Dynamic versus Static Character of the Magnetic JahnâTeller Effect

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Dynamic versus Static Character of the Magnetic Jahn−Teller Effect: Magnetostructural Studies of [Fe3O(O2CPh)6(py)3]ClO4·py Anastasia N. Georgopoulou,§ Irene Margiolaki,‡,† Vassilis Psycharis,§ and Athanassios K. Boudalis*,§,⊥ §

Institute of Nanoscience and Nanotechnology, NCSR “Demokritos”, 15310 Agia Paraskevi, Athens 15310, Greece European Synchrotron Radiation Facility, ESRF, 6 rue Jules Horowitz, B.P. 220, 38043 Grenoble cedex, France

‡

S Supporting Information *

ABSTRACT: Complex [Fe3O(O2CPh)6(py)3]ClO4·py (1) crystallizes in the hexagonal P63/m space group, and its cation exhibits a crystallographically imposed D3h symmetry due to a C3 axis passing through the oxide of its {Fe3O}7+ core. Singlecrystal unit-cell studies carried out with synchrotron radiation confirmed that this symmetry is retained down to 4.5 K; a full crystal structure determination carried out at 90 K resolved the previously reported disorder of the perchlorate anion. Magnetic susceptibility and electron paramagnetic resonance (EPR) data for complex 1 were interpreted with a model considering the retention of the threefold crystallographic symmetry while predicting a lowering of the magnetic symmetry. This model considered the effects of atomic vibrations of the central oxide on the magnetic properties of the complex by incorporating these movements into the spin Hamiltonian through angular overlap considerations of the atomic orbitals; no ad hoc magnetic Jahn−Teller effect was considered. The derived magnetostructural correlations achieved an improvement in the interpretation of the magnetic susceptibility data using the same number of free variables. They also improved the simulations of the EPR data, which exhibit a complicated set of at least five axial resonances; improved simulations were achieved using only two spectral components. Due to the thermal effects on the oxide vibrations, the model predicts a temperature dependence of the magnetic coupling J, which should not be viewed as a constant but as a variable.

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INTRODUCTION Basic metal(III) carboxylates have a particular importance in the development of molecular magnetism because they were the first family of compounds for which magnetic interactions were postulated by Kambe1 two years before the seminal paper by Bleaney and Bowers on copper(II) acetate.2 The model employed by Kambe made use of a common antiferromagnetic magnetic exchange interaction among the three FeIII (S = 5/2) or CrIII (S = 3/2) ions, which correctly predicted an ST = 1/2 ground state when the constituent spins are half-integer. This prediction was subsequently confirmed by a large body of data. Thus, the study of basic metal(III) carboxylates helped set the experimental and theoretical framework for the subsequent development of molecular magnetism. From a magnetic perspective, basic metal(III) carboxylates are part of a larger family of complexes comprising antiferromagnetic triangles of half-integer spin ions (CuII, VIV, S = 1/2; MnII, S = 5/2, GdIII, S = 7/2, etc.) sharing some distinct properties, in particular, two low-lying Kramers doublets. Those half-integer spin-triangles can be analyzed within the same theoretical framework irrespective of their chemical details. For example, it was found that a certain cofactor in ferredoxins, namely the 3Fe-4S cluster, exhibits strong antiferromagnetic interactions among the FeIII ions in its © XXXX American Chemical Society

oxidized form, stabilizing an S = 1/2 ground state, just like in the case of basic iron(III) carboxylates.3 Later, more similarities were discovered, as it was found that the interpretation of the EPR and Mössbauer data of this cluster required the use of antisymmetric exchange interactions4 in combination with small distributions on the exchange coupling parameters,5 thus exhibiting striking similarities with complexes containing the {Fe3O}7+ core.6 Polyoxovanadates have exhibited similar behaviors, with K6[V15As6O42(H2O)]·8H2O, better known as V15, requiring the use of antisymmetric exchange for the interpretation of its magnetic properties.7 Half-integer spin triangles have been proposed as candidates for the implementation of qubits; in one approach, one such complex could implement a two-qubit system by employing the excited ST = 1/2 state as the second qubit,8 while in another, intermolecular exchange couplings could be selectively turned on/off by simple excitations, allowing the implementation of many-qubit arrays.9 More recently, such triangles have been proposed as candidates for decoherence-free qubits, utilizing their spin chirality as a computational degree of freedom instead of their spin projection.10 Received: August 7, 2016

A

DOI: 10.1021/acs.inorgchem.6b01912 Inorg. Chem. XXXX, XXX, XXX−XXX

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Inorganic Chemistry

that influence the magnetic exchange. These distortions can be viewed as pseudorotations of identical isosceles conformations. The authors considered a distortion vector, which they correlated with the magnetic exchange through a parameter λ. They then tested their model on the data of Long13 using one exchange coupling, J, and the λ parameter as variables and reported an improved agreement to the experimental data compared to the earlier fits using the 2J isosceles model. However, consideration must be given to the fact that those data sets contain rather few (∼15) data points, restricted to temperatures above 20 K. Higher quality data sets would be required to safely evaluate this model. Moreover, the model is purely phenomenological, not explaining the origins of those isosceles distortions, which the authors accept as part of their original assumptions. In a relevant approach, Rakitin et al.20 started off from magnetostructural considerations examining simple molecular vibrations. Their main assumption was the approximation that only the central oxide is moving because the O atom is much lighter than the Fe ones, and they considered a cylindrical potential well, within which the oxide is located at any given moment. On the basis of the generalized angular overlap model and considering inelastic incoherent neutron scattering (IINS) data21 for complex [Fe3O(O2CPh)6(py)3]ClO4·py (1, Figure 1), they calculated a radius of 0.05−0.1 Å for this cylindrical

A prerequisite for the implementation of half-integer spin triangles in QIP is the precise knowledge of their spin Hamiltonian; however, questions still remain regarding their magnetic symmetries, which do not always coincide with their crystallographically determined molecular symmetries. Indeed, soon after Kambe’s work, discrepancies started to emerge from more refined analyses on newer experimental data. According to Kambe’s equilateral model, Ĥ = J(S1̂ S2̂ + S2̂ Ŝ3 + Ŝ3S1̂ ), the S = 1/2 ground state is doubly degenerate, a conclusion that was in disagreement with heat capacity data of [Cr 3 O(O2CMe)6(H2O)3]Cl.11 These data were interpreted by assuming removal of the ground state degeneracy by consideration of two different exchange coupling parameters according to the isosceles model Ĥ = J (S1̂ Ŝ2 + S2̂ Ŝ3) + J′S3̂ Ŝ1 (J ≠ J′). Actually, in his analysis, Kambe had used the equilateral model while also contemplating the isosceles one. His choice was obviously dictated by the fact that the available experimental data were so limited (2−6 data points between 200 and 360 K) that the use of a more refined model would result in overparametrization, though computational limitations might have also been a factor given the available infrastructure at the time. It was several years later that similar magnetic studies by Duncan, over a broader temperature range (80−360 K) and with more data points, were meaningfully and convincingly interpreted by the isosceles model12 and subsequently confirmed by Long, with measurements expanding to even lower temperatures (20−300 K),13 thus lending additional credibility to this model. Over the years, additional terms have been introduced to lift of the ground state degeneracy, with the most significant being the antisymmetric exchange (Dzyaloshinskii−Moriya) term; th is was first con sid er e d fo r c om pl e x es [M 3 O(O2CMe)6(H2O)3]Cl·xH2O to explain their 4.2 K electron paramagnetic resonance (EPR) spectra (MIII = CrIII,14 FeIII15) and heat capacity data down to 2 K (MIII = CrIII16). Because the ground state of strictly trigonal systems with half-integer spins is predicted to be the orbital doublet 2E, the role of antisymmetric exchange in the lifting of this degeneracy has been examined in great detail.17 However, such terms are usually used in tandem with the isosceles model because the equilateral model fails to reproduce the magnetic susceptibility data even with the use of those additional terms. Isotropic Models for Lifting of the Ground State Degeneracy. Despite the apparent success of the isosceles model, it was met by great reluctance among researchers due to the large differences it introduced between J and J′, which were in disagreement with the high symmetries of the clusters as derived from crystallographic studies. Murao attempted to reconcile the proposed C2v magnetic symmetry with the nearly (or strictly) D3h molecular symmetry by proposing a “Magnetic Jahn−Teller Effect” (MJTE), stemming from a “spontaneous distortion of the equilateral triangle” and which would therefore remove the degeneracy of the ground state.18 He even calculated the distortion Δd (= dFe1−Fe2 − dFe2−Fe3) required between Fe···Fe distances as ∼0.01 Å for every K of ΔJ difference (ΔJ = J12 − J23). This MJTE, he proposed, might be operative in combination with low-temperature deformations due to crystal packing effects. It must be stressed that to this day the MJTE remains a hypothesis not confirmed by structural data. In a different approach a decade later, a dynamic model19 assumed that the complex undergoes rapid dynamic distortions

Figure 1. POV-Ray plot of the cationic cluster [Fe3O(O2CPh)6(py)6]+ present in 1, showing the C3 axis passing through the μ3 oxide.

well. As an additional approximation, they considered a lack of directional preference for the oxide, which corresponds to a perfectly spherical thermal ellipsoid. However, they conceded that IINS experimental data (see below) might best be described by considering a directional preference toward certain angles at low temperatures; this would correspond to a freezing of preferred magnetic conformers. Indeed, IINS studies on deuterated iron(III) triangles confirmed the lifting of the ground state degeneracy by directly probing of the low-lying magnetic states. It was thus discovered that the magnetic symmetry of [Fe3O(O2CCD3)6(C5D5N)3]NO3 (2)22 can be described as isosceles, while that of the clusters in Maus’ salt K5[Fe3O(SO4)6(D2O)3]·6D2O (αmetavoltine, 3)23 and [Fe3O(O2CC(CD3)3)6(DOCD3)3]Cl (4)22 are scalene. Subsequent IINS experiments showed that the 1.5 K energy spectrum of 1 exhibits eight bands, requiring the consideration of two different distorted units, one of isosceles and one of scalene magnetic symmetry.21 These observations are in seeming disagreement with the threefold crystallographic symmetry of complexes 1, 3, and 4 (P63/m, P63, and P63/m24 space groups, respectively).25 While the B


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Inorganic Chemistry interpretation of the IINS data of complex 1 was carried out assuming two distinct cations, the authors explicitly considered the possibility of rapid pseudorotations, i.e., a dynamic model, without possessing experimental data to confirm or discount this hypothesis. Herein, we report for the first time detailed variabletemperature synchrotron crystallographic data for complex 1 and present a comparison of the static and dynamic models in the interpretation of its magnetic susceptibility and solid-state X-band EPR spectroscopic data. Variable-temperature structural studies of complex 1 were carried out down to liquidhelium temperatures (i.e., near the temperature at which IINS studies have been carried out). The selection of the complex was based on its high symmetry at room temperature, which would allow the straightforward detection of even minute distortions by simple unit-cell determinations and on the fact that it had already been studied by IINS, allowing us to compare the spectroscopic with structural data. Regarding the experimental apparatus, we chose to use synchrotron radiation to take advantage of: (i) the narrow line widths of its radiation and the absence of Kα1 and Kα2 components of conventional cathodic X-ray, conditions which serve to increase peak resolution. In particular, Mo Kα radiation (λ = 0.71073 Å) yields peaks with line widths of 0.25°/0.019 Å at intermediate (≈30°) angles where the Kα1 and Kα2 splitting is not observed, whereas the synchrotron λ = 0.7 Å beam exhibits 0.08°/0.006 Å line widths, i.e., for similar crystal-todetector distances, the resolution is three times higher and (ii) its high-intensity beam, requiring short exposure times and allowing large crystal-to-detector distances, which further increases peak resolution.

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Table 1. Data Collection and Refinement Details for Complex 1 formula Fw T (K) wavelength (radiation) crystal system, space group crystal dimensions (mm) a (Ǻ ) c (Ǻ ) V (Ǻ 3) Z ρc (g cm−3) F(000) μ (mm−1) Tmin, Tmax Θmin, Θmax h, k, l reflections collected/unique reflections used/parameters Rint reflections with I > 2σ(I) R[F2 > 2σ(F2)], wR(F2)a S Δ⟩max, Δ⟩min (e Å−3) (Δ/σ)max a

C62H50ClFe3N4O17 1326.06 90 0.7005 Å (synchrotron) hexagonal, P63/m 0.22 × 0.22 × 0.05 13.3680 (1) 19.0237 (2) 2944.14 (5) 2 1.496 1362 0.81 0.508, 1.000 3.0°, 30.0° −19→18, −19→19, −27→27 51699/3038 3038/208 0.022 2900 0.027, 0.078 1.04 0.43, −0.31 0.015

w = 1/[σ2(F02) + (0.0451P)2 + 0.9427P], where P = (F02 + 2Fc2)/3.

cryostream X-Sream 200 (RIGAKU MSC) which operates in the temperature range from RT down to −193 °C (80 K). Calculations of magnetic susceptibilities and EPR spectra on previously published data for complex 130 were carried out by diagonalization of the full matrix of the spin Hamiltonian using the Easyspin31 (v. 5.0) toolbox. The static and dynamic models were implemented with custom code written for Easyspin. For the static models, the results were confirmed through calculations using the Phi32 (v. 2.1.6) program package. For the dynamic model, a set of distorted magnetic triangles was defined by varying the Jij values for each magnetic triangle according to a periodic function of the oxide displacement angle φ (see text). The magnetic susceptibility contributions and EPR spectra were then calculated by full-matrix diagonalization for each triangle (i.e., for each φ value) using the curry and pepper functions, and these individual contributions were summed assuming equal weights. The sum of the magnetic susceptibility contributions was least-squares fitted against the χMT vs T experimental data. Agreement factors R were calculated as

EXPERIMENTAL SECTION

Materials and Methods. Complex 1 was prepared following method 2 of the published procedure.21 Synchrotron radiation experiments on a crystal with dimensions 0.05 × 0.22 × 0.22 mm were carried out on the ESRF BM01A beamline in Grenoble using a KM6 diffractometer equipped with an ONYX CCD detector and monochromated with a wavelength of 0.7005 Å (17.7 keV). For optimal temperature stability, variabletemperature unit cell determinations were carried out using data collected with a flow-cryostat controlled by a LakeShore controller with the crystal mounted inside a 0.7 mm glass capillary. Thermal conductivity with the helium flow was achieved by inserting a small quantity of vacuum grease inside the capillary. A full structural determination at 90 K was carried out using data collected with an Oxford Cryostream nitrogen cold stream with the crystal mounted on a glass fiber. Data collection (ω-scans) and processing (cell refinement, data reduction, and empirical/numerical absorption correction) were carried out using the CrysAlis PRO program package.26 The structure of complex 1 was solved by direct methods using SHELXS-9727 and refined by full-matrix least-squares techniques on F2 with SHELXL2014/6.28 All non-H atoms were refined anisotropically. As in the previous study, the perchlorate anion is located at (0,0,0) and is also heavily disordered. Using the PART −1 command with constraints, these anions were refined anisotropically. The pyridine solvate presents rotational disorder, and during refinement it was treated as previously, i.e., at each site 5/6 CH and 1/6 N atoms are assigned. Hydrogen atoms were either located by difference maps and refined isotropically or introduced at calculated positions as riding on bonded atoms. Crystallographic details are given in Table 1. For comparison, similar data were collected on a Rigaku R-AXIS SPIDER Image Plate diffractometer using graphite-monochromated Cu Kα radiation. For the data collection (ω-scans) and processing (cell refinement), the CrystalClear program package was used.29 The low temperature measurements were performed using the nitrogen

R=∑

(χexp T − χcalc T )2 N (χexp T )2

, where N is the number of experimental data

points.

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RESULTS AND DISCUSSION Variable-Temperature Unit Cell Determinations. Unit cell determinations between 260 and 4.5 K were carried out at 18 different temperatures, both upon heating and upon cooling. In all cases, the symmetry of the crystal remained unaffected (P63/m), and the only observed variation upon cooling was the thermal contraction of the cell axes (∼1.1% for the a-axis and less than 0.2% for the c-axis) and the concomitant decrease in the cell volume (by ∼2.7%). No hysteretic phenomena were observed. The thermal variation of these parameters is illustrated in Figure 2. The flow cryostat sets geometrical limitations which restrict the possible movement of the crystal, thus limiting the data that can be collected along some directions of the reciprocal space. C


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Inorganic Chemistry

Table 2. Comparison of Critical Structural Parameters between the 90 and 233 K Crystallographic Studies on Complex 4 synchrotron (90 K)

Mo Kα (233 K) 2

Ueq (Å ) Fe(1) O(1)

0.01908(7) 0.0235(2)

Fe(1)−O(1) Fe(1)−(N1) Fe(1)−O(2) Fe(1)−O(3)

1.9080(2) 2.1946(12) 2.0107(7) 2.0072(8)

0.0481(4) 0.0414(19) distances (Å) 1.9149(10) 2.206(6) 2.014(3) 2.015(3)

atomic displacement parameters, it is concluded that the ellipsoids both for Fe(1) and for (O1) are of similar magnitudes and quite spherical, in divergence from Rakitin’s assumption according to which the central oxide moves mostly on the triangle plane and the ferric atoms are fixed. As for the average displacements at 90 K, with Uii values of ∼0.0191 and 0.0201 Å2 for Fe(1) and O(1), respectively, we conclude that their average displacement from the equilibrium position (defined as √Uii along each cell axis) will be about 0.14 Å in both cases, in qualitative agreement with Rakitin’s estimations. This point will be elaborated further on. At higher temperatures, this displacement will be more pronounced, e.g., at 233 K,21 the Fe(1) and O(1) Uiso values are 0.0481 and 0.0414 Å2, respectively, corresponding to mean displacements of ∼0.22 and 0.20 Å, respectively. Interpretation of Magnetic Properties Using Static and Dynamic Models. Previous IINS studies of 1 had been interpreted assuming the presence of two magnetic types of molecules exhibiting different magnetic symmetries and widely varying Jav values (defined as (2J + J′)/3). Half of the spectral peaks were interpreted assuming an isosceles magnetic symmetry with Jav = −27.5 cm−1 (J = −28.5 ± 0.3, J′ = −25.6 ± 0.08 cm−1, hereafter 1A), while the other half were interpreted assuming a scalene magnetic symmetry with equally spaced J values and a Jav = −21.4 cm−1 (J1 = −18.9 ± 0.2, J2 = −23.9 ± 0.2 and J3 = −21.4 ± 0.2 cm−1, hereafter 1B).21,33 According to Murao’s calculations, these differences in J values should be reflected in Fe···Fe distances varying as much as 0.04 Å in the case of 1A and 0.07 Å in the case of 1B, distortions significantly above the detection limit of single-crystal crystallography. It is clear from the crystal data collected on the synchrotron beam (see above) that no static symmetry decrease is observed down to 4.5 K, the lower temperature limit of the experimental setup employed. Therefore, we conclude that this lifting in degeneracy, i.e., the lowering of the magnetic symmetry, is not associated with a static decrease in the cluster’s molecular symmetry. Having thus precluded that possibility, we may consider a dynamic process similar to the one proposed by Jones et al.19 or Rakitin et al.20 We will consider Rakitin et al.’s model because it makes no prior assumptions regarding an isosceles distortion and provides an explicit magnetostructural correlation to predict the J values. As we mentioned, this correlation assumes that the oxide moves along a circle with a constant radius of 0.05−0.1 Å. However, our crystallographic studies reveal the following: (i) The thermal ellipsoids of the central oxide are quite spherical, extending above and below the triangle plane. (ii) The average radii exhibit important differences for the structures at 90 and

Figure 2. Thermal variation of the unit cell dimensions of 1 between 4.5 and 260 K, as revealed by single-crystal unit-cell determinations on a synchrotron beamline and a conventional X-ray diffractometer. The lines are a guide for the eye.

Due to the positioning of the particular crystal inside the flow cryostat, this direction was along c*. Despite those limitations, the collected data were enough (∼900 unique reflections out of ∼4000 redundant data) for the calculation of the Laue group of the crystal structure. This was assigned as 6/m because the Rint value for this group was less than 4% at all temperatures. In view of this result, and because no peak splitting was observed within the measured 2θ range and the resolution range given above, it is clear that all peaks must be indexed based on a hexagonal cell. This means that no transition to a lower symmetry structure, e.g., of an orthorhombic or monoclinic cell, is observed in our case. We also examined the possibility for this structure to transform to a monoclinic one with pseudohexagonal cell dimensions (a ≈ b ≈ 13.4 Å, c ≈ 19 Å, α = β = 90° and γ ≈ 120.0°). With this in mind, during integration of all data sets, the cell dimensions were treated as corresponding to a triclinic crystal system (a procedure suggested for all data sets). The highest difference in the a and b values was 0.03 Å, and in the angles 0.06° from the ideal one. Taking into account, as mentioned above, that in all studied temperatures the Rint value for 6/m group was less than 4%, it is further corroborated that no symmetry decrease is observed. High-Resolution Structural Determination at 90 K. The structure of 1 has been previously reported for data collected at 233 K on a conventional diffractometer. The numbering convention has been retained. An interesting aspect of this structural determination concerns the thermal displacement of the atoms around their mean positions, which has direct bearing on the validity of the dynamic model (Table 2). From examination of the anisotropic D


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Inorganic Chemistry 233 K. (iii) The O(1) ellipsoid is similar in size to that of Fe(1). Taken together, these points mean that in reality, all atoms move equally, along spherical and not circular trajectories and at amplitudes that are a function of temperature. While these assumptions are quite reasonable, they must be clearly understood when assessing the agreement of their model to our experimental data (see below). During our study, we considered only two types of exchange, isotropic and antisymmetric, while we also took into account the Zeeman interaction. The static model we used can be described by the Hamiltonian: Ĥ = −2J(S1̂ Ŝ 2 + S1̂ Ŝ 2) − 2J ′S1̂ Ŝ3 − 2G(S1̂ ×Ŝ 2 + Ŝ 2 ×Ŝ 3 3

+ Ŝ3 ×S1̂ ) + μB g H ∑ Sî

(1)

i=1

where G is the antisymmetric exchange vector parameter, for which it is assumed that d = Gz ≫ (Gx, Gx) ≈ 0,34 and μB is the Bohr magneton. The dynamic model, based on the model of Rakitin et al., can be described by the Hamiltonian:

Figure 3. Dynamic model discussed in the text for a complex comprising three iron(III) ions (orange circles). The oxide (red circle) moves along a circular trajectory of radius r0 around the triangle’s center with an angular deviation φ from the position vector Fe(1). Blue dashed lines indicate φ angles that correspond to fully scalene conformations. Red and orange dashed lines correspond to isosceles conformations with |J| > |J′| and |J| < |J′|, respectively. The meanings of the various symbols are analyzed in the text (an animated version of the figure is provided in the Supporting Information).

3

Ĥ = −2

∑ Jij (φ)Sî Sĵ − 2G(S1̂ × Ŝ2 + Ŝ2 ×Ŝ3 i,j=1 3

+ Ŝ3 × S1̂ ) + μB g H ∑ Sî

(2)

i=1

where the interaction Jij(φ) between ions i and j is a periodic function of the angle φ between the position vectors of the oxide and Fe(1), as illustrated in Figure 3:

becoming equilateral. Such a progression could be quantified by a scelenity index defined as σ(φ) = (J12(φ) − J23(φ))(J23(φ) − J13(φ))(J13(φ) − J12(φ))/Jav, which is zero in the isosceles cases and acquires its maximum and minimum values in the fully scalene case (Figure 4). The model is visualized through an animation provided in the Supporting Information. Fits to the χMT vs T data were carried out over the 2−310 K range (Figure 5) with an isotropic single-ion g parameter fixed to 2 for all ferric sites. For the purposes of comparison, all fits

Jij (φ) = J {1 + a[cos(φ − ϑi) + cos(φ + ϑj)]} (θ1 = 0, θ2 = −2π /3 and θ3 = +2π /3)

(3)

In the above expression, J is the static isotropic coupling constant assuming an equilateral magnetic model and a = 2βr0/ R0, where is R0 is the Fe−Oox distance, r0 is the average radial displacement of the central oxide (minimum potential energy position), and β is a constant depending on the energies of the H S*k(d)

metal and ligand orbitals. In particular, βλk (d) = − Hd −λ Δ d

L

describes the overlap of type λ (= σ, πx, πy, δ1, δ2) of a metal d (= z2, yz, xz, xy, x2 − z2) orbital with the orbitals of ligand k, where Hd is the energy of the d orbital and ΔL is the average energy of the ligand orbitals.35 As the orientation of G is symmetry-dependent,36,37 and as the symmetry is bound to change by the oxide movement, we also considered the cases of variable G orientations by keeping |G| fixed and by letting (Gx, Gx) ≠ 0. However, EPR simulations showed that the g-anisotropy is almost invariant to the orientation of G and principally depends on the |G|/J ratio. In addition, calculations considering the single-ion anisotropy of the ferric ions revealed negligible contribution to ganisotropy even for relatively large values of DFe = 1 cm−1. It should be noted that, as in the case of the isosceles magnetic model, here we use two fitting parameters, a and J. The model is depicted in Figure 3. Its physical meaning is that Jij couplings vary in such a manner that the magnetic conformation varies constantly between fully scalene (i.e., J23 − J12 = J12 − J13 for J23 > J12 > J13) and isosceles, never

Figure 4. Dependence of the Jij coupling parameters and scalenity index σ in complex 1 on the oxide angular deviation φ from the position vector of Fe(1). The simulations are based on J and a best-fit values to the dynamic model. The vertical full lines indicate the isosceles systems and the vertical dashed lines the fully scalene systems (Jmax − Jmed = Jmed − Jmin). The horizontal line indicates the best-fit central J value, which is also the average of the three Jij values. E


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Inorganic Chemistry

Figure 6. M vs H data of 1 at 2 and 5 K, and simulations based on the best-fit parameters of solutions Aani and Bani. The effect of antisymmetric exchange is demonstrated by simulations on the same parameter sets, in which d = 0.

The Jav values of all solutions are in excellent agreement with the derived parameters for the magnetic conformer 1B. In both isotropic and anisotropic cases, the solution to the dynamic model is slightly better than solution A and significantly better than solution B. It is also to be noted that there seems to be a correlation between J and d, as inclusion of antisymmetric exchange is associated with lower Jav best-fit values. This is rationalized by considering that higher d values increase the energy separation between the spin multiplets, and a lowering of J and J′ compensates for this, as the intermultiplet separation scales with |J − J′|. In turn, this is associated with higher a values, which compensate for the decrease of |J − J′| by increasing the amplitude of the deformational dependence of the coupling parameters. Therefore, inclusion of the antisymmetric exchange term induces higher a values. It should also be noted that even better fits could be obtained for higher d values (e.g., J = −21.0 cm−1, a = 0.172, |d| = 2.6 cm−1, R = 5.46 × 10−4), however these arithmetically better fits were associated with a poorer agreement with the hightemperature data due to the unrealistically small J value. Therefore, it was decided that due to those correlations d should be constrained to values yielding a better agreement throughout the 2−310 K range. A comment needs to be made regarding the quality of the fits. While this is particularly good, especially in the case of the dynamic model, it is clear that there is no single parameter set that achieves a perfect agreement with the experimental data throughout the 2−310 K temperature range. For example, fits carried out over the 100−310 K range yielded excellent agreement (R ∼ 2 × 10−5) with stronger Jav values (∼ −21.9 cm−1) even without the inclusion of antisymmetric exchange (d = 0), whereas fits carried out over the 2−50 K range yielded significantly higher |d| values (2−5 cm−1). The validity of these parameter ranges for d was assessed by applying the same model to EPR data (see below), but from this assessment, we conclude that there is an additional thermal contribution to our system that is not accounted for by either model and which is discussed below. A comment is also useful regarding the energy levels predicted by the dynamic model. As we mentioned, these exhibit an angular dependence on the deformation vector of the oxide, which is to be expected because the Jij values are also a

Figure 5. χMT vs T experimental data for 1 (1 T) and best-fit curves according to the static and dynamic models described in the text. Top panel: isotropic models. Bottom panel: anisotropic models. The insets show the low-temperature region, which is most visibly affected by antisymmetric exchange.

assuming a static model were carried out with both Phi and Easyspin, yielding identical results. The inclusion of an antisymmetric exchange term significantly improves the quality of the fits. Best-fit solutions with Easyspin using the static isosceles model are shown in Table 3 and refer to the two Table 3. Magnetic Susceptibility Best-Fit Values for the Parameters of Various Models over the 2−310 K Temperature Range modela

J

J′

Javb

|d|c

Aiso Biso Aani Bani dynamic dynamic

−23.1 −20.2 −23.2 −19.7

−19.0 −25.2 −18.4 −25.7

−21.7 −21.8 −21.6 −21.7 −22.1 −21.5

0 0 1.62 1.74 0 1.8

a

0.147 0.159

R 2.98 3.49 6.99 9.80 2.94 6.77

× × × × × ×

10−3 10−3 10−4 10−4 10−3 10−4

A: |J| > |J′|. B: |J| < |J′|. bJav = (2J + J′)/3 for the isosceles models. Bold: fixed values.

a c

solutions with |J| > |J′| (A) and |J| < |J′| (B). The derived d values were verified by M vs H simulations (at 2 and 5 K), which were in excellent agreement with the experimental data (Figure 6). Consideration of intermolecular interactions through the inclusion of a mean-field correction (with or without antisymmetric exchange) did not significantly improve the fits and was therefore discarded. Best-fit solutions to the dynamic model yielded solutions J = −22.1 cm−1, a = 0.147, |d| = 0 (fixed) with R = 2.94 × 10−3 and J = −21.5 cm−1, a = 0.159, |d| = 1.8 cm−1 with R = 6.77 × 10−4. F


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Inorganic Chemistry function of this deformation. However, tests with various a values reveal that for certain combinations of deformation angles and amplitudes (i.e., for certain φ and a values), the spin ladder order is affected, and we observe level crossings. In particular, for a values up to ∼0.126, no such crossings are observed. Between a ∼ 0.126−0.154, the order of the low-lying multiplets is the usual, with two low-lying doublets followed by a quadruplet, but level crossings between a pair of ST = 3/2 and 5/2 multiplets occur at certain deformation angles (Figure S1.1). Above a ∼ 0.154, the deformation amplitude is such that even the low-energy spin ladder is affected, with a quadruplet becoming the first excited state for certain angles of the deformation vector (Figure 7). These deformations will also affect the spin expectation values ⟨Siz⟩ on each ferric site (Figure S1.2).

and 2 2 2 2 ⊥z E|H 0, ±1/2⟩ = − 1/2 Δ + g⊥μβ H ± g⊥μβ Hδ 2 2 2 2 ⊥z E|H 1, ±1/2⟩ = + 1/2 Δ + g⊥μβ H ± g⊥μβ Hδ

where Δ = δ 2 + 3d 2 , and δ = J − J′ and |J| < |J′|. For |J| > |J′|, |1, ± 1/2⟩ becomes the ground state doublet, and the labels in the relations are interchanged with |0, ± 1/2⟩. For H∥z, the intradoublet transition will occur at a field where hvEPR = g∥effβH, i.e., its position, quantified by g∥eff, will be independent of d and δ and will show up at g∥eff = g∥0, where g∥0 is the intrinsic g∥ component of the single-ion g-vector. However, for any other orientation, the transition field will be a function of the relative sizes of d and δ. In the more complicated case of a scalene triangle, full-matrix diagonalization techniques are required to determine the magnetic state energies and transition fields, but it is instructive to examine the analytically solved isosceles case for H⊥z. In that case, the intradoublet separations increase with increasing δ, which means that for increasing δ, the transitions occur at lower fields, i.e. g⊥eff → 2. Conversely, if δ is sufficiently small, these will be outside the experimentally accessible magnetic field strengths and therefore unobservable (g⊥eff ≪ 2). The result is that antisymmetric exchange, acting on a nonequilateral magnetic system, induces very strong g anisotropy. For an equilateral triangle (δ = 0), no H⊥z transition is expected, which is also a consequence of the fact that for equilateral systems, intradoublet transitions are forbidden.38 Moreover, in our case, the δ values are small enough for the first excited ST = 3/2 multiplet to be close to the low-lying doublets, and it has been shown that the perpendicular resonances will also be affected by the |d|/δ ratio,6 with larger d values shifting the H⊥z transition to higher magnetic fields (lower geff values). In the case of |d| = 0, no such transition is predicted. Transposing those observations to the dynamic model considered here, high a values will qualitatively correspond to high δ values because a increases the amplitude of the Jij(φ) variations and hence the differences between highest and lowest Jij(φ) values. Thus, higher a values will shift g⊥eff to lower magnetic fields, competing with the contribution of |d| which, upon increase, will shift g⊥eff to higher magnetic fields. From the above, it is clear that the presence of a downfield (g⊥eff < 2) resonance is indicative of antisymmetric exchange operating in tandem with a magnetic asymmetry, i.e., a situation where neither d and δ are zero. The fact that the powder EPR spectrum of complex 1 exhibits several such signals suggests the possibility of the existence of several magnetic conformers of varying |d|/δ ratios, each contributing its own g⊥eff resonance.40 This possibility is in line with the dynamic model presented here, which predicts magnetic conformers ranging from isosceles to fully scalene and which will give rise to axial signals. Here, we modify the model to account for a possible deceleration of the oxide motion and the partial freezing of preferred orientations. By decreasing the number of calculated magnetic conformers, we simulate the freezing of the oxide along the respective orientations. Conversely, by increasing the number of calculated magnetic conformers, we simulate a situation where no preferential orientation exists. For example, by limiting the number of φ iterations to n = 12, we assume preferred orientations to the

Figure 7. Dependence of the energies of the ST spin manifolds in 1 (H = 0) on the oxide angular deviation φ from the position vector of Fe(1). The curves are based on best-fit values to the dynamic model (J = −21.5 cm−1 and a = 0.159). The energy levels are color-coded depending on their spin multiplicities. In the above simplified diagram with |d| = 0, states within spin manifolds with ST = 1/2, 13/2, and 15/2 are angle-independent. When |d| ≠ 0 (not shown), the degeneracies within the manifolds are lifted between the different ± MST doublets, and all resulting doublet energies become angle-dependent.

Following up on these studies, we then look at the agreement of this model with X-band EPR spectra. It is instructive to start from the simple case of an antiferromagnetic half-integer triangle described by the isosceles static Hamiltonian (1). The Zeeman effect on the low-lying doublets, in parallel and perpendicular orientations to the magnetic field, can be derived analytically by a perturbation treatment, assuming that the first excited ST = 3/2 multiplet is sufficiently high in energy:38,39 || z E|H 0, ±1/2⟩ = − 1/2(Δ ± g μβ H ) || z E|H 1, ±1/2⟩ = + 1/2(Δ ± g μβ H )

(5)

(4) G


Article

Inorganic Chemistry oxide movement arising every 30° (= 360°/12). The effects of the number of calculated magnetic conformers on the shape of the EPR spectrum are shown in Figure 8.

Figure 9. Experimental 4.2 K X-band EPR spectrum of 1 (thin black line) and simulations according to the dynamic model, assuming two components with a = 0.102 (red line) and 0.118 (blue line). Other parameters are J = −21.5 cm−1, |d| = 2 cm−1, wLor(fwhm) = 8 mT, σg⊥ = 0.055 (fwhm). Each component in the sum of magnetic conformers with φ increments of 30°. The thick black line is the sum of the two components.

susceptibility data can be viewed as an upper limit for EPR simulations because the EPR experiment has been carried out at 4.2 K, and the magnetic susceptibility measurements span the entire 2−310 K range. It is valid to assume that at 4.2 K the thermal motion of the oxide is further constrained, although we are unable to quantify this due to the experimental limitations in the X-ray diffraction experiment at liquid helium temperatures; as previously mentioned, the geometry of the flow cryostat does not allow the collection of a full data set that would permit us to determine the sizes of the thermal ellipsoids at that temperature.

Figure 8. Powder X-band EPR spectra calculated by full-matrix diagonalization of sets of magnetic conformers according to the dynamic model of the text. Each spectrum consists of n spectra differing by Δφ = 360°/n increments. Simulation parameters: a = 0.102, J = −21.5 cm−1, |d| = 2.0 cm−1, hvEPR = 9.43 GHz, wLor(fwhm) = 4 mT. The two edges of the greyed rectangle correspond to the two isosceles cases, and the center corresponds to the fully scalene conformation. All the intermediate resonances correspond to intermediate scalenities.

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DISCUSSION As discussed, magnetic susceptibility data can be interpreted by considering dynamic distortions of the magnetic cluster around an equilibrium position, which on average retains the D3h symmetry of the cluster, but which on every given moment imparts it with a symmetry between isosceles and scalene. Compared to the static isosceles model, the dynamic model yields results whose agreement factor R ranges from comparable to 25% better. From a numerical perspective, both models use the same number of free variables, i.e., two for the isotropic version of the models (J, J′ vs J, a) and three for the anisotropic ones (J, J′, d vs J, a, d). However, from a conceptual perspective, whereas the static model invokes an entirely new phenomenon, namely the magnetic Jahn−Teller effect, to justify the use of two variables, the dynamic model simply incorporates already known and well-understood phenomena into the spin Hamiltonian, i.e., the presence of atomic vibrations in matter and the geometric dependence of magnetic superexchange (magnetostructural correlations). Indeed, atomic vibrations are quantified by the thermal ellipsoids of the various atoms, whose physical meaning is precisely that atoms in crystals are not stationary. Similarly, there is an extended literature on magnetostructural correlations; the dependence of magnetic superexchange between metal ions on the atomic distances and bond angles is a well-documented phenomenon.

Simulations to the dynamic model considering the full spincoupled system for various oxide orientations are shown in Figure 9. We should note that no unique parameter set could reproduce the entire set of g⊥eff resonances, which span a 200 mT region. To reproduce with some accuracy the main characteristics of these resonances, we needed to consider two a values (0.102 and 0.118) to cover the relevant magnetic field region. The physical meaning of this may be that the oxide does not strictly move along a circular trajectory but inside a circular surface. The smaller a value would then correspond to conformations in which the oxide is found nearer to the center of the triangle and not on the rim of its trajectory. We also noted that the g⊥eff resonances are broadened with respect to the g∥eff one, which we reproduced by introducing a g-strain to the single-ion g⊥0 values. We rationalize this choice by observing that the oxide movement induces changes to the Fe−Ooxo bond lengths. Because in our model the single-ion zaxis is parallel to the molecular z-axis (normal to the triangle plane), we conclude that this movement should affect the single-ion g⊥0 values through structural deformation. A final comment is relevant regarding the lower determined a values with respect to fits to magnetic susceptibility data. On the basis of the previous discussion regarding the temperature dependence of a, the value derived from the fits to magnetic H


Article

Inorganic Chemistry

the intermediate ones being averaged out. Conversely, consideration of 30° intervals (i.e., 12 iterations) resulted in the appearance of an additional, intermediate g⊥eff resonance. This could be viewed as an indication of freezing of the system to preferred directions, i.e., to specific magnetic conformers, at low temperatures. This is in line with the IINS spectra which show two such conformers. However, the number of conformers observable through either technique depends not only on its intrinsic time scale (∼10−9 s for EPR and ∼10−11 s for IINS) but also on its intrinsic line width (∼10−4 cm−1 for X-band EPR of transition metals and ∼10 cm−1 for IINS). Because both techniques detect more than one magnetic conformers, we conclude that at the temperature of the experiment (4.2 K for EPR, 1.5 K for IINS), the movement of the oxide is sufficiently decelerated for the postulated preferred directions of the oxide to be experimentally observed. The fact that more such signals are detected by EPR spectroscopy is probably related to the much smaller intrinsic line width of the technique compared to that of IINS.

While the presented model achieves a net improvement on the interpretation of magnetic susceptibility and EPR data of half-integer antiferromagnetic triangles, it also exhibits certain limitations. As was previously mentioned, it assumes stationary metal ions and a central atom moving along a circular trajectory on the plane of the triangle with a temperature-independent radius and with no directional preference. However, crystallographic data reveal that all atoms are equally mobile and that, on average, the central oxide moves within a sphere. This points us to conformations of varying pyramidality of the oxide, in which it is found above or below the Fe3 plane. This change in pyramidality should decrease the orbital overlaps with all ferric ions and, consequently, the strength of the antiferromagnetic interactions. More importantly, this model assumes that the amplitude of the thermal motions is temperature-independent. However, as also revealed from crystallographic data, the mean radius of the vibrations increases upon heating, as exhibited by the increase of the matrix elements of the thermal ellipsoid tensors. The physical meaning of this is that the dynamic distortion parameter a is not actually a constant, but a function of temperature, a(T). In particular, comparing the structural information between the reported structure at 233 K (R0 = 1.915 Å, r0 = 0.20 Å) with the present 90 K structure (R0 = 1.908 Å, r0 = 0.14 Å) and considering that a ∝ r0/R0, then we can conclude that a(233 K)/a(90 K) = 1.41. By extension, this means that the variation amplitude of the magnetic exchange coupling is also a function of temperature. Finally, it is not just the amplitude of the J values that is expected to be a function of temperature, the central value itself should be subject to a thermal effect. Because the Fe3 plane is parallel to the ab-plane, the thermal contraction of the a- and baxes affects all bond lengths and angles that have a component parallel to that plane. The thermal variance of Fe−Ooxo bond lengths (e.g., 0.31% between 233 and 90 K) is expected to affect the orbital overlaps between metal and ligand orbitals and therefore affect the magnitude of J. Thus, the magnetic exchange is expected to be affected by two different thermally activated mechanisms. Whereas the former mechanism should affect only the amplitude of the J variations, the latter should also affect its central value. This particular point is of general importance in condensed-matter magnetism: while the variation of the J amplitudes can influence the properties of certain families of materials, examined by techniques with short time scales (case in point being EPR studies of half-integer spin triangles), the thermal variations of the central J values could affect the magnetic properties of correlated materials in a more general way by modifying the orbital overlaps between magnetic ions and diamagnetic ligands as the unit cell constants change due to thermal expansion/contraction. The treatment of the EPR data with this model reproduces qualitatively the general characteristics of the g⊥eff set of transitions. While we could not find a single set of parameters to fully account for all the resonances, we reduced the independent components from five, required by a purely phenomenological model considering fictitious S = 1/2 spins,30 to only two. We found that species with a = 0.102 and 0.118, and |d| = 2 cm−1 could reproduce the upper and lower bounds of the g⊥eff values. It should also be noted that the number of magnetic conformers considered (number of φ iterations) also affected the line shape of the composite spectrum, e.g., 1° intervals (i.e., 360 iterations) resulted in a composite spectrum exhibiting just the minimum and maximum g⊥eff transitions with

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CONCLUSIONS AND PERSPECTIVES It should be noted that this variability of J couplings is not proposed as a phenomenon specifically related to spinfrustrated triangles, as it is rooted in a more general phenomenon, namely the atomic vibrations of magnetic ions and of the bridging atoms that propagate superexchange between them. While this effect can be safely ignored in the vast majority of magnetic clusters, in the particular case of spinfrustrated triangles, the competing interactions and the resulting magnetic structure imparted by symmetry render the magnetic properties extremely sensitive to small Jij and d variations. Therefore, for this family of complexes, it is more meaningful to consider Jij couplings as variables and not as constants, as opposed to other magnetic topologies, where such a consideration would lead to overparametrization. In a sense, this sensitivity to small and rapid structural variations serves as a magnetic probe for minute structural deformations, e.g., atomic vibrations. Thus, the value of these complexes in molecular magnetism remains high, as they are simple enough to be treated with exact techniques, but complex enough to exhibit a rich variety of magnetic phenomena. In conclusion, our structural studies and the coherent treatment of magnetic and spectroscopic data with a dynamic model suggest that it is not necessary to invoke an entirely new (and experimentally unobserved) “magnetic Jahn−Teller effect” to account for the lifting of the degeneracy of the low-lying doublets in equilateral spin-frustrated triangles. Instead, we can account for this lifting by incorporating atomic vibrations into our spin Hamiltonian through the use of the angular overlap model. It is proposed that refinements to this model, to account for thermal effects on the mean J values and on the amplitudes of their variations, should further improve the agreement with experimental data.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.inorgchem.6b01912. Energy levels and spin expectation values of 1 (PDF) Crystallographic information for 1 (CIF) I


Article

Inorganic Chemistry

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Liquides pour l’Acétate Complexe [Cr3(CH3COO)6(OH)2]Cl.8H2O. Physica 1954, 20 (7−12), 721−726. (12) Duncan, J. F.; Kanekar, C. R.; Mok, K. F. Some Trinuclear iron(III) Carboxylate Complexes. J. Chem. Soc. A 1969, 480. (13) Long, G. J.; Robinson, W. T.; Tappmeyer, W. P.; Bridges, D. L. The Magnetic, Electronic, and Mössbauer Spectral Properties of Several Trinuclear iron(III) Carboxylate Complexes. J. Chem. Soc., Dalton Trans. 1973, 6, 573−579. (14) Garopenko, V. A.; Eremin, M. V.; Yablokov, Y. Sov Phys. Solid State 1973, 15, 909−913. (15) Rakitin, Y.; Yablokov, Y.; Zelentsov, V. EPR Spectra of Trigonal Clusters. J. Magn. Reson. 1981, 43 (2), 288−301. (16) Mishima, M.; Uryû, N. Low Temperature Magnetic Properties of Trimeric Clusters. Phys. Lett. A 1978, 67 (1), 64−66. (17) Kögerler, P.; Tsukerblat, B.; Müller, A. Structure-Related Frustrated Magnetism of Nanosized Polyoxometalates: Aesthetics and Properties in Harmony. Dalton Trans 2010, 39 (1), 21−36. (18) Murao, T. Jahn-Teller Effect in Trinuclear Complexes. Phys. Lett. A 1974, 49 (1), 33−35. (19) Jones, D. H.; Sams, J. R.; Thompson, R. C. The Magnetic Behavior of Clusters of the Type [M3O(RCOO)6]+. An Isotropic Model Involving Dynamic Distortions. J. Chem. Phys. 1984, 81 (1), 440. (20) Rakitin, Y. V.; Kalinnikov, V. T.; Novotortsev, V. M. Jahn-Teller Effect in Trigonal μ-Oxoclusters. Russ. Chem. Bull. 2004, 53 (11), 2478−2484. (21) Sowrey, F. E.; Tilford, C.; Wocadlo, S.; Anson, C. E.; Powell, A. K.; Bennington, S. M.; Montfrooij, W.; Jayasooriya, U. A.; Cannon, R. D. Spin Frustration and Concealed Asymmetry: Structure and Magnetic Spectrum of [Fe3O(O2CPh)6(py)3]ClO4·py. J. Chem. Soc., Dalton Trans. 2001, 6, 862−866. (22) Cannon, R. D.; Jayasooriya, U. A.; Wu, R.; arapKoske, S. K.; Stride, J. A.; Nielsen, O. F.; White, R. P.; Kearley, G. J.; Summerfield, D. Spin Frustration in High-Spin Triiron(III) Complexes: An Inelastic Neutron Scattering Study. J. Am. Chem. Soc. 1994, 116 (26), 11869− 11874. (23) Furrer, A.; Güdel, H. U. Magnetische Austauschwechselwirkung Eines Dreikernigen Eisenclusters in α-Metavoltin, K10(Fe3 O(SO4)6(H2O)32·12H2O. Helvetica Phys. Acta 1977, 50 (4), 439−446. (24) Giacovazzo, C.; Scordari, F.; Menchetti, S. Hydrous Potassium and Ferric Iron Sulphate (Maus’s Salt). Acta Crystallogr., Sect. B: Struct. Crystallogr. Cryst. Chem. 1975, 31 (8), 2171−2173. (25) To the best of our knowledge, the structure of 2 has not been reported. (26) Oxford Diffraction. CrysAlis PRO; Oxford Diffraction Ltd: Abingdon, Oxfordshire, England, 2006. (27) Sheldrick, G. M. A Short History of SHELX. Acta Crystallogr., Sect. A: Found. Crystallogr. 2008, 64 (1), 112−122. (28) Sheldrick, G. M. Crystal Structure Refinement with SHELXL. Acta Crystallogr., Sect. C: Struct. Chem. 2015, 71 (1), 3−8. (29) Rigaku/MSC. CrystalClear; Rigaku/MSC Inc.: The Woodlands, TX, 2005. (30) Georgopoulou, A. N.; Sanakis, Y.; Boudalis, A. K. Magnetic Relaxation in Basic Iron(III) Carboxylate [Fe3O(O2CPh)6(H2O)3]ClO4·py. Dalton Trans. 2011, 40 (24), 6371. (31) Stoll, S.; Schweiger, A. EasySpin, a Comprehensive Software Package for Spectral Simulation and Analysis in EPR. J. Magn. Reson. 2006, 178 (1), 42−55. (32) Chilton, N. F.; Anderson, R. P.; Turner, L. D.; Soncini, A.; Murray, K. S. PHI: A Powerful New Program for the Analysis of Anisotropic Monomeric and Exchange-Coupled Polynuclear D - and F -Block Complexes. J. Comput. Chem. 2013, 34 (13), 1164−1175. (33) Stride, J. A. Letter: Comment on the Intensities of Inelastic Neutron Scattering Spectra. J. Chem. Soc., Dalton Trans. 2001, 16, 2435−2436. (34) Yoon, J.; Mirica, L. M.; Stack, T. D. P.; Solomon, E. I. Spectroscopic Demonstration of a Large Antisymmetric Exchange Contribution to the Spin-Frustrated Ground State of a D3 Symmetric Hydroxy-Bridged Trinuclear Cu(II) Complex: Ground-to-Excited

Representation of the dynamic model (AVI)

AUTHOR INFORMATION

Corresponding Author

*E-mail: athanasios.bountalis@fiu.edu. ORCID

Athanassios K. Boudalis: 0000-0002-8797-1170 Present Addresses †

I.M.: Department of Biology, Section of Genetics, Cell Biology and Development, University of Patras, 26500 Patras, Greece. ⊥ A.K.B.: Florida International University, Department of Chemistry and Biochemistry, Modesto Maidique Campus, CP304, 11200, SW Eighth Street, Miami, Florida 33199, United States. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank Dr. Dmitry Chernyshov for help with the experimental setup at BM01 and the ESRF for provision of beamtime, Dr. Yiannis Sanakis for discussions on the magnetic properties of magnetic triangles, Dr. Bertrand Vileno for discussions on writing Matlab routines, and Professor Stefan Stoll and Dr. Joscha Nehrkorn for discussions on the Easyspin forum.

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REFERENCES

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