Large-Scale Computation of Nuclear Magnetic Resonance Shifts for Paramagnetic Solids Using CP2K

Excited-State Forces with the Gaussian and Augmented Plane Wave Method for the Tamm-Dancoff Approximation of Time-Dependent Density Functional Theory

Augmented plane wave methods enable an efficient description of atom-centered or localized features of the electronic density, circumventing high energy cutoffs and thus prohibitive computational costs of pure plane wave formulations. To complement existing implementations for ground-state properties and excitation energies, we present the extension of the Gaussian and augmented plane wave method to excited-state nuclear gradients within the Tamm-Dancoff approximation of time-dependent density functional theory and its implementation in the CP2K program package. Benchmarks for a test set of 35 small molecules demonstrate that maximum errors in the nuclear forces for excited states of singlet and triplet spin multiplicity are smaller than 0.1 eV/Å. The method is furthermore applied to the calculation of the zero-phonon line of defective hexagonal boron nitride. This spectral feature is reproduced with an error of 0.6 eV in comparison to GW-Bethe-Salpeter reference computations and 0.4 eV in comparison to experimental measurements. Accuracy assessments and applications thus demonstrate the potential use of the outlined developments for large-scale applications on excited-state properties of extended systems.

A combined solid-state 1H, 13C, 17O NMR and periodic DFT study of hyperfine coupling tensors in paramagnetic copper(II) compounds

We report solid-state 1H, 13C, and 17O NMR determination of hyperfine coupling tensors (A-tensors) in several paramagnetic Cu(II) (d9, S = 1/2) complexes: trans-Cu(DL-Ala)2·H217O, Cu([1-13C]acetate)2·H2O, Cu([2-13C]acetate)2·H2O, and Cu(acetate)2·H217O. Using these new experimental results and some A-tensor data available in the literature for trans-Cu(L-Ala)2 and K2CuCl4·2H2O, we were able to examine the accuracy of A-tensor computation from a periodic DFT method implemented in the BAND program. We evaluated A-tensors on 1H (I = 1/2), 13C (I = 1/2), 14N (I = 1), 17O (I = 5/2), 39K (I = 3/2), 35Cl (I = 3/2), and 63Cu (I = 3/2) nuclei over a range spanning more than 3 orders of magnitude. We found that the BAND code can reproduce reasonably well the experimental results for both A-tensors and nuclear quadrupole coupling tensors.

Delving into theoretical and computational considerations for accurate calculation of chemical shifts in paramagnetic transition metal systems using quantum chemical methods

The chemical shielding tensor for a paramagnetic system has been derived from the macroscopically observed magnetization using the perturbation theory. An approach to calculate the paramagnetic chemical shifts in transition metal systems based on the spin-only magnetic susceptibility directly evaluated from the ab initio Hilbert space of the electronic Zeeman Hamiltonian has been discussed. Computationally, several advantages are associated with this approach: (a) it includes the state-specific paramagnetic Curie (first-order) and Van Vleck (second-order) contributions of the paramagnetic ion to the paramagnetic chemical shifts; (b) thus it avoids the system-specific modeling and evaluating effectively in terms of the electron paramagnetic resonance (EPR) spin Hamiltonian parameters of the magnetic moment of the paramagnetic ion formulated previously; (c) it can be utilized both in the point-dipole (PD) approximation (in the long-range) and with the quantum chemical (QC) method based the hyperfine tensors (in the short-range). Additionally, we have examined the predictive performance of various density functional theory (DFT) functionals of different families and commonly used core-augmented basis sets for nuclear magnetic resonance (NMR) chemical shifts. A selection of transition metal ion complexes with and without first-order orbital contributions, namely the [M(AcPyOx)3(BPh)]+ complexes of M = Mn2+, Ni2+ and Co2+ ions and CoTp2 complex and their reported NMR chemical shifts are studied from QC methods for illustration.

Geometry and electronic structure of Yb(III)[CH(SiMe3)2]3 from EPR and solid-state NMR augmented by computations

Characterization of paramagnetic compounds, in particular regarding the detailed conformation and electronic structure, remains a challenge, and - still today it often relies solely on the use of X-ray crystallography, thus limiting the access to electronic structure information. This is particularly true for lanthanide elements that are often associated with peculiar structural and electronic features in relation to their partially filled f-shell. Here, we develop a methodology based on the combined use of state-of-the-art magnetic resonance spectroscopies (EPR and solid-state NMR) and computational approaches as well as magnetic susceptibility measurements to determine the electronic structure and geometry of a paramagnetic Yb(III) alkyl complex, Yb(III)[CH(SiMe3)2]3, a prototypical example, which contains notable structural features according to X-ray crystallography. Each of these techniques revealed specific information about the geometry and electronic structure of the complex. Taken together, both EPR and NMR, augmented by quantum chemical calculations, provide a detailed and complementary understanding of such paramagnetic compounds. In particular, the EPR and NMR signatures point to the presence of three-centre-two-electron Yb-γ-Me-β-Si secondary metal-ligand interactions in this otherwise tri-coordinate metal complex, similarly to its diamagnetic Lu analogues. The electronic structure of Yb(III) can be described as a single 4f13 configuration, while an unusually large crystal-field splitting results in a thermally isolated ground Kramers doublet. Furthermore, the computational data indicate that the Yb-carbon bond contains some π-character, reminiscent of the so-called α-H agostic interaction.

Structure Determination and Refinement of Paramagnetic Materials by Solid-State NMR

Paramagnetism in solid-state materials has long been considered an additional challenge for structural investigations by using solid-state nuclear magnetic resonance spectroscopy (ssNMR). The strong interactions between unpaired electrons and the surrounding atomic nuclei, on the one hand, are complex to describe, and on the other hand can cause fast decaying signals and extremely broad resonances. However, significant progress has been made over the recent years in developing both theoretical models to understand and calculate the frequency shifts due to paramagnetism and also more sophisticated experimental protocols for obtaining high-resolution ssNMR spectra. While the field is continuously moving forward, to date, the combination of state-of-the-art numerical and experimental techniques enables us to obtain high-quality data for a variety of systems. This involves the determination of several ssNMR parameters that represent different contributions to the frequency shift in paramagnetic solids. These contributions encode structural information on the studied material on various length scales, ranging from crystal morphologies, to the mid- and long-range order, down to the local atomic bonding environment. In this perspective, the different ssNMR parameters characteristic for paramagnetic materials are discussed with a focus on their interpretation in terms of structure. This includes a summary of studies that have explored the information content of these ssNMR parameters, mostly to complement experimental data from other methods, e.g., X-ray diffraction. The presented overview aims to demonstrate how far ssNMR has hitherto been able to determine and refine the structures of materials and to discuss where it currently falls short of its full potential. We attempt to highlight how much further ssNMR can be pushed to determine and refine structure to deliver a comprehensive structural characterization of paramagnetic materials comparable to what is to date achieved by the combined effort of electron microscopy, diffraction, and spectroscopy.

Determining Local Magnetic Susceptibility Tensors in Paramagnetic Lanthanide Crystalline Powders from Solid-State NMR Chemical Shift Anisotropies

Exploring magnetic properties at the molecular level is a challenge that has been met by developing many experimental and theoretical solutions, such as polarized neutron diffraction (PND), muon-spin rotation (μ-SR), electron paramagnetic resonance (EPR), SQUID-based magnetometry measurements, and advanced modeling on open-shell systems and relativistic calculations. These methods are powerful tools that shed light on the local magnetic response in specifically designed magnetic materials such as contrast agents, for MRI, molecular magnets, magnetic tags for biological NMR, etc. All of these methods have their advantages and disadvantages. In order to complement the possibilities offered by these methods, we propose a new tool that implements a new approach combining simulation and fitting for high-resolution solid-state NMR spectra of lanthanide-based paramagnetic species. This method relies on a rigorous acquisition thanks to short high-power adiabatic pulses (SHAP) of high-resolution solid-state NMR isotropic and anisotropic data on a powdered magnetic material. It is also based on an efficient modeling of this data thanks to a semiempirical model based on a parametrization of the local magnetism and the crystal structure provided by diffraction methods. The efficiency of the calculation relies on a thorough simplification of the electron-nucleus interactions (point-dipole interaction, no Fermi contact) which is validated by experimental analysis. By taking advantage of the efficient calculation possibilities offered by our method, we can compare a great number of simulated spectra to experimental data and find the best-matching local magnetic susceptibility tensor. This method was applied to a series of isostructural lanthanide oxalates which are used as a benchmark system for many analytical methods. We present the results of thorough solid-state NMR and extensive modeling of the hyperfine interaction (including up to 400 paramagnetic centers) that yield local magnetic susceptibility tensor measurements that are self-consistent as well as consistent with bulk susceptibility measurements.

High resolution solid state NMR in paramagnetic metal-organic frameworks

We study the metal-organic framework (MOF) ZIF-67 with ¹H and ¹³C nuclear magnetic resonance (NMR). In addition to the usual orbital chemical shifts, we observe spinning sideband manifolds in the NMR spectrum due to hyperfine interactions of the paramagnetic cobalt with ¹H and ¹³C. Both orbital and paramagnetic chemical shifts are in good agreement with values calculated from first principles, allowing high-confidence assignment of the observed peaks to specific sites within the MOF. Our measured resonance shifts, line shapes, and spin lattice relaxation rates are also consistent with calculated values. We show that molecules in the pores of the MOF can exhibit high-resolution NMR spectra with fast spin lattice relaxation rates due to dipole-dipole couplings to the Co²⁺ nodes in the ZIF-67 lattice, showcasing NMR spectroscopy as a powerful tool for identification and characterization of "guests" that may be hosted by the MOF in electrochemical and catalytic applications.

The evolution of paramagnetic NMR as a tool in structural biology

Paramagnetic NMR data contain extremely accurate long-range information on metalloprotein structures and, when used in the frame of integrative structural biology approaches, they allow for the retrieval of structural details to a resolution that is not achievable using other techniques. Paramagnetic data thus represent an extremely powerful tool to refine protein models in solution, especially when coupled to X-ray or cryoelectron microscopy data, to monitor the formation of complexes and determine the relative arrangements of their components, and to highlight the presence of conformational heterogeneity. More recently, theoretical and computational advancements in quantum chemical calculations of paramagnetic NMR observables are progressively opening new routes in structural biology, because they allow for the determination of the structure within the coordination sphere of the metal center, thus acting as a loupe on sites that are difficult to observe but very important for protein function.

A machine learning protocol for revealing ion transport mechanisms from dynamic NMR shifts in paramagnetic battery materials

Solid-state nuclear magnetic resonance (ssNMR) provides local environments and dynamic fingerprints of alkali ions in paramagnetic battery materials. Linking the local ionic environments and NMR signals requires expensive first-principles computational tools that have been developed for over a decade. Nevertheless, the assignment of the dynamic NMR spectra of high-rate battery materials is still challenging because the local structures and dynamic information of alkali ions are highly correlated and difficult to acquire. Herein, we develop a novel machine learning (ML) protocol that could not only quickly sample atomic configurations but also predict chemical shifts efficiently, which enables us to calculate dynamic NMR shifts with the accuracy of density functional theory (DFT). Using structurally well-defined P2-type Na_2/3(Mg_1/3Mn_2/3)O₂ as an example, we validate the ML protocol and show the significance of dynamic effects on chemical shifts. Moreover, with the protocol, it is demonstrated that the two experimental ²³Na shifts (1406 and 1493 ppm) of P2-type Na_2/3(Ni_1/3Mn_2/3)O₂ originate from two stacking sequences of transition metal (TM) layers for the first time, which correspond to space groups P6₃/mcm and P6₃22, respectively. This ML protocol could help to correlate dynamic ssNMR spectra with the local structures and fast transport of alkali ions and is expected to be applicable to a wide range of fast dynamic systems.

We developed a widely applicable machine learning (ML) method that can help to correlate dynamic ssNMR spectra with the local structures and transport of ions and thus expands the ssNMR application to fast chemically exchanged material systems.

Origin of the temperature dependence of 13C pNMR shifts for copper paddlewheel MOFs

An efficient protocol for the calculation of 13C pNMR shifts in metal-organic frameworks based on Cu(ii) paddlewheel dimers is proposed, which involves simplified structural models, optimised using GFN2-xTB for the high-spin state, and CAM-B3LYP-computed NMR and EPR parameters. Models for hydrated and activated HKUST-1 and hydrated STAM MOFs with one, two and three Cu dimers have been used. The electronic ground states are low-spin and diamagnetic, with pNMR shifts arising from thermal population of intermediate- and high-spin excited states. Treating individual spin configurations in a broken symmetry (BS) approach, and selecting two or more of these to describe individual excited states, the magnetic shieldings of these paramagnetic states are evaluated using the approach by Hrobárik and Kaupp. The total shielding is then evaluated from a Boltzmann distribution between the energy levels of the chosen configurations. The computed pNMR shifts are very sensitive to temperature and, therefore, to the relative energies of the BS spin states. In order to reproduce the temperature dependence of the pNMR shifts seen in experiment, some scaling of the calculated energy gaps is required. A single scaling factor was applied to all levels in any one system, by fitting to experimental results at several temperatures simultaneously. The resulting scaling factor decreases with an increasing number of dimer units in the model (e.g., from ∼1.7 for mono-dimer models to 1.2 for tri-dimer models). The approach of this scaling factor towards unity indicates that models with three dimers are approaching a size where they can be considered as reasonable models for the 13C shifts of infinite MOFs. The observed unusual temperature dependencies in the latter are indicated to arise both from the "normal" temperature dependence of the pNMR shifts of the paramagnetic states and the populations of these states in the thermal equilibrium.

A method to calculate the NMR spectra of paramagnetic species using thermalized electronic relaxation

For paramagnetic species, it has been long understood that the hyperfine interaction between the unpaired electrons and the nucleus results in a nuclear magnetic resonance (NMR) peak that is shifted by a paramagnetic shift, rather than split by the coupling, due to an averaging of the electronic magnetic moment caused by electronic relaxation that is fast in comparison to the hyperfine coupling constant. However, although this feature of paramagnetic NMR has formed the basis of all theories of the paramagnetic shift, the precise theory and mechanism of the electronic relaxation required to predict this result has never been discussed, nor has the assertion been tested. In this paper, we show that the standard semi-classical Redfield theory of relaxation fails to predict a paramagnetic shift, as does any attempt to correct for the semi-classical theory using modifications such as the inhomogeneous master equation or Levitt-di Bari thermalization. In fact, only the recently-introduced Lindbladian theory of relaxation in magnetic resonance [J.Magn.Reson., 310, 106645 (2019)] is able to correctly predict the paramagnetic shift tensor and relaxation-induced linewidth in pNMR. Furthermore, this new formalism is able to predict the NMR spectra of paramagnetic species outside the high-temperature and weak-order limits, and is therefore also applicable to dynamic nuclear polarization. The formalism is tested by simulations of five case studies, which include Fermi-contact and spin-dipolar hyperfine couplings, g-anisotropy, zero-field splitting, high and low temperatures, and fast and slow electronic relaxation.

Unravelling the Fast Alkali-Ion Dynamics in Paramagnetic Battery Materials Combined with NMR and Deep-Potential Molecular Dynamics Simulation

Solid-state nuclear magnetic resonance (ssNMR) has received extensive attention in characterizing alkali-ion battery materials because it is highly sensitive for probing the local environment and dynamic information of atoms/ions. However, precise spectral assignment cannot be carried out by conventional DFT for high-rate battery materials at room temperature. Herein, combining DFT calculation of paramagnetic shift and deep potential molecular dynamics (DPMD) simulation to achieve the converged Na⁺ distribution at hundreds of nanoseconds, we obtain the statistically averaged paramagnetic shift, which is in excellent agreement with ssNMR measurements. Two ²³ Na shifts induced by different stacking sequences of transition metal layers are revealed in the fast chemically exchanged NMR spectra of P2-type Na_2/3 (Mg_1/3 Mn_2/3 )O₂ for the first time. This DPMD simulation auxiliary protocol can be beneficial to a wide range of ssNMR analysis in fast chemically exchanged material systems.

Solution of a Puzzle: High-Level Quantum-Chemical Treatment of Pseudocontact Chemical Shifts Confirms Classic Semiempirical Theory

A recently popularized approach for the calculation of pseudocontact shifts (PCSs) based on first-principles quantum chemistry (QC) leads to different results than the classic "semiempirical" equation involving the susceptibility tensor. Studies that attempted a comparison of theory and experiment led to conflicting conclusions with respect to the preferred theoretical approach. In this Letter, we show that after inclusion of previously neglected terms in the full Hamiltonian, one can deduce the semiempirical equations from a rigorous QC-based treatment. It also turns out that in the long-distance limit, one can approximate the complete A tensor in terms of the g tensor. By means of Kohn-Sham density functional theory calculations, we numerically confirm the long-distance expression for the A tensor and the theoretically predicted scaling behavior of the different terms. Our derivation suggests a computational strategy in which one calculates the susceptibility tensor and inserts it into the classic equation for the PCS.

CP2K: An electronic structure and molecular dynamics software package - Quickstep: Efficient and accurate electronic structure calculations

CP2K is an open source electronic structure and molecular dynamics software package to perform atomistic simulations of solid-state, liquid, molecular, and biological systems. It is especially aimed at massively parallel and linear-scaling electronic structure methods and state-of-the-art ab initio molecular dynamics simulations. Excellent performance for electronic structure calculations is achieved using novel algorithms implemented for modern high-performance computing systems. This review revisits the main capabilities of CP2K to perform efficient and accurate electronic structure simulations. The emphasis is put on density functional theory and multiple post-Hartree-Fock methods using the Gaussian and plane wave approach and its augmented all-electron extension.

31P MAS NMR and DFT study of crystalline phosphate matrices

We present the study of the phosphorus local environment by using ³¹P MAS NMR in a series of seven double monophosphates M^IIM^IV(PO₄)₂ (M^II and M^IV being divalent and tetravalent cations, respectively) of yavapaiite and low-yavapaiite type crystal structures. Solid-state and cluster DFT calculations were found to be efficient for predicting the ³¹P isotropic chemical shift and chemical shift anisotropy. To achieve this performance, however, a proper computational optimisation of the experimental structural data was required. From the three optimisation methods tested, the full optimisation provided the best reference structure for the calculation of the NMR parameters of the studied phosphates. Also, a better prediction of the chemical shifts was possible by using a correction to the GIPAW calculated shielding.

Magnetic susceptibility and paramagnetism-based NMR

The magnetic interactions between the nuclear magnetic moment and the magnetic moment of unpaired electron(s) depend on the structure and dynamics of the molecules where the paramagnetic center is located and of their partners. The long-range nature of the magnetic interactions is thus a reporter of invaluable information for structural biology studies, when other techniques often do not provide enough data for the atomic-level characterization of the system. This precious information explains the flourishing of paramagnetism-assisted NMR studies in recent years. Many paramagnetic effects are related to the magnetic susceptibility of the paramagnetic metal. Although these effects have been known for more than half a century, different theoretical models and new approaches have been proposed in the last decade. In this review, we have summarized the consequences for NMR spectroscopy of magnetic interactions between nuclear and electron magnetic moments, and thus of the presence of a magnetic susceptibility due to metals, and we do so using a unified notation.

What are the methodological and theoretical prospects for paramagnetic NMR in structural biology? A glimpse into the crystal ball

NMR spectroscopy is very sensitive to the presence of unpaired electrons, which perturb the NMR chemical shifts, J splittings and nuclear relaxation rates. These paramagnetic effects have attracted increasing attention over the last decades, and their use is expected to increase further in the future because they can provide structural information not easily achievable with other techniques. In fact, paramagnetic data provide long range structural restraints that can be used to assess the accuracy of crystal structures in solution and to improve them by simultaneous refinements with the X-ray data. They are also precious for obtaining information on the conformational variability of biomolecular systems, possibly in conjunction with SAXS and/or DEER data. We foresee that new tools will be developed in the next years for the simultaneous analysis of the paramagnetic data with data obtained from different techniques, in order to take advantage synergistically of the information content of all of them. Of course, the use of the paramagnetic data for structural purposes requires the knowledge of the relationship between these data and the molecular coordinates. Recently, the equations commonly used, dating back to half a century ago, have been questioned by first principle quantum chemistry calculations. Our prediction is that further theoretical/computational improvements will essentially confirm the validity of the old semi-empirical equations for the analysis of the experimental paramagnetic data.

Quantum-chemical study of 7Li NMR shifts in the context of delithiation of paramagnetic lithium vanadium phosphate, Li3V2(PO4)3 (LVP)

The lithium NMR shifts of three paramagnetic materials important in the charging/discharging processes of lithium vanadium phosphate cathode materials have been studied by large-scale quantum-chemical methodology. Namely, the ⁷Li NMR shifts of the fully lithiated Li₃V₂(PO₄)₃ (LVP3.0), and of the partly delithiated Li_2.5V₂(PO₄)₃ (LVP2.5) and Li₂V₂(PO₄)₃ (LVP2.0), have been computed and analyzed using a recently proposed approach (A. Mondal, M. Kaupp J. Phys. Chem. C 123 (2019) 8387-8405) that accounts for the Fermi-contact, pseudo-contact, as well as orbital shifts, combining periodic computations with an incremental cluster model. LVP3.0 and LVP2.0 exhibit three and two unique Li sites, respectively, which could be assigned to their experimental ⁷Li NMR signals. In case of LVP2.0, the computations clearly assigned the signals at 143 ppm and 77 ppm to the Li(1) and Li(2) sites, respectively, even though the latter is connected to V_b^(+III d² sites and the former to V_a^(+IV) d¹ sites. LVP2.5 is the most complex of these three materials, exhibiting a 50% occupation of Li(3) sites, which generates much more complicated Li NMR spectra with seven peaks that partly are closely spaced. Exploring three different occupation patterns, the computations can clearly assign five of the seven signals to one type of Li site and give most probable assignments for the two remaining signals. Notably, the calculations support seven signals to be assigned to LVP2.5, while previous interpretations took two of the signals as being entirely due to contamination by LVP2.0. The accuracy of the computations could probably be improved further by full DFT optimization of large super-cell structures. This work suggests that first-principles computations of NMR shifts of extended paramagnetic solids provide an important tool for the analysis of even rather complex NMR spectra.

How Do Nuclei Couple to the Magnetic Moment of a Paramagnetic Center? A New Theory at the Gauntlet of the Experiments

The recent derivation, based on pure quantum chemistry (QC) first-principles, of the pseudocontact shifts (PCSs) caused by a paramagnetic metal center on far away nuclei has cast doubts on the validity of the semiempirical (SE) theory, predicting PCSs to arise from the metal magnetic susceptibility anisotropy. The SE theory has been used and applied countless times, especially in the last 2 decades, to obtain structural information on proteins containing paramagnetic metal ions. We show here that the QC and SE predictions can be directly tested against experiments, provided a suitable macromolecular system is used. The SE approach yields a good prediction of the experimental PCSs while the QC one does not. It appears that the classic theory is able to grasp satisfactorily the underlying physics.

Pseudocontact shifts and paramagnetic susceptibility in semiempirical and quantum chemistry theories

Pseudocontact shifts are traditionally described as a function of the anisotropy of the paramagnetic susceptibility tensor, according to the semiempirical theory mainly developed by Kurland and McGarvey [J. Magn. Reson. 2, 286-301 (1970)]. The paramagnetic susceptibility tensor is required to be symmetric. Applying point-dipole approximation to the quantum chemistry theory of hyperfine shift, pseudocontact shifts are found to scale with a non-symmetric tensor that differs by a factor g^T/g_e from the paramagnetic susceptibility tensor derived within the semiempirical framework. We analyze the foundations of the Kurland-McGarvey pseudocontact shift expression and recall that it is inherently based on the Russell-Saunders (LS) coupling approximation for the spin-orbit coupling. We show that the difference between the semiempirical and quantum chemistry pseudocontact shift expressions arises directly from the different treatment of the orbital contribution to the hyperfine coupling.

Unraveling the Arrangement of Al and Fe within the Framework Explains the Magnetism of Mixed-Metal MIL-100(Al,Fe)

Properties of mixed-metal MOFs depend on the distribution of different metals within their frameworks. Determination of this distribution is often very challenging. Using an example of aluminum- and iron-containing MIL-100, we demonstrate that ²⁷Al NMR spectroscopy, when combined with first-principles calculations and magnetic, X-band electron paramagnetic resonance, Fe K-edge extended X-ray absorption fine structure, and Mössbauer measurements, enables one to accurately determine the arrangement of Al and Fe within the metal trimers, which are the basic building units of MIL-100. In this particular material, the incorporation of Fe and Al on the framework metal sites is random. Crucial for deciphering the arrangement is detecting NMR signals, shifted because of the strong hyperfine interaction between the ²⁷Al nuclei and the unpaired electronic spins of Fe³⁺ ions, assigning the shifted signals aided by first-principles calculations of hyperfine couplings, and quantitatively evaluating the NMR intensities and the measured effective magnetic moment.

Paramagnetic NMR in solution and the solid state

The field of paramagnetic NMR has expanded considerably in recent years. This review addresses both the theoretical description of paramagnetic NMR, and the way in which it is currently practised. We provide a review of the theory of the NMR parameters of systems in both solution and the solid state. Here we unify the different languages used by the NMR, EPR, quantum chemistry/DFT, and magnetism communities to provide a comprehensive and coherent theoretical description. We cover the theory of the paramagnetic shift and shift anisotropy in solution both in the traditional formalism in terms of the magnetic susceptibility tensor, and using a more modern formalism employing the relevant EPR parameters, such as are used in first-principles calculations. In addition we examine the theory first in the simple non-relativistic picture, and then in the presence of spin-orbit coupling. These ideas are then extended to a description of the paramagnetic shift in periodic solids, where it is necessary to include the bulk magnetic properties, such as magnetic ordering at low temperatures. The description of the paramagnetic shift is completed by describing the current understanding of such shifts due to lanthanide and actinide ions. We then examine the paramagnetic relaxation enhancement, using a simple model employing a phenomenological picture of the electronic relaxation, and again using a more complex state-of-the-art theory which incorporates electronic relaxation explicitly. An additional important consideration in the solid state is the impact of bulk magnetic susceptibility effects on the form of the spectrum, where we include some ideas from the field of classical electrodynamics. We then continue by describing in detail the solution and solid-state NMR methods that have been deployed in the study of paramagnetic systems in chemistry, biology, and the materials sciences. Finally we describe a number of case studies in paramagnetic NMR that have been specifically chosen to highlight how the theory in part one, and the methods in part two, can be used in practice. The systems chosen include small organometallic complexes in solution, solid battery electrode materials, metalloproteins in both solution and the solid state, systems containing lanthanide ions, and multi-component materials used in pharmaceutical controlled-release formulations that have been doped with paramagnetic species to measure the component domain sizes.

Blind spheres of paramagnetic dopants in solid state NMR

Solid-state NMR on paramagnetically doped crystal structures gives information about the spatial distribution of dopants in the host. Paramagnetic dopants may render NMR active nuclei virtually invisible by relaxation, paramagnetic broadening or shielding. In this contribution blind sphere radii r0 have been reported, which could be extracted through fitting the NMR signal visibility function f(x) = exp(-ar03x) to experimental data obtained on several model compound series: La1-xLnxPO4 (Ln = Nd, Sm, Gd, Dy, Ho, Er, Tm, Yb), Sr1-xEuxGa2S4 and (Zn1-xMnx)3(PO4)2·4H2O. Radii were extracted for 1H, 31P and 71Ga, and dopants like Nd3+, Gd3+, Dy3+, Ho3+, Er3+, Tm3+, Yb3+ and Mn2+. The observed radii determined differed in all cases and covered a range from 5.5 to 13.5 Å. While these radii were obtained from the amount of invisible NMR signal, we also show how to link the visibility function to lineshape parameters. We show under which conditions empirical correlations of linewidth and doping concentration can be used to extract blind sphere radii from second moment or linewidth parameter data. From the second moment analysis of La1-xSmxPO431P MAS NMR spectra for example, a blind sphere size of Sm3+ can be determined, even though the visibility function remains close to 100% over the entire doping range. Dependence of the blind sphere radius r0 on the NMR isotope and on the paramagnetic dopant could be suggested and verified: for different nuclei, r0 shows a -dependence, γ being the gyromagnetic ratio. The blind sphere radii r0 for different paramagnetic dopants in a lanthanide series could be predicted from the pseudo-contact term.