Solid-state Nuclear Magnetic Resonance Experiments Research Articles

1H and 13C chemical shift-structure effects in anhydrous β-caffeine and four caffeine-diacid cocrystals probed by solid-state NMR experiments and DFT calculations.

By using density functional theory (DFT) calculations, we refined the H atom positions in the structures of β-caffeine (C), α-oxalic acid (OA; (COOH)2), α-(COOH)2·2H2O, β-malonic acid (MA), β-glutaric acid (GA), and I-maleic acid (ME), along with their corresponding cocrystals of 2 : 1 (2C-OA, 2C-MA) or 1 : 1 (C-GA, C-ME) stoichiometry. The corresponding 13C/1H chemical shifts obtained by gauge including projector augmented wave (GIPAW) calculations agreed overall very well with results from magic-angle-spinning (MAS) nuclear magnetic resonance (NMR) spectroscopy experiments. Chemical-shift/structure trends of the precursors and cocrystals were examined, where good linear correlations resulted for all COO1H sites against the H⋯O and/or H⋯N H-bond distance, whereas a general correlation was neither found for the aliphatic/caffeine-stemming 1H sites nor any 13C chemical shift against either the intermolecular hydrogen- or tetrel-bond distance, except for the 13COOH sites of the 2C-OA, 2C-MA, and C-GA cocrystals, which are involved in a strong COOH⋯N bond with caffeine that is responsible for the main supramolecular stabilization of the cocrystal. We provide the first complete 13C NMR spectral assignment of the structurally disordered anhydrous β-caffeine polymorph. The results are discussed in relation to previous literature on the disordered α-caffeine polymorph and the ordered hydrated counterpart, along with recommendations for NMR experimentation that will secure sufficient 13C signal-resolution for reliable resonance/site assignments.

NMR methods to detect fluoride binding and transport by membrane proteins.

Solid-state nuclear magnetic resonance (NMR) methods can probe the motions of membrane proteins in liposomes at the atomic level, and propel the understanding of biomolecular processes for which static structures cannot provide a satisfactory description. High-resolution crystallography snapshots have provided a structural basis for fluoride channels. NMR is a powerful tool to build upon these snapshots and depict a dynamic picture of fluoride channels in native-like lipid bilayers. In this contribution, we discuss solid-state and solution NMR experiments to detect fluoride binding and transport by fluoride channels. Ongoing developments in membrane protein sample preparation and ssNMR methodology, particularly in using 1H, 19F and 13C-detection schemes, offer additional opportunities to study structure and functional aspects of fluoride channels.

The influence of crystal chemistry on lithium isotopic fractionation in pegmatite minerals

The understanding of lithium (Li) isotopic fractionation controlled by mineral species, especially those containing trace or minor amounts of Li (e.g., a few to tens of ppm), is hindered by lack of knowledge about the crystal chemistry of Li-bearing minerals. In this study, we performed solid-state Nuclear Magnetic Resonance (NMR) experiments, employing the Li7 Magic Angle Spinning (MAS) to investigate the local coordination environments of Li in minerals and their impact on Li isotopic fractionation. Our investigation focused on four coexisting minerals sourced from the Koktokay No. 3 pegmatite dike in the Xinjiang, China, namely quartz, K-feldspar, muscovite, and tourmaline, which has Li content and δLi7 value of 14.6 ppm and 3.90‰, 19.3 ppm and 7.09‰, 430 ppm and − 14.2‰, and 108 ppm and 13.42‰, respectively. Muscovite displayed the lowest δLi7 value, whereas tourmaline exhibited the highest value, and a considerable variation of 27.62‰ in δLi7 value was observed between muscovite and tourmaline. The analysis of Li7 MAS-NMR spectra further revealed distinct Li coordination environments among these minerals. A notable counterintuitive negative correlation exists between the δLi7 value and the chemical shift for the four Li-bearing minerals. This negative correlation contrasts to the expected Li isotopic fractionation under equilibrium conditions, in which enrichment of heavy Li isotopes is accompanied with large chemical shifts in Li-bearing minerals. This unexpected observation can be potentially attributed to nonequilibrium conditions of crystallization of the investigated minerals, coupled with fast diffusion of Li6 in minerals, boasting abnormally high coordination numbers due to weak LiO chemical bonds. Essentially, our study suggests that Li isotopic fractionation among Li-bearing minerals in pegmatite can be significant and is intricately influenced by both crystal chemistry and chemical diffusion.

Internal Vibrations of Pyridinium Cation in One-Dimensional Halide Perovskites and the Corresponding Halide Salts.

We investigate vibrations of the pyridinium cation PyH+ = C5H5NH+ in one-dimensional lead halide perovskites PyPbX3 and pyridinium halide salts PyHX (X- = I-, Br-), combining infrared absorption and Raman scattering methods at room temperature. Internal vibrations of the cation were assigned based on density functional theory modeling. Some of the vibrational bands are sensitive to perovskite or the salt environment in the solid state, while halide substitution has only a minor effect on them. These findings have been confirmed by 1H, 13C and 207Pb solid-state nuclear magnetic resonance (NMR) experiments. Narrower vibrational bands in perovskites indicate less disorder in these materials. The splitting of NH-group vibrational bands in perovskites can be rationalized the presence of nonequivalent crystal sites for cations or by more exotic phenomena such as quantum tunneling transition between two molecular orientations. We have shown how organic cations in hybrid organic-inorganic crystals could be used as spectators of the crystalline environment that affects their internal vibrations.

Digital quantum simulation of NMR experiments.

Simulations of nuclear magnetic resonance (NMR) experiments can be an important tool for extracting information about molecular structure and optimizing experimental protocols but are often intractable on classical computers for large molecules such as proteins and for protocols such as zero-field NMR. We demonstrate the first quantum simulation of an NMR spectrum, computing the zero-field spectrum of the methyl group of acetonitrile using four qubits of a trapped-ion quantum computer. We reduce the sampling cost of the quantum simulation by an order of magnitude using compressed sensing techniques. We show how the intrinsic decoherence of NMR systems may enable the zero-field simulation of classically hard molecules on relatively near-term quantum hardware and discuss how the experimentally demonstrated quantum algorithm can be used to efficiently simulate scientifically and technologically relevant solid-state NMR experiments on more mature devices. Our work opens a practical application for quantum computation.

Opportunities and Challenges in Applying Solid-State NMR Spectroscopy in Organic Mechanochemistry.

In recent years it is shown that mechanochemical strategies can be beneficial in directed conversions of organic compounds. Finding new reactions proved difficult, and due to the lack of mechanistic understanding of mechanochemical reaction events, respective efforts have mostly remained empirical. Spectroscopic techniques are crucial in shedding light on these questions. In this overview, the opportunities and challenges of solid-state nuclear magnetic resonance (NMR) spectroscopy in the field of organic mechanochemistry are discussed. After a brief discussion of the basics of high-resolution solid-state NMR under magic-angle spinning (MAS) conditions, seven opportunities for solid-state NMR in the field of organic mechanochemistry are presented, ranging from ex situ approaches to structurally elucidated reaction products obtained by milling to the potential and limitations of in situ solid-state NMR approaches. Particular strengths of solid-state NMR, for instance in differentiating polymorphs, in NMR-crystallographic structure-determination protocols, or in detecting weak noncovalent interactions in molecular-recognition events employing proton-detected solid-state NMR experiments at fast MAS frequencies, are discussed.

Automated Fragmentation Quantum Mechanical Calculation of 15N and 13C Chemical Shifts in a Membrane Protein.

In this work, we developed an accurate and cost-effective automated fragmentation quantum mechanics/molecular mechanics (AF-QM/MM) method to calculate the chemical shifts of 15N and 13C of membrane proteins. The convergence of the AF-QM/MM method was tested using Krokinobacter eikastus rhodopsin 2 as a test case. When the distance threshold of the QM region is equal to or larger than 4.0 Å, the results of the AF-QM/MM calculations are close to convergence. In addition, the effects of selected density functionals, basis sets, and local chemical environment of target atoms on the chemical shift calculations were systematically investigated. Our results demonstrate that the predicted chemical shifts are more accurate when important environmental factors including cross-protomer interactions, lipid molecules, and solvent molecules are taken into consideration, especially for the 15N chemical shift prediction. Furthermore, with the presence of sodium ions in the environment, the chemical shift of residues, retinal, and retinal Schiff base are affected, which is consistent with the results of the solid-state nuclear magnetic resonance (NMR) experiment. Upon comparing the performance of various density functionals (namely, B3LYP, B3PW91, M06-2X, M06-L, mPW1PW91, OB95, and OPBE), the results show that mPW1PW91 is a suitable functional for the 15N and 13C chemical shift prediction of the membrane proteins. Meanwhile, we find that the improved accuracy of the 13Cβ chemical shift calculations can be achieved by the employment of the triple-ζ basis set. However, the employment of the triple-ζ basis set does not improve the accuracy of the 15N and 13Cα chemical shift calculations nor does the addition of a diffuse function improve the overall prediction accuracy of the chemical shifts. Our study also underscores that the AF-QM/MM method has significant advantages in predicting the chemical shifts of key ligands and nonstandard residues in membrane proteins than most widely used empirical models; therefore, it could be an accurate computational tool for chemical shift calculations on various types of biological systems.

Nature of GaOx Shells Grown on Silica by Atomic Layer Deposition.

Gallia-based shells with a thickness varying from a submonolayer to ca. 2.5 nm were prepared by atomic layer deposition (ALD) using trimethylgallium, ozone, and partially dehydroxylated silica, followed by calcination at 500 °C. Insight into the atomic-scale structure of these shells was obtained by high-field 71Ga solid-state nuclear magnetic resonance (NMR) experiments and the modeling of X-ray differential pair distribution function data, complemented by Ga K-edge X-ray absorption spectroscopy and 29Si dynamic nuclear polarization surface enhanced NMR spectroscopy (DNP SENS) studies. When applying one ALD cycle, the grown submonolayer contains mostly tetracoordinate Ga sites with Si atoms in the second coordination sphere ([4]Ga(Si)) and, according to 15N DNP SENS using pyridine as the probe molecule, both strong Lewis acid sites (LAS) and strong Brønsted acid sites (BAS), consistent with the formation of gallosilicate Ga-O-Si and Ga-μ2-OH-Si species. The shells obtained using five and ten ALD cycles display characteristics of amorphous gallia (GaOx), i.e., an increased relative fraction of pentacoordinate sites ([5]Ga(Ga)), the presence of mild LAS, and a decreased relative abundance of strong BAS. The prepared Ga1-, Ga5-, and Ga10-SiO2-500 materials catalyze the dehydrogenation of isobutane to isobutene, and their catalytic performance correlates with the relative abundance and strength of LAS and BAS, viz., Ga1-SiO2-500, a material with a higher relative fraction of strong LAS, is more active and stable compared to Ga5- and Ga10-SiO2-500. In contrast, related ALD-derived Al1-, Al5-, and Al10-SiO2-500 materials do not catalyze the dehydrogenation of isobutane and this correlates with the lack of strong LAS in these materials that instead feature abundant strong BAS formed via the atomic-scale mixing of Al sites with silica, leading to Al-μ2-OH-Si sites. Our results suggest that [4]Ga(Si) sites provide strong Lewis acidity and drive the dehydrogenation activity, while the appearance of [5]Ga(Ga) sites with mild Lewis activity is associated with catalyst deactivation through coking. Overall, the atomic-level insights into the structure of the GaOx-based materials prepared in this work provide a guide to design active Ga-based catalysts by a rational tailoring of Lewis and Brønsted acidity (nature, strength, and abundance).

Atomic-level structure determination of amorphous molecular solids by NMR

Structure determination of amorphous materials remains challenging, owing to the disorder inherent to these materials. Nuclear magnetic resonance (NMR) powder crystallography is a powerful method to determine the structure of molecular solids, but disorder leads to a high degree of overlap between measured signals, and prevents the unambiguous identification of a single modeled periodic structure as representative of the whole material. Here, we determine the atomic-level ensemble structure of the amorphous form of the drug AZD4625 by combining solid-state NMR experiments with molecular dynamics (MD) simulations and machine-learned chemical shifts. By considering the combined shifts of all 1H and 13C atomic sites in the molecule, we determine the structure of the amorphous form by identifying an ensemble of local molecular environments that are in agreement with experiment. We then extract and analyze preferred conformations and intermolecular interactions in the amorphous sample in terms of the stabilization of the amorphous form of the drug.

Catalytic conversion of n-Dodecane to lower olefins hydrogen carriers over bran-shaped modified MCM-22 zeolite catalyst: SiO2/Al2O3 ratio effects

The demand-supply gap for light olefins and aromatics has been enormous in recent years. This imbalance is anticipated to grow exponentially due to their extensive use in manufacturing plastics, textiles, pharmacological intermediates, climate-neutral synthetic fuels, and hydrogen carriers. Production of light olefins through the catalytic cracking of naphtha is widely regarded as one of the most prestigious methods. Despite this, selecting an effective and efficient catalyst is one of the most challenging aspects of the scaling-up process. In this study, a highly active, modified MCM-22 catalyst has been synthesized by a hydrothermal method. Various techniques are used for the characterization of catalysts, including scanning electron microscopy (SEM), X-ray diffraction (XRD), N2 adsorption-desorption, temperature-programmed desorption of ammonia (NH3-TPD), and solid-state nuclear magnetic resonance experiments (NMR) with Al and Si. The cracking of n-dodecane to lower olefins was performed, and it was found that the conversion rates were highly correlated with the SiO2/Al2O3 ratio of the MCM-22 samples. Sample Si0.5 (SiO2/Al2O3 ∼ 28.9) showed a typical photograph of the MWW phase with a bran or flake shape in a stacking mode. The Si0.5 sample has a superior external surface area than other samples (up to ca. 130 m2/g) caused by a smaller particle with an average crystal size of around 15 nm, resulting in maximum catalytic performance towards light olefins. It was discovered that Si0.5 had a diminishing tendency in the paraffinic selectivity but had a robust selectivity for olefins (propylene and butylene in particular). This research may provide academics and the industry with new prospects for improving the catalytic cracking of n-dodecane into light olefins.

Dynamic Nuclear Polarization Illuminates Key Protein-Lipid Interactions in the Native Bacterial Cell Envelope.

Elucidating the structure and interactions of proteins in native environments is a fundamental goal of structural biology. Nuclear magnetic resonance (NMR) spectroscopy is well suited for this task but often suffers from low sensitivity, especially in complex biological settings. Here, we use a sensitivity-enhancement technique called dynamic nuclear polarization (DNP) to overcome this challenge. We apply DNP to capture the membrane interactions of the outer membrane protein Ail, a key component of the host invasion pathway of Yersinia pestis. We show that the DNP-enhanced NMR spectra of Ail in native bacterial cell envelopes are well resolved and enriched in correlations that are invisible in conventional solid-state NMR experiments. Furthermore, we demonstrate the ability of DNP to capture elusive interactions between the protein and the surrounding lipopolysaccharide layer. Our results support a model where the extracellular loop arginine residues remodel the membrane environment, a process that is crucial for host invasion and pathogenesis.

Development of High-Temperature Superconducting Coil for Solid-State NMR Experiments

Abstract A high-temperature superconducting (HTS) coil, in which YBa2Cu3O7-δ is vapor-deposited on the surface of a sapphire plane, has been developed to increase signal-to-noise ratios (SNR) in solid-state nuclear magnetic resonance (NMR) measurements. The Q-factor of the NMR probe is found to be 16,125 at the resonant frequency of 38.525 MHz. A single sharp signal, full width at half maximum of 20 ppm, can be detected by the HTS coil in a stationary solid-state 79Br NMR spectrum of a polycrystalline potassium bromide at the temperature of 9 K. Compared with the results for an isomorphic copper coil, an improvement of a factor of 6.6 is obtained for SNR, indicating that accumulation time of NMR experiments can be reduced to 1/44.

Stabilization of a flavoprotein for solid-state photo-CIDNP MAS NMR at room temperature by embedding in a glassy sugar matrix

Hyperpolarization via the solid-state photochemically induced dynamic nuclear polarization (photo-CIDNP) effect can be detected in frozen solutions of electron transfer proteins generating a radical-pair upon illumination. The effect has been observed in various natural photosynthetic reaction centers and in light-oxygen-voltage (LOV) sensing domains incorporating a flavin mononucleotide (FMN) as chromophore. In LOV domains, where a highly conserved cysteine is mutated to a flavin to interrupt its natural photochemistry, a radical-pair is generated by electron transfer from a nearby tryptophan to the photoexcited triplet state of FMN. During the photocycle, both the LOV domain and the chromophore are photochemically degraded, e.g., by the formation of singlet oxygen. This limits the time for collection of hyperpolarized nuclear magnetic resonance (NMR) data. We show that embedding of the protein into a trehalose sugar glass matrix stabilizes the protein for 13C solid-state photo-CIDNP NMR experiments which can be conducted at room temperature in a powder sample. Additionally, this preparation allows for incorporation of high amounts of protein further boosting the intensity of the detected signals from FMN and tryptophan at natural abundance. Signal assignment is aided by quantum chemical calculations of absolute shieldings. The underlying mechanism for the surprising absorption-only signal pattern is not yet understood. Comparison to calculated isotropic hyperfine couplings imply that the enhancement is not due to the classical radical-pair mechanism (RPM). Analysis of the anisotropic hyperfine couplings associated with solid-state photo-CIDNP mechanisms also show no simple correlation, suggesting a more complex underlying mechanism.

Do Models beyond Hybrid Density Functionals Increase the Agreement with Experiment for Predicted NMR Chemical Shifts or Electric Field Gradient Tensors in Organic Solids?

Ab initio predictions of chemical shifts and electric field gradient (EFG) tensor components are frequently used to help interpret solid-state nuclear magnetic resonance (NMR) experiments. Typically, these predictions employ density functional theory (DFT) with generalized gradient approximation (GGA) functionals, though hybrid functionals have been shown to improve accuracy relative to experiment. Here, the performance of a dozen models beyond the GGA approximation are examined for the prediction of solid-state NMR observables, including meta-GGA, hybrid, and double-hybrid density functionals and second-order Møller-Plesset perturbation theory (MP2). These models are tested on organic molecular crystal data sets containing 169 experimental 13C and 15N chemical shifts and 114 17O and 14N EFG tensor components. To make these calculations affordable, gauge-including projector augmented wave (GIPAW) Perdew-Burke-Ernzerhof (PBE) calculations with periodic boundary conditions are combined with a local intramolecular correction computed at the higher level of theory. Within the context of typical NMR property calculations performed on a static, DFT-optimized crystal structure, the benchmarking finds that the double-hybrid DFT functionals produce errors versus experiment that are no smaller than those of hybrid functionals in the best cases, and they can be larger. MP2 errors versus experiment are even bigger. Overall, no practical advantages are found for using any of the tested double-hybrid functionals or MP2 to predict experimental solid-state NMR chemical shifts and EFG tensor components for routine organic crystals, especially given the higher computational cost of those methods. This finding likely reflects error cancellation benefiting the hybrid functionals. Improving the accuracy of the predicted chemical shifts and EFG tensors relative to experiment would probably require more robust treatments of the crystal structures, their dynamics, and other factors.

Probing a Hydrogen-π Interaction Involving a Trapped Water Molecule in the Solid State.

The detection and characterization of trapped water molecules in chemical entities and biomacromolecules remains a challenging task for solid materials. We herein present proton-detected solid-state Nuclear Magnetic Resonance (NMR) experiments at 100 kHz magic-angle spinning and at high static magnetic-field strengths (28.2 T) enabling the detection of a single water molecule fixed in the calix[4]arene cavity of a lanthanide complex by a combination of three types of non-covalent interactions. The water proton resonances are detected at a chemical-shift value close to zero ppm, which we further confirm by quantum-chemical calculations. Density Functional Theory calculations pinpoint to the sensitivity of the proton chemical-shift value for hydrogen-π interactions. Our study highlights how proton-detected solid-state NMR is turning into the method-of-choice in probing weak non-covalent interactions driving a whole branch of molecular-recognition events in chemistry and biology.

New Approach To Understanding the Experimental 133Cs NMR Chemical Shift of Clay Minerals via Machine Learning and DFT-GIPAW Calculations.

Structural determination of adsorbed atoms on layered structures such as clay minerals is a complex subject. Radioactive cesium (Cs) is an important element for environmental conservation, so it is vital to understand its adsorption structure on clay. The nuclear magnetic resonance (NMR) parameters of 133Cs, which can be determined from solid-state NMR experiments, are sensitive to the local neighboring structures of adsorbed Cs. However, determining the Cs positions from NMR data alone is difficult. This paper describes an approach for identifying the expected atomic positions on clay minerals by combining machine learning (ML) with experimentally observed chemical shifts. A linear ridge regression model for ML is constructed from the smooth overlap of atomic position descriptor and gauge-including projector augmented wave (GIPAW) ab initio data. The constructed ML model predicts the GIPAW data to within a 3 ppm root-mean-squared error. At this stage, the 133Cs chemical shifts can be instantaneously calculated from the Cs positions on any clay layers using ML. The inverse analysis, which derives the atomic positions from experimentally observed chemical shifts, is developed from the ML model. The input data for the inverse analysis are the layer structure and the experimentally observed chemical shifts. The Cs positions for the targeted chemical shifts are then output. Inverse analysis is applied to montmorillonite, and the resultant Cs positions are found to be consistent with previous results (Ohkubo, T.; et al. J. Phys. Chem. A 2018, 122, 9326-9337). The Cs positions on saponite clay are also clarified from experimentally observed chemical shifts and inverse analysis.

Solid-state NMR spectroscopy of roasted and ground coffee samples: Evidences for phase heterogeneity and prospects of applications in food screening

The advancement in the use of spectroscopic techniques to investigate coffee samples is of high interest especially considering the widespread problems with coffee adulteration and counterfeiting. In this work, the use of solid-state nuclear magnetic resonance (NMR) is investigated as a means to probe the various chemically-distinct phases existent in roasted coffee samples and to detect the occurrence of counterfeiting or adulterations in coffee blends. Routine solid-state 1H and 13C NMR spectra allowed the distinction between different coffee types (Arabica/Robusta) and the evaluation of the presence of these components in coffee blends. On the other hand, the use of more specialized solid-state NMR experiments revealed the existence of phases with different molecular mobilities (e.g., associated with lipids or carbohydrates). The results illustrate the usefulness of solid-state NMR spectroscopy to examine molecular mobilities and interactions and to aid in the quality control of coffee-related products.

Field-Stepwise-Swept Solid-State 35Cl NMR and NQR of trichloroisocyanuric acid

Field-stepwise-swept solid-state 35Cl nuclear magnetic resonance (NMR) and nuclear quadrupole resonance (NQR) experiments of trichloroisocyanuric acid (TCICA) are carried out. The quadrupolar frequencies, vQ, were found to be 57.9 – 58.8 MHz in the temperature range between 180 and 299 K. The analysis of the field-stepwise-swept solid-state 35Cl NMR spectrum, observed at 240 K and acquired at 43.25 MHz, yields the quadrupolar coupling constant (CQ) of 116.6(2) MHz and the asymmetric parameter (ηQ) of 0.05(5). The tensor orientation of the 35Cl Electric-Field-Gradient (EFG) tensor is briefly discussed.

Rapid Quantification of Pharmaceuticals via 1H Solid-State NMR Spectroscopy.

The physicochemical properties of active pharmaceutical ingredients (APIs) can depend on their solid-state forms. Therefore, characterization of API forms is crucial for upholding the performance of pharmaceutical products. Solid-state nuclear magnetic resonance (SSNMR) spectroscopy is a powerful technique for API quantification due to its selectivity. However, quantitative SSNMR experiments can be time consuming, sometimes requiring days to perform. Sensitivity can be considerably improved using 1H SSNMR spectroscopy. Nonetheless, quantification via 1H can be a challenging task due to low spectral resolution. Here, we offer a novel 1H SSNMR method for rapid API quantification, termed CRAMPS-MAR. The technique is based on combined rotation and multiple-pulse spectroscopy (CRAMPS) and mixture analysis using references (MAR). CRAMPS-MAR can provide high 1H spectral resolution with standard equipment, and data analysis can be accomplished with ease, even for structurally complex APIs. Using several API species as model systems, we show that CRAMPS-MAR can provide a lower quantitation limit than standard approaches such as fast MAS with peak integration. Furthermore, CRAMPS-MAR was found to be robust for cases that are inapproachable by conventional ultra-fast (i.e., 100 kHz) MAS methods even when state-of-the-art SSNMR equipment was employed. Our results demonstrate CRAMPS-MAR as an alternative quantification technique that can generate new opportunities for analytical research.

A continuous approach to Floquet theory for pulse-sequence optimization in solid-state NMR.

We present a framework that uses a continuous frequency space to describe and design solid-state nuclear magnetic resonance (NMR) experiments. The approach is similar to the well-established Floquet treatment for NMR, but it is not restricted to periodic Hamiltonians and allows the design of experiments in a reverse fashion. The framework is based on perturbation theory on a continuous Fourier space, which leads to effective, i.e., time-independent, Hamiltonians. It allows the back-calculation of the pulse scheme from the desired effective Hamiltonian as a function of spin-system parameters. We show as an example how to back-calculate the rf irradiation in the MIRROR experiment from the desired chemical-shift offset behavior of the sequence.