A combined high-field EPR and quantum chemical study on a weakly ferromagnetically coupled dinuclear Mn(III) complex. A complete analysis of the EPR spectrum beyond the strong coupling limit

Abstract

The electronic and magnetic properties of polynuclear complexes, in particular the magnetic anisotropy (zero field splitting, ZFS), the leading term of the spin Hamiltonian (SH), are commonly analyzed in a global manner and no attempt is usually made to understand the various contributions to the anisotropy at the atomic scale. This is especially true in weakly magnetically coupled systems. The present study addresses this problem and investigates the local SH parameters using a methodology based on experimental measurements and theoretical calculations. This work focuses on the challenging mono μ-oxo bis μ-acetato dinuclear Mn^III complex: [Mn₂^III(μ-O)(μ-OAc)₂L₂](PF₆)₂ (with L = trispyrrolidine-1,4,7-triazacyclononane) (1), which is particularly difficult for EPR spectroscopy because of its large magnetic anisotropy and the weak ferromagnetic interaction between the two Mn^III ions. High field (up to 12 T) and high frequency (190–345 GHz) EPR experiments have been recorded for 1 between 5 and 50 K. These data have been analyzed by employing a complex Hamiltonian, which encompasses terms describing the local and inter-site interactions. Density functional theory and multireference correlated ab initio calculations have been used to estimate the ZFS of the Mn^III ions (D_Mn = +4.29 cm⁻¹, E_Mn/D_Mn = 0.19) and the Euler angles reflecting the relative orientation of the ZFS tensor for each Mn^III (α = −52°, β = 28°, γ = 3°). This analysis allowed the accurate determination of the local parameters: D_Mn = +4.50 cm⁻¹, E_Mn/D_Mn = 0.07, α = −35°, β = 23°, γ = 2°. The spin ladder approach has also been applied, but only the parameters of the ground spin state of 1 have been accurately determined (D₄ = +1.540 cm⁻¹, E₄/D₄ = 0.107). This is not sufficient to allow for the determination of the local parameters. The validity and practical performance of both approaches have been discussed.

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1. Introduction

Transition metal ion complexes are involved in many domains such as biology, catalysis, condensed matter physics and materials science. In view of understanding their chemical reactivity and/or their physical properties a large number of experimental techniques are used to characterize their electronic and magnetic properties. Although EPR spectroscopy is probably the best tool for measuring the magnetic anisotropy, conventional EPR spectrometers are usually inoperative in the case of paramagnetic metallic compounds because of too large D values (axial part of the zero field splitting, ZFS) or/and weak magnetic exchange interaction (J) in polynuclear systems (|J| ≈ or ≪ |ZFS|). Either the EPR signal is absent or the obtained data are too complex to be analyzed. Yet weakly coupled dimers can be found in areas as diverse as biochemistry and molecular magnetism, in addition to inorganic and bio-inorganic chemistry.

On the other hand, high field and high frequency EPR spectroscopy (HF-EPR) has been successful in elucidating the magnetic anisotropy of mononuclear transition metal ion complexes with large ZFS^1–7 and of strongly coupled high spin polynuclear clusters, including single molecule magnets.^8–11 However, for these latter compounds, the goal of the reported studies was mainly the determination of the ZFS of the entire system, and no attempt was generally made to understand the various contributions to the anisotropy at the single-ion level. It can still be noticed that with the aim of performing rational synthesis, a few investigations using ligand field approaches have been reported to assess the contribution of the local metallic ion anisotropy.^12–14

Several X- and/or Q-band-EPR investigations have been also performed on magnetically coupled dinuclear complexes with S > 1/2 metallic ions in order to reach such information, especially the ZFS of each metallic ion. These systems have been described as a Heisenberg spin ladder, for which |J| ≫ |ZFS|, and the ZFS of the paramagnetic spin states involved in the EPR spectra have been determined.^15–23 The advantage of this approach is that a detailed analysis of the EPR spectra gives access to an estimate of these ZFS values. From these spin ladder parameters, the ZFS of each metallic ion can be also estimated by means of empirical equations.²⁴ However, in most cases, the condition |J| ≫ |ZFS| was not met which makes the results of the analysis questionable.

To properly analyze EPR spectra of polynuclear complexes with relatively weak magnetic exchange interaction (relative to the ZFS), the spin ladder description cannot be used as the zero-field splitting interaction will mix the various multiplets. The complexes are then best described in terms of local spin-Hamiltonian parameters together with the parameters characterizing the magnetic interaction (foremost the isotropic Heisenberg exchange J, but also terms such as the anisotropic exchange, or, in mixed-valence cases, the double exchange B). Consequently, the parameters used to simulate the EPR spectra, especially the ZFS of each metallic ion, cannot be estimated by a simple analysis of the data. However, during this last decade, quantum chemistry made significant progresses to predict the ZFS of number of transition metal ions.^2,25–34 The relative importance of the distinct contributions to the ZFS (spin–spin and spin–orbit coupling) can vary substantially between the different metallic ions. The complexity of the problem therefore requires the use of theoretical tools adapted to each case. We have been able to demonstrate that DFT calculations correctly reproduce the measured ZFS of the Mn^II ion.^35,36 For the Mn^III ion, on the other hand, we have shown that ab initio complete active space self-consistent field (CASSCF) calculations are required because they are able to explicitly treat the excited spin states of different spin than the ground state. Unlike predicted by popular ligand field arguments, the spin–orbit coupling of these spin-flip states with the ground state makes the dominant contribution to the zero-field splitting in this case.^4,37–39

In the present work, our main objective was to properly analyze EPR spectra of a weakly exchanged coupled polynuclear complex by means of a methodology based on the crucial use of theoretical calculations. A second focus of this work was to gain insight into the conditions under which the spin ladder approach is no longer valid. We will focus on the challenging example of the mono μ-oxo bis μ-acetato dinuclear manganese(III) complex: [Mn₂^III(μ-O)(μ-OAc)₂L₂](PF₆)₂ (with L = trispyrrolidine-1,4,7-triazacyclononane) (1). This system is particularly difficult for EPR spectroscopy because it shows large magnetic anisotropy as well as a weakly ferromagnetic interaction between the two Mn^III ions (Fig. 1).⁴⁰ Indeed, the high-spin d⁴ configuration of Mn^III together with its relatively high oxidation state thus provides particularly large Jahn–Teller distortions with |D| values expected in the range of 2–5 cm⁻¹. Our strategy was to obtain a complete set of HF-EPR spectra at variable temperature using a multifrequency approach, corresponding to the first EPR data obtained on a dinuclear Mn^III complex. Then, by quantum chemistry, the ZFS parameters of each Mn^III ion were calculated and used as initial values in fitting the experimental HF-EPR data. These data were also fitted using the spin ladder approach in order to reveal the success and limitations of this approach. Finally, the quality of the theoretical predictions will be critically assessed.

Fig. 1 Principal axes of the D_A and D_B-tensors for the two dimers (1a and 1b) present in the X-ray structure (dashed lines). D_xx, D_yy, D_zz are red, green and blue respectively.

2. Theory

2.1. The spin Hamiltonian

For the present manganese(III) dinuclear, the appropriate phenomenological spin Hamiltonian (excluding hyperfine coupling terms) is:

Ĥ = μ_BB·g_A·Ŝ_A + Ŝ_A·D_A·Ŝ_A + μ_BB·g_B·Ŝ_B + Ŝ_B·D_B·Ŝ_B + Ŝ_A·D_AB·Ŝ_B (1)

where the individual terms correspond to the Zeeman perturbation and the local anisotropy for each Mn center that are parameterized by the g_A, g_B matrices and D_A, D_B tensors. The last term describes the Mn–Mn coupling, and can be decomposed into its trace, a symmetric and an antisymmetric tensor:

Ŝ_A·D_AB·Ŝ_B = −2JŜ_A·Ŝ_B + Ŝ_A·D·Ŝ_B + d⃑·Ŝ_A × Ŝ_B (2)

where J is the isotropic exchange, D is the second order symmetric tensor and d⃑ is the Dzyaloshinskii–Moriya vector.⁴¹

There are two experimentally undistinguishable contributions to the symmetrical part of the interaction: (i) the dipole–dipole interaction, which is a direct through space interaction between the two magnetic centers, and generally regarded as the most important contribution, and (ii) the spin–orbit contribution, also called the anisotropic exchange. If the separation between the two ions is large compared to the spatial extension of the dipoles themselves, we can approximate the spin–spin dipolar component by a classical interaction. The matrix elements of the D_AB tensor are given by:

where g_e is the free electron g-value (2.002319…), α (∼1/137 in atomic units) is the fine structure constant and r_AB is the vector between the two magnetic moments.

Following Moriya, the antisymmetric coupling in a complex that presents C₂ symmetry does not vanish and is perpendicular to the two-fold rotation axis of the molecule.⁴¹

2.2. The spin ladder approach

In the strong exchange limit, i.e. the isotropic interaction outweighs the local anisotropic terms; the total spin quantum number S remains a good quantum number for the AB pair. In this case, the dinuclear complex is described as a spin ladder defined in terms of total spin S, where S can take values:

|S_A − S_B| ≤ S ≤ S_A + S_B (4)

Each spin manifold is separated by an energy (E_S) dependent on J:

E_S = −J[S(S + 1) − S_A(S_A + 1) − S_B(S_B + 1)] (5)

From the spin ladder description, each spin manifolds of the system can be considered independently and characterized by the following effective spin Hamiltonian:

Ĥ_S = μ_BB·g_S·Ŝ + Ŝ·D_S·Ŝ (6)

with g_S and D_S describing the electronic Zeeman and zero field splitting of each of the spin manifolds of the dinuclear complex. It follows that in this treatment the EPR spectra can be regarded as a superposition of the spectra obtained for the different total spin states. The total spin dependent tensors for each multiplet, g_S and D_S, are typically expressed as a linear combination of the local ones, g_A, g_B and D_A, D_B:²⁴

g_S = c_Ag_A + c_Bg_B (7)

D_S = d_AD_A + d_BD_B + d_ABD_AB (8)

whereThe expressions for c, c₊ and c₋ can be found in the ESI.† For a dinuclear Mn^III complex (S_A = S_B = 2), a ferromagnetic interaction leads to an S = 4 ground state. The coefficients used to calculate the total tensors for all spin manifolds are summarized in Table S1 (ESI†). The existence of the local tensors g_A, g_B, D_A and D_B is of appealing plausibility. However, they do not have a rigorous quantum mechanical definition as the g- and D-tensors are always ‘global’ properties of the entire system. Their decomposition into additive local contributions necessarily involves a specific electronic structure model that one forces onto the system together with a number of more or less well controlled approximations.

2.3. The on-site spectroscopic parameters

The interaction of the molecular magnetic dipole moment with an external magnetic field, i.e. the molecular Zeeman effect, is described by the g-tensor, and can be computed by solving the coupled-perturbed self-consistent field (CP-SCF) equations.

The elements of the g-tensor can be calculated as a sum of four contributions:

g_kl = g_eδ_kl + Δg^RMCδ_kl + Δg^GC_kl + Δg^OZ/SOC_kl (11)

The first term is isotropic and is given by the free electron g-value (g_e = 2.002419…). The second and third terms are first-order relativistic mass and “gauge correction” (diamagnetic correction). The final term is of second-order and represents orbital Zeeman (OZ)/spin–orbit coupling (SOC) cross term, and is in almost all cases the dominant term. Details about the calculation of each of the individual terms can be found elsewhere.⁴²

The zero-field splitting, which is parameterized by the D-tensor, describes the removal of the state degeneracy for systems with S > 1/2 in the absence of an applied magnetic field, and enters the following phenomenological Hamiltonian:

H_ZFS = Ŝ·D·Ŝ (12)

Assuming a diagonal and traceless D-tensor, the ZFS Hamiltonian can be rewritten:

H_ZFS = D[Ŝ_z² − ⅓S(S + 1)] + E(Ŝ_x² − Ŝ_y²) (13)

with

D = ½(−D_xx − D_yy + 2D_zz) = [/]D_zz (14)

and

E = ½(D_xx − D_yy) (15)

By an appropriate choice of the coordinate system, we can satisfy the Blumberg convention,⁴³ commonly used in EPR studies of transition metal compounds, which requires that

|D_zz| > |D_yy| > |D_xx| (16)

and yields E/D between 0 and 1/3.

From first principles, the D-tensor can be written as a sum of first- and second-order perturbational terms. For a single determinant ground state, the spin–spin coupling (SSC) contribution to the D-tensor is given by the equation of McWeeny and Mizuno:⁴⁴

where g_e is the free electron g-value, α is the fine structure constant, S is the total spin of the electronic ground state, P^α−β the spin density matrix in the atomic orbital basis, and μ, ν, κ, and τ are the basis functions. In the present paper, the calculation of the SSC contribution is based on an unrestricted natural orbital (UNO) determinant.

The treatment of the SOC part is somewhat more complicated. This is due to the fact that SOC mixes states of different multiplicity. The quantum mechanical form of these equations has been derived in 1998.²⁵ The translation into a linear response formalism has only been attempted much more recently.⁴⁵ Following the same line of thoughts, Van Wüllen has recently suggested a minor modification of the original formalism which amounts to using the same prefactor 1/S(2S − 1) for all contributions to the ZFS tensor.⁴⁶ In our experience, either DFT based treatment is of limited predictive power. From a number of previous studies,^{3,37,39,47–49} we came to the conclusion that the calculation of the ZFS in transition metal complexes is best done on the basis of multireference ab initio treatments together with quasi-degenerate perturbation theory (QDPT) for the treatment of the SOC and SS interactions. This is a more suitable approach since, in contrast to DFT methods, the magnetic sublevels of all multiplets can be explicitly taken into account. The most efficient variant that incorporates at least the leading effects of dynamic electron correlation employs second order N-electron valence perturbation theory (NEVPT2), a strongly contracted form of second-order many body multireference perturbation theory.^50–52 In cases where the ZFS is substantial, this treatment has been proven to be successful if at least the d–d excited multiplets are included in the treatment.⁵³

2.4. The exchange coupling constant

The isotropic exchange interaction was modeled by the Heisenberg–Dirac–van Vleck (HDvV) Hamiltonian:

Ĥ = −2JŜ_A·Ŝ_B (18)

which describes the coupling of two spin-operators Ŝ_A and Ŝ_B. The Heisenberg exchange coupling constant (J) can be computed within the framework of the broken-symmetry formalism proposed by Noodleman.^54,55 A modification of the initial formulation, that consistently covers the whole range of coupling situations, has been proposed by Yamaguchi et al.:^56,57where 〈Ŝ²〉_HS and 〈Ŝ²〉_BS are the total spin angular momentum expectation values for the high-spin and broken-symmetry states.

3. Materials and methods

3.1. Sample preparation

Complex 1 was synthesized according to the published procedure.⁴⁰ The HF-EPR spectra have been recorded on a polycrystalline powder pellet to afford magnetic orientation of crystals.

3.2. EPR measurements

HF-EPR spectra were recorded on a laboratory made spectrometer.⁵⁸ Gunn diodes operating at 95 GHz and 115 GHz and equipped with a second- and third-harmonic generator have been used as the radiation source. The magnetic field was produced by a superconducting magnet (0–12 T).

3.3. EPR simulations

The HF-EPR spectra were simulated with the Easyspin program.⁵⁹ Due to the high complexity of the experimental spectra the fitting procedure was conducted on a limited number of peaks. Consequently, we have taken into account only the peaks for which our initial guess from quantum mechanical calculations provided an unambiguous determination (see main text for details). Subsequently we minimized the root mean square error of these peaks relative to experimental values. Finally the entire spectra were simulated.

3.4. Computational details

Geometry optimizations. All computations have been performed with the ORCA program.⁶⁰ The BP86^61,62 functional, the Karlsruhe polarized triple-ζ basis set (TZVP),⁶³ and the auxiliary def2-TZV/J basis set⁶⁴ for resolution of identity (RI) approximation were employed for the geometry optimizations. If not otherwise specified, the same basis set was used for all subsequent calculations. Additionally denser integration grids (Grid4 in ORCA convention) and tighter SCF convergence criteria were used. In these optimizations we also added the latest dispersion correction of Grimme et al.⁶⁵

Calculation of spectroscopic parameters. For the calculations of the on site g- and D-tensors of each Mn^III ion forming the dimer, we employed the same theoretical method as used during the geometry optimizations. Additionally, for the D-tensor, state-averaged complete active space self-consistent field (SA-CASSCF) calculations were performed, with a minimal active space comprised of four electrons in five metal d-orbitals (SA-CASSCF(4,5)), as described before.³⁷ This minimal active space should be considered satisfactory in the cases of complexes with metal–ligand bonds with predominant ionic character, and is bound to fail if there is significant metal–ligand orbital mixing. On the basis of the functions obtained from the above multireference calculations, the SSC and SOC effects are treated through quasi-degenerate perturbation theory (QDPT). Additionally, the effects of the dynamic correlation on the computed values were evaluated by substituting the diagonal elements of the QDPT matrix with the NEVPT2 corrected state energies. The complete manifold of 5 quintet and 35 triplet states was included in these calculations. No singlet state was taken into account, since it was previously established that their contribution to D is negligible in the case of Mn^III.³⁷

Broken-symmetry DFT calculations were performed with the hybrid meta-GGA TPSSh functional.⁶⁶ The coulomb and exchange parts were evaluated with the “chain-of-spheres” (RIJCOSX) algorithm.⁶⁷ Increased integration grids (Grid4 and Gridx4 in ORCA convention) and tight SCF convergence criteria were used. Subsequently, the exchange coupling constants were computed with the Yamaguchi equation.

The exchange pathways are discussed by an analysis of the magnetic orbitals. Unambiguous identification of these orbitals was achieved on the basis of the corresponding orbital transformation (COT) of Amos and Hall.^28,68,69 This transformation of canonical orbitals into corresponding orbitals leaves the broken symmetry determinant invariant and has the additional property that each spin-up orbital has a nonzero overlap with only a single spin-down orbital. The magnitude of the spatial overlap can serve as an indicator of the strength of the interactions in the system.

4. Results

4.1. Multifrequency high field EPR study

As no signal could be observed at 4.2 K on the powder X- and Q-band EPR spectra of complex 1, a high-field and high frequency EPR (HF-EPR) investigation has been carried out. The multifrequency HF-EPR study was achieved over a temperature range of 5–50 K, between 190 and 345 GHz, using a magnetic field up to 12 T. Fig. 2a, 3a and 4a illustrate the temperature and the frequency dependence of the resulting HF-EPR spectra. EPR transitions are spread over the entire range from 0 to 12 T at all frequencies. The shape of the transitions and their resolution point the fact that, due to the design of the home-built multifrequency HF-EPR spectrometer, the spectra correspond to a mixture of in-phase and out-of-phase signals. Therefore no quantitative analysis of the line intensity can be properly done but qualitative analysis can be performed which allows determination of the spin Hamiltonian parameters. It should be noted that above 25 K, the general appearance of the spectra no longer changes, only the overall intensity decreases. A weak transition located at g = 2 (10.2 T at 285 GHz and 8.2 T at 230 GHz) corresponding to a small amount of mononuclear Mn^II impurities is observed at low temperature.

Fig. 2 Experimental (a) and simulated (b–c) EPR powder spectra recorded at 285 GHz and at different temperatures. The simulated b spectra have been calculated by diagonalizing the 25 × 25 matrix (eqn (1)), and the simulated c spectra with the spin ladder approach taking into account the S = 4 and S = 3 spin states.

Fig. 3 Experimental (a) and simulated (b–c) EPR powder spectra recorded at 5 K and at different frequencies. The simulated b spectra have been calculated by diagonalizing the 25 × 25 matrix (eqn (1)), and the simulated c spectra with the spin ladder approach taking into account the S = 4 spin state.

Fig. 4 Experimental (a) and simulated (b–c) EPR powder spectra recorded at 25 K and at different frequencies. The simulated b spectra have been calculated by diagonalizing the 25 × 25 matrix (eqn (1)), and the simulated c spectra with the spin ladder approach taking into account the S = 4 and S = 3 spin states.

The most intense and resolved EPR features are observed at low temperature in the low field region (below 4 T). They are assigned to “forbidden” transitions because (i) their intensity is enhanced as the EPR frequency decreases, (ii) they are located in the low field part of the spectra where strong-field limit conditions are not reached (ZFS ≫ gβB) and, (iii) more importantly, their field position dependency as a function of the frequency matches to a g value much greater than 2.

4.2. Theoretical calculations

Structural parameters. The X-ray structure of 1 reveals the presence of two independent dimers (1a and 1b) in the asymmetric cell (Fig. 1).⁴⁰ While the ligand environment is similar for all four manganese centers, their Mn–N/O distances are quite different (see Table 1 for details), which results in dimers that deviate quite significantly from a C₂ symmetry point group. It has been shown previously that the calculated spectroscopic parameters can significantly deteriorate if an optimized geometry is used in favor of the X-ray structure.^70,71 The problem of this approach is that, in light of the structural differences, the on-site spectroscopic properties could be quite different (as will be shown later). This will lead to large number of parameters to take into account in our EPR spectral simulation procedure. Therefore, we performed a geometry optimization of a structure that has a C₂ symmetry axis perpendicular to the vector connecting the two metals. The geometrical parameters for the resulting optimized structure (1^opt) are presented in Table 1. Overall good agreement was found with the experimental data, considering that none of the counterions present in the crystal structure were included in our calculations. Interestingly, the asymmetrical binding of the carboxylato bridges is kept in the optimized structure, the Mn–O_Ac distances being equal to 2.163 Å and 2.019 Å, respectively. In both 1 and 1^opt, each Mn^III ion is in the center of an octahedron with the oxygen from the μ-oxo bridge and the nitrogen in the trans position to this oxygen along the tetragonal axis. The marked axial compression around each Mn^III ion is retained in 1^opt, the sum of Mn–N/O distances along the tetragonal axis is on average approximately 0.35 Å smaller when compared to the sets of the Mn–N/O distances in the equatorial plane.

Table 1 Structural parameters in Å (bond distances) and ° (angles) for the complexes 1a, 1b and 1^opt

	Mn–Ooxo	Mn–N^a	Mn–OAc	Mn–N^b	Mn–OAc*	Mn–N^c	Mn–Ooxo–Mn
a N in trans position from Ooxo.b N in trans position from OAc.c N in trans position from OAc*.d Structural parameters issued from ref. 36.e This work.
Mn1^d	1.785	2.141	2.053	2.228	2.063	2.217	122.93
Mn2^d	1.821	2.111	2.021	2.193	2.130	2.223
Mn3^d	1.796	2.106	1.996	2.152	2.136	2.274	121.60
Mn4^d	1.804	2.146	2.025	2.160	2.166	2.221
Mn (opt)^e	1.827	2.145	2.019	2.186	2.163	2.272	120.62

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Prediction of on-site spectroscopic parameters (g and ZFS values). In order to estimate the parameters for the single Mn^III ion in 1, a theoretical model has been designed in which one manganese ion is substituted by a diamagnetic Ga^III ion, to give a heterodinuclear Mn^IIIGa^III complex. This approach can be regarded as theoretical parallel to the experimental method of determining the spectroscopic properties of the single paramagnetic ion in a dinuclear complex by doping the complex with a diamagnetic ion. Accordingly a total of five optimized structures were used for the calculation of the local parameters. Four structures resulted from the two independent dimers presented in the crystal structure for which the position of the hydrogen atoms was relaxed prior to the in silico atom substitution, and one structure from the previously optimized structure (1^opt). In the last case, the g and ZFS parameters are the same for the two manganese ions due to the C₂ symmetry axis, albeit rotated.

For high spin Mn^III complexes, the anisotropy of the Zeeman interaction is very small leading to g-values close to 2. Already at the DFT level of theory the predicted g-values confirmed this lack of anisotropy with g_x, g_y and g_z being equal to 1.998, 2.000, and 2.004, respectively, thus no further ab initio calculations were performed for this property.

Our previous systematic investigation on mononuclear Mn^III complexes has shown that the best DFT-based approach for the calculation of the ZFS parameters corresponds to the combination of the coupled-perturbed (CP) approach for the spin–orbit coupling (SOC) part of the D-tensor, and the spin unrestricted natural orbital (UNO) variant for the calculation of the spin–spin coupling (SSC) part of D.³⁹ This combination has been used in this work for predicting the ZFS parameters of the heterodinuclear models. All results are reported in Table 2. The DFT calculations predict a positive sign of D in all cases, consistent with the compressed tetragonal distortion of the octahedra observed in the X-ray structure of 1. This is quite unusual for Mn^III, which is predominantly found in elongated octahedral geometry. Only two mononuclear Mn^III complexes with a positive D value combined with a compressed tetragonal distortion have been described yet.^4,72 As expected for a Mn^III ion, the D_SOC value represents the main contribution to D. On the other hand, the contribution of the spin–spin term (D_SSC ∼ +0.35 cm⁻¹) is not negligible since it represents about 16% of the total D value. Nevertheless, even with the corrected prefactors for the spin-flip term that enter the evaluation of the D_SOC contribution,³⁷ previous studies have shown that the DFT results consistently underestimate the calculated value of D.^4,39 Consequently, the calculation of the ZFS parameters was also be handled at a multiconfigurational ab initio level. The resulting D values (between +3.69 and +4.10 cm⁻¹) are noticeably larger than that calculated by DFT. We have also included dynamic correlation into the calculation by dressing the QDPT matrix with diagonal energies calculated with the NEVPT2 scheme. In the present case, the use of NEVPT2 energies increases the SOC contribution by about 20%, in agreement with previous reports,^25,73 leading to total D values in the range of +4.14 to +4.30 cm⁻¹. It should be noted that this effect was of minor importance in our previous systematic study on Mn^III.³⁹ In the absence of a macrocycle ligand, it has been experimentally observed by HF-EPR that the magnitude of D increases with the number of oxygenated-based ligands: 3.29 < |D_N6| < 3.50; |D_N4O2| = 3.52; |D_N2O4| = 4.50; 4.35 < |D_O6| < 4.52 cm⁻¹.^4,72,74–77 The predicted D values for the different optimized heterodinuclear Mn^IIIGa^III structures are therefore consistent with a Mn^III ion in a N₃O₃ environment.

Table 2 Local zero field splitting parameters D (cm⁻¹) and E/D, spin–spin D_SSC (cm⁻¹) and spin orbit D_SOC (cm⁻¹) contributions to D, and Euler angles (°) for the manganese atoms from complexes 1a, 1b and 1^opt. For each centers the parameters were calculated with the BP86, SA-CASSCF and NEVPT2 methods by substituting one of the Mn^III atoms by a Ga^III one. For 1^opt the calculated values are the same for the two centers due to the imposed C₂ symmetry

Complex	Center	Method	D	E/D	DSSC	DSOC	α	β	γ
1a	Mn1	BP86	2.39	0.03	0.37	2.02	−21	27	1
		SA-CASSCF	4.14	0.02	0.46	3.67	−28	28	0
		NEVPT2	4.76	0.02	0.46	4.30	−28	28	0
	Mn2	BP86	2.31	0.12	0.36	1.95	43	27	180
		SA-CASSCF	4.02	0.12	0.45	3.56	46	28	180
		NEVPT2	4.60	0.11	0.45	4.14	46	28	180
1b	Mn3	BP86	2.17	0.23	0.34	1.83	−51	27	2
		SA-CASSCF	3.72	0.24	0.43	3.29	−53	28	1
		NEVPT2	4.23	0.22	0.43	3.80	−52	28	1
	Mn4	BP86	2.17	0.22	0.34	1.83	50	27	178
		SA-CASSCF	3.84	0.21	0.43	3.41	52	27	178
		NEVPT2	4.39	0.19	0.43	3.96	52	27	177
1^opt	Mn	BP86	2.15	0.20	0.35	1.80	−51	27	3
		SA-CASSCF	3.83	0.21	0.44	3.39	−52	28	3
		NEVPT2	4.29	0.19	0.44	3.85	−52	28	3

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In addition to the magnitude of the D tensor components, the calculations also provided the orientation of their eigenframe with respect to the molecular frame, which can be expressed a set of three Euler angles, α, β and γ, summarized in Table 2. Interestingly the angle α, which specifies the orientation of the x and y principal axes of the ZFS tensor with respect to the molecular frame, presents a large variation among the metal centers, as can be seen more clearly from Fig. 1. Most probably this can be related to the particular local coordination environment of each Mn^III ion in 1 that leads to principal axes of the anisotropy tensor that are not aligned along the metal–ligand bond, in contrast to Mn(H₂O)₆, in which it is the case. No attempts were made to further investigate the origin of this difference for both the magnitude or the orientation of the D tensor. Indeed, in our previous systematic study on mononuclear Mn^III complexes, we have shown that the SA-CASSCF method permits an excellent prediction of D,³⁹ we thus deemed that the values obtained for the symmetric dimer can be considered as a reasonable guess for our fitting procedure, while restraining all parameters within the limits of the data calculated from X-ray geometries. As previously emphasized this considerably reduces the parameter space employed in the spectral simulations which is an essential ingredient of our approach.

Calculation of the exchange coupling constant. It has been experimentally measured that the magnetic exchange interaction in complex 1 is weakly ferromagnetic, J = +4.6 cm⁻¹.⁴⁰ The exact magnitude of J given in this previous study has to be taken into caution because the method used to fit the magnetic data has not been detailed. From a theoretical point of view, J can be readily calculated by employing the broken symmetry formalism developed by Noodleman. The two energy values and the two spin expectation values that enter the Yamaguchi formula can either be all calculated using the high spin optimized geometry or they can individually be calculated from the high spin and broken symmetry optimized geometries, i.e. an “adiabatic” approach. In the present study, both approaches were employed. For the broken-symmetry computations, we have used the TPSSh functional, which has shown the smallest deviation from experiment for the calculation of the exchange coupling constants for a series of manganese dimers.⁷⁸ Additionally, during the optimization procedure, we made use of the latest van der Waals correction of Grimme's group, which has shown to improve the quality of the calculated J values for the “adiabatic” approach.⁷⁹

At the chosen theoretical level, both approaches give a ferromagnetic complex, with J equal to 8.5 cm⁻¹ and 15.1 cm⁻¹ for the “single geometry” and the “adiabatic” approach, respectively. Given the previously reported errors of approximately 30 cm⁻¹ for the calculation of J,⁷⁸ the present results are quite satisfactory. Interestingly, previous calculations on dinuclear Mn^III complexes that present the same bridges (μ-O, 2μ-OAc) were unable to predict the correct sign for the coupling constant, most probably owing to the fact that J is very sensitive to structural parameters (vide infra).^80,81

Inspection of the pairs of magnetic orbitals, resulted from a corresponding orbital transformation, offers a pictorial representation of the exchange interactions in the system. The subset of the magnetic orbitals for the current system gives rise to four superexchange pathways, one with significantly larger overlap integral (S_αβ = 0.206) (Fig. 5). In line with previous studies,⁸² we notice that the μ-O bridge facilitates the magnetic coupling mainly through an out-of-plane π interaction, while the μ-OAc bridge impedes the superexchange interaction.

Fig. 5 Magnetic orbital pairs of 1^opt and their overlap integral.

Given the nature of the superexchange pathways, it is expected for the coupling constant to be sensitive to structural modifications involving the μ-O bridge, i.e. the Mn–O_oxo distance and Mn–O_oxo–Mn angle. Indeed, the nature of the coupling constant in oxo-bridged manganese(III) dinuclears varies from weakly ferromagnetic, as in the case of the current complex (r(Mn–O_oxo) = 1.80 Å, ∠(Mn–O_oxo–Mn) = 122°), to strongly antiferromagnetic, J = −120 cm⁻¹, in a quasi linear dinuclear (r(Mn–O_oxo) = 1.76 Å, ∠(Mn–O_oxo–Mn) = 168°).⁸³ In order to evaluate the magneto-structural correlations for such Mn^III dinuclears, we calculated the coupling constant for a series of geometries in which only the position of the bridging oxygen atom was moved along the C₂ axis of the complex. The plot of J as obtained for the “single geometry” approach versus δ, which represents the ratio of the Mn–O_oxo–Mn angle and Mn–O_oxo distance for the different optimized geometries, is presented in Fig. 6. The magnitude of J is smoothly reduced as δ becomes larger, which eventually means that a shorter Mn–O_oxo distance or a larger Mn–O_oxo–Mn angle tends to decrease the magnitude of the ferromagnetic coupling, in agreement with the experimental data. Interestingly, for the largest value of δ, which corresponds to an Mn–O_oxo–Mn angle of 123.9° and an Mn–O_oxo distance equal to 1.798 Å, the calculated J values pass into the antiferromagnetic regime (Table S2, ESI†). In light of these results, it becomes evident that the coupling constant is highly sensitive to small structural variations, and hence the theoretical results cannot hope to lead to better than qualitative agreement with experiment. However, the structural differences between the two dimers observed in the X-ray structure are not sufficient to lead to J values, which would cause the splitting of EPR lines, but they may explain in part the significant width of the experimental EPR lines.

Fig. 6 Representation of the variation of the magnetic exchange coupling interaction J as a function of the δ parameter, defined in the text, for various geometries derived from 1^opt.

4.3. Analysis and simulation of the HF-EPR spectra by diagonalizing the full Hamiltonian

In the case of complex 1, the appropriate spin Hamiltonian operator is described by eqn (1). Since the predicted ZFS values of each Mn^III ion are of the same order of magnitude as the isotropic magnetic exchange interaction, the simulations were performed by diagonalization of the full 25 × 25 complex Hamiltonian matrix. Since all interactions that enter the spin Hamiltonian must be expressed in the same coordinate system, the tensors that parameterize these interactions had to be rotated into a common frame. We chose the axes of the dipolar part of the D tensor which, in the present case, are related to the molecular geometry, i.e. the D_zz component is aligned with the vector that connects the two metal centers, and D_yy is perpendicular to the (Mn, Mn, O_oxo) plane.

Initially, the parameters used for simulating the HF-EPR spectra given in Table 3 are: (i) the D_Mn and E_Mn values and the Euler angles reflecting the relative orientation of the ZFS tensor for each Mn^III ion calculated with NEVPT2 corrected SA-CASSCF, (ii) the isotropic exchange interaction experimentally measured, and (iii) the dipole–dipole interaction described by the eigenvalues of the D tensor calculated from eqn (3). Concerning the description of the magnetic interaction between the two Mn^III ions (eqn (2)), the spin–orbit contribution to its symmetrical part and the antisymmetric exchange described by Dzyaloshinskii–Moriya have been neglected since these contributions should be weak in magnitude with respect to the dipole–dipole interaction, which in turn is the smallest contribution among the ones included. Furthermore, we have chosen to use the experimental J value, which corresponds to the average of the J values corresponding to the two dimers characterized by X-ray diffraction present in the sample. High-order anisotropy terms to describe the Mn^III ions have not been considered to avoid over-parameterization. Besides, their magnitude should be negligible in energy compared to the other contributions considered in the simulation process.

Table 3 Initial ZFS (cm⁻¹) parameters and Euler angles (°) used for the fitting procedure and resulting parameters to simulate the HF-EPR spectra of 1 using the full spin Hamiltonian approach (25 × 25 matrix)

	DMn	EMn/DMn	α	β	γ
Initial	4.29	0.19	−52	28	3
Fitted	4.50	0.07	−35	23	2

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Although the resulting simulated spectra reproduce the overall shape of the experimental data (Fig. S1, ESI†), an optimization of the set of parameters was performed. The fitting procedure requires a qualitative analysis of the spectra, performed based on the energy diagrams calculated with the initial parameters (Table 3). Several transitions have been considered, namely the Zeeman levels that are most populated at low temperature under the magnetic field along the x and y axes (identified in Fig. 2a). Notice that since in complex 1 the sign of D is positive, the fundamental transitions along x and y are expected to be located at the lower field compared to the center of the spectra (g = 2).⁸⁴ In contrast, at low temperature, with the magnetic field range used in the present HF-EPR experiments, the fundamental transition along the z axis cannot be observed.

During the fitting process, we noticed that, as expected, the position of the transitions is sensitive to the ZFS values of the Mn^III ions. However, the transition positions are also noticeably dependent on the Euler angle α, thus highlighting the importance of properly describing the relative orientation of the local anisotropy tensors of each metallic ion. Conversely, the interaction parametrized by D_AB is too weak to have a noticeable effect on the simulated spectra (Fig. S2, ESI†).

To assess the quality of a simulation, two criteria have been considered, namely the transition position and the evolution of its intensity as a function of temperature. Fig. 2b, 3b and 4b present simulated HF-EPR spectra at different frequencies and temperatures using the best set of fit parameters given in Table 3. The simulated spectra reproduce the overall shape of the experimental data well. The only exception is one broad feature located at 5.0 T, 6.6 T and 8.5 T at 230 GHz, 285 GHz and 345 GHz, respectively. All of our attempts to reproduce or to interpret this feature have failed and thus it has not been taken into account in our analysis. At low temperature, the first transitions along the y and x axes (1x and 1y in Fig. 2a) are at 3.88 and 7.62 T, respectively. As soon as the temperature increases, their intensity decreases for the benefit of the features at 5.36 and 7.92 T assigned to the second transitions along the y and x axes, respectively. From the energy diagrams (Fig. S3 and S4, ESI†) calculated with the fitted parameters (Table 3), the main features of the spectra can then be attributed as far as transitions between “pure” Zeeman, namely with a magnetic field above 3.5 T, are concerned. In addition, at 230 GHz, it was possible at 5 K to observe a z transition at 10.2 T (Fig. 3a and b), which is nicely reproduced in the simulated spectra, confirming the accuracy of the optimized parameters. The sign of D_Mn has been also confirmed. A negative D_Mn-value leads to HF-EPR spectra inconsistent with the experimental data, while a positive value allows to nicely reproduce the evolution of the intensity of the transitions as a function of temperature.

4.4 Analysis and simulation of the HF-EPR spectra using the spin ladder approach

In general, the spin ladder approach should be used only if the isotropic magnetic exchange interaction is the dominant parameter, i.e. |J| ≫ |D_Mn|. Although it is not the case in complex 1 (J ≈ D_Mn), we will use this approximation to conclude the pertinence of this description in such a case.

Complex 1, in which the two high spin Mn^III ions (S = 2) are coupled via a ferromagnetic interaction, is characterized by a ground spin state S = 4, implying that, in the spin ladder description, the EPR transitions observed at low temperature arise from transitions belonging to the S = 4 spin state. From eqn (8), the EPR parameters of the S = 4 spin state can be calculated using the D_Mn, E_Mn and D_AB values predicted by the theoretical study. Three sets of ZFS parameters for S = 4 (D₄ and E₄/D₄) are then obtained (Table 4), leading to close values (less than 5% of difference for D₄). These parameters have been used to simulate the HF-EPR spectra and to calculate the energy diagrams of the nine Zeeman levels of the nonet along the three magnetic axes in order to give access to an assignment of the different transitions on the experimental spectra. To perform a fit of the experimental data, the first three transitions (|4,−4〉 → |4,−3〉; |4,−3〉 → |4,−2〉; |4,−2〉 → |4,−1〉) belonging to the y axis have been taking into account using their field position at 285 and 345 GHz (3.88, 5.36 and 6.99 T/5.80, 7.31 and 9.04 T, respectively). The values that have been obtained for D₄ and E₄/D₄ given in Table 4 are close to initial values and lead to spectra that well-simulated the low-temperature data in agreement with the fact that only the S = 4 ground spin state is involved. When increasing temperatures, new transitions are observed whose origin does not belong to the S = 4 state. To interpret the appearance of these features, the first excited S = 3 spin state has to be taken into account. By the same procedure, the D₃ and E₃/D₃ values have been determined (Table 4). Spectra calculated with the appropriate weight for each spin state (using the experimental J value) as a function of the temperature are shown in Fig. 2c and 4c. While the low temperature experimental data were quite well reproduced, it is not the case for the data recorded at higher temperatures. To improve the simulation, we adjusted both the ZFS parameters of the S = 3 spin state and the weight of each spin state in a reasonable range of values, but the resulting spectra did not lead to a better agreement with the experimental data.

Table 4 ZFS parameters of the S = 4 and S = 3 parameters extracted from eqn (8) used as initial parameters for the fitting procedure and resulting parameters to simulate the HF-EPR spectra of 1 using the spin ladder approach

	Initial param.^a		Fitted param.
a Three sets of ZFS parameters for S = 4 have been obtained using eqn (8). See the text for more details.
D4	1.791	1.862	1.862	1.540
E4/D4	0.195	0.175	0.200	0.107
D3	0.792	0.891	0.891	0.770
E3/D3	0.206	0.146	0.220	0.200

---

5. Discussion and conclusion

In the present manuscript, we report on the determination of the electronic structure of a dinuclear Mn^III compound, characterized by large magnetic anisotropy and weak ferromagnetic interaction between the two metallic ions. Two methods have been considered to analyze the multifrequency HF-EPR data in order to extract the local spin Hamiltonian parameters. In the first one, a complex Hamiltonian has been employed that includes terms describing the local and inter-site interactions. This method involves the evaluation of all matrix elements of the spin Hamiltonian in the basis set of uncoupled spin states. The resulting 25 × 25 complex matrix was subsequently diagonalized in order to obtain all its eigenvalues, i.e. energy levels. This should be appropriate whatever the magnitude of the exchange coupling constant J, and in the present case allows for an excellent interpretation of all HF-EPR spectra recorded in a large range of temperatures.

In addition, the spin ladder approach has been investigated. For the present system it is only successful for the analysis of the experimental data that were recorded at the lowest temperatures. Nevertheless as soon as the temperature increases, it fails since the transitions arising from the excited state manifolds overlap with the ones from the ground state. Under these circumstances, only the parameters of the ground spin state of 1 can be precisely determined, which means that the determination of the local parameters of the metal centers is impossible. Thus, our investigation clearly shows that although the spin ladder approach is probably the more convenient and straightforward, it cannot be applied with confidence to the case of weakly exchanged coupled polynuclear complexes.

The zero field splitting parameter (D_Mn) of the Mn^III ion obtained from fitting the experimental spectra was +4.50 cm⁻¹, consistent with a N₃O₃ environment in a compressed octahedral geometry.^4,72,74–76 This value does not deviate significantly from our best estimate obtained from theoretical calculations, despite the fact that we relied on a pragmatic and rather approximate method for its calculations, i.e. by substituting one of the Mn^III atoms with a Ga^III. Therefore, the local ZFS values in a polynuclear complex can be in principle evaluated via theoretical calculations by considering each metallic center separately.

Finally we also found that the prediction of the exchange coupling remains very challenging in the case of weakly coupled systems. At most, one can expect to rationalize a trend for a series of polynuclear complexes, while an accurate prediction of the absolute value of the Heisenberg coupling constant cannot be expected, especially in the framework of density functional theory. In the special case of dinuclear μ-oxo Mn^III complexes, it has been proposed that the J value is sensitive to the nature of the octahedral distortion around the Mn^III ions: for an octahedron that is compressed in the direction of the oxo-bridge a ferromagnetic interaction should be expected between the Mn^III ions, while for an elongated octahedron, depending on the rhombicity of the system, antiferro- or ferro-magnetic interactions could result.^85–87 In addition, recent studies reported by the group of Corbella have shown that the π- or σ-acid character of the ligand can also affect J.⁸⁸ However, this correlation appears to be invalid for a nearly linear dinuclear μ-oxo Mn^III complex, characterized by an antiferromagnetic coupling (J = −120 cm⁻¹) in which the two metallic ions display compressed octahedral geometry.⁸³ Based on the nature of the superexchange pathways evidenced in complex 1, the present work demonstrates that another important structural factor that can notably influence the nature and the magnitude of J is the Mn–O_oxo–Mn angle together with the Mn–O_oxo distance. A systematic study taking into account all available experimental data would be necessary to define the minimal set of structural parameters required to predict J.

The present work clearly shows how fruitfully experimental and theoretical techniques can be combined. In the present case the theoretical calculations proved to be an essential tool to predict very good initial values of all EPR parameters and their eigenframes. This was necessary in order to arrive at successful fits to the experimental data. The importance of previous systematic studies in order to determine the appropriate methodology to calculate the ZFS of the metallic ion is also highlighted. Work is in progress in our groups to generalize this approach to other dinuclear complexes with different metallic ions or Mn ions at various oxidation states.

Acknowledgements

MR, MNC and CD thank the Agence Nationale pour la Recherche (Grant No. ANR-05-JCJC-0171-01) for financial support.

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Footnote

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c2cp42955a

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