How the Ligand Field in Lanthanide Coordination Complexes Determines Magnetic Susceptibility Anisotropy, Paramagnetic NMR Shift, and Relaxation Behavior

Conspectus Complexes of lanthanide(III) ions are being actively studied because of their unique ground and excited state properties and the associated optical and magnetic behavior. In particular, they are used as emissive probes in optical spectroscopy and microscopy and as contrast agents in magnetic resonance imaging (MRI). However, the design of new complexes with specific optical and magnetic properties requires a thorough understanding of the correlation between molecular structure and electric and magnetic susceptibilities, as well as their anisotropies. The traditional Judd–Ofelt–Mason theory has failed to offer useful guidelines for systematic design of emissive lanthanide optical probes. Similarly, Bleaney’s theory of magnetic anisotropy and its modifications fail to provide accurate detail that permits new paramagnetic shift reagents to be designed rather than discovered. A key determinant of optical and magnetic behavior in f-element compounds is the ligand field, often considered as an electrostatic field at the lanthanide created by the ligands. The resulting energy level splitting is a sensitive function of several factors: the nature and polarizability of the whole ligand and its donor atoms; the geometric details of the coordination polyhedron; the presence and extent of solvent interactions; specific hydrogen bonding effects on donor atoms and the degree of supramolecular order in the system. The relative importance of these factors can vary widely for different lanthanide ions and ligands. For nuclear magnetic properties, it is both the ligand field splitting and the magnetic susceptibility tensor, notably its anisotropy, that determine paramagnetic shifts and nuclear relaxation enhancement. We review the factors that control the ligand field in lanthanide complexes and link these to aspects of their utility in magnetic resonance and optical emission spectroscopy and imaging. We examine recent progress in this area particularly in the theory of paramagnetic chemical shift and relaxation enhancement, where some long-neglected effects of zero-field splitting, magnetic susceptibility anisotropy, and spatial distribution of lanthanide tags have been accommodated in an elegant way.

■ KEY REFERENCES

■ ELECTRONIC STRUCTURE INTRODUCTION
The unique electronic structure of trivalent 4f ions determines the distinctive properties of their coordination complexes. The electrostatic shielding of the electrons in 4f orbitals by fully occupied 5s and 5p orbitals makes the effects from surrounding ligands and other molecules far smaller than the interelectron repulsion and spin−orbit coupling ( Figure 1). Due to these order-of-magnitude differences, electronic transitions in lanthanide(III) complexes are often independent of the ligand environment, and the ligand field splitting can be considered on the basis of the ground-state total angular momentum, J. The energy of m J sublevels can be calculated using the crystal field theory that neglects mixing of f-orbitals with the orbitals of the ligands. For a given J multiplet, the model Hamiltonian has the form given in eq 1:  (1) where B q k are ligand field parameters, O k q are Stevens operators, and θ k are operator equivalent coefficients (Table 1), defined for each term and multiplet in a given configuration. 5,6 The B q k parameters are defined in a particular reference frame; in symmetric molecules, the z-axis is usually aligned with the principal axes of the symmetry group, in which case the number of nonzero parameters is reduced. 7,8 In the absence of symmetry, the expansion in eq 1 has 27 independent parameters. However, given sufficiently high symmetry or enough spectroscopic data, all nonzero ligand field parameters may be determined by luminescence spectroscopy. 9,10 The principal parameter of interest to the NMR community is B 0 2 , due to the prevalence of Bleaney's theory. 11 As an example, for Eu(III), it may be extracted directly from the 5 D 0 to 7 F 1 transition ( Figure 2). 12−14 The B q k parameters can be estimated from experimental data but are nowadays commonly obtained from multireference ab initio electronic structure methods, such as complete active space selfconsistent field (CASSCF) calculations. 15 Because the emissive state 5 D 0 is nondegenerate, the splitting of the transition must arise from the ligand field splitting of the 7 F 1 multiplet (excluding vibrational effects). Since J = 1, the series in eq 1 terminates at k = 2, and when the complex has symmetry higher than C 2 , only B 0 2 is nonzero and the spectrum exhibits two bands corresponding to the degenerate m J = ±1 pair and the m J = 0 singlet, whose separation is ∝B 0 2 . In lower symmetry, the degeneracy of the m J = ±1 states is lifted and B ±2 2 is nonzero. Therefore, the 5 D 0 → 7 F 1 band can be modeled with band-specific B 0 2 and B 2 2 , which may differ slightly from the parameters determined by fitting all observable bands. 14,16 The splittings are given as Δ = 3θ 2 B 0 2 and δ = 2θ 2 B 2 2 , and θ 2 = −1/5 (Table 1), where ligand field parameters are defined for Stevens operators, and the renormalization for more commonly used spherical tensors is given in the Figure 2 caption. 13 The sign of Δ is positive if the m J = 0 component of 7 F 1 is lower in energy than the barycenter of m J = ±1 components, giving a singlet transition at higher energy than the doublet. Comparing the azaphosphinate complexes [EuL 8b ] and [EuL 9 ] + (Figure 2), there is a change in the sign of B 0 2 , which is positive for the latter. The sign of these crystal field parameters is tightly linked to the local symmetry at the Eu(III) ion. 12,14,17,18 Even though, B q k parameters determined for Eu(III) complexes can be very similar to isostructural complexes of other lanthanide ions, B q k Figure 1. Schematic representation of electronic states for Eu(III) (4f 6 ): six electrons occupy seven degenerate 4f orbitals giving a 7 F ground term in the Russell−Saunders coupling scheme (spectroscopic notation, 2S+1 L J ), with total spin S = 3 and total orbital angular momentum L = 3. Spin−orbit coupling splits this term into seven J multiplets separated by about 10 3 cm −1 . Each J state is (2J + 1)-fold degenerate for the free-ion; this degeneracy is partially removed upon loss of spherical symmetry. The separation of m J states due to the ligand field is around 10 2 cm −1 but can be much larger. Accounts of Chemical Research pubs.acs.org/accounts Article depends on the radial part of the f-electron wave function, which changes with nuclear charge, and small changes in bond lengths and angles may also affect the angular part of B q k unexpectedly.
When the ligand field splitting is comparable to the splitting between spin−orbit multiplets, J is no longer a good quantum number, and the coupling scheme breaks down, for example, for Sm(III), 19 leading to the phenomenon of "J mixing", commonly invoked to explain unusual oscillator strengths and odd transitions in polarized emission spectra. 20−22 Despite this, many other spectral phenomena defy explanation, and "J mixing" is often cited as a "catch-all", highlighting limitations in current understanding. 12,14,23 Lanthanide magnetic moments, 24 which are often assumed to be independent of coordination environment 25,26 also routinely show reductions in room-temperature susceptibility values compared to the free-ion due to the ligand field effect; a notable 11% reduction was found for Ho(III). 27 Apart from the reduction of the average magnetic susceptibility, the ligand field also induces magnetic anisotropy that is the origin of paramagnetic NMR shifts and dramatically alters nuclear spin relaxation.

FIELD SPLITTING
The spectral behavior of several series of macrocyclic lanthanide(III) complexes [LnL 1−9 ] has been studied, owing to their interest as emissive probes in optical spectroscopy and microscopy 28−30 or contrast agents in magnetic resonance imaging. 13,17,18 Design of complexes with desired optical and magnetic properties requires an understanding of correlations between molecular structure and the electromagnetic susceptibility tensors. 31 These correlations are often assumed to follow simple models. However, Judd−Ofelt−Mason theory fails to offer guidelines for the design of emissive lanthanide optical probes, 32−34 and similarly Bleaney's theory of magnetic anisotropy 11,35,36 has been widely used for NMR spectral fitting but provides no guidance for paramagnetic shift reagent design. 37,38  Accounts of Chemical Research pubs.acs.org/accounts Article Examination of the emission spectral properties of the [LnL 1−9 ] series permits a dissection of the key factors determining the most important contribution to the ligand field. The size and sign of B 0 2 varies widely across these series of complexes (Chart 1 and Table 2).  44 When the axial donor is replaced by fluoride, B 0 2 inverts sign causing a large change in magnetic susceptibility anisotropy, as the order of the m J sublevels switches. 31 The importance of the "axial component" of the ligand field was highlighted by Di Bari, 45 examining spectral behavior of [YbL 5 ] − complexes. Another example of switching sign in B 0 2 for Yb(III) complexes combined NMR, EPR, and computational studies to track changes in the anisotropy of the magnetic susceptibility tensor. 46 (iii) For [EuL 1a−e ] (Chart 1), B 0 2 changes as the para substituent in the pyridine ring varies. A linear correlation between the Hammett parameter, σ p , and B 0 2 (R 2 = 0.97, in acetonitrile), is consistent with the strongly dipolar nature of the Ln−N py interaction. 46 The variation of overall ligand polarizability and its directionality, involving the electrostatic interaction between induced dipoles on the ligand and the quadrupole moment on the Ln 3+ ion, is important in determining the "allowedness" of f−f electronic transitions. 47,48 Thus, it is the overall ligand molecular polarizability that is important in determining the ligand field.

Variation of Complex Constitution and Symmetry
(iv) Other examples of switches in the sign of B 0 2 can be identified when complex constitution and local symmetry vary. The difference between the emission spectra of [EuL 8b ] and [EuL 9 ] + ( Figure 2) is consistent with a change in sign, as symmetry changes from C 1 to C 2 . 3,14 Other cases have been reported, including systems involving reversible coordination of a polarizable N atom, which following protonation is replaced by water. 49−51

Polyhedral Distortion
In point-charge ligand field theory, the geometric position and charge of each atom determine contributions to the ligand field potential. An axial anionic donor gives a positive contribution to B 0 2 , which becomes negative if it is in an equatorial position (switching at the "magic angle" θ ≈ 54.7°or 125.3°), 52−55 leading to sensitivity of the ligand field potential to polyhedral distortion. The tricapped trigonal prismatic geometry is particularly sensitive, as noted by Binnemans, if all nine ligands are equivalent and the two sets of axial donors have polar angles 45°and 135°, leading to exact cancellation of all contributions and hence B 0 2 = 0. 56 The situation with [LnL 1−3 ] is different. The first coordination sphere has three sets of donors: nitrogen atoms from the macrocycle (N ax ) lie in axial positions (polar angle θ ≈ 142°); pyridyl N atoms in equatorial positions (N eq , θ ≈ 90°); carboxylate oxygens in axial sites (θ ≈ 50°). 2 In [LnL 1a ], the two sets of N donors (N ax , N eq ) give contributions to B 0 2 of similar magnitude but opposite sign and cancel out; this is because the opposite of a ligand in an axial position is a ring of donors in the equatorial plane, and here the 3-fold equatorial disposition of N eq balances the N ax contribution. However, the oxygen donors lie close to the magic angle, and thus the ligand field is almost entirely ascribed to the oxygen atoms, resulting in an exquisite sensitivity of ligand field and magnetic anisotropy to very small variations in their angular position ( Figure 3). 2 Emission studies with [EuL 1a ] showed a pronounced dependence of B 0 2 on solvent, suggesting that hydrogen bonding interactions with the oxygen donors could alter their effective polar angle θ. Indeed, the X-ray structure of [YbL 1b ] shows hydrogen bonding of water to the coordinated carboxylate oxygen, demonstrating this "tugging" on the donor oxygen. For [YbL 1b ], [YbL 1e ], and [EuL 1e ], both carbonyl and carboxylate oxygen atoms served as hydrogen bond acceptors to the water hydrogen atom. 46 In 9-coordinate lanthanide complexes based on 12-N 4 (e.g., DOTA 18 ), the most common geometries are a monocapped square antiprism (SAP) and a twisted version (TSAP). The twist angle between upper and lower planes of four donor atoms found in X-ray analyses vary around 40°and 25°, respectively. Values of B 0 2 for [EuL 10−15 (OH 2 )] (Chart 2, Table 3) show larger parameters in the SAP series. [40][41][42][43][44]57 These variations relate to polyhedral distortion but may also be ascribed to changes in the axial water distances that are systematically longer (<0.3 Å) in the TSAP series due to increased steric demand. Such behavior is consistent with the concept of nonintegral metal ion hydration states, reducing in value between unity (Eu) and zero (Yb), through certain TSAP series, as the bond length to the water oxygen increases. 40   In summary, ligand field splitting of lanthanide complexes is a sensitive function of several factors: the nature and polarizability of the ligand and its donors; the type and degree of polyhedral distortion; the presence and extent of solvent dipolar interactions; hydrogen bonding effects and the degree of supramolecular order. Each factor may be non-negligible in defining the ligand field, and their relative importance varies for different lanthanides.
■ PSEUDOCONTACT SHIFT AND BLEANEY'S THEORY OF MAGNETIC ANISOTROPY When a lanthanide is treated as a point with second-rank magnetic susceptibility and infinitely fast magnetic relaxation, an additional isotropic shielding experienced by nearby nuclei is given by 63 where θ, ϕ, and r are nuclear coordinates in the eigenframe of the magnetic susceptibility tensor. The eigenvalues of the traceless susceptibility tensor are labeled to satisfy the relation |χ x | < |χ y | < |χ z |, with axiality χ ax = 3χ z /2 and rhombicity χ rh = (χ x − χ y )/2. Below we also use terms of χ av = Tr(χ)/3 and χ ∥ = χ z + χ av . Bleaney's theory of magnetic anisotropy 11,35,36,64 shows that for a well-isolated J multiplet in the high temperature approximation, the anisotropy of the susceptibility tensor depends only on the second rank B 0 2 and B 2 2 ligand field parameters:

Approximations and Their Limits
Assuming that the ligand field parameters do not vary between lanthanide ions, eq 3 suggests that χ ax /χ rh remains constant within the series and PCS only varies due to the change in the value of C J . However, if the overall ligand field splitting is greater than kT ( Figure 5), 28,63,65 the Bleaney formula is no longer valid, and χ ax /χ rh and the eigenframe of the susceptibility tensor will depend on temperature. 66 Figure 5) clearly show that in all cases the splitting is larger than kT. Thus, if eqs 2 and 3 are used to determine B 0 2 and B 2 2 from PCS data, the parameters may appear to be very different for each lanthanide simply because Bleaney's approximations do not hold. 72 Equation 2 also assumes a point magnetic source at the nuclear position of the lanthanide ion; a revised approach has recently emerged where the distribution of 4f electron density can be accounted for. 73 There are two distinct reasons for such a distribution to occur: (i) spin delocalization and (ii) fast tag

Contact Contribution to Paramagnetic Shift
In most of the cases considered in this review, the proton contact shift is negligible compared to the PCS, and the point-dipole approximation in eq 2 is valid. The contact shift is proportional to the isotropic hyperfine coupling (itself related to spin density at the nucleus) and the isotropic magnetic moment of the lanthanide ion. Accounting for admixture of excited states with different J to the ground term, the isotropic magnetic moment can be corrected, 75 and the ratio of contact contribution to the PCS can be estimated for different lanthanides provided that all other parameters in the series stay the same. 76 Such estimations suggest that in the Tb−Yb series the most pronounced effect of a contact contribution is expected for Ho/Er.

MAGNETIC ANISOTROPY
Detailed analyses of PCS data have been undertaken for isostructural complexes, with known solution speciation. 28,38,77−81 A semiautomated combinatorial assignment procedure using PCS data, XRD structure and NMR relaxation rates to limit the combinatorial space (in Spinach 82 ) was deployed for [LnL 8a ], enabling assignment of almost every proton resonance. 1 Subsequently, the traceless part of the magnetic susceptibility tensor was obtained by fitting eq 2 to experimental data, giving excellent agreement (R 2 > 0.99). The experimentally determined susceptibility tensor can be displayed as a PCS field (Figure 6), revealing significant variations in the amplitude, shape, and orientation for the [LnL 8a ] series. Bleaney's theory predicts that only the amplitude and sign should vary. However, the tensors change from almost fully rhombic (Dy and Tb; PCS field resembles d xz orbital) to near axial (Tm, PCS field resembles d z 2 orbital). Critically the tilt angle β of the main anisotropy axis, relative to the molecular pseudosymmetry axis, varies significantly between complexes: Tb 8°; Dy 20°; Ho 22°; Er 8°; Tm 6°; Yb 23°.
To illustrate the sensitivity of magnetic susceptibility anisotropy to structural change, consider PCS shifts for [YbL 8b ] and [YbL 9 ] + (Chart 3). The t Bu NMR chemical shifts vary markedly across the series but appear in the same order, notwithstanding the B 0 2 sign inversion ( Figure 7 and Figure 2); 3 Bleaney's theory predicts the shift sense should be inverted.
The explanation lies in the magnetic susceptibility tensors, expressed in their very different PCS shift fields. While the second-order magnetic anisotropy changes sign, the negative PCS lobe is still oriented in the "equatorial plane", because along with the change in sign of B 0 2 , there is a 90°rotation in orientation of the principal magnetic axis. Thus, the combined effect of the change in sign and orientation of the ligand field were shown to give rise to similar PCS fields for the t Bu protons, explaining the "hidden" changes in PCS behavior. 3

■ NMR SHIFT BEHAVIOR OF SYSTEMS WITH SMALL LIGAND FIELD SPLITTINGS
Complexes [LnL 1−3 ] adopt tricapped trigonal prismatic structures and possess small ligand field splittings close to kT. Yet, their PCS values do not conform to Bleaney's theory. 31,37,38 Both the sign and magnitude of their ligand field parameters are sensitive to local polarity changes and polyhedral distortion. They are particularly sensitive to perturbation of the polar angle of oxygen donor atoms, θ, defining the angle subtended by the Ln−O vector compared to the C 3 axis. As θ lies close to the "magic" angle, small variations cause major changes in magnetic susceptibility anisotropy. 2,46 For [YbL 1b ], DFT was used to determine a pseudosolution structure with imposed C 3 symmetry and CASSCF-SO calculations gave the anisotropy of the susceptibility tensor (squares, Figure 8). The experimental values of χ ∥ − χ av were determined assuming a fixed structural model based on experimental PCS, referenced to the diamagnetic Y(III) complex. A comparison was then made with the CASSCF-SOcalculated susceptibility anisotropy, to determine the "spectroscopic" average value of θ in solution. In [YbL 1b ], the diastereotopic methyl groups of the isopropyl substituent serve as a local probe of magnetic anisotropy. The PCS fields in acetone, water, and methanol highlight the sensitivity to solvent. The PCS field changes sign as the magnetic susceptibility anisotropy switches from "easy axis" to "easy plane" in D 2 O. 46 Similar solvent dependences were found for [LnL 1a ] (Dy, Er, Eu). 2 The sensitivity of magnetic anisotropy in these lanthanide complexes with small ligand field splittings was shown to have a major impact on solid-state EPR behavior. 84 The magnetic and spectroscopic properties depend upon a number of factors that cannot be disentangled: a distribution of structural parameters generates a range of B 0 2 values; an electronic structure sensitive to thermal changes of the ligand structure; thermally accessible EPR-active excited states; disordered solvation influencing the local ligand field. Each effect is present across the [LnL 1−3 ] series, making interpretation of EPR spectra very difficult for systems with small magnetic anisotropy. 84 ■ NMR SHIFT BEHAVIOR OF MOBILE LANTHANIDE TAGS ON PROTEINS Complexes with large magnetic anisotropies are often used as tags in protein NMR to provide structural constraints, 85,86 where large magnetic anisotropy is preferred so that PCS is measurable even at distances of 40 Å. The tag is often attached by a flexible linker, but mobility results in big deviations from the pointdipole approximation, at <15 Å.
A generalization of McConnell's expression, eq 4, was derived for lanthanide tag mobility in protein NMR: 73,74,87 where ∇ ⃗ is the gradient operator, ρ(r) is the probability distribution of the spatial position of the lanthanide tag, and subscript t indicates the traceless part of the magnetic susceptibility tensor. The susceptibility is assumed to be the same in every point of the probability density. 74 The latter assumption may be lifted, but the corresponding equation is considerably harder to solve. The partial differential eq 4 can be solved using three-dimensional Fourier transforms: 74 where FFT + refers to the forward fast-Fourier transform and FFT − the inverse. If the probability density is defined on a grid, numerical solution of eq 5 gives the PCS values. The solution of the inverse problem is possible; one can extract probability density from PCS data. Numerical solvers for both direct and inverse problems are available. 82 The resulting lanthanide probability densities from PCS are in agreement with Double Electron−Electron Resonance (DEER) spectroscopy, and PCS fits are significantly improved near the tag (Figure 9). 87

■ LANTHANIDE RELAXATION AND ITS ANISOTROPY
Common MRI contrast agents contain magnetically isotropic Gd 3+ ions. Their long electron relaxation times mean that PCS is absent, and the effect is only to accelerate nuclear relaxation. 88 Likewise, relaxation enhancement experiments in NMR often use magnetically isotropic Mn 2+ or Gd 3+ complexes to maximize the volume affected by the metal and minimize PCS. 89 Following nuclear relaxation enhancement models designed for these ions, it has often been assumed that the enhancements show a simple 1/r 6 dependence on the electron−nuclear distance, without angular terms in the molecular frame of reference. Nuclear relaxation enhancement by unpaired f electrons of lanthanide complexes has two principal components. One ("dipolar relaxation") comes from stochastic modulation of the electron−nuclear dipolar interaction and the other ("Curie relaxation") from rotational modulation of extra nuclear shielding caused by the presence of the unpaired electron. The angular dependence in non-Gd lanthanides 4 was first acknowledged for Curie relaxation. 90 The reasons are twofold. First, magnetic susceptibility tensor anisotropy can be as large as the isotropic part, contradicting the assumption made by Gueron Accounts of Chemical Research pubs.acs.org/accounts Article when he derived Curie relaxation theory. 91 Second, zero field splitting can be much stronger than the electron Zeeman interaction, the opposite limit from the classical Solomon− Bloembergen−Morgan theory of lanthanide-induced dipolar relaxation. 92 Experimental proof came from relaxation rate measurements in complexes where all nuclei in the ligand cages could be unambiguously assigned, and atomic coordinate estimates were available from DFT calculations 28,37,38,93−95 (Chart 1 and Figure 10).
It is obvious from Figure 10 that nuclear relaxation enhancements at low magnetic field (1 T) do not depend simply on the distance to the lanthanide ion. The relaxation rates also appear to depend on the sign of the magnetic anisotropy: in complexes with easy-plane anisotropy (Tb, Dy, Ho), ligand arm protons relax faster than macrocyclic ring protons; the opposite is true for the complexes with easy-axis anisotropy (Er, Tm, Yb).
Encouraged by these findings, we updated the dipolar relaxation theory 4 to include the direction of the Ln−H vector in the molecular frame: that includes diamagnetic shielding tensor σ 0 and paramagnetic shielding tensor, which is proportional to the dipolar matrix D and magnetic susceptibility tensor χ. This is necessary because the antisymmetric part is significant here; the product of two symmetric matrices is only symmetric when they commute. With the relevant extra terms in place, the Curie relaxation rates become R R  (10) here, Λ σ 2 is the first and Δ σ 2 the second rank invariant of the chemical shielding tensor. These equations have been incorporated into Spinach; 82 for [LnL 8a ], the modifications yielded a much better agreement with experiment ( Figure 11). 4 In summary, the presence of magnetic anisotropy required a fundamental update of the relevant nuclear relaxation theories. These updates revealed strong molecular-frame angular dependencies in paramagnetic relaxation enhancements. In systems with large magnetic anisotropy and at short electron−nuclear distances, the classical Solomon−Bloembergen−Morgan and Gueron expressions should not be used.

■ SUMMARY AND CONCLUSIONS
The ligand field for a lanthanide complex varies with the nature of the ligand, metal ion, and its environment. The size and sign of ligand field parameters are difficult to determine experimentally, but information can be gained using optical spectroscopy with Eu(III) complexes. 9 They are sensitive to several factors including the nature and polarizability of the overall ligand and donor atoms, the type and degree of geometric distortion, the extent of solvent dipolar interactions, and specific hydrogen bonding effects and the degree of supramolecular order.
Bleaney's magnetic anisotropy theory provided guidance in rationalizing NMR PCS data. However, its crude approximations and limitations are apparent. In explaining the nature and magnitude of PCS data, both the ligand field splitting and the type, size, and orientation of the principal component of the magnetic susceptibility tensor are key. The latter can be determined by careful magneto-structural correlations 27  Accounts of Chemical Research pubs.acs.org/accounts Article assessed by VT magnetic susceptibility measurements, low temperature EPR studies, and modern computational methods. Considerable caution is needed using PCS data for structural refinement. Such methods are used in biomolecular analyses but may fail when the lanthanide ion is permuted. Delving more deeply, the ordering, nature, and relative Boltzmann population of the m J sublevels for a given lanthanide ion complex is key to understanding the overall magnetic susceptibility and its directional dependences.
The nuclear relaxation induced by lanthanide ions can be anisotropic in the molecular frame, and accounting for this anisotropy can drastically improve agreement between experiment and theoretical models. Analyses based only on distance variations are a crude approximation for both dipolar and Curie relaxation mechanisms. Biomolecular structural refinement using lanthanide spin tags must account for this anisotropy or risk significant errors; any work using simple 1/r 6  The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript.

Notes
The authors declare no competing financial interest.

Biographies
David Parker was born in Leadgate, England, on 30th July 1956. Educated in the state sector, he read Chemistry at Oxford University (1974−1978) and remained there to study with John Brown for a DPhil, which he gained in 1980. Following a NATO Fellowship in Strasbourg with Jean-Marie Lehn, he returned to the NE of England in early 1982 as a Lecturer in Chemistry at Durham University. He gained several national and international awards and prizes for work that embraces several aspects of contemporary chemistry, including the design of sensors, targeted imaging probes, and therapeutic agents, as well as major contributions to chiral analysis and mechanistic studies. He is a Professor of Chemistry at Durham University. Elizaveta Suturina studied physics at Novosibirsk State University where she obtained her undergraduate degree and DPhil in chemical physics working in the group of Nina Gritsan. She then moved to work in Max-Planck Institute with Frank Neese and Mihail Atanasov on ab initio modeling of molecular magnetic properties and then joined Ilya Kuprov's group to work on NMR theory and modelling of paramagnetic systems. In 2018, she was awarded the Prize fellowship from the University of Bath where she is currently establishing her research group.
Ilya Kuprov is a magnetic resonance specialist and the leading author of Spinach, an advanced spin dynamics simulation package. He did his undergraduate degrees at Novosibirsk State University before moving to Oxford for a DPhil. He was a Lecturer at Durham University and is presently an Associate Professor at Southampton. He is the Secretary of the Electron Spin Resonance Group of the Royal Society of Chemistry and an Associate Editor at Science Advances. He has published about a hundred papers, principally on the theoretical and computational aspects of magnetic resonance spectroscopy and imaging.