Magnetic exchange, anisotropy and excitonic fluctuations in a [Ni^II₇] Anderson wheel†

Emily H. Payne ^a, Lucinda R. B. Wilson ^a, Mukesh K. Singh ^a, Gary S. Nichol ^a, J. R. Stewart ^b, V. Garcia-Sakai ^b, R. A. Ewings ^b, Harry Lane ^c, Jürgen Schnack *^d, Chris Stock *^c and Euan K. Brechin *^a
aEaStCHEM School of Chemistry, The University of Edinburgh, David Brewster Road, Edinburgh, EH9 3FJ, Scotland, UK. E-mail: E.Brechin@ed.ac.uk
^bISIS Pulsed Neutron and Muon Source, STF Rutherford Appleton Laboratory, Harwell Campus, Didcot, Oxon OX11 0QX, UK
^cSchool of Physics and Astronomy, The University of Edinburgh, Edinburgh, EH9 3JZ, Scotland, UK. E-mail: C.Stock@ed.ac.uk
^dFakultät für Physik, Universität Bielefeld, Postfach 100131, D-33501 Bielefeld, Germany. E-mail: jschnack@uni-bielefeld.de

First published on 23rd November 2023

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Introduction

Complexes of Ni^II have long played a prominent role in the field of molecular magnetism, from the early studies of magneto-structural correlations in Ni dimers¹ and cubes² and the emergence of Single-Molecule Magnets (SMMs),³ to the design of molecules exhibiting very large magnetic anisotropies⁴ and possessing long coherence times.⁵ During this journey coordination chemists have developed a breadth of synthetic strategies in order to construct complexes with a variety of nuclearities and topologies. Amongst these, [Ni₄] SMMs have proven to be a particularly fruitful source of information on exchange bias effects,⁶ antisymmetric exchange,⁷ quantum tunnelling,⁸ transverse anisotropy,⁹ and Berry Phase interference.¹⁰ Other common structural motifs demonstrating fascinating magnetic properties include wheels¹¹ and discs.¹² The latter offer interest from a variety of perspectives, including as molecular analogues of 2D lattices,¹³ model complexes to examine spin frustration¹⁴ and as robust platforms to quantitatively analyse magneto-structural correlations.¹⁵

Planar heptanuclear discs conforming to centred hexagons are often referred to as Anderson or Anderson-Evans wheels,¹⁶ and are a common structure type in both polyoxometalate chemistry¹⁷ and clusters of paramagnetic 3d ions.¹⁸ A search of the Cambridge Structural Database (CSD) reveals that there are twenty nine [Ni^II₇] Anderson wheels.^19–38 Of these, fifteen have had their magnetic behaviour reported, but only five have been subject to quantitative analysis.^{21,26,30,36,37} This is perhaps not surprising. Given the potential for multiple (≤12) exchange interactions, and numerous g and D values, even greatly simplified models can prove computationally demanding. In general, extracting exchange coupling constants (J) has been achieved using a model which assumes just two interactions (Fig. 1), one between the central Ni ion and the Ni ions in the outer ring (J_cr) and one between neighbouring Ni ions around the ring (J_rr). However, there is no agreement regarding the nature and magnitude of these values for those complexes exhibiting competing ferro- and antiferromagnetic exchange, despite the presence of similar Ni–X–Ni angles. Some reports suggest J_cr is ferromagnetic and J_rr is antiferromagnetic, and others state J_cr is antiferromagnetic and J_rr is ferromagnetic. The only clear-cut case is when all bridges are μ_2/3-N₃⁻ ions which promote ferromagnetic exchange for both J_cr and J_rr.³⁰ Based on magneto-structural correlations developed for [Ni₄O₄] cubanes,² one might expect J_cr to be ferromagnetic and J_rr antiferromagnetic in all O-bridged species given that the former are mediated by μ₃-O atoms with smaller Ni–O–Ni angles than the latter, which are bridged by μ₂-O atoms. Indeed, this is what is observed in a [Al^III₂Ni^II₅] Anderson wheel reported recently, where J_cr = +10 cm⁻¹ and J_rr = −5 cm⁻¹ ( formalism).¹⁵ There is also no agreement on the sign and magnitude of the zero-field splitting (D_Ni) for the central or ring Ni^II ions. In order to investigate the magneto-structural relationship in this family of complexes further, and to unequivocally extract the relevant spin-Hamiltonian parameters, we have synthesised and characterised the complex [Ni₇(hmp)_7.55(pic)_4.45](ClO₄)₂·6MeOH (1·6MeOH; Fig. 1, hmpH = 2-(hydroxymethyl)pyridine, picH = picolinic acid) using a combination of susceptibility and magnetisation measurements, magnetic neutron spectroscopy and first principles calculations. This complementary trio of techniques, employed for the first time in the analysis of [Ni₇] Anderson wheels, reveals the presence of competing ferro- and antiferromagnetic exchange interactions and both easy plane and easy axis anisotropy.

Experimental

Experimental procedures

All chemicals were procured from commercial suppliers and used as received (reagent grade).

Synthesis of [Ni₇(hmp)_7.55(pic)_4.45](ClO₄)₂·6MeOH (1·6MeOH)

Ni(ClO₄)₂·6H₂O (0.366 g, 1 mmol) was dissolved in MeOH (20 mL), with NaOMe (0.162 g, 3 mmol) yielding a pale blue solution. To this, hmpH (0.142 mL, 1.5 mmol) was added dropwise followed by picH (0.184 g, 1.5 mmol), which resulted in a deep blue solution. The solution was stirred for 12 hours and then transferred to a Teflon-lined, stainless steel autoclave which was heated for 24 hours at 140 °C. This was then allowed to cool for a further 24 hours, yielding translucent pale blue plate-shaped crystals. Anal. calcd (%) C, 43.91; H, 3.23; N, 8.54. Found: C, 43.81; H, 3.39; N, 8.49. Yield: 56%.

Note: caution should be taken when using/heating the potentially explosive perchlorate anion.

Crystallographic details

A suitable crystal with dimensions 0.48 × 0.14 × 0.04 mm³ was selected and mounted on a MITIGEN holder in Paratone oil on a Rigaku Oxford Diffraction XCalibur diffractometer. The crystal was kept at a steady T = 120.0 K during data collection. The structure was solved with the ShelXT solution program using dual methods, employing Olex2 as the graphical interface. The model was refined with ShelXL 2018/3 using full matrix least squares minimisation on F².^39–41 Three of the ligands were modelled as a disordered mixture of hmp and pic, as evidenced by a residual peak in a difference map which corresponded to the additional O site in the pic ligand. Each disorder was refined with a separate free variable. Specific occupancy values are given in the CIF; this explains the fractional values in the formula. Other than constructing an appropriate disorder model for the CH₂ group the ring atoms of each hmp/pic disorder were not split as this was not justified by either the displacement ellipsoids or residual electron density peaks.

Crystal data. C₇₈H_87.09Cl₂N₁₂Ni₇O_30.45, M_r = 2161.75, triclinic, P (No. 2), a = 12.1951(3) Å, b = 13.4801(4) Å, c = 15.0616(4) Å, α = 110.946(3)°, β = 97.395(2)°, γ = 106.809(2)°, V = 2137.68(10) ų, T = 120.0 K, Z = 1, Z′ = 0.5, μ(MoKα) = 1.662, 57226 reflections measured, 10646 unique (R_int = 0.0441) which were used in all calculations. The final wR₂ was 0.1078 (all data) and R₁ was 0.0465 (I ≥ 2σ(I)). CCDC 2286832.† Powder XRD data of compound 1 (Fig. S1†) were collected on a polycrystalline powder using a Bruker D8 ADVANCE in transmission mode fitted with a LynxEye detector using Cu radiation at 40 kV, 40 mA and a Johansson monochromator, 2 mm divergence slit and 2.5° Soller slits on the incident beam side. Bruker's DIFFRAC software was used to control the diffractometer. Diffraction data were measured from 2θ = 5°–40°; step size, 0.0101°.

Magnetic thermodynamic measurements

Magnetic susceptibility and magnetisation measurements were performed on a powdered crystalline sample restrained in eicosane in the temperature and fields ranges T = 2–300 K, B = 0.1 T and T = 2–6 K, B = 0.5–7.0 T, respectively, using a Quantum Design MPMS XL SQUID magnetometer equipped with a 7 T dc magnet. The observed paramagnetic susceptibilities were corrected for diamagnetic contributions using Pascal's constants.

Computational details

Density Functional Theory (DFT) calculations in the Gaussian16 suite⁴² have been performed to estimate the magnetic exchange interactions between the Ni^II centres on a [Ni₅Zn₂] model, constructed by replacing two Ni^II ions with two Zn^II ions (Fig. S2a and b†) in the XRD structure of complex 1. The hybrid B3LYP functional⁴³ together with the TZV basis set⁴⁴ have been employed for the Ni and Zn atoms, the 6-31G* basis set has been used for the O and N atoms, while the 6-31G basis set has used for the C and H atoms.⁴⁵ Noodleman's broken symmetry approach⁴⁶ has been employed for estimating the magnetic exchange interactions. Spin configurations used to estimate the exchange coupling constants are summarised in Table S1.† Pairwise interactions have been calculated to estimate the magnetic interaction between (a) the central Ni^II ion and the outer ring Ni^II ions (J_cr), and (b) between the ring Ni^II ions (J_rr), in dimeric models 1A–1B (Fig. S2c and d†), respectively. We have also developed magneto-structural correlations on model 1C (Fig. S2e†) to check the dependence of the magnetic exchange interaction on the average Ni–O distance and average Ni–O–Ni angle. We have made several mono- and dimetallic models (1D–1G, Fig. S2f–i†) from complex 1 to estimate the zero field splitting parameters for each Ni^II centre using the ORCA software suite (version ORCA 4.0).⁴⁷ The zeroth-order regular approximation (ZORA) method together with the ZORA contracted version of the basis set (ZORA-def2-TZVP for Ni, Zn and ZORA-def2-SVP for rest of the elements) have been used. The calculation further utilised the resolution of identity (RI) approximation with the auxiliary def2-TZVP and def2-SVP Coulomb fitting basis sets.⁴⁸ During state-average complete active space self-consistent field (SA-CASSCF) calculations, we have included eight electrons in five d-orbitals (CAS (8 electrons/5 3d-orbitals)) in the active space. Ten triplet and fifteen singlet roots are considered during CASSCF calculations. To consider the dynamic correlation, we have performed 2^nd order N-electron valence perturbation theory (NEVPT2) calculations.⁴⁹ We have employed integration Grid 6 for the Ni and Zn ions, and Grid 5 for remaining elements.

Neutron spectroscopy, data correction and background subtraction

Neutron spectroscopy measurements were performed on the LET (ISIS, Didcot) cold chopper spectrometer. The sample was fully protonated for these measurements, resulting in a large background from the incoherent cross section from H-atoms, as discussed below. Given the large amount of neutron incoherent scattering resulting from H and Ni, there was a significant temperature independent background in the spectroscopic data taken on LET. This primarily originated from elastic scattering leaking into inelastic channels due to instrumental effects such as finite resolution. To account for this, we have applied the principle of detailed balance to remove this background, following previous scattering studies where this has been successfully used.⁵⁰ This analysis depends on the fact that at a fixed momentum and energy transfer, the measured intensities at the neutron energy gain (negative energy transfer) and energy loss (positive energy transfer) are related by the following expression with the energy transfer defined as E ≡ħω = E_i − E_f

	(1)

where B₁ and B₂ are temperature-independent background constants for positive and negative energy transfers. S(|E|, T) is proportional to the structure factor which contains both lattice and magnetic contributions. Note that ± energies are related through the e^−E/kT term which is the Boltzmann factor. An assumption in this analysis is that the resolution function does not change substantially over the energy range of interest and therefore the measured intensities at ±|E| are related solely by detailed balance.

In our experiment on LET, data were collected at four different temperatures, T = 2, 20, 50, and 80 K. With data at ±|E| at each temperature, this gives eight data points for a given momentum Q and energy transfer |E|. Assuming temperature independent background parameters of B₁ and B₂ and four values of S(|E|, T = 2, 20, 50, and 80 K), this is a closed problem of fitting six parameters to eight data points for each momentum and energy transfer. To obtain the temperature independent background we performed a least-squares fit to each Q and |E| to obtain the temperature independent background parameters B_1,2. We then used this to subtract off the background from the data, leaving the structure factor S(|E|, T) which can be computed theoretically and compared to first principle calculations as discussed below.

Results and discussion

Synthesis and structure

The solvothermal reaction of Ni(ClO₄)₂·6H₂O with hmpH and picH in a basic MeOH solution affords blue plate shaped crystals of formula [Ni₇(hmp)_7.55(pic)_4.45](ClO₄)₂·6MeOH (1·6MeOH) directly upon cooling the mother liquor. The crystals were found to be in a triclinic cell and structure solution was carried out in the space group P. The asymmetric unit (ASU) comprises half of the formula. The metallic skeleton describes a [Ni^II₇] centred hexagon, commonly referred to as an Anderson wheel. The central Ni ion is connected to the six ring Ni ions through a total of six μ₃-O atoms deriving from the hmp ligands with Ni–O–Ni angles in range ∼96.4–99.0°. These O-atoms also provide the connectivity between neighbouring Ni ions around the inner rim of the [Ni^II₆] ring with Ni–O–Ni angles in the range ∼97.2–98.4°. The outer rim of the ring is connected by six μ₂-O atoms derived from the pic ligands with Ni–O–Ni angles in range ∼100.3–100.5°. Note the latter are larger than the former. All six Ni^II ions are six-coordinate and in distorted octahedral geometries. The central Ni^II ion possesses a {NiO₆} coordination sphere, while those in the ring are {NiO₄N₂}. Note that the hmp/pic disorder in three of the organic ligands, as described in the crystallography details section, has no effect upon the Ni coordination spheres. The two ClO₄ anions sit above and below the [Ni₇] plane, with the MeOH molecules of crystallisation H-bonded to the non-bonded O-arm of the pic ligands (O⋯(H)O, ∼2.75–2.81 Å). In the extended structure this leads to a brickwork-like arrangement of cluster cations. The closest intermolecular Ni⋯Ni distances are >7.7 Å.

Dc magnetic susceptibility and magnetisation measurements

Dc magnetic susceptibility (χ) and magnetisation (M) measurements were performed on a powdered microcrystalline sample of 1 in the T = 300–2 K, B = 0.1 T and T = 2–6 K, B = 0.5–7.0 T ranges, respectively. These data are shown in Fig. 2 (and Fig. S3†) in the form of the χT product versus T and M vs. B. The χT product at T = 300 K is 8.6 cm³ K mol⁻¹, similar to the Curie constant expected for seven uncorrelated S = 1 ions with g = 2.2 (8.5 cm³ K mol⁻¹). As temperature decreases the χT product slowly increases to a maximum value of 12.0 cm³ K mol⁻¹ at T = 20 K, before decreasing at lower temperatures to a value of 10.5 cm³ K mol⁻¹ at T = 2.0 K. The magnetisation data initially increase rapidly with increasing magnetic field (M = 7μ_B at B = 1 T and T = 2 K) T and then rise more gradually thereafter to a maximum value of M = 11μ_B at B = 7 T and T = 2 K. The susceptibility and magnetisation data are therefore indicative of the presence of both ferro- and antiferromagnetic exchange interactions. The susceptibility data can be simulated nicely (Fig. S3†) with an isotropic spin-Hamiltonian to a model that assumes just two independent exchange interactions, J_cr and J_rr, describing the interaction between the central Ni ion and ring Ni ions, and between ring Ni ions, respectively.

The fit can be improved by adding additional J_cr and J_rr interactions (J₁–J₄, Fig. S3†), permitted given the presence of just inversion symmetry in 1 (i.e. half the cluster is in the ASU). In all cases the J_cr interaction is found to be ferromagnetic and the J_rr interaction is found to be antiferromagnetic. However, this simple isotropic model does not fit the magnetization data in the full temperature and field range, particularly at high fields. Introduction of an anisotropic term to account for the zero-field splitting associated with the octahedral Ni^II ions is therefore required, one for the central Ni ion with an {NiO₆} coordination sphere (D₁) and one for the ring Ni ions with {NiO₄N₂} coordination spheres (D₂). The best fit of the susceptibility and magnetisation data with this model affords J₁ = +4.13 cm⁻¹, J₂ = −4.20 cm⁻¹, J₃ = +3.93 cm⁻¹, J₄ = −0.80 cm⁻¹, D₁ = +4.71 cm⁻¹ and D₂ = −3.02 cm⁻¹. Note the opposite signs of D₁ and D₂, positive for the central Ni^II ion and negative for the ring Ni^II ions.

DFT calculations

To further support the sign and magnitude of the experimentally determined magnetic exchange interactions above, DFT calculations have been performed on a [Ni₅Zn₂] model created from the XRD structure of complex 1 (Fig. S2a and Table S1†). Considering symmetry and structural parameters, complex 1 has a total of five unique magnetic exchange interactions (J₁ − J₄, J_1′ Fig. S2b†). The exchange between the central and ring Ni^II ions (J_cr = J₁, J_1′, J₃) is calculated to be ferromagnetic and in the range +3.1 cm⁻¹ ≤ J_cr ≤ +5.1 cm⁻¹. The exchange between neighbouring ring Ni^II ions (J_rr = J₂, J₄) is calculated to be antiferromagnetic and in the range −1.6 cm⁻¹ ≤ J_rr ≤ −6.0 cm⁻¹. Thus, the signs and magnitudes of the DFT calculated values are in good agreement with the experimentally fitted values derived from the magnetic susceptibility and magnetisation measurements. They are also in agreement with previous magneto-structural studies performed on polymetallic Ni^II complexes in which larger average Ni–O–Ni angles result in an antiferromagnetic interaction and smaller average Ni–O–Ni angles result in a ferromagnetic interaction, with the crossover angle being ∼97–98°.^1,2,15,51 Further DFT calculations on dimetallic models (Fig. S2c and d†) also confirm the ferromagnetic J_cr interaction (J_1A = +8.2 cm⁻¹) and antiferromagnetic J_rr interaction (J_1B = −3.1 cm⁻¹).

To rationalise the magnitude and sign of the magnetic interactions, overlap integral calculations⁵² have been performed on the dimetallic models using the singly occupied molecular orbitals (SOMOs) of the Ni^II ions. Stronger overlap values between SOMOs leads to an antiferromagnetic interaction, whereas orthogonal/weak overlap values contribute to a ferromagnetic interaction. The overlap integral calculations suggest one moderate and three weak interactions for 1A and two moderate and two weak interactions for 1B. This leads to a ferromagnetic interaction for 1A and an antiferromagnetic interaction for 1B (Fig. S4 and 5†). Spin density analysis suggests strong mixing and delocalisation onto the coordinating ligand atoms followed by weak spin polarisation. The ring Ni^II ions have a spin density between 1.693–1.698 and the central Ni^II ion has a spin density of 1.735 (Fig. S6 and 7†). The smaller spin densities on the ring Ni^II ions can be attributed to the stronger spin delocalisation from the Ni^II ion to the N atom of the bridging ligand (between 0.050–0.063). As expected, the μ₃-O bridging atoms have a larger spin density compared to the μ₂-O bridging atoms.⁵¹ A magneto-structural correlation developed for model 1C (Fig. S2e†) with respect to the average Ni–O–Ni angle and average Ni–O distance follows the same trends observed for previously published O-bridged polymetallic Ni^II complexes, suggesting a strong dependence of the magnetic exchange interaction on the average Ni–O–Ni angle in particular (Fig. S8†).⁵¹

SHAPE analysis (Table S2†) suggests distorted octahedral geometries for all the Ni^II ions present, with the ring Ni^II ions [NiO₄N₂] being more distorted than the central Ni^II ion [NiO₆] on account of the chelating hmp/pic ligands. For small distortions away from ideal O_h symmetry, as seen here, one would expect either easy axis or easy plane anisotropy with a magnitude of approximately |D| < 10 cm⁻¹.⁵² Single ion anisotropy parameters for the four unique Ni^II ions present in 1 have been estimated using ab initio CASSCF/NEVPT2 calculations in the ORCA suite on model complexes 1D–1G (Fig. S2f–i†) derived from the XRD structure of complex 1.⁴⁷Ab initio NEPVT2 calculations suggest easy axis anisotropy for the ring Ni^II ions in the range −5.4 cm⁻¹ ≤ D₂ ≤ −7.7 cm⁻¹, and easy plane anisotropy, D₁ = +7.3 cm⁻¹, for the central Ni^II ion (Fig. S9 and Table S3†). The D_zz axes for Ni1–3 lie along the shortest Ni-μ₃O distance, whereas for Ni4, D_zz is oriented along the longest Ni-μ₃O distance (Fig. S10†). The negative and positive signs of D are thus associated with the compressed and elongated character of the octahedron, respectively.⁵² The calculated values are in agreement with the experimentally fitted values from the susceptibility and magnetization data, and consistent with values in the literature.⁵² The ground state electronic arrangements for the ring and the central Ni^II ions are found to be {(d_yz)² (d_xz)² (d_xy)² (d_z²)¹(d_x²−y²)¹} and {(d_yz)²(d_xy)²(d_xz)²(d_z²)¹(d_x²−y²)¹} with the dominant spin conserved electronic transitions d_xy → d_x²−y² and d_xz → d_x²−y²/d_z², respectively.

Excitonic magnetism

Above, a series of first principle calculations have been outlined producing a quantitative range for model parameters in the magnetic Hamiltonian of [Ni₇]. In this section we validate these parameters through a comparison with magnetic neutron scattering data. We first discuss the definition of the structure factor in terms of correlations of magnetic spins. We then discuss a formalism to calculate this dynamic structure applying Green's functions formalism that links the measured neutron scattering cross section with mean field response theory. We then consider the magnetic Hamiltonian of [Ni₇] and use this formalism to solve the case of an isolated [Ni₇] molecule for both the excitation energy levels and their corresponding neutron intensities determined by dipolar selection rules. The intensity measured in an experiment of scattering neutrons off magnetic Ni^II ions is directly proportional to the structure factor ,

where g_L is the Landé factor and f(Q) is the Ni^II form factor. The term is the polarization factor, reflecting the fact that neutron scattering is only sensitive to the spin component perpendicular to the momentum transfer, . is the dynamic structure factor which is the Fourier transform of the spin–spin correlation function:

where α, β = +,−, z. The dynamic structure factor is related to the Green's response function by

To relate the Greens function to the microscopic magnetic Hamiltonian, we note that in linear response theory, it is related to the Fourier transform of the retarded Green's function that is given by

with ij representing site indices with the heaviside function Θ(t) forcing causality by excluding negative values of t. By taking time derivatives of both sides of the equation and Fourier transforming, an equation of motion for the Green's function can be developed and hence related directly back to the neutron scattering cross section by taking the imaginary part, as outlined in previous studies.^53–56

The approach followed in previous works⁵⁷ applying the Green's functions to neutron spectroscopy is to divide the magnetic Hamiltonian into two parts – a local single-ion part for the magnetic ion of interest (including terms like the crystalline electric field, spin–orbit coupling, and molecular fields), and an interion part describing interaction between magnetic ions (usually modelled in terms of a Heisenberg exchange). Here, we simplify this by considering 1 as a single ion entity with the eigenstates of the total magnetic Hamiltonian defined by {|n〉} with corresponding energy eigenvalues {ω_n}. The Green's function is then reduced to the single-ion term,

where the matrix elements are defined by S_αmn ≡ 〈m|S_α|n〉 and the thermal Boltzmann factors . For calculation purposes, we set ħω = E + iΓ where Γ can be physically interpreted as an energy broadening (or dampening) of the magnetic excitations sampled at an energy transfer of E. The final neutron cross section is then derived summing over all combinations of g^αβ(ω). Note that in the low temperature T → 0 limit, this reduces to the simplified expression,
where ω_n0 ≡ ω_n − ω₀. This reduced expression avoids the need to sum over thermally excited inter-level transitions. The problem of calculating the neutron cross section then reduces to deriving the single ion response for 1. Given that each Ni^II ion is in a S = 1 state, from Hund's rules and the Pauli principle, each of the seven Ni^II sites have a three-fold multiplicity associated with different z-axis projections. This results in 1 having 3⁷ = 2187 basis states. In this study, we have used these 2187 basis states to solve the following magnetic Hamiltonian exactly,
with H_rr, H_CR, H_aniso corresponding to ring, central, and anisotropic terms of the magnetic Hamiltonian. The Ni^II labelling and notation is schematically illustrated in Fig. S11.† These are explicitly given by,


where the indices 1–6 refer to the Ni^II ions on the ring and 7 the central Ni^II ion. This magnetic Hamiltonian divides up 1 into a problem with two exchange constants, J_rr and J_CR, and two anisotropic terms, D_1,2. Here we have used the result above that the spins on the outer wheel have uniaxial anisotropy while the inner spin has easy-plane anisotropy. Note that in the analysis discussed below, we have performed a fit without anisotropy, however this produces exchange constants that are unphysically large (for example J_CR ∼ −4 meV = −32 cm⁻¹) in comparison to the first principles calculations and experimental fits discussed above. To constrain the parameters used to describe the neutron response, it is important to consider the magnetic susceptibility which is related to the real part of the Green's response function and to the imaginary part via Kramers–Kronig relations.

In the high temperature paramagnetic limit the susceptibility scales as . In mean field theory, a plot of χ⁻¹ as a function of temperature has an x-axis intercept defined by the Curie–Weiss temperature, Θ_CW, and is related to the exchange constants by,

which provides a way of constraining the Heisenberg exchange through fitting the high temperature susceptibility. Fig. S12† plots the inverse of the magnetic susceptibility where a fit of the data for T > 100 K (in the high temperature paramagnetic regime) affords Θ_CW = −18.5 K ≡ −12.89 cm⁻¹, which fixes the sum J_rr + J_CR in our analysis discussed below with regards to the parameterization of the magnetic Hamiltonian. We note that this implies, based on the sign of the sum of the exchange constants, that the dominant exchange is antiferromagnetic.

Neutron scattering

Neutron spectroscopy results from LET (ISIS) are presented in Fig. 3. Fig. 3(a and b) illustrate the background corrected data at T = 2 K using the detailed balance analysis to extract for incident energies of E_i = 12 and 3 meV. In both panels, we have corrected for a phonon background using the extracted at T = 80 K by,

thereby estimating the purely magnetic cross section at T = 2 K. The scattering in Fig. 3a and b displays a weak peak at ∼6 meV and strong scattering at low energies below ∼3 meV. This is further displayed in Fig. 3c which plots the momentum integrated intensity as a function of energy for the two experimental configurations. For comparison, the data are normalized to agree at low energy transfers. The dashed line is discussed below in terms of a parameterization of the microscopic magnetic Hamiltonian.

Parameterization

To parameterize the results, we have combined the first principles DFT results with the thermodynamic magnetic susceptibility to constrain excitonic calculations for the neutron spectroscopy results.

The methodology we have used involves first fitting the high temperature magnetic susceptibility to obtain a Curie–Weiss constant thereby fixing the sum of the two exchange constants J_rr and J_CR with the signs of these exchange constants fixed through the DFT first principles calculations described above. Applying the results from neutron spectroscopy and our excitonic calculations described above, we then parameterize the two excitations observed through varying the anisotropy parameters, also guided by first principles calculations.

This parameterization was done by performing a least squares fit to the E_i = 12 meV data illustrated in Fig. 3. The results of this are shown in Fig. 4a. This fit produces a weak high energy peak as found in the data, however it fails to capture the significant intensity present at low-energies found in the E_i = 3 meV data in Fig. 3b and c. The parameters were then varied to bring the spectra down to lower energies while preserving a weak peak at ∼6 meV and keeping the values consistent with DFT estimates and within the constraints set by the high temperature Curie Weiss temperature (Θ_CW). Based on this iterative procedure, we derive the following parameters for the microscopic magnetic Hamiltonian, J_cr = +4.0 cm⁻¹, J_rr = −5.66 cm⁻¹, D₁ = +8.0 cm⁻¹ and D₂ = −1.6 cm⁻¹. The results of this analysis are displayed in Fig. 3d and 4b with differences in values from Fig. 4(a) and (b) representative of the underlying error bars for the parameterized values. These are in good agreement with fits of the susceptibility/magnetization data and theoretical calculations.

Conclusions

Despite several [Ni^II₇] Anderson wheels being previously reported, there has to date been no consensus on the magnitude and sign of the pairwise magnetic exchange interactions present, nor on the sign/magnitude of the single ion anisotropies of the constituent metal centres. In order to address this we have reported the synthesis, structure and characterization of the Anderson wheel [Ni₇(hmp)_7.55(pic)_4.45](ClO₄)₂·6MeOH (1·6MeOH) through a combination of first principles calculations, magnetic susceptibility and magnetisation measurements, and neutron scattering. To the best of our knowledge this combination of techniques is employed for the first time on any [Ni₇] species. To consistently describe the magnetic parameters and neutron response we applied an excitonic formalism using a Green's function response theory. The combined conclusions from these techniques and data analysis are: (a) the magnetic exchange between the central Ni ion and the ring Ni ions is ferromagnetic, (b) the magnetic exchange between the ring Ni ions is antiferromagnetic, (c) the central Ni ion possesses easy plane anisotropy, (d) the ring Ni ions possess easy axis anisotropy. Given the structural similarity between 1 and other [Ni₇] in the literature, we believe these conclusions may also be valid more generally.

Author contributions

EHP and LRBW performed the synthesis, and collected the PXRD/SQUID data; GSN collected and solved the single crystal XRD data; MKS performed the theoretical calculations; JS fitted the susceptibility and magnetisation data; JRS, VGS, RAE, HL and CS collected/analysed the neutron scattering data; EKB conceived the idea. All authors contributed to writing and editing the manuscript.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

We thank the European Union Horizon 2020 Research and Innovation Programme under the Marie Skłodowska-Curie grant agreement no. 832488 (MKS).

References

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Footnote

† Electronic supplementary information (ESI) available. CCDC 2286832. For ESI and crystallographic data in CIF or other electronic format see DOI: https://doi.org/10.1039/d3qi02307a

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