A general theoretical description of the influence of isotropic chemical shift in dipolar recoupling experiments for solid-state NMR

Bridging Dynamic Nuclear Polarization and Solid-State NMR Dipolar Recoupling: From Static Single Crystal to Spinning Powders

Spin engineering of advanced pulse sequences has had a transformative impact on the development of nuclear magnetic resonance (NMR) spectroscopy, to an extending degree also electron paramagnetic resonance (EPR), and the hybrid between the two, dynamic nuclear polarization (DNP). Based on a simple formalism, we demonstrate that (i) single-crystal static-sample optimizations may tremendously ease design and understanding of experiments for rotating powders and (ii) pulse sequences may readily be exchanged between these distinct spectroscopies. Specifically, we design broadband heteronuclear solid-state NMR magic-angle-spinning (MAS) dipolar recoupling experiments based on the recently developed PLATO (PoLarizAtion Transfer via nonlinear Optimization) microwave (MW) pulse sequence optimized on a single crystal for powder static-sample DNP. Using this concept, we demonstrate design of ultrabroadband 13C-15N and 2H-13C cross-polarization experiments, using PLATO on the 13C radio frequency (RF) channel and square/ramped or RESPIRATION (Rotor Echo Short Pulse IRrAdiaTION) RF irradiation on the 15N and 2H RF channels, respectively.

Optimal control design strategies for pulsed dynamic nuclear polarization

We present optimal control methods for the optimization of periodic pulsed dynamic nuclear polarization (DNP) sequences. Specifically, we address the challenge of the optimization of a basic and repeated pulse sequence element which, apart from being easily adaptable to spin systems with different coupling interaction sizes, also proves beneficial in terms of performance. It is demonstrated that matrix power and matrix logarithm functions combined with an auxiliary matrix formalism can be used to derive expressions for gradient ascent pulse engineering (GRAPE) optimization. We illustrate how different implementations provide effective and intuitive control of DNP experiments by tailoring the effective Hamiltonian governing polarization transfer and, in this manner, addressing some of the limitations of prevailing optimal control based pulse design strategies.

Dipolar Recoupling in Rotating Solids

Magic angle spinning (MAS) nuclear magnetic resonance (NMR) has evolved significantly over the past three decades and established itself as a vital tool for the structural analysis of biological macromolecules and materials. This review delves into the development and application of dipolar recoupling techniques in MAS NMR, which are crucial for obtaining detailed structural and dynamic information. We discuss a variety of homonuclear and heteronuclear recoupling methods which are essential for measuring spatial restraints and explain in detail the spin dynamics that these sequences generate. We also explore recent developments in high spinning frequency MAS, proton detection, and dynamic nuclear polarization, underscoring their importance in advancing biomolecular NMR. Our aim is to provide a comprehensive account of contemporary dipolar recoupling methods, their principles, and their application to structural biology and materials, highlighting significant contributions to the field and emerging techniques that enhance resolution and sensitivity in MAS NMR spectroscopy.

Dynamic nuclear polarization pulse sequence engineering using single-spin vector effective Hamiltonians

Dynamic nuclear polarization (DNP) has proven to be a powerful technique to enhance nuclear spin polarization by transferring the much higher electron spin polarization to nuclear spins prior to detection. While major attention has been devoted to high-field applications with continuous microwave irradiation, the introduction of fast arbitrary waveform generators is gradually increasing opportunities for the realization of pulsed DNP. Here, we describe how static-powder DNP pulse sequences may systematically be designed using single-spin vector effective Hamiltonian theory. Particular attention is devoted to the intricate interplay between two important parts of the effective first-order Hamiltonian, namely, linear field (single-spin) terms and Fourier coefficients determining scaling of the bilinear coupling terms mediating polarization transfer. We address two cases. The first case operates in the regime, where the microwave field amplitude is lower than the nuclear Larmor frequency. Here, we illustrate the predictive strength of a single-spin vector model by comparing analytical calculations with experimental DNP results at 9.8 GHz/15 MHz on trityl radicals at 80 K. The second case operates in the high-power regime, where we combine the underlying single-spin vector design principles with numerical non-linear optimization to optimize the balance between the linear terms and the bilinear Fourier coefficients in a figure of merit function. We demonstrate, numerically and experimentally, a broadband DNP pulse sequence PLATO (PoLarizAtion Transfer via non-linear Optimization) with a bandwidth of 80 MHz and optimized for a microwave field with a maximum (peak) amplitude of 32 MHz.

Homonuclear J-couplings and heteronuclear structural constraints

In magic angle spinning (MAS) experiments involving uniformly 13C/15N labeled proteins, 13C-13C and 13C-15N dipolar recoupling experiments are now routinely used to measure direct dipole-dipole couplings that constrain distances and torsion angles and determine molecular structures. When the distances are short (<4 Å), the direct couplings dominate the evolution of the spin system, and the 13C-13C and 13C-15N J-couplings (scalar couplings) are ignored. However, for structurally interesting >4 Å distances, the dipolar and J-couplings are generally of comparable magnitude, and the variation in J must be included in order to optimize the precision of the experiment. This problem is circumvented in cases with well resolved spectra by using frequency-selective dipolar recoupling methods where the effects of J-couplings are refocused. However, for larger molecules with more spectral crowding, the requisite pulse length to achieve selectivity becomes long and leads to unacceptable sensitivity losses during the pulse or the spectral overlap precludes selective excitation. In this paper, we address this problem with two approaches aimed at facilitating higher precision internuclear distance measurements in systems that are not fully resolved. Namely, (1) we describe an approach for high precision measurements of specific J-couplings using the in-phase anti-phase (IPAP) sequence which is integrated into a non-selective dipolar recoupling technique and (2) we utilize the measured J-couplings to implement a double quantum filter experiment capable of providing the resolution necessary for frequency selective dipolar recoupling techniques without resorting to multidimensional spectroscopy. We illustrate these methods using a 7-peptide segment from the amyloidogenic Sup-35p protein, U-13C/15N-GNNQQNY, where we have measured 25 of the 27 possible one bond 13C-13C J-couplings.

Observation of dynamic nuclear polarization echoes

It is demonstrated that the time evolution of the electron-nuclear polarization transfer process during pulsed dynamic nuclear polarization (DNP) can be reversed on a microsecond timescale, leading to the observation of DNP echoes. The DNP echoes are induced by consecutive application of two pulse trains that produce effective Hamiltonians that differ only in the sign of the effective hyperfine coupling. The experiments have been performed on a frozen solution of trityl radicals in water/glycerol on a homebuilt X-band electron paramagnetic resonance/DNP spectrometer at 80 kelvins. We envisage that DNP echoes will play an important role in future development of pulsed DNP for sensitivity-enhanced nuclear magnetic resonance, hyperfine spectroscopy, and quantum sensing.

Continuous Floquet theory in solid-state NMR

This article presents the application of continuous Floquet theory in solid-state nuclear magnetic resonance (NMR). Continuous Floquet theory extends the traditional Floquet theory to non-continuous Hamiltonians, enabling the description of observable effects not fully captured by the traditional Floquet theory due to its requirement for a periodic Hamiltonian. We present closed-form expressions for computing first- and second-order effective Hamiltonians, streamlining integration with the traditional Floquet theory and facilitating application in NMR experiments featuring multiple modulation frequencies. Subsequently, we show examples of the practical application of continuous Floquet theory by investigating several solid-state NMR experiments. These examples illustrate the importance of the duration of the pulse scheme regarding the width of the resonance conditions and the near-resonance behavior.

Residual proton line width under refocused frequency-switched Lee-Goldburg decoupling in MAS NMR

Despite many decades of research, homonuclear decoupling in solid-state NMR under magic-angle spinning (MAS) has yet to reach a point where the achievable proton line widths become comparable to the resolution obtained in solution-state NMR. This makes the precise determination of isotropic chemical shifts difficult and thus presents a limiting factor in the application of proton solid-state NMR to biomolecules and small molecules. In this publication we analyze the sources of the residual line width in refocused homonuclear-decoupled spectra in detail by comparing numerical simulations and experimental data. Using a hybrid analytical/numerical approach based on Floquet theory, we find that third-order effective Hamiltonian terms are required to realistically characterize the line shape and line width under frequency-switched Lee-Goldburg (FSLG) decoupling under MAS. Increasing the radio-frequency field amplitude enhances the influence of experimental rf imperfections such as pulse transients and the MAS-modulated radial rf-field inhomogeneity. While second- and third-order terms are, as expected, reduced in size at higher rf-field amplitudes, the line width becomes dominated by first-order terms which severely limits the achievable line width. We expect, therefore, that significant improvements in the line width of FSLG-decoupled spectra can only be achieved by reducing the influence of MAS-modulated rf-field inhomogeneity and pulse transients.

Interaction frames in solid-state NMR: A case study for chemical-shift-selective irradiation schemes

Interaction frames play an important role in describing and understanding experimental schemes in magnetic resonance. They are often used to eliminate dominating parts of the spin Hamiltonian, e.g., the Zeeman Hamiltonian in the usual (Zeeman) rotating frame, or the radio-frequency-field (rf) Hamiltonian to describe the efficiency of decoupling or recoupling sequences. Going into an interaction frame can also make parts of a time-dependent Hamiltonian time independent like the rf-field Hamiltonian in the usual (Zeeman) rotating frame. Eliminating the dominant term often allows a better understanding of the details of the spin dynamics. Going into an interaction frame can also reduces the energy-level splitting in the Hamiltonian leading to a faster convergence of perturbation expansions, average Hamiltonian, or Floquet theory. Often, there is no obvious choice of the interaction frame to use but some can be more convenient than others. Using the example of frequency-selective dipolar recoupling, we discuss the differences, advantages, and disadvantages of different choices of interaction frames. They always include the complete radio-frequency Hamiltonian but can also contain the chemical shifts of the spins and may or may not contain the effective fields over one cycle of the pulse sequence.

Designing broadband pulsed dynamic nuclear polarization sequences in static solids

Dynamic nuclear polarization (DNP) is a nuclear magnetic resonance (NMR) hyperpolarization technique that mediates polarization transfer from unpaired electrons with large thermal polarization to NMR-active nuclei via microwave (mw) irradiation. The ability to generate arbitrarily shaped mw pulses using arbitrary waveform generators allows for remarkable improvement of the robustness and versatility of DNP. We present here novel design principles based on single-spin vector effective Hamiltonian theory to develop new broadband DNP pulse sequences, namely, an adiabatic version of XiX (X-inverse X)-DNP and a broadband excitation by amplitude modulation (BEAM)-DNP experiment. We demonstrate that the adiabatic BEAM-DNP pulse sequence may achieve a ¹H enhancement factor of ∼360, which is better than ramped-amplitude NOVEL (nuclear spin orientation via electron spin locking) at ∼0.35 T and 80 K in static solids doped with trityl radicals. In addition, the bandwidth of the BEAM-DNP experiments (~50 MHz) is about three times the ¹H Larmor frequency. The generality of our theoretical approach will be helpful in the development of new pulsed DNP sequences.

Efficient Pulsed Dynamic Nuclear Polarization with the X-Inverse-X Sequence

Pulsed dynamic nuclear polarization (DNP) is a promising new approach to enhancing the sensitivity of high-resolution magic-angle spinning (MAS) NMR. In pulsed DNP, the transfer of polarization from unpaired electrons to nuclei (usually ¹H) is induced by a sequence of microwave pulses. Enhancement factors of the thermal ¹H polarization are expected to be independent of the magnetic field, and sample heating by absorption of microwave irradiation will be strongly reduced. The development of DNP pulse sequences is still in its infancy. Of the two basic sequences in existence, NOVEL and TOP DNP, the former is, due to an extremely high power requirement, incompatible with high-resolution MAS NMR, while the latter displays a relatively slow transfer of polarization from electrons to ¹H. We introduce here a new pulse sequence for DNP of solids, termed X-inverse-X (XiX) DNP. In experiments at 1.2 T, XiX DNP produces, compared to TOP DNP, a 2-fold higher gain in sensitivity. Our data suggest that a faster transfer of polarization from electrons to ¹H is behind the superior performance of XiX DNP. Numerical simulations and experiments indicate that microwave pulse lengths can be chosen across a broad range, without loss of efficiency. These findings are a substantial step toward the implementation of pulsed DNP at high magnetic fields.

Accuracy of 1H-1H distances measured using frequency selective recoupling and fast magic-angle spinning

Selective recoupling of protons (SERP) is a method to selectively and quantitatively measure magnetic dipole-dipole interaction between protons and, in turn, the proton-proton distance in solid-state samples at fast magic-angle spinning. We present a bimodal operator-based Floquet approach to describe the numerically optimized SERP recoupling sequence. The description calculates the allowed terms in the first-order effective Hamiltonian, explains the origin of selectivity during recoupling, and shows how different terms are modulated as a function of the radio frequency amplitude and the phase of the sequence. Analytical and numerical simulations have been used to evaluate the effect of higher-order terms and offsets on the polarization transfer efficiency and quantitative distance measurement. The experimentally measured ¹H-¹H distances on a fully protonated thymol sample are ∼10%-15% shorter than those reported from diffraction studies. A semi-quantitative model combined with extensive numerical simulations is used to rationalize the effect of the third-spin and the role of different parameters in the experimentally observed shorter distances. Measurements at high magnetic fields improve the match between experimental and diffraction distances. The measurement of ¹H-¹H couplings at offsets different from the SERP-offset has also been explored. Experiments were also performed on a perdeuterated ubiquitin sample to demonstrate the feasibility of simultaneously measuring multiple quantitative distances and to evaluate the accuracy of the measured distance in the absence of multispin effects. The estimation of proton-proton distances provides a boost to structural characterization of small pharmaceuticals and biomolecules, given that the positions of protons are generally not well defined in x-ray structures.

Time-optimized pulsed dynamic nuclear polarization

Pulsed dynamic nuclear polarization (DNP) techniques can accomplish electron-nuclear polarization transfer efficiently with an enhancement factor that is independent of the Zeeman field. However, they often require large Rabi frequencies and, therefore, high-power microwave irradiation. Here, we propose a new low-power DNP sequence for static samples that is composed of a train of microwave pulses of length τ_p spaced with delays d. A particularly robust DNP condition using a period τ_m = τ_p + d set to ~1.25 times the Larmor period τ_Larmor is investigated which is a time-optimized pulsed DNP sequence (TOP-DNP). At 0.35 T, we obtained an enhancement of ~200 using TOP-DNP compared to ~172 with nuclear spin orientation via electron spin locking (NOVEL), a commonly used pulsed DNP sequence, while using only ~7% microwave power required for NOVEL. Experimental data and simulations at higher fields suggest a field-independent enhancement factor, as predicted by the effective Hamiltonian.