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A new heteronuclear decoupling sequence for solid-state NMR and magic angle spinning faster than 60 kHz was recently introduced [Simion et al., J. Chem. Phys. 157, 014202 (2022)]. It was dubbed ROtor-Synchronized Phase-Alternated Cycles (ROSPAC), and it offers robustness for a large range of chemical shifts and low radio-frequency (RF) powers and is almost independent of the radio-frequency power. Here, we theoretically explore the robustness of the ROSPAC sequence toward 1H offset and RF field inhomogeneities, as well as the spacing effect of the π pulses on the decoupling efficiency. We use a generalized theoretical framework based on the Floquet theory to assess these parameters. The optimum decoupling conditions, where the magnitude of the second-order cross-terms and first-order resonance conditions are small, were identified., The Journal of Chemical Physics, 158 (15), ISSN:0021-9606, ISSN:1089-7690

Selective recoupling of protons (SERP) is a method to selectively and quantitatively measure magnetic dipole-dipole interaction between protons and, in turn, the proton-proton distance in solid-state samples at fast magic-angle spinning. We present a bimodal operator-based Floquet approach to describe the numerically optimized SERP recoupling sequence. The description calculates the allowed terms in the first-order effective Hamiltonian, explains the origin of selectivity during recoupling, and shows how different terms are modulated as a function of the radio frequency amplitude and the phase of the sequence. Analytical and numerical simulations have been used to evaluate the effect of higher-order terms and offsets on the polarization transfer efficiency and quantitative distance measurement. The experimentally measured

Rotational-Echo DOuble Resonance, REDOR, is an experimentally robust and a well-established dipolar-recoupling technique to measure dipolar couplings between isolated pairs of spin-1/2 heteronuclei in solid-state nuclear magnetic resonance. REDOR can also be used to estimate motional order parameters when the bond distance is known, for example, in the case of directly bound nuclei. However, the relatively fast dipolar dephasing for strongly coupled spin-1/2 pairs, such as 13C–1H, makes the stroboscopic measurement required in this experiment challenging, even at fast Magic-Angle-Spinning (MAS) frequencies. In such cases, modified REDOR-based methods like Shifted-REDOR (S-REDOR) are used to scale the dipolar coupling compared to REDOR. This is achieved by changing the position of one of the two recoupling π-pulses in a rotor period. This feature, however, comes at the cost of mixing multiple Fourier components of the dipolar coupling and can, additionally, require high radio-frequency amplitudes to realise small scaling factors. We introduce here a general pulse scheme which involves shifting both the π pulses in the REDOR scheme to achieve arbitrary scaling factors whilst retaining the robustness and simplicity of REDOR recoupling and avoiding the disadvantages of S-REDOR. The classical REDOR is a specific case of this scheme with a scaling factor of one. We demonstrate the results on isolated 13C–15N and 1H–13C spin pairs at 20 and 62.5 kHz MAS, respectively.

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

To interpret molecular dynamics simulations of complex systems, systematic dimensionality reduction methods such as principal component analysis (PCA) represent a well-established and popular approach. Apart from Cartesian coordinates, internal coordinates, e.g., backbone dihedral angles or various kinds of distances, may be used as input data in a PCA. Adopting two well-known model problems, folding of villin headpiece and the functional dynamics of BPTI, a systematic study of PCA using distance-based measures is presented which employs distances between Cα-atoms as well as distances between inter-residue contacts including side chains. While this approach seems prohibitive for larger systems due to the quadratic scaling of the number of distances with the size of the molecule, it is shown that it is sufficient (and sometimes even better) to include only relatively few selected distances in the analysis. The quality of the PCA is assessed by considering the resolution of the resulting free energy landscape (to identify metastable conformational states and barriers) and the decay behavior of the corresponding autocorrelation functions (to test the time scale separation of the PCA). By comparing results obtained with distance-based, dihedral angle, and Cartesian coordinates, the study shows that the choice of input variables may drastically influence the outcome of a PCA.

© 2015 AIP Publishing LLC. We show a theoretical framework based on triple mode Floquet theory to analyze recoupling sequences derived from symmetry based pulse sequences which have a non vanishing effective field and are not rotor synchronized. We analyze the properties of one such sequence a homonuclear double quantum recoupling sequence derived from the C7 2 1 sequence. The new asynchronous sequence outperforms the rotor synchronized version for spin pairs with small dipolar couplings in the presence of large chemical shift anisotropy. The resonance condition of the new sequence is analyzed using triple mode Floquet theory. Analytical calculations of second order effective Hamiltonian are performed to compare the efficiency in suppressing second order cross terms. Experiments and numerical simulations are shown to corroborate the results of the theoretical analysis.

The Journal of Chemical Physics, 146 (24), ISSN:0021-9606, ISSN:1089-7690

The dipolar Hamiltonian in a rapidly rotating sample is scaled by the second Legendre polynomial of the cosine of the angle between the rotation axis and the static magnetic field. It is, therefore, possible to refocus the spatial polarization-transfer process, often termed spin diffusion, in extended spin systems by reorienting the rotor axis such that the dipolar interaction Hamiltonian changes sign. We present experimental results which demonstrate that a rapid mechanical sample reorientation leads to a time reversal of the “radio-frequency-driven” spin diffusion among 13C spins.

The source of the residual line broadening in continuous‐wave (cw) decoupled spectra under magic‐angle sample spinning conditions is reexamined. It is shown that an important contribution to the line broadening comes from a second‐order recoupling of the heteronuclear dipolar‐coupling tensor and the chemical‐shielding tensor of the irradiated spin. Such an interference between the two tensors leads to a sum of a zeroth‐rank, a second‐rank, and a fourth‐rank tensor component in the Hamiltonian. The zeroth‐rank and the fourth‐rank tensor components are not averaged out under magic‐angle sample spinning (MAS) conditions, requiring the use of higher‐order averaging such as double rotation (DOR) for obtaining narrow lines. This broadening is distinctly different from off‐resonance decoupling effects which transform as a second‐rank tensor and can be averaged out by MAS. The properties of this second‐order recoupling as a source of structural information are explored, and the conditions for removing the broadening in systems with weak homonuclear dipolar‐coupling networks are discussed.

An experiment is described that utilizes the truncation of the Hamiltonian in the rotating frame by a radio‐frequency field designed to yield an isotropic shift for the dipolar coupling. This approach allows the measurement of a normally orientation‐dependent coupling constant by a single isotropic value. The dipolar isotropic shift is closely related to the field‐dependent chemical shift in solids due to the second‐order dipolar perturbation observed in magic‐angle spinning experiments. In the rotating frame, larger shifts of up to 1000 Hz can be observed for the case of a one‐bond C–H coupling compared to a shift of a few Hertz in the laboratory‐frame experiment. In addition to the isotropic shift, a line broadening due to the P4(cos β) terms is observed when the experiment is carried out under magic‐angle sample spinning (MAS) conditions, leading to the requirement of higher‐order averaging such as double rotation (DOR) for obtaining narrow lines. As an application of this new experiment the separation...

Generating dipolar order under magic-angle spinning (MAS) conditions is explored for different pulse sequences and dipolar-coupling networks. It is shown that under MAS second-order dipolar order can be generated reliably with 10% to 30% efficiency using the Jeener-Broekaert sequence in systems where the second-order average Hamiltonian is a (near) constant of the motion. When using adiabatic demagnetization and remagnetization, second-order dipolar order can be generated and reverted back to Zeeman order with up to 60% efficiency. This requires a maximum field strength with a nutation frequency that is less than one-quarter of the rotor frequency, and that the spin system can be properly spinlocked under such conditions. A simple coherent description accounts for the principal features of the spin dynamics, even using the smallest possible system of three coupled spins. For the systems investigated, the lifetime of second-order dipolar order under MAS was found to be on the order of T(1).

Short-range DFT/long-range ab initio methods allow for a combination of the weak basis-set dependency of DFT with an accurate ab initio treatment of long-range effects like van der Waals interaction. In order to improve existing short-range LDA and GGA density functionals, we developed a TPSS-like short-range meta-GGA exchange-correlation functional and checked its performance in long-range CCSD(T) calculations for thermodynamical properties of the G2 set of molecules.

A theoretical description of the two-pulse phase-modulated (TPPM) decoupling sequence in magic-angle spinning NMR is presented using a triple-mode Floquet approach. The description is formulated in the radio-frequency interaction-frame representation and is valid over the entire range of possible parameters leading to the well-known results of continuous-wave (cw) decoupling and XiX decoupling in the limit of a phase change of 0 degrees and 180 degrees , respectively. The treatment results in analytical expressions for the heteronuclear residual coupling terms and the homonuclear spin-diffusion terms. It also allows the characterization of all resonance conditions that can contribute in a constructive or a destructive way to the residual linewidth. Some of the important resonance conditions are described for the first time since they are not accessible in previous treatments. The combination of the contributions from the residual couplings and the resonance conditions to the effective Hamiltonian, as obtained in a Floquet description, is shown to be required to describe the decoupling behavior over the full range of parameters. It is shown that for typical spin system and experimental parameters a (13)C linewidth of approximately 12 Hz can be obtained for TPPM decoupling in an organic solid or a protein. This is a major contribution to the experimentally observed linewidths of around 20 Hz and indicates that decoupling techniques are still one of the limiting factors in the achievable linewidths.

Quantitative solid-state NMR distance measurements in strongly coupled spin systems are often complicated due to the simultaneous presence of multiple noncommuting spin interactions. In the case of zeroth-order homonuclear dipolar recoupling experiments, the recoupled dipolar interaction between distant spins is attenuated by the presence of stronger couplings to nearby spins, an effect known as dipolar truncation. In this article, we quantitatively investigate the effect of dipolar truncation on the polarization-transfer efficiency of various homonuclear recoupling experiments with analytical theory, numerical simulations, and experiments. In particular, using selectively (13)C-labeled tripeptides, we compare the extent of dipolar truncation in model three-spin systems encountered in protein samples produced with uniform and alternating labeling. Our observations indicate that while the extent of dipolar truncation decreases in the absence of directly bonded nuclei, two-bond dipolar couplings can generate significant dipolar truncation of small, long-range couplings. Therefore, while alternating labeling alleviates the effects of dipolar truncation, and thus facilitates the application of recoupling experiments to large spin systems, it does not represent a complete solution to this outstanding problem.

Many solid-state NMR experiments exploit interference effects between time dependencies in the system Hamiltonian to design an effective time-independent Hamiltonian with the desired properties. Effective Hamiltonians can be designed such that they contain only selected parts of the full system Hamiltonian while all other parts are averaged to zero. A general theoretical description of such experiments has to accommodate several time-dependent perturbations with incommensurate frequencies. We describe an extension of the analytical operator-based Floquet description of NMR experiments to situations with three incommensurate frequencies. Experiments with three time dependencies are quite common in solid-state NMR. Examples include experiments which combine magic-angle spinning and radio-frequency irradiation on two nuclei or asynchronous multiple-pulse sequences on a single spin species. The Floquet description is general in the sense that the resulting effective Hamiltonians can be calculated without a detailed knowledge of the spin-system Hamiltonian and can be expressed fully as a function of the Fourier components of the time-dependent Hamiltonian. As a prototype experiment we treat the application of two continuous-wave (cw) radio-frequency fields under magic-angle spinning. Experiments that are included in such a description are Hartmann-Hahn cross polarization or rotary-resonance recoupling experiments with simultaneous cw decoupling.

The application of two or more different time-dependent coherent perturbations with, in general, incommensurable frequencies occurs quite commonly in NMR experiments. Here we develop a unified description of the entire class of experiments using bimodal Floquet theory and van Vleck-Primas perturbation theory. This treatment leads to a time-independent effective Hamiltonian in Hilbert space and can be looked at as a generalization of average Hamiltonian theory to several incommensurate time dependencies. As a prototype experiment we treat the application of continuous-wave (cw) radio-frequency irradiation in combination with magic-angle sample spinning. Practically relevant examples of this type of experiments are heteronuclear spin decoupling and recoupling experiments using cw irradiation, e.g., rotary-resonance recoupling. Perturbations up to the third order must be taken into account to explain all experimentally observed resonance conditions.

Relaxation in nuclear magnetic resonance (NMR) results from stochastic motions that modulate anisotropic NMR interactions. Therefore, measurement of relaxation-rate constants can be used to characterize molecular-dynamic processes. The motion is often characterized by Markov processes using an auto-correlation function, which is assumed to be a sum of multiple decaying exponentials. We have recently shown that such a model can lead to severe misrepresentation of the real motion, when the real correlation function is more complex than the model. Furthermore, multiple distributions of motion may yield the same set of dynamics data. Therefore, we introduce optimized dynamics “detectors” to characterize motions which are linear combinations of relaxation-rate constants. A detector estimates the average or total amplitude of motion for a range of motional correlation times. The information obtained through the detectors is less specific than information obtained using an explicit model, but this is necessary because the information contained in the relaxation data is ambiguous, if one does not know the correct motional model. On the other hand, if one has a molecular dynamics trajectory, one may calculate the corresponding detector responses, allowing direct comparison to experimental NMR dynamics analysis. We describe how to construct a set of optimized detectors for a given set of relaxation measurements. We then investigate the properties of detectors for a number of different data sets, thus gaining an insight into the actual information content of the NMR data. Finally, we show an example analysis of ubiquitin dynamics data using detectors, using the DIFRATE software. ISSN:0021-9606 ISSN:1089-7690