Matthias Ernst

An accurate theoretical treatment of periodically driven quantum systems is crucial for various fields in the exact sciences, for instance Nuclear Magnetic Resonance (NMR) spectroscopy. Conventionally, either average Hamiltonian theory or Floquet theory is used to predict or describe experimental outcomes, such as the time evolution or the spectra yielding the information of the sample under study. A detailed analysis of the equivalence of these two approaches with an emphasis on applications in NMR will help to improve the theoretical understanding of NMR experiments. In this work, we identify the Floquet–Magnus expansion as essential to prove the mathematical equivalence of Floquet theory and average Hamiltonian theory. We advocate a calculation scheme which is less prone to algebraic mistakes because explicit integration is avoided. On this basis, we provide the first four orders of both theories. We further examine their applicability to some experiments in NMR. As examples, we investigate the Bloch–Siegert shift and dipolar coupled spin systems under magic-angle spinning. Based on our analysis, we recommend the use of the Floquet–Van Vleck approach including both the effective Hamiltonian and the kick operator. The consistent separation of secular and non-secular contributions appears to be especially advantageous for numerical robustness. Its accuracy is about three times better than the one provided by average Hamiltonian theory despite their formal equivalence. Our findings provide important insights into the theoretical background of Floquet theory and average Hamiltonian theory. This includes the extent of their algebraic and perturbative equivalence, with an emphasis on how these findings are of relevance in the analysis of solid-state NMR experiments.

Establishing carbon–carbon connectivities by NMR at natural isotope abundance is intrinsically challenging due to the low sensitivity of 13C – 13C correlation experiments. Conventional strategies rely on polarization transfer from light nuclei, i.e. 1H or 19F, to enhance sensitivity, rendering systems devoid of such nuclei particularly difficult to study. We recently developed an NMR-optimized Overhauser-effect dynamic nuclear polarization (OE-DNP) setup that provides signal enhancements of up to two orders of magnitude for carbons which are chlorinated or iodinated, but lack 1H and 19F, in small organic molecules dissolved in organic solvent. Here, we demonstrate that in such systems, hyperpolarization can be coherently transferred between 13C nuclei using 1D and pseudo-2D isotropic mixing (IM) experiments. This allows 13C – 13C satellite signals to be detected, at natural isotopic abundance, from which one-bond carbon–carbon scalar coupling (1JCC) constants can be read off. These OE-DNP-enhanced IM experiments are shown to even outperform conventional 1D INADEQUATE experiments in these challenging systems. Furthermore, we demonstrate that IM can be used as a preparatory block in OE-DNP-enhanced multiple pulse NMR experiments, as exemplified by a 2D DNP-IM-HETCOR sequence. Finally, the transfer of polarization in hyperpolarized systems is described using both an approximate analytical treatment, based on the product operator formalism, and numerical simulations employing an effective Hamiltonian theory.

ZUSAMMENFASSUNG: Guanidinium spielt eine zentrale Rolle in zahlreichen biologischen und chemischen Prozessen. Insbesondere die Eigenschaft, über verschiedenste nicht‐kovalente Wechselwirkungen mit anderen Molekülen in Wechselwirkung zu treten, begründet die einzigartige Rolle des Guanidiniumkations in einer Vielzahl von molekularen Erkennungsprozessen. In Salzen oder Perowskiten ist die Rotation dieser Kationen entlang ihrer Symmetrieachsen bekannt und findet sogar im Festkörper auf Zeitskalen in der Größenordnung von ps statt. In dieser Arbeit berichten wir über die Stabilisierung eines Guanidiniumkations innerhalb der aromatischen Kavität eines π‐Containers durch Kation‐π‐Wechselwirkungen. Röntgenkristallographische Untersuchungen und Festkörper‐Kernspinmagnetresonanzspektroskopie (NMR) bei schneller Rotation um den magischen Winkel (MAS) wurden verwendet, um die Wirt‐Gast‐Wechselwirkungen in diesen Komplexen zu bestimmen. Die Bewegung des Guanidiniumkations wurde durch temperaturabhängige MAS‐NMR‐Experimente bis 100 K und verschiedene weitere Festkörper‐NMR‐Messungen, sowie weiterführenden quantenchemischen Rechnungen und Molekulardynamik‐Simulationen, vollständig charakterisiert. Die Ergebnisse deuten darauf hin, dass die Bewegung des Guanidiniumkations auf eine Rotation entlang der lokalen C ₃ ‐Achse beschränkt ist und mit einer Korrelationszeit im Bereich von ns erfolgt. Hiermit konnten wir zeigen, dass das Calixarengrundgerüst sich als Drehlager für die Rotation eines Guanidiniumkations eignet, was dem Traum von Chemikerinnen und Chemikern näherkommt, solche molekularen Rotationen gezielt zu steuern.

Guanidinium plays an essential role in many disciplines of biology and chemistry, particularly due to its unique role of being engaged in a variety of molecular‐recognition events that are facilitated by the ability to participate in a broad range of noncovalent interactions. Guanidinium cations in salts or perovskite materials are known to rotate along their local symmetry axes rather fast in the order of ps , even in the solid state. We herein employ a π‐container to trap a guanidinium cation inside the aromatic cavity by cation–π interactions. X‐ray crystallography and solid‐state nuclear magnetic resonance (NMR) spectroscopy at fast magic‐angle spinning (MAS) frequencies have been utilized to probe the underlying host–guest interactions. The guanidinium motion has been fully characterized by temperature‐dependent MAS‐NMR experiments down to 100 K, as well as by a variety of further solid‐state NMR experiments, and supplemented by quantum‐chemical calculations and molecular dynamics (MD) simulations. Our data point to a restriction of the guanidinium cation rotation about the local C ₃ ‐axis with correlation times in the order of ns . Our study, therefore, showcases that using the calixarene framework as a bearing for trapping a guanidinium cation, we are getting closer to the chemist's dream of controlling molecular rotations.

The dynamics of disordered nuclear spin ensembles are the subject of nuclear magnetic resonance studies. Because of the through-space long-range dipolar interaction, generically, many spins are involved in the time evolution, so that exact brute force calculations are impossible. The recently established spin dynamic mean-field theory (spinDMFT) represents an efficient and unbiased alternative to overcome this challenge. The approach only requires the dipolar couplings as input, and the only prerequisite for its applicability is that each spin interacts with a large number of other spins. Here, we show that spinDMFT can be used to describe spectral spin diffusion in static samples and to simulate zero-quantum line shapes which eluded an efficient quantitative simulation so far to the best of our knowledge. We perform benchmarks for two test substances that establish an excellent match with published experimental data. As spinDMFT combines low computational effort with high accuracy, we suggest to use it for large-scale simulations of spin diffusion, which are important in various areas of magnetic resonance.

Reverse DNP combined with a band-selective inversion pulse enabled electron-detected proton spectroscopy. A 5.4–6.8 Å spin-diffusion barrier was determined and hole burning on the proton line revealed a three-spin electron diffusion mechanism.

Dynamic nuclear polarization (DNP) enhances the intensity of NMR signals by transferring polarization from electron spins to nuclei via microwave irradiation. Pulsed DNP methods offer more control on the spin dynamics than conventional continuous-wave approaches. Here, we report on-resonance and off-resonance DNP sequences optimized using effective Hamiltonians derived from continuous Floquet theory. Experiments at 80 K and 0.35 T using a sample of 5 mM Trityl OX063 in a glycerol-d₈/D₂O/H₂O matrix (60:30:10, v/v/v) demonstrate that the optimized on-resonance sequence achieves 100 MHz electron offset bandwidth, while the off-resonance sequence centered at an electron offset of 50 MHz can cover 20 MHz, with 25 and 20 MHz of microwave power, respectively. These results demonstrate that continuous Floquet theory is a useful framework for the optimization of pulsed DNP sequences.

Dynamic nuclear polarization (DNP) enhances the intensity of NMR signals by transferring polarization from electron spins to nuclei via microwave irradiation. Pulsed DNP methods offer more control on the spin dynamics than conventional continuous-wave approaches. Here, we report on-resonance and off-resonance DNP sequences optimized using effective Hamiltonians derived from continuous Floquet theory. Experiments at 80 K and 0.35 T using a sample of 5 mM Trityl OX063 in a glycerol-d8/D2O/H2O matrix (60:30:10, v/v/v) demonstrate that the optimized on-resonance sequence achieves 100 MHz electron offset bandwidth, while the offresonance sequence cantered at an electron offset of 50 MHz can cover 20 MHz, with 25 MHz and 20 MHz of microwave power, respectively. These results demonstrate that continuous Floquet theory is a useful framework for the optimization of pulsed DNP sequences.