Spinor Double-Quantum Excitation in the Solution NMR of Near-Equivalent Spin-1/2 Pairs

TL;DR

This work develops and validates spinor-based double-quantum excitation methods for solution NMR in near-equivalent spin-1/2 pairs, achieving rapid DQ coherence by either geometric (GeoDQ) or spinor (PulsePol/SLIC) routes. The Spinor-DQ framework uses either symmetry-based PulsePol or SLIC, with a compensated variant cSLIC to mitigate rf-amplitude errors, offering performance well above the conventional INADEQUATE approach. Experimental demonstrations on diastereotopic 19F nuclei show substantial DQ-filtered signal enhancements (GeoDQ and PulsePol-DQ) and robust rf-error compensation (cSLIC-DQ), highlighting potential applications in diastereotopic proton and fluorine pairs and broader multi-spin systems. The results suggest practical routes to selective double-quantum filtering and enhanced spectral editing in complex biomolecules and materials where near-equivalence limits conventional methods. The methods hold promise for extending double-quantum techniques to more spins and enabling new sensing modalities in solution NMR.

Abstract

A family of double-quantum excitation schemes is described for the solution nuclear magnetic resonance (NMR) of near-equivalent spin-1/2 pairs. These new methods exploit the spinor behaviour of 2-level systems, whose signature is the change of sign of a quantum state upon a $2π$ rotation. The spinor behaviour is used to manipulate the phases of single-quantum coherences, in order to prepare a double-quantum precursor state which is rapidly converted into double-quantum coherence by a straightforward $π/2$ rotation. One set of spinor-based methods exploits symmetry-based pulse sequences, while the other set exploits SLIC (spin-lock-induced crossing), in which the nutation frequency under a resonant radiofrequency field is matched to the spin-spin coupling. A variant of SLIC is introduced which is well-compensated for deviations in the radiofrequency field amplitude. The methods are demonstrated by performing double-quantum-filtered $^{19}$F NMR on a molecular system containing a pair of diastereotopic $^{19}$F nuclei. The new methods are compared with existing techniques.

Figures (14)

• Figure 1: Double-quantum filtering amplitude for the INADEQUATE sequence of Equation \ref{['eq:INADQ']}. The square magnitude of the double-quantum excitation amplitude, $\vert a_\mathrm{DQ}^\mathrm{INADQ}\vert^2$, is plotted against the singlet-triplet mixing angle $\theta_\mathrm{ST}$ (horizontal) and the interpulse delays $\tau_1$ (vertical), using the expression in Equation \ref{['eq:aDQINADQ']}. The top axis shows the values of the ratio $\Delta/J$. Near-ideal double-quantum excitation is achieved for $\tau_1=(4J)^{-1}$ (horizontal dashed line), but only in the weakly-coupled regime $\theta_\mathrm{ST}\gtrsim70\degree$. Note the weak double-quantum excitation in the near-equivalence regime ($\theta_\mathrm{ST}\lesssim 30 \degree$).
• Figure 2: (a) Double-quantum-filtering pulse sequence, using PulsePol/$R4_3^1$ for double-quantum excitation, including a $z$-filtering step before signal acquisition. The phases $\Phi_\mathrm{A}$, $\Phi_\mathrm{B}$, and $\Phi_\mathrm{rec}$ are cycled in 4 steps to implement double-quantum filtering. (b) Structure of the $R4_3^1$ sequence, as shown in Equation \ref{['eq:R431']}.
• Figure 3: Double-quantum-filtering pulse sequence, using SLIC for double-quantum excitation. A $z$-filtering step is included before acquisition. The phases $\Phi_\mathrm{A}$, $\Phi_\mathrm{B}$ and $\Phi_\mathrm{rec}$ are cycled in 4 steps to implement double-quantum filtering. The SLIC element is applied with phase $\phi=0$, such that the SLIC field has the same phase as the subsequent $90\degree$ pulse. The compensated cSLIC sequence may be used as the SLIC element, as described in Section \ref{['sec:cSLIC']}.
• Figure 4: Double-quantum excitation schemes for spin-1/2 pairs. (a) The INADEQUATE three-pulse method. (b) The geometric double-quantum method. The element $R_z^{13}(\pi)$ indicates a $\pi$ rotation about the $z$-axis of the zero-quantum subspace, as described in Refs.bengs_aharonov_2023heramun_singlet_2025. (c) The PulsePol/symmetry-based
• Figure 5: Dependence of double-quantum-filtered signals on sequence duration $T$ for different excitation schemes: Analytical functions neglecting relaxation (blue curves), simulated points neglecting relaxation (open circles), and experimental data for the solution of I (solid black lines). (a): INADEQUATE (blue curve: $|a_\mathrm{DQ}^\mathrm{INADQ}(T)|^2$, where $a_\mathrm{DQ}^\mathrm{INADQ}(T)$ is given by equation \ref{['eq:aDQINADQ']}); (b): PulsePol-DQ sequence (blue curve: equation \ref{['eq:aDQFpulsepol']}); (c): SLIC-DQ sequence, using the implementation in Figure \ref{['fig:SLIC-DQ-sequences']}(b) (blue curve: equation \ref{['eq:aDQFSLICb']}); (d): cSLIC-DQ sequence (blue curve: equation \ref{['eq:aDQFcSLIC']}). All simulations use $J = 255.94$ Hz and $\Delta=17.8$ Hz. The experimentally optimised values of $T$ are shown by vertical dashed lines.