Symmetry rules for the indirect nuclear spin-spin coupling tensor revisited

Impact of the electric field on isotropic and anisotropic spin Hamiltonian parameters

One may obviously think that the best way to control magnetic properties relies on using a magnetic field. However, it is not convenient to focus a magnetic field on a small object, whereas it is much easier to do so with an electric field. Magnetoelectric coupling allows one to control the magnetization with the electric field and the polarization with the magnetic field and could therefore provide a solution to this problem. This paper aims at quantifying the impact of the electric field on both the isotropic magnetic exchange and the Dzyaloshinskii–Moriya interaction in the case of a binuclear system of S = 1/2 spins. This study follows previous studies that showed that very high Dzyaloshinskii–Moriya interaction, i.e., the antisymmetric exchange, can be generated when close to first order spin orbit coupling. We will, therefore, explore this regime in a model Cu(II) complex that exhibits a quasi-degeneracy of the [Formula: see text] and d xy orbitals. This situation is indeed the one that allows us to obtain the largest spin orbit couplings in transition metal complexes. We will show that both the magnetic exchange and the Dzyaloshinskii–Moriya interaction are very sensitive to the electric field and that it would therefore be possible to modulate and control magnetic properties by the electric field. Finally, rationalizations of the obtained results will be proposed.

Antisymmetric Exchange in a Real Copper Triangular Complex

The antisymmetric exchange, also known as the Dzyaloshinskii-Moriya interaction (DMI), is an effective interaction that may be at play in isolated complexes (with transition metals or lanthanides, for instance), nanoparticles, and highly correlated materials with adequate symmetry properties. While many theoretical works have been devoted to the analysis of single-ion zero-field splitting and to a lesser extent to symmetric exchange, only a few ab initio studies deal with the DMI. Actually, it originates from a subtle interplay between weak electronic interactions and spin-orbit couplings. This article aims to highlight the origin of this interaction from theoretical grounds in a real tri-copper(II) complex, capitalizing on previous methodological studies on bi-copper(II) model complexes. By tackling this three-magnetic-center system, we will first show that the multispin model Hamiltonian is appropriate for trinuclear (and likely for higher nuclearity) complexes, then that the correct application of the permutation relationship is necessary to explain the outcomes of the ab initio calculations, and finally, that the model parameters extracted from a binuclear model transfer well to the trinuclear complex. For a more theory-oriented purpose, we will show that the use of a simplified structural model allows one to perform more demanding electronic structure calculations. On this simpler system, we will first check that the previous transferability is still valid, prior to performing more advanced calculations on the derived two-magnetic-center model system. To this end, we will explain in detail the physics of the DMI in the copper triangle of interest, before advocating further theory/experiment efforts.

Antisymmetric Spin Exchange in a μ-1,2-Peroxodicopper(II) Complex with an Orthogonal Cu-O-O-Cu Arrangement and S = 1 Spin Ground State Characterized by THz-EPR

A unique type of Cu₂/O₂ adduct with orthogonal (close to 90°) Cu-O-O-Cu arrangement has been proposed for initial stages of O₂ binding at biological type III dicopper sites, and targeted ligand design has now allowed us to emulate such an adduct in a pyrazolate-based μ-η¹ :η¹-peroxodicopper(II) complex (2) with Cu-O-O-Cu torsion φ of 87°, coined ^⊥ P intermediate. Full characterization of 2, including X-ray diffraction (d _O-O = 1.452 Å) and Raman spectroscopy (ν̃_O-O = 807 cm^-1), completes a series of closely related Cu₂/O₂ intermediates featuring μ-η¹ :η¹-peroxodicopper(II) cores with φ ranging from 55° (A, cis-peroxo ^C P; Brinkmeier A.et al., J. Am. Chem. Soc.2021, 143, 10361) via 87° (2, ^⊥ P type) up to 104° (B, approaching trans-peroxo ^T P; Kindermann N.et al., Angew. Chem., Int. Ed.2015, 54, 1738). SQUID magnetometry revealed ferromagnetic interaction of the Cu^II ions and a triplet (S _t = 1) ground state in 2. Frequency-domain THz-EPR has been employed to quantitatively investigate the spin systems of 2 and B. Magnetic transitions within the triplet ground states confirmed their substantial zero-field splittings (ZFS) suggested by magnetometry. Formally forbidden triplet-to-singlet transitions at 56 (2) and 157 cm^-1 (B), which are in agreement with the exchange coupling strengths J _iso inferred from SQUID data, are reported for the first time for coupled dicopper(II) complexes. Rigorous analysis by spin-Hamiltonian-based simulations attributed the corresponding nonzero transition probabilities and the ZFS to substantial antisymmetric (Dzyaloshinskii-Moriya) exchange d and provided robust values and orientations for the d , J , and g tensors. These interactions can be correlated with the Cu-O-O-Cu geometries, revealing a linear increase of J _iso with the Cu-O-O-Cu torsion and a strong linear decrease with the Cu-O-O angle. Relevance of the ^⊥ P intermediate for O₂ activation at type III dicopper sites and a potential role of antisymmetric exchange in the concomitant intersystem crossing are proposed.

Extraction of giant Dzyaloshinskii-Moriya interaction from ab initio calculations: First-order spin-orbit coupling model and methodological study

The Dzyaloshinskii-Moriya interaction is expected to be at the origin of interesting magnetic properties, such as multiferroicity, skyrmionic states, and exotic spin orders. Despite this, its theoretical determination is far from being established, neither from the point of view of ab initio methodologies nor from that of the extraction technique to be used afterward. Recently, a very efficient way to increase its amplitude has been demonstrated near the first-order spin-orbit coupling regime. Within the first-order regime, the anisotropic spin Hamiltonian involving the Dzyaloshinskii-Moriya operator becomes inappropriate. Nevertheless, in order to approach this regime and identify the spin Hamiltonian limitations, it is necessary to characterize the underlying physics. To this end, we have developed a simple electronic and spin-orbit model describing the first-order regime and used ab initio calculations to conduct a thorough methodological study.

How to create giant Dzyaloshinskii-Moriya interactions? Analytical derivation and ab initio calculations on model dicopper(II) complexes

This paper is a theoretical "proof of concept" on how the on-site first-order spin-orbit coupling (SOC) can generate giant Dzyaloshinskii-Moriya interactions in binuclear transition metal complexes. This effective interaction plays a key role in strongly correlated materials, skyrmions, multiferroics, and molecular magnets of promising use in quantum information science and computing. Despite this, its determination from both theory and experiment is still in its infancy and existing systems usually exhibit very tiny magnitudes. We derive analytical formulas that perfectly reproduce both the nature and the magnitude of the Dzyaloshinskii-Moriya interaction calculated using state-of-the-art ab initio calculations performed on model bicopper(II) complexes. We also study which geometrical structures/ligand-field forces would enable one to control the magnitude and the orientation of the Dzyaloshinskii-Moriya vector in order to guide future synthesis of molecules or materials. This article provides an understanding of its microscopic origin and proposes recipes to increase its magnitude. We show that (i) the on-site mixings of 3d orbitals rule the orientation and magnitude of this interaction, (ii) increased values can be obtained by choosing more covalent complexes, and (iii) huge values (∼1000 cm^-1) and controlled orientations could be reached by approaching structures exhibiting on-site first-order SOC, i.e., displaying an "unquenched orbital momentum."

Electric Dipole-Magnetic Dipole Polarizability and Anapole Magnetizability of Hydrogen Peroxide as Functions of the HOOH Dihedral Angle

The trace of tensors that account for chiroptical response of the H₂O₂ molecule is a function of the HO-OH dihedral angle. It vanishes at 0° and 180°, due to the presence of molecular symmetry planes, but also for values in the range 90-100° of this angle, in which the molecule is unquestionably chiral. Such an atypical effect is caused by counterbalancing contributions of diagonal tensor components with nearly maximal magnitude but opposite sign, determined by electron flow in open or closed helical paths, and associated with induced electric and magnetic dipole moments and anapole moments. For values of dihedral angle external to the 90-100° interval, the helical paths become smaller in size, thus reducing the amount of cancellation among diagonal components. Shrinking of helical paths determines the appearance of extremum values of tensor traces approximately at 50° and 140° dihedral angles.

Chiral discrimination in nuclear magnetic resonance spectroscopy

Chirality is a fundamental property of molecules whose spatial symmetry is characterized by the absence of improper rotations, making them not superimposable to their mirror image. Chiral molecules constitute the elementary building blocks of living species and one enantiomer is favoured in general (e.g. L-aminoacids and D-sugars pervade terrestrial homochiral biochemistry) because most chemical reactions producing natural substances are enantioselective. Since the effect of chiral chemicals and drugs on living beings can be markedly different between enantiomers, the quest for practical spectroscopical methods to scrutinize chirality is an issue of great importance and interest. Nuclear magnetic resonance (NMR) is a topmost analytical technique, but spectrometers currently used are 'blind' to chirality, i.e. unable to discriminate the two mirror-image forms of a chiral molecule, because, in the absence of a chiral solvent, the spectral parameters, chemical shifts and spin-spin coupling constants are identical for enantiomers. Therefore, the development of new procedures for routine chiral recognition would offer basic support to scientists. However, in the presence of magnetic fields, a distinction between true and false chirality is mandatory. The former epitomizes natural optical activity, which is rationalized by a time-even pseudoscalar, i.e. the trace of a second-rank tensor, the mixed electric dipole/magnetic dipole polarizability. The Faraday effect, magnetic circular dichroism and magnetic optical activity are instead related to a time-odd axial vector. The present review summarizes recent theoretical and experimental efforts to discriminate enantiomers via NMR spectroscopy, with the focus on the deep connection between chirality and symmetry properties under the combined set of fundamental discrete operations, namely charge conjugation, parity (space inversion) and time (motion) reversal.

Antisymmetric Couplings Enable Direct Observation of Chirality in Nuclear Magnetic Resonance Spectroscopy

Here we demonstrate that a term in the nuclear spin Hamiltonian, the antisymmetric J-coupling, is fundamentally connected to molecular chirality. We propose and simulate a nuclear magnetic resonance (NMR) experiment to observe this interaction and differentiate between enantiomers without adding any additional chiral agent to the sample. The antisymmetric J-coupling may be observed in the presence of molecular orientation by an external electric field. The opposite parity of the antisymmetric coupling tensor and the molecular electric dipole moment yields a sign change of the observed coupling between enantiomers. We show how this sign change influences the phase of the NMR spectrum and may be used to discriminate between enantiomers.

A ZORA-DFT and NLMO study of the one-bond fluorine–X indirect nuclear spin-spin coupling tensors for various VSEPR geometries

Zeroth-order regular approximation (ZORA) density functional theory (DFT) calculations of one-bond X–19F indirect nuclear spin-spin coupling (J) tensors were performed on a series of fluorine-containing compounds covering several valence shell electron pair repulsion (VSEPR) theory geometries for which J, by symmetry, is not required to be axially symmetric. The calculations show that the antisymmetric components of J are only of the same order of magnitude as the principal components of the symmetric J-coupling tensor for a few geometries, and that in cases of approximate axial symmetry along the bond, J remains nearly axially symmetric with its unique component along the bond. In general, different species having the same nominal geometry tend to have similar tensor orientations, magnitudes of anisotropy of J relative to the isotropic coupling constant, as well as the same dominant contributions from the different coupling mechanisms. Structures are also systematically modified to determine how the tensor components depend on geometrical parameters. The isotropic coupling constants are subsequently interpreted using a natural localized molecular orbital (NLMO) approach. Our results could prove to be useful for future experimental characterizations of J tensors in systems having symmetry properties that do not force J to be axially symmetric or coincident with the dipolar coupling tensor.

Prediction of NMR J-coupling in solids with the planewave pseudopotential approach

We review the calculation of NMR J-coupling in solid materials using the planewave pseudopotential formalism of Density Functional Theory. The methodology is briefly summarised and an account of recent applications is given. We discuss various aspects of the calculations which should be taken into account when comparing results with solid-state NMR experiments including anisotropy and orientation of the J tensors, the reduced coupling constant, and the relation between J and crystal structure.

Measurement of delta(1)J((199)Hg, (31)P) in [HgPCy3(OAc)2]2 and relativistic ZORA DFT investigations of mercury-phosphorus coupling tensors

Using 31P solid-state NMR spectroscopy, anisotropy in the indirect 199Hg-31P spin-spin coupling tensor (DeltaJ) for powdered [HgPCy3(OAc)2]2 (1) has been measured as 4700 +/- 300 Hz. Zeroth-order regular approximation (ZORA) density functional theory (DFT) calculations, including scalar and spin-orbit relativistic effects, performed on 1 and a series of other related compounds show that DeltaJ(199Hg, (31)P) arises entirely from the ZORA Fermi-contact-spin-dipolar cross term. The calculations validate assumptions made in the spectral analysis of 1 and in previous determinations of DeltaJ in powder samples, namely that J is axially symmetric and shares its principal axis system with the direct dipolar coupling tensor (D). Agreement between experiment and theory for various 199Hg, 31P spin-spin coupling anisotropies is reasonable; however, experimental values of 1J(199Hg, 31P)(iso) are significantly underestimated by the calculations. The most important improvements in the agreement were obtained as a result of including more of the crystal lattice in the model used for the calculations, e.g., a change of 43% was noted for 1J(199Hg, 31P)(iso) in [HgPPh3(NO3)2]2 depending on whether the two or three nearest nitrate ions are included in the model. Finally, we have written a computer program to simulate the effects of non-axial symmetry in J and of non-coincidence of the J and D on powder NMR spectra. Simulations clearly show that both of these effects have a pronounced impact on the 31P NMR spectrum of 199Hg-31P spin pairs, suggesting that the effects should be observable experimentally if a suitable compound can be identified.

NMR line shapes from AB spin systems in solids — The role of antisymmetric spin–spin coupling

NMR parameters such as indirect nuclear spin–spin coupling (J), nuclear magnetic shielding (σ), direct dipolar coupling (D), and electric field gradient (V) are properly described by second-rank tensors. Each may be decomposed into isotropic, symmetric, and antisymmetric components; the number of these three components which may be nonzero is a distinguishing attribute of each interaction tensor. The rank-1 antisymmetric portion of J (Janti) holds the distinction of remaining the only nonzero part of these fundamental NMR interaction tensors which has never been observed experimentally. Accordingly, effects from Janti are usually ignored, but it is important to consider when this is valid. An experimental strategy for observing Janti in powdered samples of tightly coupled homonuclear spin pairs, based on ideas originally presented by Andrew and Farnell ( Mol. Phys. 1968, 15, 157 ), is described. The theory of Andrew and Farnell is extended to powder samples, and methods for analyzing NMR spectra from powdered samples are presented. It is found that, in certain rare cases, Janti has the potential to affect the NMR line shapes from AB spin systems, but that even in these systems, the most intense features of the spectra are not affected and may be analyzed independently of Janti. Furthermore, Janti will only have an observable effect on the NMR spectra when its magnitude is comparable with that of Jiso and with the difference in chemical shifts (in Hz) between the two sites. Finally, the first experimental attempts to measure Janti are reported, and experimental proof that no elements of Janti(119Sn,119Sn) in hexa(p-tolyl)ditin are larger than 2900 Hz is given. The benefits of modern double-quantum filtering NMR pulse sequences in isolating effects from Janti are also illustrated.

Anisotropy in the 31P, 63/65Cu indirect spin-spin coupling and 31P nuclear shielding tensors of linear copper(I) phosphines

Phosphorus-31 CP/MAS NMR spectra of two linear bis(tribenzylphosphine)cuprate(I) salts reveal significant anisotropy in the one-bond 31P, 63,65Cu J tensors. The parallel and perpendicular components of 1J(31P, 63Cu), for example, are +2.0 and +1.3 kHz, respectively. The well-characterized space group and point symmetry in one of these compounds greatly simplifies the 31P lineshape analysis, thus dispensing with the need to make assumptions about the interaction tensors involved. This offers an unique opportunity to study DeltaJ by calculating dipolar coupling constants from known internuclear distances, and by using zero-field nuclear quadrupole resonance spectroscopy to obtain the magnitudes of the 63/65Cu nuclear quadrupole coupling constants, CQ. Copper-63 CQ values in these complexes exceed 80 MHz, proving to be the largest reported for copper(I) phosphines. Phosphorus-31 NMR spectra of non-spinning samples at three applied magnetic fields are also presented, along with lineshape calculations based on full-matrix Zeeman-quadrupolar Hamiltonian diagonalization. It is shown that exact 31P lineshape calculations provide the relative signs of CQ, the isotropic J-coupling, and the effective dipolar coupling constant. This appears to be the first unambiguous determination of Delta1J(31P, 63/65Cu).