Efficient orientational averaging by the extension of Lebedev grids via regularized octahedral symmetry expansion

Numerical evaluation of orientation averages and its application to molecular physics

In molecular physics, it is often necessary to average over the orientation of molecules when calculating observables, in particular when modeling experiments in the liquid or gas phase. Evaluated in terms of Euler angles, this is closely related to integration over two- or three-dimensional unit spheres, a common problem discussed in numerical analysis. The computational cost of the integration depends significantly on the quadrature method, making the selection of an appropriate method crucial for the feasibility of simulations. After reviewing several classes of spherical quadrature methods in terms of their efficiency and error distribution, we derive guidelines for choosing the best quadrature method for orientation averages and illustrate these with three examples from chiral molecule physics. While Gauss quadratures allow for achieving numerically exact integration for a wide range of applications, other methods offer advantages in specific circumstances. Our guidelines can also be applied to higher-dimensional spherical domains and other geometries. We also present a Python package providing a flexible interface to a variety of quadrature methods.

Low-power synchronous helical pulse sequences for large anisotropic interactions in MAS NMR: Double-quantum excitation of 14N

We develop a theoretical framework for a class of pulse sequences in the nuclear magnetic resonance (NMR) of rotating solids, which are applicable to nuclear spins with anisotropic interactions substantially larger than the spinning frequency, under conditions where the radiofrequency amplitude is smaller than or comparable to the spinning frequency. The treatment is based on average Hamiltonian theory and allows us to derive pulse sequences with well-defined relationships between the pulse parameters and spinning frequency for exciting specific coherences without the need for any detailed calculations. This framework is applied to the excitation of double-quantum spectra of ¹⁴N and is used both to evaluate the existing low-power pulse schemes and to predict the new ones, which we present here. It is shown that these sequences can be designed to be γ-encoded and therefore allow the acquisition of sideband-free spectra. It is also shown how these new double-quantum excitation sequences are incorporated into heteronuclear correlation NMR, such as ¹H-¹⁴N dipolar double-quantum heteronuclear multiple-quantum correlation spectroscopy. The new experiments are evaluated both with numerical simulations and experiments on glycine and N-acetylvaline, which represent cases with "moderate" and "large" quadrupolar interactions, respectively. The analyzed pulse sequences perform well for the case of a "moderate" quadrupolar interaction, however poorly with a "large" quadrupolar interaction, for which future work on pulse sequence development is necessary.

Large-scale magnetic resonance simulations: A tutorial

Computational modeling is becoming an essential tool in magnetic resonance to design and optimize experiments, test the performance of theoretical models, and interpret experimental data. Recent theoretical research and software development made possible simulations of large spin systems, for example, proteins with thousands of spins, in reasonable time. In the last few years, the Fokker-Planck formalism also re-emerged due to its ability to handle spatial dynamics. The purpose of this tutorial is to describe advantages and disadvantages of the most common formalisms, the latest developments and strategies to improve the computational efficiency, and to guide users in the setting up of a simulation using the Spinach software.

Assessment of new symmetry-based dipolar recoupling schemes for homonuclear magnetization exchange between quadrupolar nuclei in two-dimensional correlation MAS NMR

We provide an extensive experimental and numerical evaluation of MQ-phase (S)M supercycles with M={3,4} of three groups of symmetry-based homonuclear dipolar recoupling rf-pulse sequences, [Formula: see text] , for establishing proximities among half-integer spin quadrupolar nuclei under moderately fast magic-angle-spinning (MAS) conditions in single-quantum-single-quantum (1Q-1Q) correlation NMR experiments. The relative merits of the (S)M schemes for variations in resonance offsets and rf-amplitude errors were assessed by numerically simulated magnetization transfers in spin-3/2 pairs with variable isotropic chemical shifts and quadrupolar coupling constants. Experimental demonstrations of ²³Na (spin-3/2) NMR on Na₂MoO₄·2H₂O and ²⁷Al (spin-5/2) NMR on AlPO-CJ19 [(NH₄)₂Al₄(PO₄)₄HPO₄·H₂O] are presented at 14.1 T and 24 kHz MAS. We recommend using the (SR2₂¹)3 or (SR2₂¹)4 supercycles for samples that exhibit small chemical-shift dispersions (<3 kHz), and any (SRN_N^N/2)3 scheme with N⩾10 for larger spreads of isotropic chemical shifts. However, because the (SRN_N^N/2)3 sequences recouple heteronuclear dipolar interactions, their application to proton-bearing samples requires high-power proton decoupling during the mixing period. Alternatively, the (SR2₄¹)3 and (SR2₄¹)4 schemes may be employed in the absence of proton decoupling, but with poorer compensation to resonance-offsets and rf-amplitude errors.

Improved Magnetization Transfers among Quadrupolar Nuclei in Two-Dimensional Homonuclear Correlation NMR Experiments Applied to Inorganic Network Structures

We demonstrate that supercycles of previously introduced two-fold symmetry dipolar recoupling schemes may be utilized successfully in homonuclear correlation nuclear magnetic resonance (NMR) spectroscopy for probing proximities among half-integer spin quadrupolar nuclei in network materials undergoing magic-angle-spinning (MAS). These (SR221)M, (SR241)M, and (SR281)M recoupling sequences with M=3 and M=4 offer comparably efficient magnetization transfers in single-quantum–single-quantum (1Q–1Q) correlation NMR experiments under moderately fast MAS conditions, as demonstrated at 14.1 T and 24 kHz MAS in the contexts of 11B NMR on a Na2O–CaO–B2O3–SiO2 glass and 27Al NMR on the open framework aluminophosphate AlPO-CJ19 [(NH4)2Al4(PO4)4HPO4·H2O]. Numerically simulated magnetization transfers in spin–3/2 pairs revealed a progressively enhanced tolerance to resonance offsets and rf-amplitude errors of the recoupling pulses along the series (SR221)M< (SR241)M< (SR281)M for increasing differences in chemical shifts between the two nuclei. Nonetheless, for scenarios of a relatively minor chemical-shift dispersions (≲3 kHz), the (SR221)M supercycles perform best both experimentally and in simulations.

Artefact-free broadband 2D NMR for separation of quadrupolar and paramagnetic shift interactions

Two new two-dimensional, broadband, solid-state NMR experiments for separating and correlating the quadrupolar and shift interactions of spin I=1 nuclei in paramagnetic systems are proposed. The new pulse sequences incorporate the short, high-power adiabatic pulses (SHAPs) into the shifting d-echo experiment of Walder et al. [J. Chem. Phys., 142, 014201 (2015)], in two different ways, giving double and quadruple adiabatic shifting d-echo sequences. These new experiments have the advantage over previous methods of both suppressing spectral artefacts due to pulse imperfections, and exhibiting a broader excitation bandwidth. Both experiments are analysed with theoretical calculations and simulations, and are applied experimentally to the ²H NMR of deuterated CuCl₂ ⋅2H₂O, and two deuterated samples of the ion conductor oxyhydride BaTiO_3-xH_y prepared using two different methods. For the CuCl₂ ⋅2H₂O sample, both new methods obtain very high-quality spectra from which the parameters describing the shift and quadrupolar interaction tensors, and their relative orientation, were extracted. The two BaTiO_3-xH_y samples exhibited different local hydride environments with different tensor parameters. The ²H spectra of these oxyhydrides exhibit inhomogeneous broadening of the ²H shifts, and so whilst the quadrupolar interaction parameters were easily extracted, the measurement of the shift parameters was more complex. However, effective shift parameters were extracted, which combine the effects of both the paramagnetic shift tensor and the inhomogeneous broadening.

Fokker-Planck formalism in magnetic resonance simulations

This paper presents an overview of the Fokker-Planck formalism for non-biological magnetic resonance simulations, describes its existing applications and proposes some novel ones. The most attractive feature of Fokker-Planck theory compared to the commonly used Liouville - von Neumann equation is that, for all relevant types of spatial dynamics (spinning, diffusion, stationary flow, etc.), the corresponding Fokker-Planck Hamiltonian is time-independent. Many difficult NMR, EPR and MRI simulation problems (multiple rotation NMR, ultrafast NMR, gradient-based zero-quantum filters, diffusion and flow NMR, off-resonance soft microwave pulses in EPR, spin-spin coupling effects in MRI, etc.) are simplified significantly in Fokker-Planck space. The paper also summarises the author's experiences with writing and using the corresponding modules of the Spinach library - the methods described below have enabled a large variety of simulations previously considered too complicated for routine practical use.

CW-EPR Spectral Simulations: Solid State

This chapter summarizes the core concepts underlying the simulation of EPR spectra from biological samples in the solid state, from a user perspective. The key choices and decisions that have to be made by a user when simulating an experimental EPR spectrum are outlined. These include: the choice of the simulation model (the network of spins and the associated spin Hamiltonian), the dynamic regime (solid, liquid, slow motion), the level of theory used in the simulation (matrix diagonalization, perturbation theory, etc.), the treatment of orientational order and disorder (powder, crystal, partial ordering), the inclusion of the effects of structural disorder (strains), the effects of other line broadening mechanisms (unresolved hyperfine couplings, relaxation), and the inclusion of experimental distortions (field modulation, power saturation, filtering). Additionally, the salient aspects of utilizing least-squares fitting algorithms to aid the analysis of experimental spectra with the help of simulations are outlined. Although drawing from the experience gained from implementing EasySpin and from interacting with EasySpin's user base, this chapter applies to any EPR simulation software.

Computer-intensive simulation of solid-state NMR experiments using SIMPSON

Conducting large-scale solid-state NMR simulations requires fast computer software potentially in combination with efficient computational resources to complete within a reasonable time frame. Such simulations may involve large spin systems, multiple-parameter fitting of experimental spectra, or multiple-pulse experiment design using parameter scan, non-linear optimization, or optimal control procedures. To efficiently accommodate such simulations, we here present an improved version of the widely distributed open-source SIMPSON NMR simulation software package adapted to contemporary high performance hardware setups. The software is optimized for fast performance on standard stand-alone computers, multi-core processors, and large clusters of identical nodes. We describe the novel features for fast computation including internal matrix manipulations, propagator setups and acquisition strategies. For efficient calculation of powder averages, we implemented interpolation method of Alderman, Solum, and Grant, as well as recently introduced fast Wigner transform interpolation technique. The potential of the optimal control toolbox is greatly enhanced by higher precision gradients in combination with the efficient optimization algorithm known as limited memory Broyden-Fletcher-Goldfarb-Shanno. In addition, advanced parallelization can be used in all types of calculations, providing significant time reductions. SIMPSON is thus reflecting current knowledge in the field of numerical simulations of solid-state NMR experiments. The efficiency and novel features are demonstrated on the representative simulations.

Singularities in the lineshape of a second-order perturbed quadrupolar nucleus and their use in data fitting

Even for large quadrupolar interactions, the powder spectrum of the central transition for a half-integral spin is relatively narrow, because it is unperturbed to first order. However, the second-order perturbation is still orientation dependent, so it generates a characteristic lineshape. This lineshape has both finite step discontinuities and singularities where the spectrum is infinite, in theory. The relative positions of these features are well-known and they play an important role in fitting experimental data. However, there has been relatively little discussion of how high the steps are, so we present explicit formulae for these heights. This gives a full characterization of the features in this lineshape which can lead to an analysis of the spectrum without the usual laborious powder average. The transition frequency, as a function of the orientation angles, shows critical points: maxima, minima and saddle points. The maxima and minima correspond to the step discontinuities and the saddle points generate the singularities. Near a maximum, the contours are ellipses, whose dimensions are determined by the second derivatives of the frequency with respect to the polar and azimuthal angles. The density of points is smooth as the contour levels move up and down, but then drops to zero when a maximum is passed, giving a step. The height of the step is determined by the Hessian matrix-the matrix of all partial second derivatives. The points near the poles and the saddle points require a more detailed analysis, but this can still be done analytically. The resulting formulae are then compared to numerical simulations of the lineshape. We expand this calculation to include a relatively simple case where there is chemical shielding anisotropy and use this to fit experimental (139)La spectra of La2O3.

Grid-free powder averages: on the applications of the Fokker-Planck equation to solid state NMR

We demonstrate that Fokker-Planck equations in which spatial coordinates are treated on the same conceptual level as spin coordinates yield a convenient formalism for treating magic angle spinning NMR experiments. In particular, time dependence disappears from the background Hamiltonian (sample spinning is treated as an interaction), spherical quadrature grids are avoided completely (coordinate distributions are a part of the formalism) and relaxation theory with any linear diffusion operator is easily adopted from the Stochastic Liouville Equation theory. The proposed formalism contains Floquet theory as a special case. The elimination of the spherical averaging grid comes at the cost of increased matrix dimensions, but we show that this can be mitigated by the use of state space restriction and tensor train techniques. It is also demonstrated that low correlation order basis sets apparently give accurate answers in powder-averaged MAS simulations, meaning that polynomially scaling simulation algorithms do exist for a large class of solid state NMR experiments.

First principles calculation of inhomogeneous broadening in solid-state cw-EPR spectroscopy

We present a scheme for the first-principles calculation of EPR lineshapes for continuous-wave-EPR spectroscopy (cw-EPR) of spin centers in complex chemical environments. We specifically focus on poorly characterized systems, e.g. powders and frozen glasses with variable microsolvation structures. Our approach is based on ab initio molecular dynamics simulations and ab initio calculations of the ensemble of g- and A-tensors along the trajectory. The method incorporates temperature effects as well as the full anharmonicity of the intra- and intermolecular degrees of freedom of the system. We apply this scheme to compute the lineshape of a prototypical spin probe, the nitrosodisulfonate dianionic radical (Fremy's salt), dissolved in a 50 : 50 mixture of water and methanol. We are able to determine the specific effect of variations of local solvent composition and microsolvation structure on the cw-EPR lineshape. Our molecular dynamics reveal a highly anisotropic solvation structure with distinct spatial preferences for water and methanol around Fremy's salt that can be traced back to a combination of steric and polar influences. The overall solvation structure and conformational preferences of Fremy's salt as found in our MD simulations agree very well with the results obtained from EPR and orientation-selective ENDOR spectroscopy performed on the frozen glass. The simulated EPR lineshapes show good agreement with the experimental spectra. When combined with our MD results, they characterize the lineshape dependence on local morphological fluctuations.

C-REDOR curves of extended spin systems

The convergence of simulated C-REDOR curves of (infinitely) large spin systems is investigated with respect to the number of spins considered in the calculations. Taking a sufficiently large number of spins (>20,000 spins) into account enables the simulation of converged C-REDOR curves over the entire time period and not only the initial regime. The calculations are based on an existing approximation within first order average Hamiltonian theory (AHT), which assumes the absence of homonuclear dipole-dipole interactions. The C-REDOR experiment generates an average Hamiltonian close to the idealized AHT behavior even for multiple spin systems including multiple homonuclear dipole-dipole interactions which is shown from numerically exact calculations of the spin dynamics. Experimentally it is shown that calculations accurately predict the full, experimental C-REDOR curves of the multi-spin systems (31)P-(19)F in apatite, (31)P-(1)H in potassium trimetaphosphimate and (1)H-(31)P in potassium dihydrogen phosphate. We also present (13)C-(1)H and (15)N-(1)H data for the organic compounds glycine, l-alanine and l-histidine hydrochloride monohydrate which require consideration of molecular motion. Furthermore, we investigated the current limits of the method from systematic errors and we suggest a simple way to calculate errors for homogeneous and heterogeneous samples from experimental data.

Theoretical aspects of dynamic nuclear polarization in the solid state - the solid effect

Dynamic nuclear polarization has gained high popularity in recent years, due to advances in the experimental aspects of this methodology for increasing the NMR and MRI signals of relevant chemical and biological compounds. The DNP mechanism relies on the microwave (MW) irradiation induced polarization transfer from unpaired electrons to the nuclei in a sample. In this publication we present nuclear polarization enhancements of model systems in the solid state at high magnetic fields. These results were obtained by numerical calculations based on the spin density operator formalism. Here we restrict ourselves to samples with low electron concentrations, where the dipolar electron-electron interactions can be ignored. Thus the DNP enhancement of the polarizations of the nuclei close to the electrons is described by the Solid Effect mechanism. Our numerical results demonstrate the dependence of the polarization enhancement on the MW irradiation power and frequency, the hyperfine and nuclear dipole-dipole spin interactions, and the relaxation parameters of the system. The largest spin system considered in this study contains one electron and eight nuclei. In particular, we discuss the influence of the nuclear concentration and relaxation on the polarization of the core nuclei, which are coupled to an electron, and are responsible for the transfer of polarization to the bulk nuclei in the sample via spin diffusion.

Two-dimensional MAS NMR correlation protocols involving double-quantum filtering of quadrupolar spin-pairs

Three two-dimensional (2D) NMR homonuclear correlation techniques invoking double-quantum (2Q) filtration of the central transitions of half-integer spins are evaluated numerically and experimentally. They correlate directly detected single-quantum (1Q) coherences in the t(2) domain with either of 1Q, two-spin 2Q or single-spin multiple-quantum coherence-evolutions in the indirect (t(1)) dimension. We employ experimental (23)Na and (27)Al NMR on sodium sulfite and the natural mineral sillimanite (SiAl(2)O(5)), in conjunction with simulated 2D spectra from pairs of dipolar-recoupled spins-3/2 and 5/2 at different external magnetic fields, to compare the correlation strategies from the viewpoints of 2D spectral resolution, signal sensitivity, implementational aspects and their relative merits for establishing internuclear proximities and quadrupolar tensor orientations.

Isotropic filtering using polyhedral phase cycles: application to singlet state NMR

A technique is described for filtering out the components of an NMR signal that have passed through an isotropic spin order term. The method involves a coordinated cycle of three radiofrequency phase angles, where two of the phases correspond to the polar angles describing the vertices of a regular polyhedron, and the third angle is stepped around a circle. The most economical isotropic filtering scheme involves a 12-step phase cycle based on tetrahedral symmetry. The method is used to filter out NMR signals that have passed through singlet populations in a solution NMR experiment.