Optimal control in NMR spectroscopy: numerical implementation in SIMPSON

Red Shift in the Absorption Spectrum of Phototropin LOV1 upon the Formation of a Semiquinone Radical: Reconstructing the Orbital Architecture

Flavin mononucleotide (FMN) is a ubiquitous blue-light pigment due to its ability to drive one- and two-electron transfer reactions. In both light-oxygen-voltage (LOV) domains of phototropin from the green algae Chlamydomonas reinhardtii, FMN is noncovalently bound. In the LOV1 cysteine-to-serine mutant (C57S), light-induced electron transfer from a nearby tryptophan occurs, and a transient spin-correlated radical pair (SCRP) is formed. Within this photocycle, nuclear hyperpolarization is created by the solid-state photochemically induced dynamic nuclear polarization (photo-CIDNP) effect. In a side reaction, a stable protonated semiquinone radical (FMNH·) forms undergoing a significant bathochromic shift of the first electronic transition from 445 to 591 nm. The incorporation of phototropin LOV1-C57S into an amorphous trehalose matrix, stabilizing the radical, allows for application of various magnetic resonance experiments at ambient temperatures, which are combined with quantum-chemical calculations. As a result, the bathochromic shift of the first absorption band is explained by lifting the degeneracy of the molecular orbital energy levels for electrons with alpha and beta spins in FMNH· due to the additional electron.

Controlling NMR spin systems for quantum computation

Nuclear magnetic resonance is arguably both the best available quantum technology for implementing simple quantum computing experiments and the worst technology for building large scale quantum computers that has ever been seriously put forward. After a few years of rapid growth, leading to an implementation of Shor's quantum factoring algorithm in a seven-spin system, the field started to reach its natural limits and further progress became challenging. Rather than pursuing more complex algorithms on larger systems, interest has now largely moved into developing techniques for the precise and efficient manipulation of spin states with the aim of developing methods that can be applied in other more scalable technologies and within conventional NMR. However, the user friendliness of NMR implementations means that they remain popular for proof-of-principle demonstrations of simple quantum information protocols.

High-efficiency low-power 13C-15N cross polarization in MAS NMR

Biomolecular solid-state magic angle spinning (MAS) NMR spectroscopy frequently relies on selective 13C-15N magnetization transfers, for various kinds of correlation experiments. Introduced in 1998, spectrally induced filtering in combination with cross polarization (SPECIFIC-CP) is a selective heteronuclear magnetization transfer experiment widely used for biological applications. At MAS frequencies below 20 kHz, commonly used for 13C-detected MAS NMR experiments, SPECIFIC-CP transfer between amide 15N and 13Cα atoms (NCA) is typically performed with radiofrequency (rf) fields set higher than the MAS frequency for both 13C and 15N channels, and high-power 1H decoupling rf field is simultaneously applied. Here, we experimentally explore a broad range of NCA zero-quantum (ZQ) SPECIFIC-CP matching conditions at the MAS frequency of 14 kHz and compare the best high- and low-power matching conditions with respect to selectivity, robustness, and sensitivity at lower 1H decoupling rf fields. We show that low-power NCA SPECIFIC-CP matching condition gives rise to 20% sensitivity enhancement compared to high-power conditions, in 2D NCA spectra of microcrystalline assemblies of HIV-1 CACTD-SP1 protein with inositol hexakis-phosphate (IP6).

Selectively Probing the Magnetic Resonance Signals of γ-Aminobutyric Acid in Human Brains In Vivo

BACKGROUND

Gamma-aminobutyric acid (GABA) is an inhibitory neurotransmitter in human brains, playing a role in the pathogenesis of various psychiatric disorders. Current methods have some non-neglectable shortcomings and noninvasive and accurate detection of GABA in human brains is long-term challenge.

PURPOSE

To develop a pulse sequence capable of selectively detecting and quantifying the 1 H signal of GABA in human brains based on optimal controlled spin singlet order.

STUDY TYPE

Prospective.

SUBJECTS/PHANTOM

A phantom of GABA (pH = 7.3 ± 0.1) and 11 healthy subjects (5 females and 6 males, body mass index: 21 ± 3 kg/m2 , age: 25 ± 4 years).

FIELD STRENGTH/SEQUENCE

7 Tesla, 3 Tesla, GABA-targeted magnetic resonance spectroscopy (GABA-MRS-7 T, GABA-MRS-3 T), magnetization prepared two rapid acquisition gradient echoes sequence.

ASSESSMENT

By using the developed pulse sequences applied on the phantom and healthy subjects, the signals of GABA were successfully selectively probed. Quantification of the signals yields the concentration of GABA in the dorsal anterior cingulate cortex (dACC) in human brains.

STATISTICAL TESTS

Frequency.

RESULTS

The 1 H signals of GABA in the phantom and in the human brains of healthy subjects were successfully detected. The concentration of GABA in the dACC of human brains was 3.3 ± 1.5 mM.

DATA CONCLUSION

The developed pulse sequences can be used to selectively probe the 1 H MR signals of GABA in human brains in vivo.

EVIDENCE LEVEL

1 TECHNICAL EFFICACY STAGE: 1.

Accelerated acquisition of wideline solid-state NMR spectra of spin 3/2 nuclei by frequency-stepped indirect detection experiments

73% of all NMR-active nuclei are quadrupolar nuclei with a nuclear spin I > 1/2. The broadening of the solid-state NMR signals by the quadrupolar interaction often leads to poor sensitivity and low resolution. In this work we present experimental and theoretical investigations of magic angle spinning (MAS) 1H{X} double-echo resonance-echo saturation-pulse double-resonance (DE-RESPDOR) and Y{X} J-resolved solid-state NMR experiments for the indirect detection of spin 3/2 quadrupolar nuclei (X = spin 3/2 nuclei, Y = spin 1/2 nuclei). In these experiments, the spectrum of the quadrupolar nucleus is reconstructed by plotting the observed dephasing of the detected spin as a function of the transmitter offset of the indirectly detected spin. Numerical simulations were used to investigate the achievable levels of dephasing and to predict the lineshapes of indirectly detected NMR spectra of the quadrupolar nucleus. We demonstrate 1H, 31P and 207Pb detection of 35Cl, 81Br, and 63Cu (I = 3/2) nuclei in trans-Cl2Pt(NH3)2 (transplatin), (CH3NH3)PbCl3 (methylammonium lead chloride, MAPbCl3), (CH3NH3)PbBr3 (methylammonium lead bromide, MAPbBr3) and CH3C(CH2PPh2)3CuI (1,1,1-tris(diphenylphosphinomethyl)ethane copper(I) iodide, triphosCuI), respectively. In all of these experiments, we were able to detect megahertz wide central transition or satellite transition powder patterns. Significant time savings and gains in sensitivity were attained in several test cases. Additionally, the indirect detection experiments provide valuable structural information because they confirm the presence of dipolar or scalar couplings between the detected nucleus and the quadrupolar nucleus of interest. Finally, numerical simulations suggest these methods are also potentially applicable to abundant spin 5/2 and spin 7/2 quadrupolar nuclei.

Pushing the Detection Limit of Static Wideline NMR Spectroscopy Using Ultrafast Frequency-Swept Pulses

We report a simple design strategy for wideband uniform-rate smooth truncation (WURST) pulses that enables ultrafast frequency sweeps to maximize the sensitivity of Carr-Purcell-Meiboom-Gill (CPMG) acquisition in static wideline nuclear magnetic resonance (NMR). Three compelling examples showcase the advantage of ultrafast frequency sweeps over currently employed WURST-CPMG protocols, demonstrating the potential of investigating materials that are typically inaccessible to static wideline NMR techniques, e.g., paramagnetic solids with short homogeneous transverse relaxation times.

Low-field NMR with multilayer Halbach magnet and NMR selective excitation

This study introduces a low-field NMR spectrometer (LF-NMR) featuring a multilayer Halbach magnet supported by a combined mechanical and electrical shimming system. This setup offers improved field homogeneity and sensitivity compared to spectrometers relying on typical Halbach and dipole magnets. The multilayer Halbach magnet was designed and assembled using three nested cylindrical magnets, with an additional inner Halbach layer that can be rotated for mechanical shimming. The coils and shim-kernel of the electrical shimming system were constructed and coated with layers of zirconia, thermal epoxy, and silver-paste resin to facilitate passive heat dissipation and ensure mechanical and thermal stability. Furthermore, the 7-channel shim coils were divided into two parts connected in parallel, resulting in a reduction of joule heating temperatures from 96.2 to 32.6 °C. Without the shimming system, the Halbach magnet exhibits a field inhomogeneity of approximately 140 ppm over the sample volume. The probehead was designed to incorporate a solenoidal mini coil, integrated into a single planar board. This design choice aimed to enhance sensitivity, minimize [Formula: see text] inhomogeneity, and reduce impedance discrepancies, transmission loss, and signal reflections. Consequently, the resulting linewidth of water within a 3 mm length and 2.4 mm inner diameter sample volume was 4.5 Hz. To demonstrate the effectiveness of spectral editing in LF-NMR applications at 29.934 MHz, we selectively excited hydroxyl and/or methyl protons in neat acetic acid using optimal control pulses calculated through the Krotov algorithm.

Optimal control pulses for the 1.2-GHz (28.2-T) NMR spectrometers

The ability to measure nuclear magnetic resonance (NMR) spectra with a large sample volume is crucial for concentration-limited biological samples to attain adequate signal-to-noise (S/N) ratio. The possibility to measure with a 5-mm cryoprobe is currently absent at the 1.2-GHz NMR instruments due to the exceedingly high radio frequency power demands, which is four times compared to 600-MHz instruments. Here, we overcome the high-power demands by designing optimal control (OC) pulses with up to 20 times lower power requirements than currently necessary at a 1.2-GHz spectrometer. We show that multidimensional biomolecular NMR experiments constructed using these OC pulses can bestow improvement in the S/N ratio of up to 26%. With the expected power limitations of a 5-mm cryoprobe, we observe an enhancement in the S/N ratio of more than 240% using our OC sequences. This motivates the development of a cryoprobe with a larger volume than the current 3-mm cryoprobes.

Simulation and design of shaped pulses beyond the piecewise-constant approximation

Response functions of resonant circuits create ringing artefacts if their input changes rapidly. When physical limits of electromagnetic spectroscopies are explored, this creates two types of problems. Firstly, simulation: the system must be propagated accurately through every response transient, this may be computationally expensive. Secondly, optimal control: circuit response must be taken into account; it may be advantageous to design pulses that are resilient to such distortions. At the root of both problems is the popular piecewise-constant approximation for control sequences in the rotating frame; in magnetic resonance it has persisted since the earliest days and has become entrenched in the commercially available hardware. In this paper, we report an implementation and benchmarks of recent Lie-group methods that can efficiently simulate and optimise smooth control sequences.

Optimal control flow encoding for time-efficient magnetic resonance velocimetry

Phase contrast velocimetry relies on bipolar gradients to establish a direct and linear relationship between the phase of the magnetic resonance signal, and the corresponding fluid motion. Despite its utility, several limitations and drawbacks have been reported, the most important being the extended echo time due to the encoding after the excitation. In this study, we elucidate a new approach based on optimal control theory that circumvents some of these disadvantages. An excitation pulse, termed FAUCET (flow analysis under controlled encoding transients), is designed to encode velocity into phase already during the radiofrequency excitation. As a result of concurrent excitation and flow encoding, and hence elimination of post-excitation flow encoding, FAUCET achieves a shorter echo time than the conventional method. This achievement is a matter of significance not only because it decreases the loss of signal due to spin-spin relaxation and B₀ inhomogeneity, but also because a shorter echo time is always preferred in order to reduce the dimensionless dephasing parameter and the required residence time of the flowing sample in the detection coil. The method is able to establish a non-linear bijective relationship between phase and velocity, which can be employed to enhance the resolution over a specific range of velocities, for example along flow boundaries. A computational comparison between the phase contrast and optimal control methods reveals that the latter's encoding is more robust against remnant higher-order-moment terms of the Taylor expansion for faster voxels, such as acceleration, jerk, and snap.

Distinguishing glutamate and glutamine in in vivo 1 H MRS based on nuclear spin singlet order filtering

PURPOSE

The signals of glutamate (Glu) and glutamine (Gln) are often significantly overlapped in routine ¹ H-MR spectra of human brain in vivo. Selectively probing the signals of Glu and Gln in vivo is very important for the study of the metabolisms in which Glu and Gln are involved.

METHODS

The Glu-/Gln- targeted pulse sequences are developed to selectively probe the signals of Glu and Gln. The core part of the Glu-/Gln- targeted pulse sequences lies on the preparation of the nuclear spin singlet orders (SSOs) of the five-spin systems of Glu and Gln. The optimal control method is used to prepare the SSOs of Glu and Gln with high efficiency.

RESULTS

The Glu-/Gln- targeted pulse sequences have been applied on phantoms to selectively probe the signals of Glu and Gln. Moreover, in the in vivo experiments, the signals of Glu and Gln in human brains of healthy subjects have been successfully probed separately.

CONCLUSION

The developed Glu-/Gln- targeted pulse sequences can be used to distinguish the ¹ H-MR signals of Glu and Gln in human brains in vivo. The optimal control method provides an effective way to prepare the SSO of a specific spin system with high efficiency and in turn selectively probe the signals of a targeted molecule.

Comparison of methods for 14N-1H recoupling in 14N-1H HMQC MAS NMR

¹H-detected ¹⁴N heteronuclear multiple-quantum coherence (HMQC) magic-angle-spinning (MAS) NMR experiments performed at fast magic-angle spinning (≥50 kHz) are finding increasing application, e.g., to pharmaceuticals. Of importance to the efficacy of these techniques is the recoupling technique applied to reintroduce the ¹H-¹⁴N dipolar coupling. In this paper, we compare, by experiment and 2-spin density matrix simulations, two classes of recoupling scheme: first, those based on n = 2 rotary resonance, namely R³ and spin-polarisation inversion SPI-R³, and the symmetry based SR4₁² method and, second, the TRAPDOR method. Both classes require optimisation depending on the magnitude of the quadrupolar interaction, and thus there is a compromise choice for samples with more than one nitrogen site, as is the case for the studied dipeptide β-AspAla that contains two nitrogen sites with a small and large quadrupolar coupling constant. Considering this, we observe better sensitivity for the TRAPDOR method, though noting the marked sensitivity of TRAPDOR to the ¹⁴N transmitter offset, with both SPI-R³ and SR4₁² giving similar recoupling performance.

Acceleration of indirect detection 195Pt solid-state NMR experiments by sideband selective excitation or alternative indirect sampling schemes

The measurement of the of chemical shift (CS) tensors via solid-state NMR (ssNMR) spectroscopy has proven to be a powerful probe of structure for organic molecules, biomolecules, and inorganic materials. However, when measuring the NMR spectra of heavy spin-1/2 isotopes the chemical shift anisotropy (CSA) is commonly on the order of thousands of parts per million, which makes acquisition of NMR spectra difficult due to the low NMR sensitivity imposed by the breadth of the signals and challenges in uniformly exciting the NMR spectrum. We have recently shown that complete ¹⁹⁵Pt NMR spectra could be rapidly measured by using ¹⁹⁵Pt saturation or excitation selective long pulses (SLP) with multiple rotor-cycle durations and RF fields less than 50 kHz into ¹H{¹⁹⁵Pt} or ¹H-³¹P{¹⁹⁵Pt} PE S-RESPDOR, TONE D-HMQC-4, J-resolved, and J-HMQC pulse sequences. The SLP only provide signal or dephasing when they are applied on resonance with a spinning sideband. The magic angle spinning ¹⁹⁵Pt NMR spectrum is reconstructed in the sideband selective NMR experiments by acquiring 1D NMR spectra at variable ¹⁹⁵Pt pulse offsets. In this work, we present a detailed investigation of the specific pulse conditions required for the ideal performance of sideband selective experiments. Sideband selective experiments are shown to be able to accurately reproduce MAS NMR spectra with minimal distortions of relative sideband intensities. It is also demonstrated that a ¹⁹⁵Pt NMR spectrum indirectly detected with HMQC can be rapidly obtained by acquiring a single rotor cycle of indirect dimension evolution points. We dub this method One Rotor Cycle of Acquisition (ORCA) HMQC. Sideband selective experiments and ORCA HMQC experiments are shown to provide a one order of magnitude improvement in experiment times as compared to conventional wideline HMQC experiments.

Sensitivity-Enhanced Multidimensional Solid-State NMR Spectroscopy by Optimal-Control-Based Transverse Mixing Sequences

Recently, proton-detected magic-angle spinning (MAS) solid-state nuclear magnetic resonance (NMR) spectroscopy has become an attractive tool to study the structure and dynamics of insoluble proteins at atomic resolution. The sensitivity of the employed multidimensional experiments can be systematically improved when both transversal components of the magnetization are transferred simultaneously after an evolution period. The method of preservation of equivalent pathways has been explored in solution-state NMR; however, it does not find widespread application due to relaxation issues connected with increased molecular size. We present here for the first time heteronuclear transverse mixing sequences for correlation experiments at moderate and fast MAS frequencies. Optimal control allows to boost the signal-to-noise ratio (SNR) beyond the expected factor of 2 for each indirect dimension. In addition to the carbon-detected sensitivity-enhanced 2D NCA experiment, we present a novel proton-detected, doubly sensitivity-enhanced 3D hCANH pulse sequence for which we observe a 3-fold improvement in SNR compared to the conventional experimental implementation. The sensitivity gain turned out to be essential to unambiguously characterize a minor fibril polymorph of a human lambda-III immunoglobulin light chain protein that escaped detection so far.

Molecular and Electronic Structure of Isolated Platinum Sites Enabled by the Expedient Measurement of 195Pt Chemical Shift Anisotropy

Techniques that can characterize the molecular structures of dilute surface species are required to facilitate the rational synthesis and improvement of Pt-based heterogeneous catalysts. ¹⁹⁵Pt solid-state NMR spectroscopy could be an ideal tool for this task because ¹⁹⁵Pt isotropic chemical shifts and chemical shift anisotropy (CSA) are highly sensitive probes of the local chemical environment and electronic structure. However, the characterization of Pt surface-sites is complicated by the typical low Pt loadings that are between 0.2 and 5 wt% and broadening of ¹⁹⁵Pt solid-state NMR spectra by CSA. Here, we introduce a set of solid-state NMR methods that exploit fast MAS and indirect detection using a sensitive spy nucleus (¹H or ³¹P) to enable the rapid acquisition of ¹⁹⁵Pt MAS NMR spectra. We demonstrate that high-resolution wideline ¹⁹⁵Pt MAS NMR spectra can be acquired in minutes to a few hours for a series of molecular and single-site Pt species grafted on silica with Pt loading of only 3-5 wt%. Low-power, long-duration, sideband-selective excitation, and saturation pulses are incorporated into t₁-noise eliminated dipolar heteronuclear multiple quantum coherence, perfect echo resonance echo saturation pulse double resonance, or J-resolved pulse sequences. The complete ¹⁹⁵Pt MAS NMR spectrum is then reconstructed by recording a series of 1D NMR spectra where the offset of the ¹⁹⁵Pt pulses is varied in increments of the MAS frequency. Analysis of the ¹⁹⁵Pt MAS NMR spectra yields the ¹⁹⁵Pt chemical shift tensor parameters. Zeroth order approximation density functional theory calculations accurately predict ¹⁹⁵Pt CS tensor parameters. Simple and predictive orbital models relate the CS tensor parameters to the Pt electronic structure and coordination environment. The methodology developed here paves the way for the detailed structural and electronic analysis of dilute platinum surface-sites.

Genetic algorithm-based optimization of pulse sequences

PURPOSE

The performance of pulse sequences in vivo can be limited by fast relaxation rates, magnetic field inhomogeneity, and nonuniform spin excitation. We describe here a method for pulse sequence optimization that uses a stochastic numerical solver that in principle is capable of finding a global optimum. The method provides a simple framework for incorporating any constraint and implementing arbitrarily complex cost functions. Efficient methods for simulating spin dynamics and incorporating frequency selectivity are also described.

METHODS

Optimized pulse sequences for polarization transfer between protons and X-nuclei and excitation pulses that eliminate J-coupling modulation were evaluated experimentally using a surface coil on phantoms, and also the detection of hyperpolarized [2-13 C]lactate in vivo in the case of J-coupling modulation-free excitation.

RESULTS

The optimized polarization transfer pulses improved the SNR by ~50% with a more than twofold reduction in the B1 field, and J-coupling modulation-free excitation was achieved with a more than threefold reduction in pulse length.

CONCLUSION

This process could be used to optimize any pulse when there is a need to improve the uniformity and frequency selectivity of excitation as well as to design new pulses to steer the spin system to any desired achievable state.

Multiple-targeting NMR signal selection by optimal control of nuclear spin singlet

Selectively probing specific molecules in complex mixtures with nuclear magnetic resonance promises new insights into molecular structures or molecular interaction. Such a study often can be further facilitated when two or more objects in chemical moieties of interest can be precisely targeted. Herein, we proposed a novel method to implement the multiple-targeting signal selection by optimal control of the spin singlets of two or more targeted spin systems from one or more molecules. This method can endow the conventional nuclear magnetic resonance (NMR), magnetic resonance image (MRI) and magnetic resonance spectrum (MRS) with the multiple-targeting signal selectivity to selectively probe several targeted molecules and/or chemical groups simultaneously.

Double echo symmetry-based REDOR and RESPDOR pulse sequences for proton detected measurements of heteronuclear dipolar coupling constants

¹H{X} symmetry-based rotational echo double resonance pulse sequences (S-REDOR) and symmetry-based rotational echo saturation pulse double resonance (S-RESPDOR) solid-state NMR experiments have found widespread application for ¹H detected measurements of difference NMR spectra, dipolar coupling constants, and internuclear distances under conditions of fast magic angle spinning (MAS). In these experiments the supercycled R4₁² (SR4₁²) symmetry-based recoupling pulse sequence is typically applied to the ¹H spins to reintroduce heteronuclear dipolar couplings. However, the timing of SR4₁² and other symmetry-based pulse sequences must be precisely synchronized with the rotation of the sample, otherwise, the evolution of ¹H CSA and other interactions will not be properly refocused. For this reason, significant distortions are often observed in experimental dipolar dephasing difference curves obtained with S-REDOR or S-RESPDOR pulse sequences. Here we introduce a family of double echo (DE) S-REDOR/S-RESPDOR pulse sequences that function in an analogous manner to the recently introduced t₁-noise eliminated (TONE) family of dipolar heteronuclear multiple quantum coherence (D-HMQC) pulse sequences. Through numerical simulations and experiments the DE S-REDOR/S-RESPDOR sequences are shown to provide dephasing difference curves similar to those obtained with S-REDOR/S-RESPDOR. However, the DE sequences are more robust to the deviations of the MAS frequency from the ideal value that occurs during typical solid-state NMR experiments. The DE sequences are shown to provide more reliable ¹H detected dipolar dephasing difference curves for nuclei such as ¹⁵N (with isotopic labelling), ¹⁸³W and ³⁵Cl. The double echo sequences are therefore recommended to be used in place of conventional S-REDOR/S-RESPDOR sequences for measurement of weak dipolar coupling constants and long-range distances.

Maximizing efficiency of dipolar recoupling in solid-state NMR using optimal control sequences

Dipolar recoupling is a central concept in the nuclear magnetic resonance spectroscopy of powdered solids and is used to establish correlations between different nuclei by magnetization transfer. The efficiency of conventional cross-polarization methods is low because of the inherent radio frequency (rf) field inhomogeneity present in the magic angle spinning (MAS) experiments and the large chemical shift anisotropies at high magnetic fields. Very high transfer efficiencies can be obtained using optimal control–derived experiments. These sequences had to be optimized individually for a particular MAS frequency. We show that by adjusting the length and the rf field amplitude of the shaped pulse synchronously with sample rotation, optimal control sequences can be successfully applied over a range of MAS frequencies without the need of reoptimization. This feature greatly enhances their applicability on spectrometers operating at differing external fields where the MAS frequency needs to be adjusted to avoid detrimental resonance effects.

Frequency-Swept Ultra-Wideline Magic-Angle Spinning NMR Spectroscopy

Recent years have witnessed the development of solid-state NMR techniques that allow the direct investigation of extremely wide inhomogeneously broadened resonance lines. To date, this typically involves the application of frequency sweeps as offered by wideband uniform rate smooth truncation (WURST) pulses. While the effects of such advanced irradiation schemes on static samples are well understood, the interference between the varying carrier frequency and the time-dependent evolution of the spin system under magic-angle spinning (MAS) conditions is more complex. Herein, we introduce the well-known WURST-Carr-Purcell-Meiboom-Gill (WCPMG) pulse sequence for spinning samples. Using numerical spin-density matrix analysis, an ideal design based on fast frequency sweeps and high truncation of the incorporated WURST pulses is presented that enables uniform excitation/refocusing under MAS conditions with low-to-moderate radio-frequency power requirements. This permits the acquisition of ultra-wideline MAS NMR lines exceeding 500 kHz with chemical shift resolution in a single transmitter step.

Proton-detected solid-state NMR spectroscopy of spin-1/2 nuclei with large chemical shift anisotropy

Constant-time (CT) dipolar heteronuclear multiple quantum coherence (D-HMQC) has previously been demonstrated as a method for proton detection of high-resolution wideline NMR spectra of spin-1/2 nuclei with large chemical shift anisotropy (CSA). However, ¹H transverse relaxation and t₁-noise often reduce the sensitivity of D-HMQC experiments, preventing the theoretical gains in sensitivity provided by ¹H detection from being realized. Here we demonstrate a series of improved pulse sequences for ¹H detection of spin-1/2 nuclei under fast MAS, with ¹⁹⁵Pt SSNMR experiments on cisplatin as an example. First, a t₁-incrementation protocol for D-HMQC dubbed Arbitrary Indirect Dwell (AID) is demonstrated. AID allows the use of arbitrary, rotor asynchronous t₁-increments, but removes the constant time period from CT D-HMQC, resulting in improved sensitivity by reducing transverse relaxation losses. Next, we show that short high-power adiabatic pulses (SHAPs), which efficiently invert broad MAS sideband manifolds, can be effectively incorporated into ¹H detected symmetry-based resonance echo double resonance (S-REDOR) and t₁-noise eliminated (TONE) D-HMQC experiments. The S-REDOR experiments with SHAPs provide approximately double the dipolar dephasing, as compared to experiments with rectangular inversion pulses. We lastly show that sensitivity and resolution can be further enhanced with the use of swept excitation pulses as well as adiabatic magic angle turning (aMAT).

Optimization of phase dispersion enables broadband excitation without homonuclear coupling artifacts

In NMR spectroscopy, many specialized shaped pulses are available for broadband excitation, beyond the bandwidth of conventional high-powered hard pulses. These shaped pulses typically have long duration. However, long-duration pulses are unsuitable for spectra containing significant homonuclear couplings, such as polyfluorinated compounds in 19F NMR. J-coupling evolution during the excitation pulse leads to spectral artifacts and incorrect peak integrals. Here, we report an approach to optimal control pulse design which significantly reduces the pulse length required to excite large bandwidths of chemical shift frequencies. The target state phase is not chosen beforehand but is instead only constrained to be linearly dependent on offset frequency. The first-order phase of the target state is then treated as a free-variable, to be optimized at the same time as the RF waveform itself. The resulting spectra are easily phased using standard NMR processing software. We observe that the required pulse length is significantly shorter than for currently available in-phase excitation schemes. Spectral artifacts from homonuclear couplings are avoided. We also demonstrate that pure in-phase excitation can be obtained over the same bandwidth by appending two inversion pulses, at the expense of increased overall duration.

Solid-state NMR spectroscopy

Solid-state nuclear magnetic resonance (NMR) spectroscopy is an atomic-level method used to determine the chemical structure, three-dimensional structure, and dynamics of solids and semi-solids. This Primer summarizes the basic principles of NMR as applied to the wide range of solid systems. The fundamental nuclear spin interactions and the effects of magnetic fields and radiofrequency pulses on nuclear spins are the same as in liquid-state NMR. However, because of the anisotropy of the interactions in the solid state, the majority of high-resolution solid-state NMR spectra is measured under magic-angle spinning (MAS), which has profound effects on the types of radiofrequency pulse sequences required to extract structural and dynamical information. We describe the most common MAS NMR experiments and data analysis approaches for investigating biological macromolecules, organic materials, and inorganic solids. Continuing development of sensitivity-enhancement approaches, including 1H-detected fast MAS experiments, dynamic nuclear polarization, and experiments tailored to ultrahigh magnetic fields, is described. We highlight recent applications of solid-state NMR to biological and materials chemistry. The Primer ends with a discussion of current limitations of NMR to study solids, and points to future avenues of development to further enhance the capabilities of this sophisticated spectroscopy for new applications.

Sensitivity-enhanced multiple-quantum MAS NMR for half-integer spin quadrupolar nuclei using WURST-amplitude shaped pulses

Two-dimensional multiple-quantum MAS (MQMAS) NMR spectroscopy is one of the most widely used solid-state NMR techniques for resolving multiple overlapping central-transition lineshapes for half-integer spin quadrupolar nuclei. In particular when relying on nutation-driven MQ coherence transfers, this technique suffers from low sensitivity that can only be improved by increasing the rf-amplitude of the involved radio-frequency (rf) pulses, which are therefore typically operated at the rf-limit. In such situations, frequently encountered for the three-pulse z-filtered and split-t₁ shifted-echo MQMAS NMR sequences, we introduce the advantages of rf-pulses with smoothly truncated amplitude profiles, which we have termed WURST-Amplitude Shaped Pulses (WASPs). When considering the NMR spectrometer hardware, we demonstrate that WASPs feature more suitable properties in comparison to conventional rectangular pulses, enabling a substantial reduction of voltage reflections and transient effects under identical rf-conditions. By employing extensive numerical simulations and experimental validation, we further show that WASPs intrinsically possess a higher potential for nutation-based 3Q excitation involving spin-3/2 and 3Q and 5Q excitation for spin-5/2 quadrupolar nuclei, specifically when large nutation frequencies are available. The concept of smoothly truncating rf-amplitudes is also extended to Fast Amplitude Modulation (FAM) pulses, normally incorporated for rotor-driven 1Q conversion. We additionally evaluate the potential of employing WASPs with peak rf-amplitudes beyond the rf-limit for conventional rectangular rf-pulses.

Effective Hamiltonian and 1H-14N cross polarization/double cross polarization at fast MAS

In this work we investigate in detail the underlying spin-dynamics associated with ¹H-¹⁴N double CP experiments under fast MAS, recently demonstrated by Carnevale et al. We employ matrix logarithm and Floquet theory to compute numerically the effective Hamiltonian associated to the time-dependent problem. Certain common features related to construction of effective Hamiltonians by both approaches are discussed. The main observations related to ¹H-¹⁴N CPMAS/double CP transfer are: (a) various spin terms of the effective Hamiltonian strongly depend on the crystallite orientation; (b) significant CP transfer occurs only when the magnitudes of the effective¹H and ¹⁴N RF strengths are comparable, and simultaneously all pure ¹⁴N terms in the effective Hamiltonian are small, except for the longitudinal and the RF terms; (c) the sign of ¹⁴N CPMAS signal follows the sign of ¹⁴N effective RF strength; (d) sign of the double CP signal is largely independent of crystallite orientation. We predict and verify matching conditions employing multiples of the spinning frequency or involving different ¹⁴N RF strengths. We provide an analytical proof for (d). The proof also provides an estimate for the ratio of ¹H-¹⁴N and ¹⁴N-¹H transfer amplitudes which is further substantiated through simulations. In addition, we find that double CP signals include contributions from several single-quantum coherences present after the first CP process. The uneven contribution from different coherences leads to a reversal of signal at very short contact times, a feature noted experimentally by Carnevale et al. The connection between CPMAS transfer and efficient spin-lock is discussed and illustrated. The factors affecting second-order quadrupolar lineshapes in double CP experiment are examined. With a linear ramp of ¹H RF amplitude we have observed that significant CP transfer occurs for more crystallite orientations resulting in improved sensitivity.

Preparation of Long-Lived States in a Multi-Spin System by Using an Optimal Control Method

The lifetime Ts of a long-lived nuclear spin state (LLS) could be much longer than the longitudinal order T₁ . Many spin systems were used to produce long-lived states, including two or more homonuclear spins that couple to each other. For multiple homonuclear spins with rather small chemical shift difference, normally it is difficult to selectively control the spins and then to prepare a LLS. Herein, we present a scheme that prepares different spin orders in a multi-spin system by using optimal control and numerical calculation. By experimentally measuring the lifetime of the states, we find that for a three-spin physical system, although there are many forms of state combinations with different spin orders, each component has its own lifetime.

Suppressing 1H Spin Diffusion in Fast MAS Proton Detected Heteronuclear Correlation Solid-State NMR Experiments

Fast magic angle spinning (MAS) and indirect detection by high gyromagnetic ratio (γ) nuclei such as proton or fluorine are increasingly utilized to obtain 2D heteronuclear correlation (HETCOR) solid-state NMR spectra of spin-1/2 nuclei by using cross polarization (CP) for coherence transfer. However, one major drawback of CP HETCOR pulse sequences is that ¹H spin diffusion during the back X→¹H CP transfer step may result in relayed correlations. This problem is particularly pronounced for the indirect detection of very low-γ nuclei such as ⁸⁹Y, ¹⁰³Rh, ¹⁰⁹Ag and ¹⁸³W where long contact times on the order of 10-30 ms are necessary for optimal CP transfer. Here we propose two methods that eliminate relayed correlations and allow more reliable distance information to be obtained from 2D HETCOR NMR spectra. The first method uses Lee-Goldburg (LG) CP during the X→¹H back-transfer step to suppress ¹H spin diffusion. We determine LG conditions compatible with fast MAS frequencies (ν_rot) of 40-95 kHz and show that ¹H spin diffusion can be efficiently suppressed at low effective radiofrequency (RF) fields (ν_1,eff ≪ 0.5ν_rot) and also at high effective RF fields (ν_1,eff ≫ 2ν_rot). We describe modified Hartmann-Hahn LG-CP match conditions compatible with fast MAS and suitable for indirect detection of moderate-γ nuclei such as ¹³C, and low-γ nuclei such as ⁸⁹Y. The second method uses D-RINEPT (dipolar refocused insensitive nuclei enhanced by polarization transfer) during the X→¹H back-transfer step of the HETCOR pulse sequence. The effectiveness of these methods for acquiring HETCOR spectra with reduced relayed signal intensities is demonstrated with ¹H{¹³C} HETCOR NMR experiments on l-histidine⋅HCl⋅H₂O and ¹H{⁸⁹Y} HETCOR NMR experiments on an organometallic yttrium complex.

t1-Noise eliminated dipolar heteronuclear multiple-quantum coherence solid-state NMR spectroscopy

t₁-Noise eliminated (TONE) heteronuclear multiple quantum correlation (HMQC) solid-state nuclear magnetic resonance pulse sequences improve the sensitivity of 2D ¹H{X} heteronuclear correlation experiments with X = ¹⁷O, ²⁵Mg, ²⁷Al and ³⁵Cl.

On the use of frequency-swept pulses and pulses designed with optimal control theory for the acquisition of ultra-wideline NMR spectra

Frequency-swept (FS) pulses, such as wideband uniform-rate smooth-truncation (WURST) pulses, have found much success for the acquisition of ultra-wideline (UW) solid-state NMR spectra. In this preliminary study, new pulses and pulse sequences are explored in simulation and experimentally for several nuclei exhibiting UWNMR powder patterns under static conditions, including ¹¹⁹Sn (I = 1/2), ¹⁹⁵Pt (I = 1/2), ²H (I = 1), and ⁷¹Ga (I = 3/2). First, hyperbolic secant (HS) and tanh/tan (THT) pulses are tested and implemented as excitation and refocusing pulses in spin-echo and Carr-Purcell/Meiboom Gill (CPMG)-type sequences, and shown to have comparable performances to analogous WURST pulses. Second, optimal control theory (OCT) is utilized for the design of new Optimal Control Theory Optimized Broadband Excitation and Refocusing (OCTOBER) pulses, using carefully parameterized WURST, THT, and HS pulses as starting points. Some of the new OCTOBER pulses used in spin-echo sequences are capable of efficient broadband excitation and refocusing, in some cases resulting in spectra with increased signal enhancements over those obtained in experiments using conventional FS pulses. Finally, careful consideration of the spin dynamics of several systems, by monitoring of the time evolution of the density matrix via the Liouville-von Neumann equation and analysis of the time-resolved Fourier transforms of the pulses, lends insight into the underlying mechanisms of the FS and OCTOBER pulses. This is crucial for understanding their performance in terms of generating uniformly excited patterns of high signal intensity, and for identifying trends that may offer pathways to generalized parameterization and/or new pulse shapes.

Probing the Surface Structure of Semiconductor Nanoparticles by DNP SENS with Dielectric Support Materials

Surface characterization is crucial for understanding how the atomic-level structure affects the chemical and photophysical properties of semiconducting nanoparticles (NPs). Solid-state nuclear magnetic resonance spectroscopy (NMR) is potentially a powerful technique for the characterization of the surface of NPs, but it is hindered by poor sensitivity. Dynamic nuclear polarization surface enhanced NMR spectroscopy (DNP SENS) has previously been demonstrated to enhance the sensitivity of surface-selective solid-state NMR experiments by 1-2 orders of magnitude. Established sample preparations for DNP SENS experiments on NPs require the dilution of the NPs on mesoporous silica. Using hexagonal boron nitride (h-BN) to disperse the NPs doubles DNP enhancements and absolute sensitivity in comparison to standard protocols with mesoporous silica. Alternatively, precipitating the NPs as powders, mixing them with h-BN, and then impregnating the powdered mixture with radical solution leads to further 4-fold sensitivity enhancements by increasing the concentration of NPs in the final sample. This modified procedure provides a factor of 9 improvement in NMR sensitivity in comparison to previously established DNP SENS procedures, enabling challenging homonuclear and heteronuclear 2D NMR experiments on CdS, Si, and Cd₃P₂ NPs. These experiments allow NMR signals from the surface, subsurface, and core sites to be observed and assigned. For example, we demonstrate the acquisition of DNP-enhanced 2D ¹¹³Cd-¹¹³Cd correlation NMR experiments on CdS NPs and natural isotropic abundance 2D ¹³C-²⁹Si HETCOR of functionalized Si NPs. These experiments provide a critical understanding of NP surface structures.

Optimization of band-selective homonuclear dipolar recoupling in solid-state NMR by a numerical phase search

Spin polarization transfers among aliphatic ¹³C nuclei, especially ¹³Cα-¹³Cβ transfers, permit correlations of their nuclear magnetic resonance (NMR) frequencies that are essential for signal assignments in multidimensional solid-state NMR of proteins. We derive and demonstrate a new radio-frequency (RF) excitation sequence for homonuclear dipolar recoupling that enhances spin polarization transfers among aliphatic ¹³C nuclei at moderate magic-angle spinning (MAS) frequencies. The phase-optimized recoupling sequence with five π pulses per MAS rotation period (denoted as PR5) is derived initially from systematic numerical simulations in which only the RF phases are varied. Subsequent theoretical analysis by average Hamiltonian theory explains the favorable properties of numerically optimized phase schemes. The high efficiency of spin polarization transfers in simulations is preserved in experiments, in part because the RF field amplitude in PR5 is only 2.5 times the MAS frequency so that relatively low ¹H decoupling powers are required. Experiments on a microcrystalline sample of the β1 immunoglobulin binding domain of protein G demonstrate an average enhancement factor of 1.6 for ¹³Cα → ¹³Cβ polarization transfers, compared to the standard ¹³C-¹³C spin-diffusion method, implying a two-fold time saving in relevant 2D and 3D experiments.

Spin-Scenario: A flexible scripting environment for realistic MR simulations

In this paper we present a new open source package, Spin-Scenario, aimed at developing an intuitive, flexible and unique scripting framework able to cover many aspects of simulations in both MR imaging and MR spectroscopy. For this purpose, we adopted the Liouville space model as the standard computing engine and let the consequent computational burden be afforded by parallel computing techniques. Benefitting from the powerful Lua scripting language, the pulse sequence programming syntax was specially designed to offer an extremely concise way of scripting. Moreover, the built-in dataflow graph based optimal control scheme enables an efficient optimization of shaped pulses or multiple cooperative pulses for real-life experiment evaluations. As the name states, the users are expected to be able to realize their creative ideas like a scenarist that creates a scenario script and looks at the spin actors acting accordingly. The validation of the framework was demonstrated with several examples within MR imaging and MR spectroscopy. Spin-Scenario is available for download at https://github.com/spin-scenario.

Optimal control theory enables homonuclear decoupling without Bloch-Siegert shifts in NMR spectroscopy

The Bloch-Siegert shift is a phenomenon in NMR spectroscopy and atomic physics in which the observed resonance frequency is changed by the presence of an off-resonance applied field. In NMR, it occurs especially in the context of homonuclear decoupling. Here we develop a practical method for homonuclear decoupling that avoids inducing Bloch-Siegert shifts. This approach enables accurate observation of the resonance frequencies of decoupled nuclear spins. We apply this method to increase the resolution of the HNCA experiment. We also observe a doubling in sensitivity for a 30 kDa protein. We demonstrate the use of band-selective Cβ decoupling to produce amino acid-specific line shapes, which are valuable for assigning resonances to the protein sequence. Finally, we assign the backbone of a 30 kDa protein, Human Carbonic Anhydrase II, using only HNCA experiments acquired with band-selective decoupling schemes, and instrument time of one week.

Quantitative description of 1H SQ and DQ coherences for the hydroxyl disorder within hydrous ringwoodite

Proton-containing point defects in solid materials are important for a variety of properties ranging from ionic transport over thermal conductivity up to compressibility. Ultrafast magic-angle spinning techniques nowadays offer high-resolution solid-state NMR spectra, even for 1H, and thus open up possibilities to study the underlying defect chemistry. Nevertheless, disorder within such defects again leads to heavy spectral overlap of 1H resonances, which prevents quantitative analysis of defect concentrations, if several defect types are present. Here, we present a strategy to overcome this limitation by simulating the 1H lineshape as well as 1H-1H double-quantum buildup curves, which we then validate against the experimental data in a joint cost function. To mimic the local structural disorder, we use molecular dynamics simulations at the DFT level. It turned out to be advantageous for the joint refinement to put the computational effort into the structural optimisation to derive accurate proton positions and to use empirical correlations for the relation between isotropic and anisotropic 1H chemical shifts and structural elements. The expressiveness of this approach is demonstrated on ringwoodite's (γ-Mg2SiO4) OH defect chemistry containing four different defect types in octahedral and tetrahedral voids with both pure Mg and mixed Si and Mg cation environments. Still, we determine the ratio for each defect type with an accuracy of about 5% as a result of the minimization of the joint cost function. We expect that our approach is generally applicable for local proton disorder and might prove to be a valuable alternative to the established AIRSS and Monte Carlo methods, respectively.

Fast simulations of multidimensional NMR spectra of proteins and peptides

To simulate full multidimensional nuclear magnetic resonance spectra of peptides and proteins in a reasonable time frame, a strategy for separating the time-consuming full-density matrix calculations from the chemical shift prediction and calculation of coupling patterns is presented. The simulation setup uses SIMulation Program for SOlid-state NMR (SIMPSON) to calculate total correlation spectroscopy transfer amplitudes and average distances as a source for nuclear Overhauser effect spectroscopy transfer amplitudes. Simulated ¹ H 1D, 2D total correlation spectroscopy, and 2D nuclear Overhauser effect spectroscopy nuclear magnetic resonance spectra of peptides with sequence Pro─Ala─Gly─Tyr─Asn and Asn─Phe─Gly─Ala─Ile─Leu and of ubiquitin are presented. In all cases, the simulations lasted from a few seconds to tens of seconds on a normal laptop computer.

Bang-bang optimal control of large spin systems: Enhancement of 13C-13C singlet-order at natural abundance

Using a bang-bang optimal control technique, we transfer polarization from a set of abundant high-γ nuclei directly to singlet order of a low-γ spin-pair. This approach is analogous to algorithmic cooling (AC) procedure used in quantum state purification. Specifically, we apply this method for enhancing the singlet order in a natural abundant ¹³C- ¹³C spin pair by exploiting nine equivalent protons of an 11-spin system. Compared to the standard method not involving polarization transfer, we find an enhancement of singlet order by about 3.4 times. In addition, since the singlet magnetization is contributed by the faster relaxing protons, the recycle delay is halved. Thus effectively we observe a reduction in the overall experimental time by a factor of 23. We also discuss a possible extension of AC, known as heat-bath algorithmic cooling (HBAC).

Radiofrequency fields in MAS solid state NMR probes

We present a detailed analysis of the radiofrequency (RF) field over full volume of a rotor that is generated in a solenoid coil. On top of the usually considered static distribution of amplitudes along the coil axis we describe dynamic radial RF inhomogeneities induced by sample rotation. During magic angle spinning (MAS), the mechanical rotation of the sample about the magic angle, a spin packet travels through areas of different RF fields and experiences periodical modulations of both the RF amplitude and the phase. These modulations become particularly severe at the end regions of the coil where the relative RF amplitude varies up to ±25% and the RF phase changes within ±30°. Using extensive numerical simulations we demonstrate effects of RF inhomogeneity on pulse calibration and for the ramped CP experiment performed at a wide range of MAS rates. In addition, we review various methods to map RF fields using a B₀ gradient along the sample (rotor axis) for imaging purposes. Under such a gradient, a nutation experiment provides directly the RF amplitude distribution, a cross polarization experiment images the correlation of the RF fields on the two channels according to the Hartmann-Hahn matching condition, while a spin-lock experiment allows to calibrate the RF amplitude employing the rotary resonance recoupling condition. Knowledge of the RF field distribution in a coil provides key to understand its effects on performance of a pulse sequence at the spectrometer and enables to set robustness requirements in the experimental design.

An optimal control approach to design entire relaxation dispersion experiments

A general approach is introduced to optimize experiments for the analysis of spin systems in the presence of chemical exchange. Rather than optimizing individual pulse sequence elements, such as refocusing pulses, entire relaxation dispersion sequences are optimized in the form of a single shaped pulse. This is achieved by defining a performance index that is only based on the remaining signal after the relaxation dispersion sequence for a range of exchange, relaxation, offset, and rf inhomogeneity parameters. The approach is demonstrated by optimizing energy-limited broadband relaxation dispersion sequences that closely approach the overall effect of ideal CPMG sequences. As illustrated both theoretically and experimentally, significant improvements are found compared to standard amplitude or energy-limited CPMG sequences.

Rapid convergence of optimal control in NMR using numerically-constructed toggling frames

We present a numerical method for rapidly solving the Bloch equation for an arbitrary time-varying spin-1/2 Hamiltonian. The method relies on fast, vectorized computations such as summation and quaternion multiplication, rather than slow computations such as matrix exponentiation. A toggling frame is constructed in which the Hamiltonian is time-invariant, and therefore has a simple analytical solution. The key insight is that constructing this frame is faster than solving the system dynamics in the original frame. Rapidly solving the Bloch equations for an arbitrary Hamiltonian is particularly useful in the context of NMR optimal control. Optimal control theory can be used to design pulse shapes for a range of tasks in NMR spectroscopy. However, it requires multiple simulations of the Bloch equations at each stage of the algorithm, and for each relevant set of parameters (e.g. chemical shift frequencies). This is typically time consuming. We demonstrate that by working in an appropriate toggling frame, optimal control pulses can be generated much faster. We present a new alternative to the well-known GRAPE algorithm to continuously update the toggling-frame as the optimal pulse is generated, and demonstrate that this approach is extremely fast. The use and benefit of rapid optimal pulse generation is demonstrated for 19F fragment screening experiments.

Hidden Oceans? Unraveling the Structure of Hydrous Defects in the Earth's Deep Interior

High-pressure silicates making up the main proportion of the earth's interior can incorporate a significant amount of water in the form of OH defects. Generally, they are charge balanced by removing low-valent cations such as Mg²⁺. By combining high-resolution multidimensional single- and double-quantum ¹H solid-state NMR spectroscopy with density functional theory calculations, we show that, for ringwoodite (γ-Mg₂SiO₄), additionally, Si⁴⁺ vacancies are formed, even at a water content as low as 0.1 wt %. They are charge balanced by either four protons or one Mg²⁺ and two protons. Surprisingly, also a significant proportion of coupled Mg and Si vacancies are present. Furthermore, all defect types feature a pronounced orientational disorder of the OH groups, which results in a significant range of OH···O bond distributions. As such, we are able to present unique insight into the defect chemistry of ringwoodite's spinel structure, which not only accounts for a potentially large fraction of the earth's entire water budget, but will also control transport properties in the mantle. We expect that our results will even impact other hydrous spinel-type materials, helping to understand properties such as ion conduction and heterogeneous catalysis.

Perspectives of shaped pulses for EPR spectroscopy

This article describes current uses of shaped pulses, generated by an arbitrary waveform generator, in the field of EPR spectroscopy. We show applications of sech/tanh and WURST pulses to dipolar spectroscopy, including new pulse schemes and procedures, and discuss the more general concept of optimum-control-based pulses for applications in EPR spectroscopy. The article also describes a procedure to correct for experimental imperfections, mostly introduced by the microwave resonator, and discusses further potential applications and limitations of such pulses.