Temperature dependence of contact and dipolar NMR chemical shifts in paramagnetic molecules

Development of Ln(III) Derivatives as 19F Parashift Probes

19F parashift probes with paramagnetically shifted reporter nuclei provide attractive platforms to develop molecular imaging probes. These probes enable ratiometric detection of molecular disease markers using a direct detection technique. Here, we describe a series of trivalent lanthanide (Ln(III)) complexes that are structural analogues of the clinically approved MR contrast agent (CA) ProHance to obtain LnL 19F parashift probes. We evaluated trans-gadolinium paramagnetic lanthanides compared to diamagnetic YL for 19F chemical shift and relaxation rate enhancement. The paramagnetic contribution to chemical shift (δPCS) for paramagnetic LnL exhibited either shifts to lower frequency (δPCS < 0 for TbL, DyL, and HoL) or shifts to higher frequency (δPCS > 0 for ErL, TmL, and YbL) compared to YL 19F spectroscopic signal. Zero-echo time pulse sequences achieved 56-fold sensitivity enhancement for DyL over YL, while developing probe-specific pulse sequences with fast delay times and acquisition times achieved 0.6-fold enhancement in limit of detection for DyL. DyL provides an attractive platform to develop 19F parashift probes for ratiometric detection of enzymatic activity.

Paramagnetic Nuclear Magnetic Resonance Shifts for Triplet Systems and Beyond with Modern Relativistic Density Functional Methods

An efficient framework for the calculation of paramagnetic NMR (pNMR) shifts within exact two-component (X2C) theory and (current-dependent) density functional theory (DFT) up to the class of local hybrid functionals (LHFs) is presented. Generally, pNMR shifts for systems with more than one unpaired electron depend on the orbital shielding contribution and a temperature-dependent term. The latter includes zero-field splitting (ZFS), hyperfine coupling (HFC), and the g-tensor. For consistency, we calculate these three tensors at the same level of theory, i.e., using scalar-relativistic X2C augmented with spin-orbit perturbation theory. Results for pNMR chemical shifts of transition-metal complexes reveal that this X2C-DFT framework can yield good results for both the shifts and the individual tensor contributions of metallocenes and related systems, especially if the HFC constant is large. For small HFC constants, the relative error is often large, and sometimes the sign may be off. 4d and 5d complexes with more complicated structures demonstrate the limitations of a fully DFT-based approach. Additionally, a Co-based complex with a very large ZFS and pronounced multireference character is not well described. Here, a hybrid DFT-multireference framework is necessary for accurate results. Our results show that X2C is sufficient to describe relativistic effects and computationally cheaper than a fully relativistic approach. Thus, it allows use of large basis sets for converged HFCs. Overall, current-dependent meta-generalized gradient approximations and LHFs show some potential; however, the currently available functionals leave a lot to be desired, and the predictive power is limited.

Zero-field splitting parameters within exact two-component theory and modern density functional theory using seminumerical integration

An efficient implementation of zero-field splitting parameters based on the work of Schmitt et al. [J. Chem. Phys. 134, 194113 (2011)] is presented. Seminumerical integration techniques are used for the two-electron spin-dipole contribution and the response equations of the spin-orbit perturbation. The original formulation is further generalized. First, it is extended to meta-generalized gradient approximations and local hybrid functionals. For these functional classes, the response of the paramagnetic current density is considered in the coupled-perturbed Kohn-Sham equations for the spin-orbit perturbation term. Second, the spin-orbit perturbation is formulated within relativistic exact two-component theory and the screened nuclear spin-orbit (SNSO) approximation. The accuracy of the implementation is demonstrated for transition-metal and diatomic main-group compounds. The efficiency is assessed for Mn and Mo complexes. Here, it is found that coarse integration grids for the seminumerical schemes lead to drastic speedups while introducing clearly negligible errors. In addition, the SNSO approximation substantially reduces the computational demands and leads to very similar results as the spin-orbit mean field Ansatz.

Unravelling the effect of paramagnetic Ni2+ on the 13C NMR shift tensor for carbonate in Mg2-xNixAl layered double hydroxides by quantum-chemical computations

Structural disorder and low crystallinity render it challenging to characterise the atomic-level structure of layered double hydroxides (LDH). We report a novel multi-step, first-principles computational workflow for the analysis of paramagnetic solid-state NMR of complex inorganic systems such as LDH, which are commonly used as catalysts and energy storage materials. A series of 13CO32--labelled Mg2-xNixAl-LDH, x ranging from 0 (Mg2Al-LDH) to 2 (Ni2Al-LDH), features three distinct eigenvalues δ11, δ22 and δ33 of the experimental 13C chemical shift tensor. The δii correlate directly with the concentration of the paramagnetic Ni2+ and span a range of |δ11 - δ33| ≈ 90 ppm at x = 0, increasing to 950 ppm at x = 2. In contrast, the isotropic shift, δiso(13C), only varies by -14 ppm in the series. Detailed insight is obtained by computing (1) the orbital shielding by periodic density-functional theory involving interlayer water, (2) the long-range pseudocontact contribution of the randomly distributed Ni2+ ions in the cation layers (characterised by an ab initio susceptibility tensor) by a lattice sum, and (3) the close-range hyperfine terms using a full first-principles shielding machinery. A pseudohydrogen-terminated two-layer cluster model is used to compute (3), particularly the contact terms. Due to negative spin density contribution at the 13C site arising from the close-by Ni2+ sites, this step is necessary to reach a semiquantitative agreement with experiment. These findings influence future NMR investigations of the formally closed-shell interlayer species within LDH, such as the anions or water. Furthermore, the workflow is applicable to a variety of complex materials.

Creating Solid Solutions of Metallocenes: Migration of Nickelocene into the Ferrocene Crystal Lattice in the Absence of a Solvent

Ferrocene and nickelocene do not react with each other in solution; however, the large impact of the paramagnetic component on the ferrocene 1H NMR signal linewidth and relaxation times has been quantified. Co-crystallization of ferrocene and nickelocene at any ratio from a solvent can be explained with both pure substances crystallizing in the same space group P21/n. As a new phenomenon, when a ferrocene single crystal is exposed to polycrystalline nickelocene in the absence of a solvent, the nickelocene migrates into the ferrocene crystal lattice and a mixed crystal is formed that retains its macroscopic shape. This process has been proven visually by cutting the single crystal. Mixing polycrystalline ferrocene with polycrystalline nickelocene at different molar ratios with a mortar and pestle leads to crystalline solid solutions with the corresponding molar ratios of both components. This migration of one organometallic component into an existing crystal lattice of another at ambient temperature in the absence of a solvent has not been described previously. Paramagnetic 1H solid-state NMR spectroscopy of static and rotating samples of dry ferrocene/nickelocene mixtures at varying ratios is used to prove and quantify the mixing of both metallocenes at the molecular level. A single-crystal X-ray structure of a 50/50 mixed crystal corroborates the NMR results that nickelocene and ferrocene are randomly distributed in the lattice and that the space group P21/n is retained. All ferrocene molecules in the mixed crystal lattice show a broadening of their 1H wideline signals and residual magic-angle spinning (MAS) lines at ambient temperature. The broadening of the ferrocene signals correlates with the nickelocene content. 1H T1 relaxation time measurements for the signals of ferrocene in samples with different amounts of nickelocene corroborate the assumption that the signal broadening is due to paramagnetic dipole-dipole relaxation of ferrocene molecules in the vicinity of nickelocene. Spatially separated ferrocene and nickelocene powders in one rotor show the solid-state NMR characteristics of the individual polycrystalline metallocenes. The described formation of solid solutions of metallocenes in the absence of a solvent will open new pathways to homogeneously mixed nanoparticles with desired metal ratios and dual-atom catalysts.

Computational NMR of the iron pyrazolylborate complexes [Tp2Fe]+ and Tp2Fe including solvation and spin-crossover effects

Transition metal complexes have important roles in many biological processes as well as applications in fields such as pharmacy, chemistry and materials science. Paramagnetic nuclear magnetic resonance (pNMR) is a valuable tool in understanding such molecules, and theoretical computations are often advantageous or even necessary in the assignment of experimental pNMR signals. We have employed density functional theory (DFT) and the domain-based local pair natural orbital coupled-cluster method with single and double excitations (DLPNO-CCSD), as well as a number of model improvements, to determine the critical hyperfine part of the chemical shifts of the iron pyrazolylborate complexes [Tp₂Fe]⁺ and Tp₂Fe using a modern version of the Kurland-McGarvey theory, which is based on parameterising the hyperfine, electronic Zeeman and zero-field splitting interactions via the parameters of the electron paramagnetic resonance Hamiltonian. In the doublet [Tp₂Fe]⁺ system, the calculations suggest a re-assignment of the ¹³C signal shifts. Consideration of solvent via the conductor-like polarisable continuum model (C-PCM) versus explicit solvent molecules reveals C-PCM alone to be insufficient in capturing the most important solvation effects. Tp₂Fe exhibits a spin-crossover effect between a high-spin quintet (S = 2) and a low-spin singlet (S = 0) state, and its recorded temperature dependence can only be reproduced theoretically by accounting for the thermal Boltzmann distribution of the open-shell excited state and the closed-shell ground-state occupations. In these two cases, DLPNO-CCSD is found, in calculating the hyperfine couplings, to be a viable alternative to DFT, the demonstrated shortcomings of which have been a significant issue in the development of computational pNMR.

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.

Halide Effects in Reductive Splitting of Dinitrogen with Rhenium Pincer Complexes

Transition metal halide complexes are used as precursors for reductive N₂ activation up to full splitting into nitride complexes. Distinct halide effects on the redox properties and yields are frequently observed yet not well understood. Here, an electrochemical and computational examination of reductive N₂ splitting with the rhenium(III) complexes [ReX₂(PNP)] (PNP = N(CH₂CH₂PtBu₂)₂ and X = Cl, Br, I) is presented. As previously reported for the chloride precursor ( J. Am. Chem. Soc. 2018, 140, 7922), the heavier halides give rhenium(V) nitrides upon (electro-)chemical reduction in good yields yet with significantly anodically shifted electrolysis potentials along the halide series. Dinuclear, end-on N₂-bridged complexes, [{ReX(PNP)}₂(μ-N₂)], were identified as key intermediates in all cases. However, while the chloride complex is exclusively formed via 2-electron reduction and Re^III/Re^I comproportionation, the iodide system also reacts via an alternative Re^II/Re^II-dimerization mechanism at less negative potentials. This alternative pathway relies on the absence of the potential inversion after reduction and N₂ activation that was observed for the chloride precursor. Computational analysis of the relevant Re^III/II and Re^II/I redox couples by energy decomposition analysis attributes the halide-induced trends of the potentials to the dominating electrostatic Re-X bonding interactions over contributions from charge transfer.

Paramagnetic NMR Shielding Tensors Based on Scalar Exact Two-Component and Spin-Orbit Perturbation Theory

The temperature-dependent Fermi-contact and pseudocontact terms are important contributions to the paramagnetic NMR shielding tensor. Herein, we augment the scalar-relativistic (local) exact two-component (X2C) framework with spin-orbit perturbation theory including the screened nuclear spin-orbit correction for the EPR hyperfine coupling and g tensor to compute these temperature-dependent terms. The accuracy of this perturbative ansatz is assessed with the self-consistent spin-orbit two-component and four-component treatments serving as reference. This shows that the Fermi-contact and pseudocontact interaction is sufficiently described for paramagnetic NMR shifts; however, larger deviations are found for the EPR spectra and the principle components of the EPR properties of heavy elements. The impact of the perturbative treatment is further compared to that of the density functional approximation and the basis set. Large-scale calculations are routinely possible with the multipole-accelerated resolution of the identity approximation and the seminumerical exchange approximation, as shown for [CeTi₆O₃(OiPr)₉(salicylate)₆].

Hyperion: A New Computational Tool for Relativistic Ab Initio Hyperfine Coupling

Herein we describe Hyperion, a new program for computing relativistic picture-change-corrected magnetic resonance parameters from scalar relativistic active space wave functions, with or without spin–orbit coupling (SOC) included a posteriori. Hyperion also includes a new orbital decomposition method for assisting active space selection for calculations of hyperfine coupling. For benchmarking purposes, we determine hyperfine coupling constants of selected alkali metal, transition metal, and lanthanide atoms, based on complete active space self-consistent field spin–orbit calculations in OpenMolcas. Our results are in excellent agreement with experimental data from atomic spectroscopy as well as theoretical predictions from four-component relativistic calculations.

Electron correlation and vibrational effects in predictions of paramagnetic NMR shifts

Electronic structure calculations are fundamentally important for the interpretation of nuclear magnetic resonance (NMR) spectra from paramagnetic systems that include organometallic and inorganic compounds, catalysts, or metal-binding sites in proteins. Prediction of induced paramagnetic NMR shifts requires knowledge of electron paramagnetic resonance (EPR) parameters: the electronic g tensor, zero-field splitting D tensor, and hyperfine A tensor. The isotropic part of A, called the hyperfine coupling constant (HFCC), is one of the most troublesome properties for quantum chemistry calculations. Yet, even relatively small errors in calculations of HFCC tend to propagate into large errors in the predicted NMR shifts. The poor quality of A tensors that are currently calculated using density functional theory (DFT) constitutes a bottleneck in improving the reliability of interpretation of the NMR spectra from paramagnetic systems. In this work, electron correlation effects in calculations of HFCCs with a hierarchy of ab initio methods were assessed, and the applicability of different levels of DFT approximations and the coupled cluster singles and doubles (CCSD) method was tested. These assessments were performed for the set of selected test systems comprising an organic radical, and complexes with transition metal and rare-earth ions, for which experimental data are available. Severe deficiencies of DFT were revealed but the CCSD method was able to deliver good agreement with experimental data for all systems considered, however, at substantial computational costs. We proposed a more computationally tractable alternative, where the A was computed with the coupled cluster theory exploiting locality of electron correlation. This alternative is based on the domain-based local pair natural orbital coupled cluster singles and doubles (DLPNO-CCSD) method. In this way the robustness and reliability of the coupled cluster theory were incorporated into the modern formalism for the prediction of induced paramagnetic NMR shifts, and became applicable to systems of chemical interest. This approach was verified for the bis(cyclopentadienyl)vanadium(II) complex (Cp₂V; vanadocene), and the metal-binding site of the Zn²⁺ → Co²⁺ substituted superoxide dismutase (SOD) metalloprotein. Excellent agreement with experimental NMR shifts was achieved, which represented a substantial improvement over previous theoretical attempts. The effects of vibrational corrections to orbital shielding and hyperfine tensor were evaluated and discussed within the second-order vibrational perturbation theory (VPT2) framework.

Dipolar and Contact Paramagnetic NMR Chemical Shifts in AnIV Complexes with Dipicolinic Acid Derivatives

Actinide +IV complexes (An^IV = Th^IV, U^IV, Np^IV, and Pu^IV) with two dipicolinic acid derivatives (DPA and Et-DPA) have been studied by ¹H and ¹³C NMR spectroscopies and first-principles calculations. The Fermi contact and dipolar contributions to the actinide-induced shifts (AIS) are evaluated from a temperature dependence analysis, combined with ab initio results. It allows an experimental estimation of the axial anisotropy of the magnetic susceptibility Δχ_ax and of the hyperfine coupling constants of the NMR-active nuclei. Due to the compactness of the coordination sphere, the magnetic anisotropy of the paramagnetic center is small, and this makes the contact contribution to be the dominant one, even on the remote atoms. The sign of the hyperfine coupling constants and related spin densities is alternating on the nuclei of the ligand cycle, denoting a preponderant spin polarization mechanism. This is well reproduced by unrestricted density functional theory (DFT) calculations. Those values are furthermore slightly decreasing in the actinide series, which indicates a small decrease of the covalency from U^IV to Pu^IV.

Molecular Dynamics and Surface Interactions of Nickelocene Adsorbed on Silica: A Paramagnetic Solid-State NMR Study

When grinding nickelocene with silica in the absence of a solvent at room temperature, it adsorbs on the surface within the pores. This has also been demonstrated visually by adsorbing green nickelocene in the pores of a large colorless silica gel specimen. While this dry adsorption and translational mobility of nickelocene within the pores is proven visually, the site-to-site mobility of the nickelocene molecules and their orientation toward the surface are not yet understood. In this contribution, mesoporous silica is used as the support material for a systematic solid-state NMR study of these issues. Paramagnetic ¹H VT solid-state NMR and T₁ relaxation times have been powerful tools for studying the dynamics of nickelocene on the silica surface. Herewith, the mobility of the surface-adsorbed nickelocene molecules in the pores could be quantified on the molecular scale. According to the obtained data, the nickelocene molecules move like a liquid on the surface. Isotropically moving molecules exchange places rapidly with surface-attached molecular states of nickelocene in a sample with submonolayer surface coverage. This finding is corroborated by a macroscopic visualization experiment. The states of the surface-attached horizontally oriented nickelocene molecules that are prevalent at temperatures below 200 K have been quantified. The temperature dependencies of the rate k in coordinates of ln(k) versus 1/T and ln(k/T) versus 1/T form ideal straight lines that allow the determination of the kinetic parameters E_act = 5.5 kcal/mol, A = 1.1 × 10¹⁰, ΔH^‡ = 5.0 kcal/mol, and ΔS^‡ = -15 eu. Investigating a sample with equal amounts of nickelocene and ferrocene in a submonolayer amount of 80% overall surface coverage shows that the different metallocenes mix on the molecular level on the silica surface.

Theoretical analysis of the long-distance limit of NMR chemical shieldings

After some years of controversy, it was recently demonstrated how to obtain the correct long-distance limit [point-dipole approximation (PDA)] of pseudo-contact nuclear magnetic resonance chemical shifts from rigorous first-principles quantum mechanics [Lang et al., J. Phys. Chem. Lett. 11, 8735 (2020)]. This result confirmed the classical Kurland-McGarvey theory. In the present contribution, we elaborate on these results. In particular, we provide a detailed derivation of the PDA both from the Van den Heuvel-Soncini equation for the chemical shielding tensor and from a spin Hamiltonian approximation. Furthermore, we discuss in detail the PDA within the approximate density functional theory and Hartree-Fock theories. In our previous work, we assumed a relatively crude effective nuclear charge approximation for the spin-orbit coupling operator. Here, we overcome this assumption by demonstrating that the derivation is also possible within the fully relativistic Dirac equation and even without the assumption of a specific form for the Hamiltonian. Crucial ingredients for the general derivation are a Hamiltonian that respects gauge invariance, the multipolar gauge, and functional derivatives of the Hamiltonian, where it is possible to identify the first functional derivative with the electron number current density operator. The present work forms an important foundation for future extensions of the Kurland-McGarvey theory beyond the PDA, including induced magnetic quadrupole and higher moments to describe the magnetic hyperfine field.

Analysis of reduced paramagnetic shifts as an effective tool in NMR spectroscopy

A recently introduced concept of reduced paramagnetic shifts (RPS) in NMR spectroscopy is applied here to a series of paramagnetic complexes with different metal ions, such as iron(II), iron(III) and cobalt(II), in different coordination environments of N-donor ligands, including a unique trigonal-prismatic geometry that is behind some record single-molecule magnet behaviours. A simple, almost visual analysis of the chemical shifts as a function of temperature, which is at the core of this approach, allows for a correct signal assignment and evaluation of the anisotropy of the magnetic susceptibility, the key indicator of a good single molecule magnet, that often cannot be done using traditional techniques rooted in quantum chemistry and NMR spectroscopy. The proposed approach thus emerged as a powerful alternative in deciphering the NMR spectra of paramagnetic compounds for applications in data processing and storage, magnetic resonance imaging and structural biology.

Photo-Initiated Cobalt-Catalyzed Radical Olefin Hydrogenation

Outer-sphere radical hydrogenation of olefins proceeds via stepwise hydrogen atom transfer (HAT) from transition metal hydride species to the substrate. Typical catalysts exhibit M-H bonds that are either too weak to efficiently activate H2 or too strong to reduce unactivated olefins. This contribution evaluates an alternative approach, that starts from a square-planar cobalt(II) hydride complex. Photoactivation results in Co-H bond homolysis. The three-coordinate cobalt(I) photoproduct binds H2 to give a dihydrogen complex, which is a strong hydrogen atom donor, enabling the stepwise hydrogenation of both styrenes and unactivated aliphatic olefins with H2 via HAT.

Paramagnetic NMR Shielding Tensors and Ring Currents: Efficient Implementation and Application to Heavy Element Compounds

We present an efficient implementation of paramagnetic NMR shielding tensors and shifts in a nonrelativistic and scalar-relativistic density functional theory framework. For the latter, we make use of the scalar exact two-component Hamiltonian in its local approximation, and generally we apply the well established (multipole-accelerated) resolution of the identity approximation and the seminumerical exchange approximation. The perturbed density matrix of a paramagnetic NMR shielding calculation is further used to study the magnetically induced current density and ring currents of open-shell systems as illustrated for [U@Bi₁₂]^3-. [U@Bi₁₂]^3- features delocalized highest occupied molecular orbitals and sustains a net diatropic ring current of ca. 18 nA/T through the Bi₁₂ torus similar to the all-metal aromatic heavy-element cluster [Th@Bi₁₂]^4-.

Observability of Paramagnetic NMR Signals at over 10 000 ppm Chemical Shifts

We report an experimental observation of 31 P NMR resonances shifted by over 10 000 ppm (meaning percent range, and a new record for solutions), and similar 1 H chemical shifts, in an intermediate-spin square planar ferrous complex [tBu (PNP)Fe-H], where PNP is a carbazole-based pincer ligand. Using a combination of electronic structure theory, nuclear magnetic resonance, magnetometry, and terahertz electron paramagnetic resonance, the influence of magnetic anisotropy and zero-field splitting on the paramagnetic shift and relaxation enhancement is investigated. Detailed spin dynamics simulations indicate that, even with relatively slow electron spin relaxation (T1 ≈10-11  s), it remains possible to observe NMR signals of directly metal-bonded atoms because pronounced rhombicity in the electron zero-field splitting reduces nuclear paramagnetic relaxation enhancement.

Crystal and Substituent Effects on Paramagnetic NMR Shifts in Transition-Metal Complexes

Nuclear magnetic resonance (NMR) spectroscopy of paramagnetic molecules provides detailed information about their molecular and electron-spin structure. The paramagnetic NMR spectrum is a very rich source of information about the hyperfine interaction between the atomic nuclei and the unpaired electron density. The Fermi-contact contribution to ligand hyperfine NMR shifts is particularly informative about the nature of the metal–ligand bonding and the structural arrangements of the ligands coordinated to the metal center. In this account, we provide a detailed experimental and theoretical NMR study of compounds of Cr(III) and Cu(II) coordinated with substituted acetylacetonate (acac) ligands in the solid state. For the first time, we report the experimental observation of extremely paramagnetically deshielded ¹³C NMR resonances for these compounds in the range of 900–1200 ppm. We demonstrate an excellent agreement between the experimental NMR shifts and those calculated using relativistic density-functional theory. Crystal packing is shown to significantly influence the NMR shifts in the solid state, as demonstrated by theoretical calculations of various supramolecular clusters. The resonances are assigned to individual atoms in octahedral Cr(acac)₃ and square-planar Cu(acac)₂ compounds and interpreted by different electron configurations and magnetizations at the central metal atoms resulting in different spin delocalizations and polarizations of the ligand atoms. Further, effects of substituents on the ¹³C NMR resonance of the ipso carbon atom reaching almost 700 ppm for Cr(acac)₃ compounds are interpreted based on the analysis of Fermi-contact hyperfine contributions.

The ligand NMR shifts in paramagnetic acetylacetonato Cr(III) and Cu(II) complexes have been predicted and measured in the solid state and interpreted by relativistic DFT calculations. The effects of the metal atom, ligand, and crystal packing on the spin delocalization and polarization reflected in the Fermi-contact contribution to the hyperfine interaction are rationalized.

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.

Temperature Dependence of 1 H Paramagnetic Chemical Shifts in Actinide Complexes, Beyond Bleaney's Theory: The AnVI O2 2+ -Dipicolinic Acid Complexes (An=Np, Pu) as an Example

Actinide +VI complexes ( A n V I = U V I , N p V I and P u V I ) with dipicolinic acid derivatives were synthesized and characterized by powder XRD, SQUID magnetometry and NMR spectroscopy. In addition, N p V I and P u V I complexes were described by first principles CAS based and two-component spin-restricted DFT methods. The analysis of the ¹ H paramagnetic NMR chemical shifts for all protons of the ligands according to the X-rays structures shows that the Fermi contact contribution is negligible in agreement with spin density determined by unrestricted DFT. The magnetic susceptibility tensor is determined by combining SQUID, pNMR shifts and Evans' method. The SO-RASPT2 results fit well the experimental magnetic susceptibility and pNMR chemical shifts. The role of the counterions in the solid phase is pointed out; their presence impacts the magnetic properties of the N p V I complex. The temperature dependence of the pNMR chemical shifts has a strong 1 / T contribution, contrarily to Bleaney's theory for lanthanide complexes. The fitting of the temperature dependence of the pNMR chemical shifts and SQUID magnetic susceptibility by a two-Kramers-doublet model for the N p V I complex and a non-Kramers-doublet model for the P u V I complex allows for the experimental evaluation of energy gaps and magnetic moments of the paramagnetic center.

A Quantum Chemistry View on Two Archetypical Paramagnetic Pentacoordinate Nickel(II) Complexes Offers a Fresh Look on Their NMR Spectra

Quantum chemical methods for calculating paramagnetic NMR observables are becoming increasingly accessible and are being included in the inorganic chemistry practice. Here, we test the performance of these methods in the prediction of proton hyperfine shifts of two archetypical high-spin pentacoordinate nickel(II) complexes (NiSAL-MeDPT and NiSAL-HDPT), which, for a variety of reasons, turned out to be perfectly suited to challenge the predictions to the finest level of detail. For NiSAL-MeDPT, new NMR experiments yield an assignment that perfectly matches the calculations. The slightly different hyperfine shifts from the two “halves” of the molecules related by a pseudo-C₂ axis, which are experimentally divided into two well-defined spin systems, are also straightforwardly distinguished by the calculations. In the case of NiSAL-HDPT, for which no X-ray structure is available, the quality of the calculations allowed us to refine its structure using as a starting template the structure of NiSAL-MeDPT.

State-of-the-art quantum chemical methods and paramagnetism-tailored NMR experiments provide a deep insight on the relation between the spectra and the electronic structure for two paramagnetic pentacoordinate nickel(II) complexes.

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.

Remarkable reversal of 13C-NMR assignment in d1, d2 compared to d8, d9 acetylacetonate complexes: analysis and explanation based on solid-state MAS NMR and computations

13C solid-state MAS NMR spectra of a series of paramagnetic metal acetylacetonate complexes; [VO(acac)2] (d1, S = ½), [V(acac)3] (d2, S = 1), [Ni(acac)2(H2O)2] (d8, S = 1), and [Cu(acac)2] (d9, S = ½), were assigned using modern NMR shielding calculations. This provided a reliable assignment of the chemical shifts and a qualitative insight into the hyperfine couplings. Our results show a reversal of the isotropic 13C shifts, δiso(13C), for CH3 and CO between the d1 and d2versus the d8 and d9 acetylacetonate complexes. The CH3 shifts change from about -150 ppm (d1,2) to roughly 1000 ppm (d8,9), whereas the CO shifts decrease from 800 ppm to about 150 ppm for d1,2 and d8,9, respectively. This was rationalized by comparison of total spin-density plots and computed contact couplings to those corresponding to singly occupied molecular orbitals (SOMOs). This revealed the interplay between spin delocalization of the SOMOs and spin polarization of the lower-energy MOs, influenced by both the molecular symmetry and the d-electron configuration. A large positive chemical shift results from spin delocalization and spin polarization acting in the same direction, whereas their cancellation corresponds to a small shift. The SOMO(s) for the d8 and d9 complexes are σ-like, implying spin-delocalization on the CH3 and CO groups of the acac ligand, cancelled only for CO by spin polarization. In contrast, the SOMOs of the d1 and d2 systems are π-like and a large CO-shift results from spin polarization, which accounts for the reversed assignment of δiso(13C) for CH3 and CO.

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.

NMR chemical shifts of urea loaded copper benzoate. A joint solid-state NMR and DFT study

We report solid-state ¹³C NMR spectra of urea-loaded copper benzoate, Cu₂(C₆H₅CO₂)₄·2(urea), a simplified model for copper paddlewheel-based metal-organic frameworks (MOFs), along with first-principles density functional theory (DFT) computation of the paramagnetic NMR (pNMR) chemical shifts. Assuming a Boltzmann distribution between a diamagnetic open-shell singlet ground state (in a broken-symmetry Kohn-Sham DFT description) and an excited triplet state, the observed δ(¹³C) values are reproduced reasonably well at the PBE0-⅓/IGLO-II//PBE0-D3/AE1 level. Using the proposed assignments of the signals, the mean absolute deviation between computed and observed ¹³C chemical shifts is below 30 ppm over a range of more than 1100 ppm.

Probing Jahn-Teller distortions in Mn(acac)3 through paramagnetic interactions in solid-state MAS NMR

The structural and electronic properties of solid coordination compounds are of great interest in a wide variety of applications. The sensitivity of solid-state NMR to local structural environments in the presence of a transition metal with unpaired spins is demonstrated through ¹³C and ¹H MAS NMR spectra of tris(acetylacetonato)manganese (III), Mn(acac)₃. The spectral assignment is established using a combination of density-functional theory and experimental NMR methods. An analysis of the spin-density distribution throughout the ligands as probed by NMR provides an understanding of the molecular electronic structure. The paramagnetic interaction enhances the spectral resolution and offers the possibility of using these spectra as structural fingerprints. The unique character of Mn(acac)₃ to manifest various degrees of Jahn-Teller distortions allows for a discussion of the relation between the electron occupancy, crystal structure, and ¹³C NMR peak shifts.

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.

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.

Ab initio paramagnetic NMR shifts via point-dipole approximation in a large magnetic-anisotropy Co(ii) complex

Transition metal complexes can possess a large magnetic susceptibility anisotropy, facilitating applications such as paramagnetic tags or shift agents in nuclear magnetic resonance (NMR) spectroscopy. Due to its g-shift and zero-field splitting (ZFS) we demonstrate on a Co(ii) clathrochelate with an aliphatic 16-carbon chain, a modern approach for ab initio calculation of paramagnetic susceptibility. Due to its large anisotropy, large linear dimension but relatively low number of atoms, the chosen complex is especially well-suited for testing the long-range point-dipole approximation (PDA) for the pseudocontact shifts (PCSs) of paramagnetic NMR. A static structure of the complex is used to compare the limiting long-distance PDA with full first-principles quantum-mechanical calculation. A non-symmetric formula for the magnetic susceptibility tensor is necessary to be consistent with the latter. Comparison with experimental shifts is performed by conformational averaging over the chain dynamics using Monte Carlo simulation. We observe satisfactory accuracy from the rudimentary simulation and, more importantly, demonstrate the fast applicability of the ab initio PDA.

Interplay of Through-Bond Hyperfine and Substituent Effects on the NMR Chemical Shifts in Ru(III) Complexes

The links between the molecular structure and nuclear magnetic resonance (NMR) parameters of paramagnetic transition-metal complexes are still relatively unexplored. This applies particularly to the contact term of the hyperfine contribution to the NMR chemical shift. We report combining experimental NMR with relativistic density functional theory (DFT) to study a series of Ru(III) complexes with 2-substituted β-diketones. A series of complexes with systematically varied substituents was synthesized and analyzed using ¹H and ¹³C NMR spectroscopy. The NMR spectra recorded at several temperatures were used to construct Curie plots and estimate the temperature-independent (orbital) and temperature-dependent (hyperfine) contributions to the NMR shift. Relativistic DFT calculations of electron paramagnetic resonance and NMR parameters were performed to interpret the experimental observations. The effects of individual factors such as basis set, density functional, exact-exchange admixture, and relativity are analyzed and discussed. Based on the calibration study in this work, the fully relativistic Dirac-Kohn-Sham (DKS) method, the GIAO approach (orbital shift), the PBE0 functional with the triple-ζ valence basis sets, and the polarizable continuum model for describing solvent effects were selected to calculate the NMR parameters. The hyperfine contribution to the total paramagnetic NMR (pNMR) chemical shift is shown to be governed by the Fermi-contact (FC) term, and the substituent effect (H vs Br) on the through-bond FC shifts is analyzed, interpreted, and discussed in terms of spin-density distribution, atomic spin populations, and molecular-orbital theory. In contrast to the closed-shell systems of Rh(III), the presence of a single unpaired electron in the open-shell Ru(III) analogs significantly alters the NMR resonances of the ligand atoms distant from the metal center in synergy with the substituent effect.

Fe(ii) and Co(ii) N-methylated CYCLEN complexes as paraSHIFT agents with large temperature dependent shifts

Several complexes of Co(ii) or Fe(ii) with 1,4,7,10-tetraazacyclododecane (CYCLEN) appended with 1,7-(6-methyl)2-picolyl groups are studied as ¹H NMR paraSHIFT agents (paramagnetic shift agents) for the registration of temperature. Two of the complexes, [Co(BMPC)]²⁺ and [Fe(BMPC)]²⁺, contain methyl groups only on the methyl picolyl pendents. Two other complexes, [Co(2MPC)]²⁺ and [Fe(2MPC)]²⁺, contain picolyl groups and also methyl groups on the macrocyclic amines. All macrocyclic complexes are in high spin form as shown by solution magnetic moments in the range of 5.0-5.9μ_BM and 5.3-5.8μ_BM for Co(ii) and Fe(ii) complexes, respectively. The ¹H NMR spectra of both of the Fe(ii) complexes and one of the Co(ii) complexes are consistent with a predominant diastereomeric form in deuterium oxide solutions. The highly shifted methyl proton resonances for [Co(2MPC)]²⁺ appear at 164 and -113 ppm for macrocycle and pendent picolyl methyls and show temperature coefficients of -0.58 ppm °C^-1 and 0.49 ppm °C^-1, respectively. Fe(ii) complexes have less shifted methyl proton resonances and smaller temperature coefficients. The ¹H resonances of [Fe(2MPC)]²⁺ appear at 105 ppm and -46 ppm with corresponding temperature coefficients (CT) of -0.29 ppm °C^-1 and 0.22 ppm °C^-1, respectively. The relatively narrow linewidths of [Fe(2MPC)]²⁺, however, produce superior CT/FWHM values of 0.44 and 0.31 °C^-1 for the N-methyl and picolyl proton resonances where FWHM is the full width at half maximum of the ¹H resonance. The crystal structure of [Co(BMPC)]Cl₂ shows a six-coordinate Co(ii) bound to the macrocyclic amines and two pendent picolyl groups. The distorted trigonal prismatic geometry of the complex resembles that of an analogous complex containing four 6-methyl-2-picolyl groups, in which only two picolyl pendents are coordinated.

Tetranuclear and dinuclear phenoxido bridged copper complexes based on unsymmetrical thiosemicarbazone ligands

Dinuclear and tetranuclear copper(ii) complexes based on N,S ligand systems in the solid state and in solution are described.

NMR analysis of an Fe(i)–carbene complex with strong magnetic anisotropy

A paramagnetic, easy-plane anisotropic Fe^I complex, bearing cyclic-alkyl(amino) carbene (cAAC) ligands, is studied by means of NMR and DFT.

Pseudo-Contact NMR Shifts over the Paramagnetic Metalloprotein CoMMP-12 from First Principles

Long-range pseudo-contact NMR shifts (PCSs) provide important restraints for the structure refinement of proteins when a paramagnetic metal center is present, either naturally or introduced artificially. Here we show that ab initio quantum-chemical methods and a modern version of the Kurland-McGarvey approach for paramagnetic NMR (pNMR) shifts in the presence of zero-field splitting (ZFS) together provide accurate predictions of all PCSs in a metalloprotein (high-spin cobalt-substituted MMP-12 as a test case). Computations of 314 ¹³ C PCSs using g- and ZFS tensors based on multi-reference methods provide a reliable bridge between EPR-parameter- and susceptibility-based pNMR formalisms. Due to the high sensitivity of PCSs to even small structural differences, local structures based either on X-ray diffraction or on various DFT optimizations could be evaluated critically by comparing computed and experimental PCSs. Many DFT functionals provide insufficiently accurate structures. We also found the available 1RMZ PDB X-ray structure to exhibit deficiencies related to binding of a hydroxamate inhibitor. This has led to a newly refined PDB structure for MMP-12 (5LAB) that provides a more accurate coordination arrangement and PCSs.