Origin of the unusual ground-state spin S = 9 in a Cr10 single-molecule magnet

Origin of the Unusual Ground-State Spin S= 9 in a Cr10 SingleMolecule Magnet Javier Rubín,*Ana Arauzo, Elena Bartolomé,*Francesco Sedona, Marzio Rancan, Lidia Armelao, Javier Luzón, Tatiana Guidi, Elena Garlatti, Fabrice Wilhelm, Andrei Rogalev, Andreas Amann, Stefano Spagna, Juan Bartolomé, and Fernando Bartolomé* Cite This: J. Am. Chem. Soc. 2022, 144, 12520−12535 Read Online ACCESS Metrics & More Article Recommendations * sıSupporting Information ABSTRACT: The molecular wheel [Cr10(OMe)20(O2CCMe3)10], abbreviated {Cr10}, with an unusual intermediate total spin S= 9 and non-negligible cluster anisotropy, D/kB=−0.045(2) K, is a rare case among wheels based on an even number of 3d-metals, which usually present an antiferromagnetic (AF) ground state (S= 0). Herein, we unveil the origin of such a behavior. Angular magnetometry measurements performed on a single crystal confirmed the axial anisotropic behavior of {Cr10}. For powder samples, the temperature dependence of the susceptibility plotted as χT(T) showed an overall ferromagnetic (FM) behavior down to 1.8 K, whereas the magnetization curve M(H) did not saturate at the expected 30 μB/fu for 10 FM coupled 3/2 spin Cr3+ ions, but to a much lower value, corresponding to S= 9. In addition, the Xray magnetic circular dichroism (XMCD) measured at high magnetic field (170 kOe) and 7.5 K showed the polarization of the cluster moment up to 23 μB/fu. The magnetic results can be rationalized within a model, including the cluster anisotropy, in which the {Cr10} wheel is formed by two semiwheels, each with four Cr3+ spins FM coupled (JFM/kB= 2.0 K), separated by two Cr3+ ions AF coupled asymmetrically (J23/kB=J78/kB=−2.0 K; J34/kB=J89/kB=−0.25 K). Inelastic neutron scattering and heat capacity allowed us to confirm this model leading to the S= 9 ground state and first excited S= 8. Singlemolecule magnet behavior with an activation energy of U/kB= 4.0(5) K in the absence of applied field was observed through ac susceptibility measurements down to 0.1 K. The intriguing magnetic behavior of {Cr10} arises from the detailed asymmetry in the molecule interactions produced by small-angle distortions in the angles of the Cr−O−Cr alkoxy bridges coupling the Cr3+ ions, as demonstrated by ab initio and density functional theory calculations, while the cluster anisotropy can be correlated to the single-ion anisotropies calculated for each Cr3+ ion in the wheel. ■INTRODUCTION Magnetic molecular wheels are a subclass of molecular magnets that have received considerable attention for their intrinsic magnetic properties, as benchmark systems for the investigation of macroscopic quantum coherent phenomena, and in view of their possible application in quantum information processing. 1−3 The planar and high-symmetry geometry of cyclic molecules makes them in addition attractive for deposition onto substrates, a critical step toward device fabrication. 4,5 Magnetic wheels made of different 3d transition metals (Cr, Ni, Cu, V, Mo, Mn, Fe, etc.) have been extensively investigated. 6,7 For example, ferric wheels of different nuclearity (Fe6, 8 Fe10, 9−11 Fe12, 12 Fe18 13 ) have been reported presenting dominant antiferromagnetic (AF) coupling between the Fe3+ ions, with Si= 5/2 spin, and a ground state of total spin S= 0. In contrast, single-molecule magnet (SMM) behavior was found for noncyclic clusters Fe8with a ground state S= 10 and activation barrier for spin reversal U/kB= 24.5 K 14 and in Fe4propellers (with S= 5 and U/kB= 3.5 15 to 15.6 K 16 ), for which quantum tunneling of the magnetization (QTM) between the states ±mwas observed. A Ni12 cyclic cluster showing ferromagnetically (FM) coupled Si=1Ni spins giving rise to a high spin S= 12 ground state has been reported. 17 The presence of two different nearest-neighbor FM interactions, and an AF nearest-neighbor interaction could only be assessed by means of inelastic neutron scattering experiments (INS), showing the relevance of this type of experiments in the resolution of the intracluster interactions. The SMM behavior corresponded to U/kB= 9.6 K. In particular, Cr3+ wheels have been intensively studied. The best well-known molecule of this family and precursor of other cyclic systems is {Cr8}, 18,19 characterized by a perfect AF coupling between each of the eight Cr3+ ions with spin Si= 3/2, leading to a total S= 0 ground state. Four-dimensional INS allowed to directly obtain the dynamic correlation functions, confirming the existence of anisotropy and the Received: May 24, 2022 Published: June 27, 2022 Articlepubs.acs.org/JACS © 2022 American Chemical Society 12520 https://doi.org/10.1021/jacs.2c05453 J. Am. Chem. Soc. 2022, 144, 12520−12535 Downloaded via CSIC on June 26, 2023 at 10:03:27 (UTC). See https://pubs.acs.org/sharingguidelines for options on how to legitimately share published articles. presence of S-mixing, while demonstrating 20 that the lowtemperature dynamics of {Cr8} is not determined by coherent Neel vector tunneling, as it had been proposed earlier. 21 Starting from the synthesis of that molecule, a whole series of other homometallic 22,23 and heterometallic Cr-based cages were engineered. 24−28 The heterometallic counterparts were used to compare the spin dynamics of AF closed {Cr8} and open {Cr8Zn} rings, 29 the latter displaying quantum oscillations of the total spin under an applied field. 30,31 Finite-size effects were also observed on the local magnetization of open rings. 32,33 Furthermore, heterometallic {Cr7Ni} 34,35 rings containing seven Cr3+ ions and one Ni2+ ion AF coupled, with an S= 1/2 ground state, have been considered very carefully, as possible candidates for qubits. 2,4 An interesting case is the family of {Cr10} wheels of general formula [Cr10(OR)20(O2CR′)10] reported by Low et al., 36 whose magnetic properties strongly depend on the OR and O2CR′ligands. Susceptibility measurements revealed that those members of the family with ethoxy group ligands (Et) interconnecting the Cr3+ ions exhibited AF behavior (S= 0). The exchange constants could be readily obtained from the Curie−Weiss law. In contrast, when the interconnecting ligand contains a methyl group, giving rise to the methoxy group OMe, the Cr3+ spins couple ferromagnetically, as determined by the Curie−Weiss positive constant, though an AF coupling contribution, observed as a downturn in χT, appeared in the compounds with the smaller R′groups, which was assigned to small intermolecular interactions. In both the AF and FM wheels, the values of the interaction constants depend on the external ligand O2CR’. Only one of the members of this family, denoted as 5in ref 36, carrying R = Me and the largest R′group (CMe3), shows an overall FM behavior down to the lowest temperature (1.8 K), although it is peculiar, as the magnetization does not saturate at the expected 30 μB/fu for 10 FM coupled 3/2 spin Cr3+ ions but to a much lower value, indicating a ground state different from S = 15. Electron paramagnetic resonance (EPR) measurements were performed by Sharmin et al. 37 for this member of the family, with applied magnetic field along the axis perpendicular to the average {Cr10} ring’s plane. The absorption spectrum indicated a spin S= 9 ground state, whose temperature dependence suggests an excited state at ca. 10 K, proposed to be a spin S= 10 multiplet. The single-molecule uniaxial anisotropy constant was derived to be D/kB=−0.045(4) K from the spectrum taken at 7 K. The unusual intermediate ground-state spin S= 9 of this wheel has remained without explanation for a long time, apart from the assumption of displaying both FM and AF exchange interactions in the wheel. The presence of more than one value of exchange constants in this family of compounds had already been suggested by Low et al. 36 in the wheel denoted as 1with all FM interactions. We propose a model of interactions along the full wheel, which is supported by both ab initio, DFT calculations and experimental results, and shows a correlation of structural data and interaction parameters all. In the present work, we investigate in-depth the magnetic properties of wheel 5(Figure 1), from now on simply called {Cr10} to relate them to a model of exchange interactions and single-ion anisotropies along the 10 Cr3+ ions of the full wheel. This model gives a consistent interpretation of the origin of the S= 9 ground state. In the next sections, we will show angular magnetometry on a single crystal (SC), which has allowed us to directly observe the wheel’s magnetic anisotropy, in agreement with EPR data, and XANES and XMCD spectra at the Cr K-edge, which are used to determine the magnetic moment of Cr3+ ions. The model proposed to explain the existence of the S= 9 ground state is supported by low-temperature INS spectra and heat capacity measurements. Finally, the occurrence of SMM behavior, with an activation energy approximately given by DSz 2, is shown through ac susceptibility measurements down to 0.1 K. ■RESULTS Dc Magnetometry of Polycrystalline {Cr10}. The magnetization as a function of the applied field, M(H), measured for a polycrystalline sample at T= 1.8 K is shown in Figure 2a (green symbols). The magnetization does not saturate to the expected value of 30 μB/fu for 10 uncoupled Cr3+ ions but reaches a much smaller value of 16.8 μB/fu at the maximum applied field of H= 50 kOe, suggesting that the ground state is not S= 15 but possibly S=9. The temperature dependence of susceptibility, plotted as χT(T), is shown in Figure 2b for a powder sample fixed in cotton wool to prevent the orientation of the grains. At high temperature, χTapproaches the limit expected for 10 free Cr3+ ions with g=2,χT300 K =10g2Si(Si+ 1)/8 = 18.7 emu·K/mol. For decreasing temperature, the product χTincreases, reaching 32.7 emu K/mol at 1.8 K, (Figure 2b). This behavior points to an overall predominance of ferromagnetic interactions in the wheel. A similar χT(T) curve was earlier found Figure 1. Structure of {Cr10(OMe)20(O2CCMe3)10} wheel. (a) Local D5d quasi-symmetry of carboxylate ligands has been outlined by lines. Color code: Cr green, O red, C black/gray atoms above/below the plane of metal atoms; (b) {Cr10} wheels are tilted with respect to the stacking direction along the a-axis. Journal of the American Chemical Society pubs.acs.org/JACS Article https://doi.org/10.1021/jacs.2c05453 J. Am. Chem. Soc. 2022, 144, 12520−12535 12521 by Low et al., 36 although the reported value at 1.8 K was somewhat smaller (25 emu K/mol). It is to be noted that the magnetic properties of the bulk (the saturation magnetization value, the initial M(H) slope, and χTat a low T) changed when the sample was mechanically crushed into a powder or embedded in oil to fix the grains (see experiments in S1). We believe these changes may be explained by the severe dependence of the magnetic properties on even tiny distortions in the wheel Cr−O−Cr angles (vide infra). This may also be the reason for the different M(H) and χT(T) data for powder {Cr10} samples reported by previous authors. 36,38 {Cr10} Anisotropy: Single-Crystal Results. To further investigate the magnetic anisotropy of {Cr10}, we performed magnetometry measurements on an oriented single crystal. The compound crystallizes in the triclinic P1space group with Z= 1, thus all {Cr10} molecules are identically oriented in the crystal; the asymmetric unit contains five contiguous Cr3+ ions, and the other five ions in the ring are obtained by inversion. The molecule displays a close to D5d symmetry (Figure 1). Magnetization was measured as a function of the angle θ between the molecular fivefold quasi-symmetry axis and the direction of the applied magnetic field. The magnetization curves M(H) measured with the applied magnetic field H parallel (θ=0°) and perpendicular (θ=90°) to that quasifivefold ring axis clearly differ, evidencing a substantial anisotropy (Figure 2a). The measured M(H) curve with the field Hat an intermediate angle falls in between, as may be expected. No saturation of the magnetization is observed up to 50 kOe (Figure 2a). Figure 3 shows the magnetization of the SC measured as a function of the angle, M(θ), with a constant applied magnetic field H= 1 kOe, performed at three different temperatures, T= 1.8, 5, and 10 K. Though the sample is paramagnetic, an anisotropic magnetization response is clearly observed in the 1.8 K measurements. The maxima at θ=0°and minima at θ= 90°identify the quasi-symmetry axis as an easy axis (EA) of magnetization, while a direction on the wheel’s mean plane is a hard axis (HA). The amplitude of the angle dependence decreases strongly with temperature and is practically negligible at 10 K. To interpret quantitatively these results, we will assume that the strong exchange interaction approximation is applicable; therefore, the {Cr10} cluster may be described by a total cluster spin Sin the giant-spin (GS) approximation. 39 At a very low temperature, we will also assume that only the S=9 ground multiplet is thermally occupied. Thus, the cluster Hamiltonian with uniaxial anisotropy described by the zero-field splitting parameter D, under an applied external field at an angle θwith the anisotropy axis, is written as H DS SS gSH SH () 1 3(1) (cos sin) z Bzx Cl 2 Ä Ç Å Å Å Å Å Å Å Å É Ö Ñ Ñ Ñ Ñ Ñ Ñ Ñ Ñ θ μθ θ =− + ++ (1) The first term corresponds to the uniaxial anisotropy interaction, and the second to the Zeeman interaction with the applied magnetic field H. This Hamiltonian operates on the 9|,sz⟩states. From the eigenvalues and eigenfunctions, the partition function was calculated, and the magnetization M(θ, T,H) was predicted, within the Boltzman statistics, at fixed temperature and field modulus, at varying θ. The parameters D/kB=−0.045 K and g= 2 (isotropic) were used, the same previously found by EPR measurements. 37 This kind of experiment and procedure was applied earlier to a single crystal of Mn12-PrCl. 40 The calculated magnetization is shown in Figure 3 as solid curves for 1.8 K (black), 5 K (blue), and 10 K (red). The data are presented in arbitrary units because of Figure 2. Dc magnetometry. (a) Field dependence of the magnetization, M(H), at T= 1.8 K for a powder sample of {Cr10} and SC sample measured with the applied magnetic field parallel (EA) and perpendicular (HA) to the easy axis of magnetization, and at an intermediate angle (IA); (b) temperature dependence of the susceptibility-temperature, χT(T), for the same samples. The dotted line corresponds to the free-ion high Tlimit, χT300K = 18.7 emu K/ mol. Figure 3. Single-crystal angular magnetization. Magnetization of the SC as a function of the angle between the easy axis of magnetization and the applied magnetic field (H= 1 kOe), measured at three different temperatures, T= 1.8, 5.0, and 10 K, together with calculated curves obtained using the Hamiltonian in eq 1 with S=9,g= 2 and D/kB=−0.045 K. Journal of the American Chemical Society pubs.acs.org/JACS Article https://doi.org/10.1021/jacs.2c05453 J. Am. Chem. Soc. 2022, 144, 12520−12535 12522 several experimental indeterminations in the weighing of the very small sample. However, it must be noted that all of the data could be explained with the same scaling factor. The χT(T) curves for the SC with the applied field parallel and perpendicular to the easy axis of magnetization are shown in Figure 2b. We observe that while χT(T) increases with decreasing the temperature down to 1.8 K when the field is applied with H|| EA, the curve reaches a maximum and then decreases when H|| HA. So far, we can state that {Cr10} wheels at a low temperature show S= 9 as ground multiplet, uniaxial anisotropy (D/kB= −0.045 K) with the EA perpendicular to the wheel plane. Heat Capacity. The heat capacity as a function of the temperature in zero applied field was measured for the powder sample (Figure 4). The C(T) curve shows a typical Schottky shape, which is reasonably well fitted under the single-cluster Hamiltonian of eq 1, in agreement with the magnetometry results. By considering in the model also the excited S=8 multiplet, as derived in the section from INS experiments (vide infra), split by the same anisotropy constant D/kB=−0.045 K, 37 the calculated curve (red line), lies slightly above the experimental data. XANES and XMCD at Cr K-Edge. X-ray absorption spectroscopy (XANES) and X-ray magnetic circular dichroism (XMCD) experiments at the Cr K-edge were performed on a powder sample. Figure 5a shows the XANES and XMCD spectra measured at 170 kOe and 7.5 K. The photon energy range spans 5980 eV < E< 6060 eV. Three regions (separated by vertical dashed lines in Figure 5a) should be distinguished. The pre-edge region, below E≈5997 eV involves transitions from 1s to unoccupied 3d states, which may be purely quadrupolar or dipolar if the final states are symmetry-allowed hybridized 3d−4p states. Above E≈5997 eV, XANES and XMCD are dominated by transitions from the 1s to the 4p empty states. Finally, at energies above 6035 eV, superCoster−Kronig multielectron excitations take place, involving two-electron transitions from 1s to 4p and from shallow core 3p to 3d states. 41,42 In this third region, these multielectron excitations dominate the XMCD spectrum. The XANES and XMCD spectra at the pre-edge energy region are shown in expanded view in Figure 5b. Two peaks are clearly observed in XANES, with corresponding maxima and two minima in the XMCD spectrum. The feature at the lowest energy (5990.85 eV) may be ascribed to the 1s →t2g transition. 43 A direct comparison with the spectra obtained in reference compounds for Cr oxidation states ions 41,44,45 leads us to conclude that chromium ions in {Cr10} are in trivalent state, Cr(III). Since the initial state in the K-edge, 1s, is not spin−orbit split, XMCD is originated only in the orbital imbalance of the final states. As shown by Thole and Carra, 46,47 the integrated XMCD signal, IXMCD, is related through the orbital sum rule to the ground-state expectation value of the orbital moment ⟨Lz⟩ of the final state, which may be 4p for dipolar transitions or 3d Figure 4. Heat capacity. Symbols: temperature dependence of the heat capacity of the powder sample measured at H= 0. Dotted lines: magnetic contribution Cm(T) Schottky curve, calculated considering D/kB=−0.045 K and only the S= 9 (blue), and the S= 9 ground state and excited S= 8 state with the same Dvalues (red). Solid lines: total heat capacity, C(T)=Cm(T)+CL(T), where the lattice contribution is CL(T)=ATn, with A/R= 0.35 K−1.52,n= 1.52. Figure 5. XANES and XMCD spectra at Cr K-edge. (a) Top: (•) normalized X-ray absorption spectroscopy (XANES), measured at 7.5 K and 170 kOe; bottom: (•) Normalized X-ray magnetic circular dichroism (XMCD); red, right scale: integrated XMCD, (IXMCD)4p. The vertical red dashed lines signal the lower and upper limits of integration; (b) zoom of the pre-K-edge region; bottom, red, right scale: integrated XMCD, (IXMCD)pre‑edge; (c) XMCD(H) scaled to M(H) isotherms measured with SQUID magnetometry on the same sample. Note the matching of the XMCD(H) curve with M(H)atT= 7.5 K. Journal of the American Chemical Society pubs.acs.org/JACS Article https://doi.org/10.1021/jacs.2c05453 J. Am. Chem. Soc. 2022, 144, 12520−12535 12523 for quadrupolar ones. 48 As shown in Figure 5b, the integral of the pre-edge XMCD, (IXMCD)pre‑edge, is very small compared to (IXANES)pre‑edge (Figure S2.1), therefore indicating that ⟨Lz⟩3d is negligible. Likewise, in Figure 5a, we show that (IXMCD)4p =9 ×10−4eV is also very small, thus, ⟨Lz⟩4p is negligible. As we find that both orbital contributions from 3d and 4p electrons are negligible, the Cr magnetic moment is totally of spin character. Therefore, an isotropic gfactor is fully justified to describe the Zeeman effect. Besides, it is known that in compounds comprising just one magnetic element, the XMCD at the K-edge is proportional to the magnetization in an applied field, and therefore, to the magnetic moment of the absorbing atom. 41 By scaling the experimental XMCD K-edge spectra to that of a reference Cr(III) compound, namely, trans-[Cr(III)Cl2(pyridine)4]- (ClO4)·1/4H2O, in short {Cr(III)}, an estimation of the average Cr moment in the field direction may be extracted. The magnetic moment of {Cr(III)} is saturated at 3 K and 170 kOe, and amounts to 3.1 μB. 41 This value needs to be multiplied by a factor of 0.68 to scale with that of {Cr10} (see Figure S2.2), yielding a value of mCr = 2.1 μBper Cr ion, at T= 7.5 K and 170 kOe for {Cr10}. Since the coordination and distances around the Cr3+ ion are different in {Cr(III)} and {Cr10}, the spectral shapes are quite different. However, we consider this estimation correct, as we show below. The isothermal, field-dependent XMCD signal was measured by changing the helicity of the beam at a fixed photon energy of 5990.85 eV (*in Figure 5b). This incident energy corresponds to the Cr pre-edge, whose XMCD peak is sensitive to the 3d magnetic moment. When the field is varied in the range 0 < H< 170 kOe, at fixed temperature T= 7.5 K, the XMCD(H, Ephoton = 5990.85 eV) curve is proportional to M(H), i.e the magnetic moment from the 3d states, which is dominant with respect to any other contribution. In Figure 5c, the normalized XMCD(H) per Cr ion is shown in the right axis compared with the M(H) isotherms measured with SQUID magnetometry on the same powder sample. The XMCD(H) agrees perfectly with the isotherm for T= 7.5 K. The magnetization deduced at 170 kOe is M(H) = 22.6 μB, which corresponds to a value of mCr = 2.26 μB, per Cr ion which is very close to the estimation derived from the scaling of the XMCD spectrum described in the previous paragraph. It is noteworthy that the XMCD(H) curve continuously increases its value over the Ms=18μBsaturation value expected for S= 9, implying that the 10 spins of the molecule are progressively oriented by the increasing field. For even higher fields, one may expect that the limit for S= 15 i.e. Ms≈ 30 μBmay be reached, once all of the individual spins are maximally oriented along the field direction. Ab Initio and DFT Calculations. The multispin (MS) cluster Hamiltonian approximation consists of two terms, the first encompassing zero-field splitting by ligand field interactions on the single ions, Ha, and the second comprising the interion interaction, either dipolar or exchange, Hex H HH Cl a ex =+ (2) Here, Ha=∑i=1 10 Hiis a sum over the 10 Cr3+ spins, where Hi= DiSiz′ 2+Ei(Six′ 2−Siy′ 2) is the second-order single-ion zero-field splitting Hamiltonian of each Criwith axial and rhombic terms. We performed ab initio calculations to determine the local anisotropy of the Cr3+ ions in the {Cr10} ring. The axial (Di) and rhombic (Ei) local anisotropy terms were calculated for each of the five inequivalent Cr3+ ions of the asymmetric unit cell (numbered Cr1 to Cr5 as in ref 36) using the package ORCA (see S3). The values of Di/kB≈−0.25 K and Ei/Diof all ions were found to be similar, with a slight depletion around ion Cr3 (see Table 1). The anisotropy is axial (Di< 0) for all five ions with an important rhombic contribution (Ei/Diin the range 0.25−0.33). In all cases, there is a local easy anisotropy axis close to the {Cr10} ring’s mean axis (Figure 6a,b). The interion, intracluster exchange interaction may be expressed in terms of the single-ion spins, Si,witha Heisenberg−Dirac−van Vleck Hamiltonian, neglecting anisotropic exchange, dipolar and intercluster interactions H JS S2 ij ij ijex ∑ =− · ⟨⟩ (3) with ⟨ij⟩denoting all interactions between near-neighbor (n.n.) Cr3+ ions with spins Si= 3/2. The constants Jij were calculated by DFT in ORCA using B3LYP hybrid functional for each of the five Cr−Cr couples in the asymmetric cell. The theoretical results, summarized in Figure 6c and Table S4, evidence a clear asymmetry in the interaction around the Cr3 ion: indeed, while J23 is AF and J34 only slightly FM, all other interactions are FM (see Figure 6c). The calculated decreased values of Diand Jij around Cr3 can be correlated with small variations in the angles of the Cr-O-Cr alkoxide bridges connecting this ion to its n.n. It is noted that there are two types of Cr−O−Cr alkoxide bridges (see S4 and Figure 6a). The outer one points upward (alternatively downward in the next Cr−O−Cr bridge) with respect to the {Cr10} mean plane with the dihedral angle between the Cr− O−Cr and {Cr10} planes all in the range 105−108°.In contrast, the inner bridge points toward the molecule plane’s normal axis and downward (alternatively upward) with the Cr−O−Cr plane angle of ≈45°with respect to the {Cr10} plane. Notably, as shown in Figure 6a,b, both Cr−O−Cr bridges for Cr3 are peculiar: it is the only case in this molecule where the Cr−O−Cr angle of the inner bridge is larger than that of the outer one, on both sides of the ion, and this small angular difference drastically reduces the coupling constant. Although the absolute values for the interactions might somehow depend on the chosen functional, the results undoubtedly demonstrate the existence of a rupture in the symmetry of the interactions associated with small structural changes. The use of heteroleptic carboxylate/alkoxide bridging ligands in {Cr10} favors the formation of ferromagnetic Table 1. Anisotropy Constants a Cr ion Di/kB(K) Ei/Di Cr1 −0.32 0.33 Cr2 −0.25 0.28 Cr3 −0.20 0.25 Cr4 −0.27 0.26 Cr5 −0.33 0.33 {Cr10} cluster D/kB(K) E/D EPR 37,38 −0.045(4) INS −0.045(2) calculated Ea/DaEa/Da −0.033 0.15 a Ab initio calculated anisotropy constants Diand Ei/Diratio for the five Cr3+ ions in the asymmetric unit cell. Also, calculated and experimental cluster anisotropy Dand E/Dratio. Journal of the American Chemical Society pubs.acs.org/JACS Article https://doi.org/10.1021/jacs.2c05453 J. Am. Chem. Soc. 2022, 144, 12520−12535 12524 interactions between the Cr3+ ions, likely due to the orbital counter-complementarity effect introduced by the carboxylates. 49,50 However, we observe that the magnitude and sign of the exchange constant finely depend on structural parameters, in agreement with results of magneto-structural DFT studies performed on particular Cr2dimeric model systems. 50 To promote SMM behavior, it is not enough to have a large spin Sand single ions with large anisotropy, but parallel alignment between the anisotropy axes of the ions is also important. 50 Figure 6a,b shows the different deviation of the local anisotropic axes of each Criwith respect to the wheel’szaxis. The deviation angle and coupling Jij are correlated, as shown in Figure S4.1, which stems from the fact that both magnitudes depend on the local structural details around each Cri. The minimum deviation angle (∼17°) and largest AF exchange correspond to Cr3, for which the difference between the Cr−O−Cr angle of the inner and outer bridge is maximum and positive, whereas the largest deviation (26°) corresponds to Cr1, with the largest FM and maximum negative difference between the angles of the inner and outer Cr−O−Cr bridges. Adifferent approach to describe the cluster Hamiltonian is to apply the giant-spin (GS) approximation, as has been used already in eq 1,HCl =DSz 2+E(Sx 2−Sy 2), where Sis the total cluster spin, and x,y, and zare the cluster axes, with z perpendicular to the wheel plane. The isotropic exchange interaction between Cr spins is already implicitly considered when the ground state with total spin Sis assumed; therefore, the Jij parameters do not appear explicitly in this formulation. The next step is to correlate the multispin Hamiltonian description with the giant-spin one. The transformation from the local anisotropy of Cr3+ ions, Di, and interion interaction anisotropy to an equivalent effective anisotropy of the {Cr10} wheel, D, can be carried out in the strong isotropic exchange limit, and assuming collinearity of the spins, by means of a linear combination of the former, with diand dij coefficients, which may be calculated taking into account symmetry relations. Then, according to ref 51, the following relations are fulfilled DD D dD dD i i ii ij ij i j ant , ∑∑ =+=+ (4) dd21 i i ij ij , ∑∑ += (5) where Diare local anisotropy tensors, the Dij tensors contain interion anisotropy terms as, e.g., dipolar and exchange interactions, and Dis the anisotropy tensor of the full molecule. This method has been successful in several analyses of anisotropy in dinuclear clusters. 52−54 The coefficients dihave been calculated following the method described in ref 51 and are collected in Table S3.3. Once the divalues are calculated, we proceed to calculate the Da=∑idiDitensor. The Ditensors must be written in a common coordinate system, in fact, that of the Cr3 (Figure S3.1). It gives rise to a nondiagonal molecular tensor which has to be diagonalized to obtain the equivalent anisotropy constants Daand Ea, from single-ion anisotropies, i.e., excluding interion interactions. The result is Da/kB=−0.033 K and Ea/kB=−0.0048 K (Ea/Da= 0.0145). Hence, the magnetic anisotropy of the {Cr10} ring is uniaxial with a rhombic distortion smaller than that of the constituent Cr3+ ions, and the anisotropy direction (z′) is very close to the perpendicular to the wheel’s mean plane (≈2.7°out of the molecule’sz-axis). The difference between the experimental value D/kB= −0.045(2) K and the calculated value Da/kB=−0.033 K may be ascribed to the uncertainty originated by the approximations applied in the ab initio methods, or in the deviation from collinearity, but it could also have a contribution from the interion interaction, Dint term in eq 4. Considering Dint/kB= −0.012 K as an upper threshold for the interaction contribution, an estimate of the maximum Cr−Cr interion exchange anisotropy can be made using eq 4 and the calculated Figure 6. (a) Easy axes of magnetization (EAM) calculated by ab initio for the {Cr10} ring. The inner and outer Cr−O−Cr alkoxide bridges, the carboxylate bridges, and Cr ions numbering is that of the asymmetric unit as in Low et al.; 36 (b) lateral view; and (c) correlation between the angle Cr−O−Cr alkoxy bridges and DFTcalculated Jij coupling constants between spins of ions Criand Crjfor each of the five Cr3+ ions in the asymmetric cell of the {Cr10} wheel. Journal of the American Chemical Society pubs.acs.org/JACS Article https://doi.org/10.1021/jacs.2c05453 J. Am. Chem. Soc. 2022, 144, 12520−12535 12525 average coefficient dCrCr = 0.0428 (S5), yielding a negative value DCrCr/kB=−0.028 K. Hence, its effect is to reinforce the local uniaxial anisotropy, perpendicular to the wheel’s plane. All in all, these calculations allow us to conclude that the full {Cr10} ring behaves as a magnetic unit with axial anisotropy of lower rhombicity than the constituent ions, in agreement with the interpretation of EPR results, 37 which yielded the experimental value D/kB=−0.045 K (Table 1), and the M(θ) results obtained in this paper. Monte Carlo Calculations. The thermomagnetic properties can be calculated, within the Boltzman equilibrium statistics, using the cluster MS Hamiltonian in terms of the single-ion spins Si H JSS DS g S H 2 ij ij ij i ii i izN Cl 1 10 , 2 1 10 ∑∑ ∑ μ=− ·++· ⟨⟩ = = (6) where the first term describes the exchange interaction coupling, with ⟨ij⟩denoting all interactions between the n.n. Si= 3/2 spins in the {Cr10} ring; spins are numbered from 1 to 10 following the numbering of the ions in asymmetric unit from 1 to 5, and 6 to 10 for the ions obtained by inversion (Cr1 →Cr6, etc.). The second term accounts for the zero-field splitting produced by the single-ion anisotropy of each Cr3+ site (Di), and the last term is the Zeeman splitting with g=2. Since the full diagonalization of that Hamiltonian requires to work in a functions space of dimension (2 ×3/2 +1)10,an alternative approach is to use Monte Carlo (MC) simulations. Classical MC methods have proved useful to rationalize the magnetic properties of 3d clusters with a large number of ions. 58,59 We used the MC method as implemented in ALPS 55,56 for ten 3/2 spins in a ring and a distribution of exchange constants Jij as shown in Figure 7a, where the {Cr10} wheel is divided into two sets of five Cr3+ ions as in the crystallographic asymmetric unit. The results of the DFT calculations were used as a trend and simplified as follows: the n.n. interactions were taken as identical and ferromagnetic (JFM), except around two Cr3+ ions on opposite sides of the wheel, denoted Cr3 and Cr8 as in the previous section (Figure 7a), which were taken as antiferromagnetic and asymmetric (J23 =J78 ≠J34 =J89). This ensures a ground state with total spin S= 9 in zero magnetic field. For the MC simulations, the cluster anisotropy value obtained by EPR D/kB=−0.045 K, normal to the {Cr10} wheel plane and confirmed by INS (see next section), was translated into an average single-ion anisotropy Di/kB=(D/ kB)/∑idi=−0.31 K using the strong exchange limit approximation 57 with neglected interion anisotropic contributions; this single-ion anisotropy Diwas set equal for all ions and no rhombic anisotropy constant was included. The Figure 7. {Cr10} magnetic modeling. (a) {Cr10} coupling scheme and (b−e) magnetic curves calculated with a classical Monte Carlo model along the EA (blue line) and HA (red lines) directions and quantum Monte Carlo only for the EA direction (green lines), with the parameters set JFM/kB = 2.0 K, J23/kB=−2.0 K, J34/kB=−0.25 K, Di/kB=−0.31 K. Experimental data with the applied field in the EA (blue full circles) and HA (red full circles) directions. (b) Magnetic susceptibility χT(T) at 1 kOe; (c) magnetization M(H)atT= 1.8 K; (d) zoom-in of the M(H) up to 50 kOe; and (e) calculated classical MC and XMCD experimental results of M(H) for the powder sample at 7.5 K. The quantum MC simulation for the EA direction is also included (green dashed line). Journal of the American Chemical Society pubs.acs.org/JACS Article https://doi.org/10.1021/jacs.2c05453 J. Am. Chem. Soc. 2022, 144, 12520−12535 12526 classical MC method was used to derive the magnetization along the easy anisotropy axis (EA) and perpendicular to that direction (HA). The susceptibility as a function of temperature χ(T) was calculated as M/Hfor an external applied field of 1 kOe, as it was carried out experimentally. Additionally, quantum MC simulations, using the directed loop code in the stochastic series expansion representation, 60 were performed only for the component parallel to the anisotropy axis, since for the in-plane component, a sign problem 55,56 is produced. The field dependence of the magnetization at 1.8 K and the temperature dependence of the susceptibility of this {Cr10} model were calculated searching for a set of exchange coupling constants consistent with the experimental data. Figure 7b−e shows the results for the coupling constants JFM/kB= 2.0 K, J23/kB=−2.0 K, and J34/kB=−0.25 K. The anisotropy in the magnetic susceptibility as a function of temperature is explained qualitatively for both the EA and HA directions (Figure 7b). The anisotropy in the magnetization measured for the SC at 1.8 K (Figure 7c,d) is also explained qualitatively, although the predicted curve quickly diverges from the experimental data for larger fields. Besides, the MC simulation approaches the XMCD(H) curve measured for the powder at 7.5 K in the same range of fields (Figure 7e). For low magnetic field values, the classical MC simulation for the present magnetic system yields similar values to those of the quantum MC down to ≈2 K, while as the field increases the two models differ. Our model predicts that the M(H)curveincreases continuously over the maximum value expected for an S=9 total spin (18 μB), linearly approaching the 30 μBlimit for the totally polarized spin state. Thus, the S= 9 ground state holds for a limited range of magnetic field, about H= 100 kOe. For higher applied fields, magnetic states with a larger Sand Sz become the ground state. The saturation value of 30 μB, expected for the totally polarized set of ten 3/2 spins (S= 15), has not been reached in the present work up to 170 kOe. The present coupling scheme is supported by the INS measurements (vide infra). We note that other sets of parameters (Jij,Di,) are able to fit the magnetic data. Indeed, Figure 8. Inelastic neutron scattering (INS) spectra of {Cr10}. (a) Spectra measured at T= 1.8 and 5.0 K on the LET spectrometer, with Ei= 1.5 meV.(b) Low-energy loss spectrum obtained from the difference between the T= 1.8 and 5.0 K spectra shown in (a). The low-energy peak is compared to the simulation of the difference spectrum (red full line) for the same two temperatures for only intramultiplet S= 9 transitions with D/kB=−0.045 K, E/D= 0. The purple arrow indicates the energy of the calculated |9, ±9⟩to |8, ±8⟩transition as shown in (e); (c) integral intensity, I(Q), of the experimental peak at ℏω≈0.06 meV and (d) I(Q) for the peak at ≈0.13 meV. In (c) and (d), dashed lines are a guide to the eye. (e) Calculated energy levels as a function of Szcalculated by diagonalization of the exchange Hamiltonian in eq 6 for the coupling scheme shown in Figure 7a. For Sz< 8, only the first 10 levels calculated with the Lanczos method are shown, while for Sz≥8, all of the levels up to 1.3 meV are depicted. Red arrows: excitations within the S= 9 multiplet, fulfilling the selection rule ΔS=0,ΔSz=±1. Purple and green arrows: excitations complying with the selection rule ΔS=1,ΔSz=0,±1. The horizontal dotted line shows the energy corresponding to the measurement temperature. Journal of the American Chemical Society pubs.acs.org/JACS Article https://doi.org/10.1021/jacs.2c05453 J. Am. Chem. Soc. 2022, 144, 12520−12535 12527 the experimental magnetization slowly tends toward the maximum value for a ten 3/2 spins system as the magnetic field increases, and the faster drop of the HA component of χT at low temperatures suggest that the actual distribution of exchange interactions within the {Cr10} ring includes a higher antiferromagnetic character. However, as we will show in the next section, this is incompatible with the INS results. Inelastic Neutron Scattering. Although the results presented above have clearly confirmed an S= 9 ground state and a wheel’s uniaxial magnetic anisotropy D/kB= −0.045 K, which is proved to be along the symmetry axis of the {Cr10} molecule, we still lack conclusive information about the excited states. Inelastic neutron scattering (INS) is a powerful spectroscopy technique to determine the magnetic exchange splitting and reveal the energy of the excited states directly. This technique has been applied very successfully in the study of magnetic clusters (e.g., on Fe8, 61 Mn12 62 and others 63 )andrings(e.g.,{Cr 8}, 19,20 {Cr7M}, 27,28,31 {Cr8Cd} 32 ). In the INS experiment, the incoming neutron exchanges energy and momentum with the sample and the energy and momentum of the outgoing neutron are analyzed. The states of the sample are therefore determined by the difference ℏω between the incoming and outgoing neutron energies. The energy spectrum S(ℏω) is obtained by integrating the scattered intensity S(Q,ℏω) over the scattered intensity in the experimentally accessible Q-space (0 < Q< 1.6 for Ei= 1.5 meV); see Figure 8a. The T= 1.8 K spectrum displays visible peaks in the neutron energy loss region, i.e., the sample’s excitations (ℏω> 0), at the energy transfers of 0.4 meV (peak labeled as (iii) in the figure) and 0.8 meV (peak (iv)), and a small but evident feature at ∼0.15 meV (peaks (i)−(ii) on the right-hand side of the elastic peak). In the neutron energy gain region (ℏω< 0), these peaks are not discernible, except for the small shoulder of the elastic peak. This reflects transitions from the {Cr10} wheel’s magnetic ground and low-energy states or lattice excitations. By the temperature and Q-dependence of the observed excitations, we can deduce that they are predominantly of magnetic origin. On the contrary, at T=5 K, higher-energy peaks become more unstructured in the ℏω> 0 region, while they are now visible in the ℏω< 0 region, and the intensity of the shoulder on the elastic peak shifts from the neutron energy loss to the energy gain region. This is consistent with the population of excited states at 5 K. To extract this low-energy peak from the elastic contribution, we subtract the 5 K spectrum from the T= 1.8 K one. The difference between both spectra in the range 0 < ℏω< 0.6 meV, is shown in Figure 8b, which evidences that the initial shoulder of the elastic line actually comprises two well-resolved peaks ((i) and (ii)). To understand which type of magnetic states are involved in these two intensity peaks, we extracted the neutron scattering intensity as a function of the momentum transfer, I(Q), for both excitations. The I(Q) can help to understand what type of magnetic transitions occur. 63 The intensity of the peak at 0.06 meV (i) tends to a nonzero constant value for Q→0(Figure 8c), as expected for intramultiplet transitions in neutron spectra, i.e., with no change in the total spin (ΔS= 0), while for the peak at 0.13 meV (ii) that integral intensity drops to zero for Q→0(Figure 8d), as predicted for magnetic transitions between multiplets of different total spin (ΔS=±1). Therefore, these peaks are associated with transitions between states of the S= 9 ground state multiplet split by magnetic anisotropy, and transitions to excited states belonging to multiplets with either S= 8 or 10. Further insight into the magnetic transitions displayed by the neutron spectrum can be gained by calculating the energy levels, using the same Hamiltonian of eq 6 and the intracluster interactions scheme and constants as in the Monte Carlo simulations above with no applied magnetic field or dipolar interactions. However, unlike the classical Monte Carlo simulation, which only provides the ground state properties, now we need a quantum calculation to get the excited states. Although this implies to diagonalize a matrix of dimension 410, it can be factorized in smaller matrices of fixed Sz. It should be noticed that when the axial anisotropy is included, the total spin Sis not a good quantum number anymore, but Szstill is; however, we will still use states labeling as |α,S,Sz⟩as an approximation (αdenotes different single-ion spin configurations producing total spin third component Sz), which allows us to track the states from the isotropic case, where Sis a good quantum number (see Figure S6). For Sz≥8, the dimensions of the matrices to be diagonalized were small enough to render practical a standard full diagonalization method, while for total spin values Sz< 8, we chose a Lanczos sparse diagonalization method to obtain the lowest energy levels. Figure 8e shows the obtained energy levels as a function of Szfor the coupling scheme and Jij values obtained in the Monte Carlo simulations above. Since in the 1.8 K spectrum we are observing transitions from the lowest S= 9 total spin states, states in the multiplets of total spin S≥7orS≥11 are not detected because of the neutron scattering selection rules (see below). To simulate the lowest-energy peak in the spectrum, we used a simpler approach starting from the energy-level scheme of the particular case Di= 0, and then we added the cluster’s uniaxial magnetic anisotropy in the strong exchange limit approximation as a perturbation acting in an Sfixed space. That is, we take the results from the MS approximation, as starting values in the GS approximation. Then, at H=0 H HH HDS SS 1 3(1) zCl ex pert ex 2 Ä Ç Å Å Å Å Å Å Å Å É Ö Ñ Ñ Ñ Ñ Ñ Ñ Ñ Ñ =+ =+ − + (7) The anisotropy term produces spliting of the S= 9 and S=8 multiplets. The splitting on these two multiplets may be regarded with some caution since there may be some mixing between states neglected by the model. 64 The neutron scattering function for magnetic INS transitions within the ground S= 9 state (in the strong exchange approximation, GS) can be written as 62 QQSN rgf pfSi E E (, ) 2() () if Ne222 ,i 2fi i k j j jy { z z z ∑ ωγ δω ℏ= ×|⟨||⟩|[ℏ−− ] ⊥ (8) with the matrix elements between initial |i⟩and final |f⟩ states in the |S,Sz⟩spin states base, S⊥is the spin component perpendicular to the momentum transfer vector Q, and pEkT EkTexp( / )/ exp( / ) j j iiB B ∑ =− − (9) is the thermal population for the state. For a polycrystalline sample Journal of the American Chemical Society pubs.acs.org/JACS Article https://doi.org/10.1021/jacs.2c05453 J. Am. Chem. Soc. 2022, 144, 12520−12535 12528 Design and Assessment of Accuracy. Phys. Chem. Chem. Phys. 2005,7, 3297−3305. (85) Becke, A. D. A New Mixing of Hartree-Fock and Local DensityFunctional Theories. 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