Effective Floquet Hamiltonian for spin I = 1 in magic angle spinning NMR using contact transformation

Contact transformation is an operator transformation method in time-independent perturbation theory which is used successfully in molecular spectroscopy to obtain an effective Hamiltonian. Floquet theory is used to transform the periodic time-dependent Hamiltonian, to a time-independent Floquet Hamiltonian. In this article contact transformation method has been used to get the analytical representation of Floquet Hamiltonian for quadrupolar nuclei with spin I = 1 in the presence of an RF field and first order quadrupolar interaction in magic angle spinning NMR experiments. The eigenvalues of contact transformed Hamiltonian as well as Floquet Hamiltonian have been calculated and a comparison is made between the eigenvalues obtained using the two Hamiltonians.     

Effective Hamiltonians by optimal control: solid-state NMR double-quantum planar and isotropic dipolar recoupling

We report the use of optimal control algorithms for tailoring the effective Hamiltonians in nuclear magnetic resonance (NMR) spectroscopy through sophisticated radio-frequency (rf) pulse irradiation. Specifically, we address dipolar recoupling in solid-state NMR of powder samples for which case pulse sequences offering evolution under planar double-quantum and isotropic mixing dipolar coupling Hamiltonians are designed. The pulse sequences are constructed numerically to cope with a range of experimental conditions such as inhomogeneous rf fields, spread of chemical shifts, the intrinsic orientation dependencies of powder samples, and sample spinning. While the vast majority of previous dipolar recoupling sequences are operating through planar double-or zero-quantum effective Hamiltonians, we present here not only improved variants of such experiments but also for the first time homonuclear isotropic mixing sequences which transfers all I(x), I(y), and I(z) polarizations from one spin to the same operators on another spin simultaneously and with equal efficiency. This property may be exploited to increase the signal-to-noise ratio of two-dimensional experiments by a factor of square root 2 compared to conventional solid-state methods otherwise showing the same efficiency. The sequences are tested numerically and experimentally for a powder of (13)C(alpha),(13)C(beta)-L-alanine and demonstrate substantial sensitivity gains over previous dipolar recoupling experiments.     

Rotor modulations and recoupling strategies in 13C solid-state magic-angle-spinning NMR spectroscopy: probing molecular orientation and dynamics.

Recoupling strategies for anisotropic interactions enable the investigation of molecular structure, order and dynamics in a sensitive and site-specific fashion by solid-state NMR spectroscopy. Whereas magic-angle spinning (MAS) efficiently averages anisotropic interactions and enhances spectral resolution, recoupling pulse sequences selectively restore certain parts of rotor-modulated dipole-dipole couplings or chemical shift anisotropies (CSA). More specifically, it is possible to recouple either the omegaR- or the 2omegaR-modulated terms of an interaction Hamiltonian, which exhibit different orientation dependencies and, in this way, provide a means of distinguishing whether the observed NMR spectra are affected by molecular motion or by molecular orientation. Sideband patterns generated by reconversion rotor encoding allow for a precise and selective determination of coupling constants and anisotropies, which contain site-specific information on structure, orientation and/or dynamics of individual molecular segments. Corresponding recoupling schemes are presented in a common context, and the possibilities of exploiting these effects for the determination of order parameters of oriented materials, such as oriented polymer chains or extruded fibres of a discotic mesogen, are discussed. The obtained orientational order parameters are compared to results from two-dimensional wide angle X-ray scattering (WAXS).     

Characterization of H2-Splitting Products of Frustrated Lewis Pairs: Benefit of Fast Magic-Angle Spinning

Proton spectroscopy in solid-state NMR on catalytic materials offers new opportunities in structural characterization, in particular of reaction products of catalytic reactions such as hydrogenation reactions. Unfortunately, the ¹H NMR line widths in magic-angle spinning solid-state spectra are often broadened by an incomplete averaging of ¹H-¹H dipolar couplings. We herein discuss two model compounds, namely the H₂-splitting products of two phosphane-borane Frustrated Lewis Pairs (FLPs), to study potentials and limitations of proton solid-state NMR experiments employing magic-angle spinning frequencies larger than 100 kHz at a static magnetic field strength of 20.0 T. The ¹H lines are homogeneously broadened as illustrated by spin-echo decay experiments. We study two structurally similar materials which however show significant differences in ¹H line widths which we explain by differences in their ¹H-¹H dipolar networks. We discuss the benefit of fast MAS experiments up to 110 kHz to detect the resonances of the H⁺/H⁻ pair in the hydrogenation products of FLPs.     

Low-power high-resolution solid-state NMR of peptides and proteins

We demonstrate that two-dimensional solid-state NMR chemical-shift correlation spectra can be recorded under low-power conditions. Except for the cross-polarization period, no rf-field amplitudes above 40 kHz are used. Such experiments require the use of fast (>50 kHz) magic-angle spinning (MAS). A comparison with the high-power version of the experiment shows no general line broadening but some changes in the polarization-transfer dynamics.     

Solid-state NMR spectroscopic methods in chemistry

Over the last decades, NMR spectroscopy has grown into an indispensable tool for chemical analysis, structure determination, and the study of dynamics in organic, inorganic, and biological systems. It is commonly used for a wide range of applications from the characterization of synthetic products to the study of molecular structures of systems such as catalysts, polymers, and proteins. Although most NMR experiments are performed on liquid-state samples, solid-state NMR is rapidly emerging as a powerful method for the study of solid samples and materials. This Review outlines some of the developments of solid-state NMR spectroscopy, including techniques such as cross-polarization, magic-angle spinning, multiple-pulse sequences, homo- and heteronuclear decoupling and recoupling techniques, multiple-quantum spectroscopy, and dynamic angle spinning, as well as their applications to structure determination. Modern solid-state NMR spectroscopic techniques not only produce spectra with a resolution close to that of liquid-state spectra, but also capitalize on anisotropic interactions, which are often unavailable for liquid samples. With this background, the future of solid-state NMR spectroscopy in chemistry appears to be promising, indeed.     

Radio frequency-driven recoupling at high magic-angle spinning frequencies: homonuclear recoupling sans heteronuclear decoupling

We describe solid-state NMR homonuclear recoupling experiments at high magic-angle spinning (MAS) frequencies using the radio frequency-driven recoupling (RFDR) scheme. The effect of heteronuclear decoupling interference during RFDR recoupling at high spinning frequencies is investigated experimentally and via numerical simulations, resulting in the identification of optimal decoupling conditions. The effects of MAS frequency, RF field amplitude, bandwidth, and chemical shift offsets are examined. Most significantly, it is shown that broadband homonuclear correlation spectra can be efficiently obtained using RFDR without decoupling during the mixing period in fully protonated samples, thus considerably reducing the rf power requirements for acquisition of (13)C-(13)C correlation spectra. The utility of RFDR sans decoupling is demonstrated with broadband correlation spectra of a peptide and a model protein at high MAS frequencies and high magnetic field.     

Third-order effect in solid-state NMR of quadrupolar nuclei

The theory and experimental observation of the third-order effect in solid-state NMR of quadrupolar nuclei are presented. The third-order effect consists of spherical harmonic terms up to rank l=6 and shifts NMR frequencies between two spin states that are not symmetric such as satellite transitions. Two-dimensional satellite transition magic-angle spinning experiment averages both the first and the second-order quadrupolar interactions making the quantitative measurement of the third-order effect possible. The third-order quadrupolar effect in andalusite has been measured at 11.7 T and its powder patterns are fitted with numerical simulations.     

Origins of linewidth in 1H magic-angle spinning NMR

A detailed study of the factors determining the linewidth (and hence resolution) in 1H solid-state magic-angle spinning NMR is described. Although it has been known from the early days of magic-angle spinning (MAS) that resolution of spectra from abundant nuclear spins, such as 1H, increases approximately linearly with increasing sample rotation rate, the difficulty of describing the dynamics of extended networks of coupled spins has made it difficult to predict a priori the resolution expected for a given sample. Using recently developed, highly efficient methods of numerical simulation, together with experimental measurements on a variety of test systems, we propose a comprehensive picture of 1H resolution under MAS. The "homogeneous" component of the linewidth is shown to depend primarily on the ratio between an effective local coupling strength and the spin rate, modified by geometrical factors which loosely correspond to the "dimensionality" of the coupling network. The remaining "inhomogeneous" component of the natural linewidth is confirmed to have the same properties as in dilute-spin NMR. Variations in the NMR frequency due to chemical shift effects are shown to have minimal impact on 1H resolution. The implications of these results for solid-state NMR experiments involving 1H and other abundant-spin nuclei are discussed.     

Introduction of the Floquet-Magnus expansion in solid-state nuclear magnetic resonance spectroscopy

In this article, we present an alternative expansion scheme called Floquet-Magnus expansion (FME) used to solve a time-dependent linear differential equation which is a central problem in quantum physics in general and solid-state nuclear magnetic resonance (NMR) in particular. The commonly used methods to treat theoretical problems in solid-state NMR are the average Hamiltonian theory (AHT) and the Floquet theory (FT), which have been successful for designing sophisticated pulse sequences and understanding of different experiments. To the best of our knowledge, this is the first report of the FME scheme in the context of solid state NMR and we compare this approach with other series expansions. We present a modified FME scheme highlighting the importance of the (time-periodic) boundary conditions. This modified scheme greatly simplifies the calculation of higher order terms and shown to be equivalent to the Floquet theory (single or multimode time-dependence) but allows one to derive the effective Hamiltonian in the Hilbert space. Basic applications of the FME scheme are described and compared to previous treatments based on AHT, FT, and static perturbation theory. We discuss also the convergence aspects of the three schemes (AHT, FT, and FME) and present the relevant references.     

Theory of heteronuclear decoupling in solid-state nuclear magnetic resonance using multipole-multimode Floquet theory

A formal theory for heteronuclear decoupling in solid-state magic angle spinning (MAS) nuclear magnetic resonance experiments is presented as a first application of multipole-multimode Floquet theory. The method permits a straightforward construction of the multispin basis and describes the spin dynamics via effective Floquet Hamiltonians obtained using the van Vleck transformation method in the Floquet-Liouville space. As a test case, we consider a model three-spin system (I2S) under asynchronous time modulations (both MAS and rf irradiation) and derive effective Hamiltonians for describing the spin dynamics in the Floquet-Liouville space during heteronuclear decoupling. Furthermore, we describe and evaluate the origin of cross terms between the various anisotropic interactions and illustrate their exact contributions to the spin dynamics. The theory presented herein should be applicable to the design and understanding of pulse sequences for heteronuclear and homonuclear recoupling and decoupling.     

Dynamics of polypropene and propene-ethylene copolymers at temperatures above ambient

Several applications of solid-state nuclear magnetic resonance (NMR) spectroscopy to studying polyolefin mobility at temperatures ranging from room temperature to above the polymer melt are described. 13_C NMR can be used with magic-angle spinning and high-power proton decoupling to determine the fraction of mobile polymer in polypropene and to characterize the nature of the polymer chain motions as a function of sample temperature. Similar techniques can be used to characterize the local motions of complex copolymer systems such as heterophasic ethylene-propene copolymers. The practicality of low-speed magic angle spinning to observe quantitative high-resolution NMR spectra of neat, molten polymer samples is also described.     

Dual acquisition magic-angle spinning solid-state NMR-spectroscopy: simultaneous acquisition of multidimensional spectra of biomacromolecules

Fast data collection: a general method for dual data acquisition of multidimensional magic-angle spinning solid-state NMR experiments is presented. The method uses a simultaneous Hartmann-Hahn cross-polarization from (1)H to (13)C and (15)N nuclei and exploits the long-living (15)N polarization for parallel acquisition of two multidimensional experiments.     

1H and 19F ultra-fast MAS double-quantum single-quantum NMR correlation experiments using three-spin terms of the dipolar homonuclear Hamiltonian

Measuring internuclear distances through dipolar interaction is a major challenge for solid-state nuclear magnetic resonance (NMR) spectroscopy. Obtaining reliable interatomic distances provides an access to the local structure in ordered or disordered solids. We show that at magic angle spinning (MAS) frequencies larger than ca. 50 kHz, some of the three-spin terms of the homogeneous homonuclear dipolar Hamiltonian can be used to promote the creation of double-quantum coherences between neighbouring (1)H or (19)F spins without using dipolar recoupling pulse sequences in the Dipolar Homonuclear Homogeneous Hamiltonian (DH(3)) double-quantum/single-quantum correlation experiment. This makes it possible to probe inter-nuclear spatial proximity with limited risk of probe or sample damage from radio-frequency (RF) irradiation, and is fully appropriate for fast repetition rate offering sensitivity gains in favourable cases. Experimental demonstrations are supported by multi-spin numerical simulations, which points to new possibilities for the characterization of spin-system geometries.     

Broadband rotational resonance in solid state NMR spectroscopy

A new technique for restoring nuclear magnetic dipole-dipole couplings under magic-angle spinning (MAS) in solid state nuclear magnetic resonance (NMR) spectroscopy is described and demonstrated. In this technique, called broadband rotational resonance (BroBaRR), the coupling between a pair of nuclear spins with NMR frequency difference close (but not necessarily equal) to the MAS frequency is restored by the application of a train of weak radio-frequency pulses at a carrier frequency close to the average of the two NMR frequencies. Phase or amplitude modulation of the pulse train at half the MAS frequency splits the carrier into sidebands close to the two NMR frequencies. The pulse train then removes offsets from the exact rotational resonance condition, leading to dipolar recoupling over a bandwidth controlled by the amplitude of the pulse train. (13)C NMR experiments on uniformly (15)N,(13)C-labeled L-valineHClH(2)O powder validate the theoretical analysis. BroBaRR will be useful in studies of molecular structures by solid state NMR, for example in the detection of long-range couplings between carbons in uniformly labeled organic and biological materials.     

Observation of highly flexible residues in amyloid fibrils of the HET-s prion

We report the observation of undetected (until now) residues of the prion protein fragment HET-s(218-289) which give rise to well-resolved (13)C, (15)N, and (1)H NMR resonances under high-resolution magic-angle spinning (HRMAS) conditions. The observed signals belong to large polymeric units as shown by measuring the lateral diffusion constants. The amino acids identified in the spectra are compatible with their localization in the segments of the protein which could not be detected in earlier solid-state NMR experiments. The observed chemical shifts indicate that these residues are in a random-coil conformation. Complementary experiments which detect only dynamic or static residues, respectively, strongly suggest that they belong to different parts of the same molecule.     

Pure Isotropic Proton NMR Spectra in Solids using Deep Learning

The resolution of proton solid-state NMR spectra is usually limited by broadening arising from dipolar interactions between spins. Magic-angle spinning alleviates this broadening by inducing coherent averaging. However, even the highest spinning rates experimentally accessible today are not able to completely remove dipolar interactions. Here, we introduce a deep learning approach to determine pure isotropic proton spectra from a two-dimensional set of magic-angle spinning spectra acquired at different spinning rates. Applying the model to 8 organic solids yields high-resolution ¹H solid-state NMR spectra with isotropic linewidths in the 50–400 Hz range.     

High-performance selective excitation pulses for solid- and liquid-state NMR spectroscopy.

Computer-optimized selective pulses are routinely used in solution-state NMR spectroscopy. At the same time, their utility and importance for solid-state applications has yet to be fully realized. We suggest a new computational approach that makes the design of soft selective pulses with desired properties relatively straightforward. By applying this technique to the generic selective excitation problem, we have arrived at a family of high performance selective excitation pulses, dubbed E-Family, that allows more flexibility and better performance than analogous pulses previously reported in the literature. The new pulses have been successfully tested in both solid- and solution-state NMR experiments. A theoretical treatment of the effects of chemical shift anisotropy (CSA) on the selective excitation in magic-angle spinning (MAS) experiments in solids is presented. The set of heuristics that comprise our new strategy were incorporated into a general NMR simulation program SPINEVOLUTION.