A crystal of [Cr₄Dy₄(μ₃-OH)₄(μ-N₃)₄(mdea)₄(piv)₈]·3CH₂Cl₂ was prepared as previously described (Rinck et al., 2010).

Angular resolved susceptibility measurements were performed on a Quantum Design MPMS SQUID magnetometer (Superconducting Quantum Interference Device) using the commercial horizontal rotator from Quantum Design. The single crystal habit was determined by using a single crystal Oxford Xcalibur3 X-Ray diffractometer. The crystal was mounted on a square acetate foil (side ≈ 2 mm) and fixed onto the horizontal rotator using silicon grease. Details about the crystal orientation can be found in Table S1.

Torque magnetometry experiments were performed by using a homemade two-legged CuBe cantilever separated by 0.1 mm from a gold plate (Perfetti, 2017). The cantilever was inserted into an Oxford Instruments MAGLAB2000 platform with automated rotation of the cantilever chip in a vertical magnet. The capacitance of the cantilever was detected with an Andeen-Hegerling 2500. An Ultra Precision Capacitance Bridge. Details about the crystal orientation can be found in Table S1.

Ab initio calculations were performed by using MOLCAS 7.8 quantum chemistry package (Aquilante et al., 2010). Each mdea ligand deviates from mirror symmetry by a slight twisting about the Cr-N bond. Two structures of the Cr₄Dy₄ compound were therefore considered, which differ in the arrangement of the twist directions of the four ligands of the molecule: Structure 1 has D₂ point group symmetry (two Dy sites were calculated ab initio) whereas Structure 2 has an S₄ point group symmetry (one Dy site was computed ab initio since all Dy sites are equivalent). Mononuclear structures containing only one Dy site were built by replacing neighbouring metal sites by their diamagnetic equivalents: Lu was used in place of neighbouring Dy sites while Sc³⁺ ions were employed in place of Cr³⁺ in these calculations. Importantly, the entire ligand framework of the original Cr₄Dy₄ molecule was kept unaltered. All atoms were described by ANO-RCC basis sets of VTZP/VDZP quality (Roos et al., 2004, 2005, 2008). The employed basis set contractions are listed in the Supplementary Information. All calculations were of the SA-CASSCF/RASSI kind (Malmqvist et al., 2002; Chibotaru et al., 2008). The active space of the CASSCF method included the 4f ⁹ configuration of the Dy site. The spin-orbit coupling was included within RASSI method. All spin sextet states, 128 spin quartet states and 130 spin doublet states arising from the defined active space CAS(9in7) were included in the spin-orbit interaction. In the basis of the obtained spin-orbital multiplets, the g tensor, parameters of the crystal field and other related magnetic properties were evaluated within the SINGLE_ANISO module (Aquilante et al., 2010; Chibotaru and Ungur, 2012).

From the single crystal magnetometry experiment, whose results are reported in Figure 2, it appears that the out-of-plane anisotropy is, as expected for a tetragonal system, more pronounced than the in-plane anisotropy. In rotation along c (open circles in Figure 2), the resulting in-plane contribution of the anisotropy is investigated whereas for the rotation along a, the magnetic field goes from perpendicular to parallel to the molecular plane (see Table S1). Rotating the sample along a (filled circles in Figure 2) results in a strong variation of the ratio of the magnetization and the magnetic field M/B, assumed at this moderate field to coincide with the susceptibility and reported as χ. This is in agreement with previous theoretical calculations (Figure S1). Indeed, the previously proposed model (Rinck et al., 2010) suggested a strong variation between the in-plane and the out-of-plane magnetic susceptibility. However, the minimum of the magnetic susceptibility is surprisingly measured when the magnetic field is applied along the fourfold axis, and, the maximum when the magnetic field is applied in the (ab) plane. Consequently, the rotation along c, during which the magnetic field remains inside the molecular plane is almost constant at the maximum value, the small deviation being attributed to an experimental error of about 2° in the orientation of the crystal for this measurement (see Supplementary Information).

Magnetic field dependence of the magnetization and temperature (T) dependence of the susceptibility were measured on a single crystal in order to unequivocally determine the orientation of the maximum of the magnetic moment within the crystal. At all fields and temperatures, the magnetic response along the (ab) plane is higher than along c. This is in agreement with the angular dependence of the susceptibility measurements. Interestingly, the in-plane χT vs. T curve (Figure 3 green dots) exhibits a slight decrease followed by a sharp increase at low temperatures. This behaviour can be attributed to a mixture of effects, namely the depopulation of the CF levels of the Dy^III ions and the presence of coupling between the magnetic ions. Conversely, the magnetization curve obtained along c (Figure 3 pink dots) exhibits a monotonic increase from low to high temperature, with absence of saturation even at room temperature, at difference from what was expected from calculations (Rinck et al., 2010).

If we take the weighted average of the single crystal measurements according to 1/3(χT_||+ 2 χT₋) then the obtained room temperature value of 62.3 emuKmol⁻¹ is close to that expected for the randomly orientated independent ions of χT = 64.2 emuKmol⁻¹.

SCM experiments provide the magnetic anisotropy of the system but do not help to disentangle the single centres contributions. The latter are symmetry related but not necessarily coincident if the symmetry of the site is lower than the symmetry of the crystal. The symmetry of the molecule (Figure 1) constrains the main anisotropy axes to lie along the mirror planes of the molecule. Moreover, as the metal ions lie on symmetry elements, mirror planes and 2-fold axes for the Cr^III and Dy^III ions, respectively, also the individual principal anisotropies show geometrical constraints. In particular, the only free parameter, beyond anisotropy amplitude, is the Euler angle between the z magnetic axis of the single centres and the c crystallographic axis. Cantilever torque magnetometry represents an excellent technique for this purpose. The measurements were performed using high magnetic fields to overwhelm the intramolecular interactions and directly access the single ion contributions. The good alignment of the crystal is proven by the symmetry of the peaks and the position of the zeros in the angular dependence of the torque moment (Figure 4), which in the experimental setup used here is detected along the rotation axis (Table S1). The in-plane rotation exhibits a τ = 0 every 45°, i.e., when the field lies along a principal crystallographic axis and along a mirror plane. The deviation from a perfect sinusoidal curve (steeper/smoother variation of the torque when the field is perpendicular/parallel to the easy axis) is a characteristic feature of torque measurements taken at high fields and is the key to disentangle noncollinear contributions. (Perfetti et al., 2014). Moreover, the out-of-plane rotation shows two significant features: (i) a shoulder at 90° and (ii) the peculiar shape of the torque moment near 0° and 180° with τ increasing less rapidly than expected (Perfetti, 2017). The comparable magnitude between the two rotations indicates that the z axes of the lanthanides should be significantly tilted from the c axis, since the Dy^III ions are expected to be the dominant contributors to the anisotropy of the complex.

Due to the intrinsic complexity of this system, our approach to simulate the torque data included the smallest number of parameters able to reproduce the experimental data. A global simulation of all the experimental torque data (Figure 4; Figure S2) was thus obtained using the following spin Hamiltonian, which does not account for the exchange interaction between metal sites: H=∑i=14{μBgz,DyHzi·S^zi+μBgxy,DyHxyi·S^xyi+μBgCrH·S^i      +DCrS^zi2} where the summation contains the Zeeman energy (first three terms) and the zero-field splitting (ZFS, fourth term). Coupling terms were neglected due to the high applied fields (B ≥ 5T). The Dy^III ions were described as S = 12 pseudospins with axially anisotropic g factors. The Cr^III ions were modelled using an isotropic g and an axial ZFS term. The individual z_i axes of the four Dy^III and Cr^III ions are related by the symmetry elements of the molecule. The best agreement with experiments was obtained using the parameters in Table 1.

The shoulder near 90° in the out of plane rotation can only be reproduced if the easy axes of the Dy^III ions are very close to the (ab) plane (between 75 and 85° from the c axis, depending on the chosen g components). The slope between 0° and 40° (and, by symmetry, between 140° and 180°) can only be reproduced by introducing an axially anisotropic contribution D from the Cr^III ions (D = −0.7 cm⁻¹) with the z axis at 0–10° from c (depending on the value of D). The correlation between the value of D and the angle between the z axis of the Cr^III ions and c is intrinsically difficult to disentangle based on the experimental data that we collected. In Figure 5 we reported the contributions of the Dy^III and Cr^III ions to the torque in both rotations. Interestingly, the Cr^III anisotropy does not significantly affect the data in the in-plane rotation, i.e., along c, whose features are unambiguously indicative of the almost in-plane orientation of the Dy^III easy axes. The simplification of the model, i.e., neglecting the exchange interaction between the magnetic ions, was necessary for the treatment of the contributions of Dy^III and Cr^III ions to the total magnetic anisotropy. However, the development of a more complex model, also encompassing interactions and excited states, is ongoing.

In the previous work published by some of us (Rinck et al., 2010), mononuclear fragments of Dy and Cr units were computed. In the light of the new experimental data, we performed new ab initio calculations on the two full structures of Cr₄Dy₄ molecules without altering the ligand framework for the cluster fragmentation. Table 2 reports the obtained energy spectrum and g-tensor of the ground Kramers doublet on the calculated Dy sites and the angle made by the ground main magnetic axis with the c crystallographic axis for both Structure 1 and Structure 2. On each Dy site there are several excited Kramers doublets with small excitation energy. Therefore, we can expect the lowest of them to be admixed by the Dy^III-Cr^III exchange interaction.

These new computation results, corroborated by the SCM and the CTM experiments, give the main anisotropy axis of the Dy^III ions lying much closer to the Dy₄ plane than predicted by the previously published calculations. The improvement of the output can be explained by the high sensitivity of the ab initio results on the cluster fragmentation in the Cr₄Dy₄ complexes, as, due to insufficient computational resources, the applied fragmentation was more severe in the previous calculations.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.