuclear recoupling via single-spin quaternion optimal control; the central separation assumption is under-tested and the promised code release is missing. the 4 major comments →

arxiv 2607.13907 v1 pith:X3D3G5SZ submitted 2026-07-15 physics.chem-ph

Solid-State NMR Dipolar Recoupling in Presence of Large Chemical Shielding Anisotropies by Quaternion-Based Effective Hamiltonian Optimal Control

Enik\H{o} Balig\'acs , Nino Wili , Jos\'e P. Carvalho , Anders Bodholt Nielsen , Niels Chr. Nielsen This is my paper
classification physics.chem-ph PACS 76.60.-k
keywords dipolar recouplingmagic-angle spinningsolid-state NMRchemical shielding anisotropyoptimal controlquaternioneffective Hamiltoniancross polarization
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper claims that dipolar recoupling in magic-angle-spinning NMR can be made stable against large chemical-shift anisotropy by turning the two-spin design problem into a single-spin one: optimize the effective field that one radio-frequency channel exerts on its spin, with the anisotropic shielding explicitly moved into the interaction frame, and express rotations as quaternions for cheap gradient-based optimization. The payoff is a short, repeatable pulse element that produces a nearly constant effective field over a broad offset range and across crystallite orientations even when CSA is 80 kHz. In numerical two-spin simulations, sequences built from this element transfer 19F to 13C coherence more homogeneously than conventional CP and ramped CP across offset/CSA grids up to 200 kHz. Experimental measurements on octafluoronaphthalene (CSA about 70 kHz) reproduce the simulated profiles, supporting the claim that the design is predictive and modular: the second channel can be shaped independently once the first channel's field is stabilized.

Core claim

The central claim, stated on the authors' terms: including both isotropic and anisotropic shielding, together with RF irradiation, in the interaction-frame transformation makes the first-order effective Hamiltonian capture the dominant spin dynamics, so optimizing a single-spin effective field becomes sufficient to design a recoupling element. The optimized 80-microsecond, 40-step element (target rotation 120 degrees about x, effective field 4.17 kHz) maintains a stable effective field over isotropic offsets approaching ±100 kHz and CSA up to roughly 200 kHz, and two-spin simulations show that QOpt-based transfer is substantially less sensitive to offsets and CSA than conventional CP-based a

What carries the argument

The load-bearing mechanism is the combination of single-spin-vector effective Hamiltonian theory with quaternion algebra. Each piecewise-constant pulse interval is represented by a rotation quaternion built from RF amplitude, phase, offset, and CSA; the total pulse element is the quaternion product; the fidelity is the overlap between the total quaternion and a target rotation, averaged over crystallite orientations and offsets; and gradients are obtained analytically via the product rule. This turns pulse design into a compact algebraic optimization. The physical target is the resonance-matching condition between the two effective fields, ω_I_eff = ± ω_S_eff, and the design deliberately tar

Load-bearing premise

The load-bearing premise is that the two spins can be optimized independently because each spin's own interactions (RF, offset, CSA) dominate over the mutual dipolar coupling; if the dipolar coupling is not a small perturbation, the separately optimized effective field does not guarantee the partner-spin matching that transfer requires.

What would settle it

Run a two-spin simulation or experiment on a 19F–13C pair with CSA about 80 kHz but dipolar coupling reduced to a few kilohertz, and compare the QOpt transfer profile against CP-ramp across offsets: if the QOpt profile develops the same narrow matching holes as CP, the single-spin separation fails. Alternatively, in the octafluoronaphthalene system, lower the RF amplitude until the dipolar coupling becomes comparable to the linear terms and look for the predicted loss of homogeneity.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • QOpt-based transfer remains efficient over isotropic offset ranges approaching ±100 kHz and CSA up to about 200 kHz, whereas CP and ramped CP develop dead zones in the regime relevant to octafluoronaphthalene (CSA about 70 kHz).
  • Because the element is modular and repeatable, the same 19F pulse can be combined with square, ramped, or phase-toggled pulses on the 13C channel; toggled variants broaden the carbon-channel matching bandwidth without changing the fluorine-channel behavior.
  • The first-order effective-Hamiltonian description including CSA gives quantitatively predictive agreement with experiment, meaning the design strategy can be used to generate new sequences in silico.
  • The design principle generalizes: once one channel's effective field is optimized, the other channel can be tailored independently as long as the matching condition holds.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Editorial inference: the same single-spin, CSA-inclusive effective-field optimization could be applied to other anisotropic interactions, such as quadrupolar couplings, because the quaternion machinery does not depend on the interaction being a chemical shift.
  • Editorial inference: since the element is repeated to build the total mixing time, one could interleave two or more elements optimized for different offset windows to widen the effective bandwidth further; the paper does not test this.
  • Editorial inference: a testable extension is a weak-coupling limit experiment—if the dipolar coupling is reduced well below the linear single-spin terms, matching should degrade; such a measurement would separate the genuine bandwidth gain from the specific coupling strength in octafluoronaphthalene.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper presents a quaternion-based optimal-control framework for designing single-spin pulse elements that generate a prescribed effective field robust to isotropic offsets and chemical-shielding anisotropy, with the goal of improving heteronuclear dipolar recoupling (19F→13C) under MAS when the I-spin CSA is large. The optimization is performed in an interaction frame that explicitly includes the anisotropic shielding, uses analytical quaternion gradients, and targets a 120° rotation per 80 μs QOpt element. The element is characterized by single-spin effective-field analysis, evaluated in two-spin SIMPSON simulations over offset and CSA grids, and demonstrated experimentally on octafluoronaphthalene at 16.4 T with 25 kHz MAS. The paper's central claim is that the two spin channels can be designed independently once the effective-field matching condition is satisfied, and that the resulting QOpt-based sequences are substantially less sensitive to offsets and CSA than conventional CP-based approaches.

Significance. The quaternion formulation is clean, the analytical gradients are given, and the paper ships full pulse shapes and simulation parameters in the Supplementary Information, which is a strength. The exact SIMPSON simulations cover broad offset and CSA ranges, and one experimental demonstration on OFN is included. If the central design principle holds, the paper would make a useful methodological contribution to CSA-robust recoupling and to the SSV-EHT design philosophy. However, the current evidence establishes the principle for a single dipolar coupling strength only; the experimental validation is normalized and therefore not an absolute efficiency comparison; and there is an unresolved discrepancy between the RF amplitudes used in optimization/simulation and those used in the OFN experiments. These issues are load-bearing for the 'general design principle' stated in the Conclusions.

major comments (4)
  1. [§2.1, Eqs. (5)–(6), §3.2, Fig. 2] The central simplification is the separation of the two-spin problem into independent single-spin optimizations. The only justification is the sentence 'This is valid as long as the linear terms in Eq. (2) are larger than the bilinear terms in Eq. (3)' (§2.1). No expansion parameter, perturbation order, or convergence criterion for the SSV-EHT first-order effective Hamiltonian is given. This is not a formal quibble: the optimized element is designed for an effective field of ω_eff/(2π)=4.17 kHz (Section 4), while the only two-spin simulations and the OFN experiment use b_IS/(2π)=-12.1 kHz (§3.2). The linear term is therefore not obviously dominant in the effective-field frame. Because the Conclusions state a 'general design principle' that the second channel can be added independently once the first is optimized, the validity domain of this separation must be quantified. I ask for SIMPSO
  2. [§3.3, Fig. 3] The experimental validation is not a quantitative validation of the efficiency claim as it stands. All datasets are 'normalized to the transfer efficiency at zero offset' (§3.3); Fig. 3 therefore shows only relative offset profiles. The text concludes that the experiments 'confirmed the predicted improvement in recoupling performance,' but without absolute ε_FC values (or the zero-offset efficiencies before normalization) and without replicate error bars, the experiment cannot distinguish a genuinely better sequence from one whose apparent improvement is an artifact of normalization. Please report absolute transfer efficiencies and uncertainties.
  3. [§3.3, Tables S1/S2, Listing 1, Fig. 3] There is a direct mismatch between the pulse used in the optimization/simulations and the pulse used in the OFN experiment. The QOpt element was optimized and simulated with maximum RF amplitude 100 kHz (§3.1, Fig. 1B, Table S1), but the experimental QOpt-based sequences are listed with 19F RF amplitude 77.9 kHz (Table S2). The paper does not state whether the shape was rescaled, re-optimized, or recalibrated at 77.9 kHz. If it was merely scaled, the target 120° rotation and CSA-compensation properties used throughout the paper are no longer valid for the experimental pulse; if it was re-optimized, the new shape and parameters should be reported. This discrepancy must be resolved for the experimental demonstration to support the design claim.
  4. [§1, Refs. 25–27, Fig. 2] The closest prior optimal-control approach for heteronuclear recoupling in the presence of large anisotropic interactions, OPTIANS (Ref. 27), is cited but never compared. All two-spin comparisons are against conventional CP, ramped CP, and RESPIRATIONCP. Given that the paper's claimed contribution is an optimal-control design framework, omission of the state-of-the-art optimal-control baseline leaves it unclear whether the quaternion formulation improves on existing OC methods or is an alternative route to similar performance. Please include OPTIANS (or a closely related OC method) in the two-spin simulations, and ideally in the experiment, or state explicitly why it is not a meaningful comparator.
minor comments (5)
  1. [Eq. (5)] The Fourier coefficients a_p^{kI} and a_q^{kS} are introduced implicitly. Please state the ranges of kI and kS used in practice and comment on convergence of the expansion.
  2. [Fig. 1G caption] The phrase 'Solid lines link ... give corresponds' is grammatically broken and the schematic is difficult to follow at first reading; please revise the caption.
  3. [Fig. 3 caption] The caption should state explicitly that experimental and simulated curves are normalized to zero-offset efficiency; otherwise the reader may misinterpret the panels as absolute efficiencies.
  4. [Eq. (6), Table S1] The matching condition in Eq. (6) uses ω_I_eff = ω_S_eff, but the S-channel RF amplitudes in Table S1 (e.g., 16.5 kHz for QOpt) are not equal to the 4.17 kHz effective field of the QOpt element. Please clarify how modulation sidebands enter the matching condition; the current text leaves this ambiguous.
  5. [General] The names 'RESPIRATIONCP' and 'RESPIRATION-CP' are used inconsistently; also the abstract would benefit from stating that the experimental comparison is made at one coupling strength and one CSA value.

Circularity Check

0 steps flagged

No significant circularity: the QOpt pulse is produced by single-spin optimization and then validated by independent exact SIMPSON simulations and experiment.

full rationale

The paper's derivation chain is not circular. The QOpt element is obtained by numerically optimizing the single-spin quaternion fidelity in Eq. (10) over an offset/CSA grid; no two-spin transfer data or final recoupling efficiencies enter the objective. The two-spin transfer profiles in Figs. 2-3 are then computed with exact SIMPSON spin dynamics and confirmed experimentally on OFN, which are independent of the single-spin optimization target. The effective-field analysis in Fig. 1 is a post-hoc characterization of the optimized element, not a fitted input. Self-citations to SSV-EHT (refs. 28-30) and to prior DNP/recoupling work (refs. 37-38, 49-50) provide background formalism and design rationale, but the central claim that the optimized element enables robust heteronuclear recoupling is supported by external benchmarks (exact simulation and experiment), so the self-citations are not load-bearing in a circular way. The main weakness is the unquantified validity condition in §2.1, "This is valid as long as the linear terms in Eq. (2) are larger than the bilinear terms in Eq. (3)," and the fact that the two-spin validation uses a single heteronuclear coupling strength (b_IS/(2π) = −12.1 kHz). This is a generality/characterization gap, not a circular reduction: nothing in the derivation forces the two-spin transfer result to equal the single-spin objective by construction. The mixing times are optimized per experiment, but that is standard parameter adjustment and does not make the offset/CSA robustness profiles fitted predictions.

Axiom & Free-Parameter Ledger

7 free parameters · 6 axioms · 0 invented entities

The design and validation rest on standard NMR/MAS theory plus several domain assumptions about the dominance of single-spin linear terms and the validity of first-order effective Hamiltonians. The optimization also contains hand-chosen design parameters (target rotation, modulation time, RF limits, offset/CSA grids, per-experiment mixing times). No new physical entities are introduced.

free parameters (7)
  • Target rotation angle = 120° about x
    Hand-chosen to avoid 90/180/270/360 degree effective-field rotations and to set the target effective field (4.17 kHz) for the 80 µs modulation period.
  • Modulation time and discretization = 80 µs, 40×2 µs steps
    Chosen to synchronize with two rotor periods at 25 kHz MAS and to balance ZQ/DQ overlap, buildup time, and RF inhomogeneity.
  • Maximum RF amplitude constraint = ≤100 kHz
    Imposed instrument/practical limit in the optimization; constrains the reachable effective-field bandwidth.
  • Optimization offset grid = 70 kHz span, 7 points
    Defines the claimed isotropic-shift robustness band of the optimized pulse element.
  • Optimization CSA target = 80 kHz, η=0.5
    Representative/sample-motivated CSA magnitude used in the single-spin optimization and main simulations.
  • Powder averaging grid for optimization = 5 γ_CR × 10 (α,β) REPULSION
    Discretization coarser than the validation grids; affects the crystallite robustness actually achieved.
  • Per-experiment mixing times = Table S1/S2 values, e.g., QOpt 400 µs, QOpt-tog-ramp 2880 µs
    Mixing time for each pulse sequence was optimized at zero offset and CSA=80 kHz (0 for CP) and then held fixed; this is a per-method fitting step.
axioms (6)
  • domain assumption Average/effective Hamiltonian theory converges at first order for the pulse elements used (short modulation periods, fast MAS).
    Invoked in Theory (“first-order term efficiently catching dominant parts of the spin dynamics”) and throughout the SSV-EHT formulation; if higher-order terms contribute, the single-spin effective-field analysis may misrepresent two-spin dynamics.
  • domain assumption The linear single-spin terms dominate the bilinear dipolar terms during the pulse element.
    Stated in §2.1: “valid as long as the linear terms in Eq. (2) are larger than the bilinear terms in Eq. (3)”. This justifies reducing the two-spin design problem to independent single-spin optimizations.
  • domain assumption An isolated I–S two-spin system with CSA only on the I spin captures the relevant physics.
    Hamiltonian in Eqs. (1)–(4) and SIMPSON simulations assume a 19F–13C isolated pair with CSA and offset only on 19F; real OFN has multiple coupled spins and 13C CSA, so agreement relies on this simplification.
  • domain assumption MAS is ideal and RF pulses are piecewise-constant without transients or inhomogeneity in the optimization.
    The quaternion propagation (Eq. 8) and simulations assume ideal square pulses; the paper does not explicitly compensate for RF inhomogeneity/phase transients except through robustness grids.
  • domain assumption Quaternion rotation overlap fidelity (Eq. 10) is a valid proxy for heteronuclear polarization-transfer efficiency.
    The optimization maximizes single-spin rotation fidelity, not two-spin transfer; the link is the Hartmann-Hahn matching condition (Eq. 6) and stable effective fields, validated a posteriori by SIMPSON.
  • standard math Wigner rotation matrices and Fourier expansion of MAS-modulated interactions are standard and correct.
    Used in Eqs. (2)–(5); no derivation is provided and standard NMR background is assumed.

pith-pipeline@v1.3.0-alltime-deepseek · 21188 in / 13123 out tokens · 119115 ms · 2026-08-02T03:21:19.775434+00:00 · methodology

0 comments
read the original abstract

Dipolar recoupling is a key element in magic-angle-spinning (MAS) solid-state NMR spectroscopy with reintroduced dipole-dipole coupling interactions providing information about internuclear distances and enabling transfer of polarization between spins in resolution-enhancing multiple-dimensional experiments. Such methods may be challenged in many important applications by the presence of large anisotropic nuclear spin interactions such as chemical shielding anisotropy. In this paper, we address this challenge by presenting quaternion-based optimal control. This is founded in single-spin operations enabling optimization of effective Hamiltonians with reduced influences from anisotropic shielding. Along with the principles underlying such optimizations, we present numerical and experimental demonstration of 19F to 13C polarization transfer in presence of 19F chemical shielding anisotropy.

discussion (0)

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Reference graph

Works this paper leans on

50 extracted references

  1. [1]

    N.C.Nielsen,L.A.Strassø,A.B.Nielsen,DipolarRecoupling,inSolidStateNMR(Ed.Jerry C. C. Chan, Springer Berlin Heidelberg), pp. 1–45 (2012)

    2012

  2. [2]

    Y. Ji, L. Liang, X. Bao, G. Hou, Recent progress in dipolar recoupling techniques under fast MAS in solid-state NMR spectroscopy.Solid State Nuclear Magnetic Resonance112, 101711 (2021)

    2021

  3. [3]

    Chemical Reviews124(22), 12844–12917 (2024)

    V.Ladizhansky,R.S.Palani,M.Mardini,R.G.Griffin,DipolarRecouplinginRotatingSolids. Chemical Reviews124(22), 12844–12917 (2024)

    2024

  4. [4]

    Lasorsa, P

    A. Lasorsa, P. C. van Der Wel, Solid-state NMR protocols for unveiling dynamics and (drug) interactions of membrane-bound proteins.Protein Science34(4), e70102 (2025)

    2025

  5. [5]

    B. Reif, S. E. Ashbrook, L. Emsley, M. Hong, Solid-state NMR spectroscopy.Nature Reviews Methods Primers1(1), 2 (2021)

    2021

  6. [6]

    S.Ahlawat,K.R.Mote,N.-A.Lakomek,V.Agarwal,Solid-StateNMR:MethodsforBiological Solids.Chemical Reviews122(10), 9643–9737 (2022)

    2022

  7. [7]

    P. Gao, Y. Ji, G. Hou, Solid-state nuclear magnetic resonance spectroscopy for surface char- acterization of metal oxide nanoparticles: State of the art and perspectives.Journal of the American Chemical Society147(4), 2919–2937 (2025)

    2025

  8. [8]

    R.Zhang,W.Chen,T.Miyoshi,Elucidationsofstructureandmoleculardynamicsofcomplex polymersbystate-of-the-artsolid-stateNMRspectroscopy.Macromolecules57(5),1893–1918 (2024). 19

    1918

  9. [9]

    Haro Mares,et al., Solid-State NMR of Heterogeneous Catalysts.ChemCatChem16(22), e202401159 (2024)

    N. Haro Mares,et al., Solid-State NMR of Heterogeneous Catalysts.ChemCatChem16(22), e202401159 (2024)

    2024

  10. [10]

    Chien, K

    P.-H. Chien, K. J. Griffith, H. Liu, Z. Gan, Y.-Y. Hu, Recent Advances in Solid-State Nuclear MagneticResonanceTechniquesforMaterialsResearch.AnnualReviewofMaterialsResearch 50(Volume 50, 2020), 493–520 (2020)

    2020

  11. [11]

    Raleigh, M

    D. Raleigh, M. Levitt, R. Griffin, Rotational resonance in solid state NMR.Chemical Physics Letters146(1), 71–76 (1988)

    1988

  12. [12]

    Gullion, J

    T. Gullion, J. Schaefer, Rotational-echo double-resonance NMR.Journal of Magnetic Reso- nance81(1), 196–200 (1989)

    1989

  13. [13]

    Schaefer, R

    J. Schaefer, R. McKay, E. Stejskal, Double-cross-polarization NMR of solids.Journal of Magnetic Resonance34(2), 443–447 (1979)

    1979

  14. [14]

    N. C. Nielsen, H. Bildsøe, H. J. Jakobsen, M. H. Levitt, Double-quantum homonuclear rotary resonance:Efficientdipolarrecoveryinmagic-anglespinningnuclearmagneticresonance.The Journal of Chemical Physics101(3), 1805–1812 (1994)

    1994

  15. [15]

    Bennett, R

    A. Bennett, R. Griffin, J. Ok, S. Vega, Chemical shift correlation spectroscopy in rotating solids: Radio frequency-driven dipolar recoupling and longitudinal exchange.The Journal of Chemical Physics96(11), 8624–8627 (1992)

    1992

  16. [16]

    Y.Lee,etal.,EfficientdipolarrecouplingintheNMRofrotatingsolids.Asevenfoldsymmetric radiofrequency pulse sequence.Chemical Physics Letters242(3), 304–309 (1995)

    1995

  17. [17]

    M.Hohwy,H.J.Jakobsen,M.Edén,M.H.Levitt,N.C.Nielsen,Broadbanddipolarrecoupling in the nuclear magnetic resonance of rotating solids: A compensated C7 pulse sequence.The Journal of Chemical Physics108(7), 2686–2694 (1998)

    1998

  18. [18]

    Carravetta, M

    M. Carravetta, M. Eden, X. Zhao, A. Brinkmann, M. H. Levitt, Symmetry principles for the designofradiofrequencypulsesequencesinthenuclearmagneticresonanceofrotatingsolids. Chemical Physics Letters321(3-4), 205–215 (2000). 20

    2000

  19. [19]

    M. H. Levitt, Symmetry-based pulse sequences in magic-angle spinning solid-state NMR, in Encyclopedia of Nuclear Magnetic Resonance. Volume 9, Advances in NMR, D. M. Grant, R. K. Harris, Eds. (Wiley), pp. 165–196 (2002)

    2002

  20. [20]

    C. T. Kehlet,et al., Improving Solid-State NMR Dipolar Recoupling by Optimal Control. Journal of the American Chemical Society126(33), 10202–10203 (2004)

    2004

  21. [21]

    N. C. Nielsen, C. Kehlet, S. J. Glaser, N. Khaneja, Optimal Control Methods in NMR Spec- troscopy, eMagRes, John Wiley & Sons (2010)

    2010

  22. [22]

    Odgaard, M

    L. Odgaard, M. Bak, H. J. Jakobsen, N. C. Nielsen, 13C chemical shift and 13C–14N dipolar coupling tensors determined by 13C rotary resonance solid-state NMR.Journal of Magnetic Resonance148(2), 298–308 (2001)

    2001

  23. [23]

    De Paëpe, Dipolar recoupling in magic angle spinning solid-state nuclear magnetic reso- nance.Annual review of physical chemistry63(1), 661–684 (2012)

    G. De Paëpe, Dipolar recoupling in magic angle spinning solid-state nuclear magnetic reso- nance.Annual review of physical chemistry63(1), 661–684 (2012)

    2012

  24. [24]

    K. O. Tan, M. Rajeswari, P. K. Madhu, M. Ernst, Asynchronous symmetry-based sequences for homonuclear dipolar recoupling in solid-state nuclear magnetic resonance.The Journal of Chemical Physics142(6), 065101 (2015)

    2015

  25. [25]

    S. Wi, Z. Gan, R. Schurko, L. Frydman, Cross-polarization phenomena in the NMR of fast spinning solids subject to adiabatic sweeps.The Journal of Chemical Physics142(6) (2015)

    2015

  26. [26]

    T. Wolf, Y. Goobes, L. Frydman, Sensitivity Enhancement of Ultra-Wideline NMR by Pro- gressive Saturation of the Proton Reservoir Under Magic-Angle Spinning.ChemPhysChem 25(19), e202400613 (2024)

    2024

  27. [27]

    S. Ray, V. S. Redrouthu, A. Equbal, S. K. Jain, Optimal control-based nuclear spin cross- polarization in the presence of complicating anisotropic interactions.Phys. Chem. Chem. Phys.27, 7016–7027 (2025)

    2025

  28. [28]

    R. Shankar,et al., A general theoretical description of the influence of isotropic chemical shiftindipolarrecoupling experimentsforsolid-stateNMR.TheJournalofChemical Physics 146(13), 134105 (2017). 21

    2017

  29. [29]

    A. B. Nielsen, M. R. Hansen, J. E. Andersen, T. Vosegaard, Single-spin vector analysis of strongly coupled nuclei in TOCSY NMR experiments.The Journal of Chemical Physics 151(13), 134117 (2019)

    2019

  30. [30]

    A. B. Nielsen, N. C. Nielsen, Accurate analysis and perspectives for systematic design of magnetic resonance experiments using single-spin vector and exact effective Hamiltonian theory.Journal of Magnetic Resonance Open12-13, 100064 (2022)

    2022

  31. [31]

    S. W. R. Hamilton, On a new Species of Imaginary Quantities connected with a theory of Quaternions.Proceedings of the Royal Irish Academy2, 424–434 (1844)

  32. [32]

    Journal of Magnetic Resonance (1969)61(2), 356–362 (1985)

    B.Blümich,H.Spiess,Quaternionsasapracticaltoolfortheevaluationofcompositerotations. Journal of Magnetic Resonance (1969)61(2), 356–362 (1985)

    1969

  33. [33]

    N.Khaneja,T.Reiss,C.Kehlet,T.Schulte-Herbrüggen,S.J.Glaser,Optimalcontrolofcoupled spin dynamics: design of NMR pulse sequences by gradient ascent algorithms.Journal of Magnetic Resonance172(2), 296–305 (2005)

    2005

  34. [34]

    M. Bak, J. T. Rasmussen, N. C. Nielsen, SIMPSON: A General Simulation Program for Solid-State NMR Spectroscopy.Journal of Magnetic Resonance147(2), 296–330 (2000)

    2000

  35. [35]

    Caravatti, G

    P. Caravatti, G. Bodenhausen, R. Ernst, Selective pulse experiments in high-resolution solid state NMR.Journal of Magnetic Resonance (1969)55(1), 88–103 (1983)

    1969

  36. [36]

    A. J. Pell,et al., Broadband inversion for MAS NMR with single-sideband-selective adiabatic pulses.The Journal of Chemical Physics134(2), 024117 (2011)

    2011

  37. [37]

    Baligács, J

    E. Baligács, J. P. Carvalho, N. C. Nielsen, A. B. Nielsen, Accordion-like tuning of composite pulse dipolar recoupling in solid-state NMR.Journal of chemical physics164(12), 124201 (2026)

    2026

  38. [38]

    Carvalho, A

    J. Carvalho, A. B. Nielsen, E. Baligács, N. Wili, N. C. Nielsen, Bridging Dynamic Nuclear PolarizationandSolid-StateNMRDipolarRecoupling:FromStaticSingleCrystaltoSpinning Powders.The Journal of Physical Chemistry Letters16, 4363–4371 (2025). 22

    2025

  39. [39]

    Voight,Quaternion Algebras, vol

    J. Voight,Quaternion Algebras, vol. 288 ofGraduate Texts in Mathematics(Springer Interna- tional Publishing, Cham) (2021)

    2021

  40. [40]

    MATLAB Optimization Toolbox (2024), the MathWorks, Natick, MA, USA

    2024

  41. [41]

    M. Bak, N. C. Nielsen, REPULSION, A Novel Approach to Efficient Powder Averaging in Solid-State NMR.Journal of Magnetic Resonance125(1), 132–139 (1997)

    1997

  42. [42]

    D. L. Goodwin,et al., Extending numerical simulations in SIMPSON: Electron paramagnetic resonance,dynamicnuclearpolarisation,propagatorsplitting,pulsetransients,andquadrupolar cross terms.Journal of Magnetic Resonance Openp. 100218 (2026)

    2026

  43. [43]

    Stejskal, J

    E. Stejskal, J. Schaefer, J. Waugh, Magic-angle spinning and polarization transfer in proton- enhanced NMR.Journal of Magnetic Resonance (1969)28(1), 105–112 (1977)

    1969

  44. [44]

    G. Metz, X. Wu, S. Smith, Ramped-Amplitude Cross Polarization in Magic-Angle-Spinning NMR.Journal of Magnetic Resonance, Series A110(2), 219–227 (1994)

    1994

  45. [45]

    S. Jain, M. Bjerring, N. C. Nielsen, Efficient and robust heteronuclear cross-polarization for high-speed-spinning biological solid-state NMR spectroscopy.The journal of physical chemistry letters3(6), 703–708 (2012)

    2012

  46. [46]

    A. B. Nielsen, S. Jain, M. Ernst, B. H. Meier, N. C. Nielsen, Adiabatic Rotor-Echo-Short- Pulse-Irradiation mediated cross-polarization.Journal of Magnetic Resonance237, 147–151 (2013)

    2013

  47. [47]

    K.Basse,etal.,Handlingtheinfluenceofchemicalshiftinamplitude-modulatedheteronuclear dipolarrecouplingsolid-stateNMR.TheJournalofChemicalPhysics145(9),094202(2016)

    2016

  48. [48]

    A. B. Nielsen,et al., Theoretical description of RESPIRATION-CP.Chemical Physics Letters 645, 150–156 (2016)

    2016

  49. [49]

    A. B. Nielsen, J. P. A. Carvalho, D. L. Goodwin, N. Wili, N. C. Nielsen, Dynamic nuclear polarizationpulsesequenceengineeringusingsingle-spinvectoreffectiveHamiltonians.Phys. Chem. Chem. Phys.26, 28208–28219 (2024). 23

    2024

  50. [50]

    A. B. Nielsen,et al., Controlling effective Hamiltonians: Broadband pulsed dynamic nuclear polarizationbyconstrainedrandomwalkandnon-linearoptimization.TheJournalofChemical Physics163(14), 144111 (2025). Supplementary materials S1. Quaternion Formalism S2. Analytical Gradient S3. Chemical Shielding Anisotropy Fitting S4. Materials and methods S5. QOpt Pul...

    arXiv 2025