Air-stable redox-active nanomagnets with lanthanide spins radical-bridged by a metal–metal bond

Engineering intramolecular exchange interactions between magnetic metal atoms is a ubiquitous strategy for designing molecular magnets. For lanthanides, the localized nature of 4f electrons usually results in weak exchange coupling. Mediating magnetic interactions between lanthanide ions via radical bridges is a fruitful strategy towards stronger coupling. In this work we explore the limiting case when the role of a radical bridge is played by a single unpaired electron. We synthesize an array of air-stable Ln2@C80(CH2Ph) dimetallofullerenes (Ln2 = Y2, Gd2, Tb2, Dy2, Ho2, Er2, TbY, TbGd) featuring a covalent lanthanide-lanthanide bond. The lanthanide spins are glued together by very strong exchange interactions between 4f moments and a single electron residing on the metal–metal bonding orbital. Tb2@C80(CH2Ph) shows a gigantic coercivity of 8.2 Tesla at 5 K and a high 100-s blocking temperature of magnetization of 25.2 K. The Ln-Ln bonding orbital in Ln2@C80(CH2Ph) is redox active, enabling electrochemical tuning of the magnetism.

Resolution in linear mode is not high enough for analysis of isotopic distribution. In the reflector mode, strong fragmentation does not allow for detection of molecular peak, but spectral resolution is sufficient to prove correct isotopic distribution of the Gd2@C80 − fragment. In the negative ion mode, strong fragmentation does not allow for detection of molecular peak, but spectral resolution is sufficient to prove correct isotopic distribution of the Y2@C80 − fragment.

Supplementary
Supplementary Figure 10. Matrix-assisted laser desorption/ionization time-of-flight (MALDI TOF) massspectra of TbY@C80(CH2Ph). Linear negative (top) and positive (bottom) ionization modes. Resolution in positive mode is not high enough for analysis of isotopic distribution. In the negative ion mode, strong fragmentation does not allow for detection of molecular peak, but spectral resolution is sufficient to prove correct isotopic distribution of the TbY@C80 − fragment.    Supplementary Figure 17. Matrix-assisted laser desorption/ionization time-of-flight (MALDI TOF) massspectra of Ho2@C80(CH2Ph). Linear negative (top) and positive (bottom) ionization modes. Resolution in positive mode is not high enough for analysis of isotopic distribution. In the negative ion mode, strong fragmentation does not allow for detection of molecular peak, but spectral resolution is sufficient to prove correct isotopic distribution of the Ho2@C80 − fragment.

Supplementary
Supplementary Figure 18. Separation of Er2@C80(CH2Ph). Three steps HPLC separation were required to obtain the pure compound (I) HPLC profile of the mixture of benzyl-derivatized Er-EMFs. The highlighted fraction contained Er2@C80(CH2Ph) was collected for further separation. HPLC conditions: linear combination of two 4.6 × 250 mm Buckyprep columns; flow rate 1.6 mL/min; injection volume 800 µL; toluene as eluent; 40 °C. (II) Recycling HPLC separation of the fraction collected in the first step. The highlighted fraction was collected for further separation (10 × 250 mm Buckyprep column; flow rate 1.5 mL/min; injection volume 4.5 mL; toluene as eluent). (III) HPLC separation of the fraction collected in the second step. Pure Er2@C80(CH2Ph) was obtained as the highlighted fraction. (10 × 250 mm Buckyprep-D column; flow rate 1.0 mL/min; injection volume 4.5 mL; toluene as eluent).
Supplementary Figure 19. Matrix-assisted laser desorption/ionization time-of-flight (MALDI TOF) massspectra of Er2@C80(CH2Ph). Linear negative (top) and positive (bottom) ionization modes. Resolution in positive mode is not high enough for analysis of isotopic distribution. In the negative ion mode, strong fragmentation does not allow for detection of molecular peak, but spectral resolution is sufficient to prove correct isotopic distribution of the Er2@C80 − fragment.
Supplementary Figure 20. HPLC profiles of the purified M2@C80(CH2Ph). HPLC conditions: linear combination of two 4.6 × 250 mm Buckyprep columns; flow rate 1.6 mL/min; injection volume 800 µL; toluene as eluent; 40 °C. The identical retention times for TbxGd2-x@C80(CH2Ph) indicates that the separation of these compounds by HPLC is an impossible mission.

Supplementary Figure 21.
Air stability: HPLC trace of the freshly synthesized {Tb2} and 8 months after the synthesis (during this time period the sample was studied in air by SQUID magnetometry and NMR spectroscopy and stored in solution).  Figure 22. Molecular structure of Dy2@C80(CH2Ph) determined with single-crystal X-ray diffraction at temperatures from 100K to 290K. The thermal ellipsoids were shown with 50%. The hexagon belt of the C80 fullerene cage where the Dy2 locates is highlighted with yellow bonds. Color code: grey for carbon, green for Dy, the deeper the green color, the higher occupancy of the Dy site. Figure 23. Molecular structure of Dy2@C80(CH2Ph) determined with single-crystal X-ray diffraction at temperatures from 100K to 290K. The Dy sites are shown as spheres whose radii are scaling proportional to the site occupancy (the bigger the sphere, the higher the occupancy).

Supplementary Table 1 (continued). Crystal data and data collection parameters
Dy2@Ih (7) .   b Electrochemical gap is defined as the difference between the first oxidation and the first reduction

Supplementary Note 2. Axiality of the ligand field in {Ln2} molecules
As follows from ab initio calculations (Supplementary Tables 8-11), single-ion magnetic anisotropy in {Ln2} is rather high. For instance, the total LF splitting for Tb ions is ca 1000 cm −1 , and the energy of the first excited KD state is ca 260 cm −1 . Ligand field is indeed highly axial, resulting in high-spin ground states of Ising type.
The axiality of the ligand field in {Ln2} may have several reasons. First, metal atoms transfer their valence electrons to the fullerene cage, resulting in accumulation of the negative charge on carbon atoms coordinated to the endohedral lanthanide ion. Note that this interaction also has considerable covalent contribution via overlap of π-electron density of the fullerene with vacant d-orbital of the lanthanide. Next, covalent Ln-Ln bonding results in a concentration of the electron density between two Ln ions. In Ref. 1 we used a simple point-charge model to show that even relatively small negative charge located between two lanthanide ions may induce rather high axial magnetic anisotropy. Thus, metal-metal bond is important not only for exchange interactions, but also to support the axial ligand field. Finally, lanthanide ions in EMFs have no "equatorial" ligands -the situation which also facilitates imposing of the axial ligand field. Here, the Stevens factors , and are rational numbers depending on , , and and describe the angular shape of the 4f charge distribution, < > are the expectation values of taken with the radial 4f wave function, (i) are the CF coefficients describing the charge distribution around the 3+ ion at site i, and ̂ are the standard Stevens operators expressed in polynomials of ( , , , ). 14 For the axial parameters, = 0, the parameters B(l,0) given in Table S8 Table 8 shows that states with different hardly mix for Tb ions (and similar was observed for Dy in Ref. 1 ). Thus, the LF Hamiltonian is almost diagonal for the Tb and the Dy ions, i.e., only the axial LF parameters are relevant. We evaluated the LF levels of the Tb and Dy ions using only the axial LF parameters and assuming the non-axial parameters to be zero. The resulting levels (curves denoted as axial 2, 4, 6) are compared with the levels obtained by our ab-initio calculations (crosses) in Supplementary  Fig. 43. As expected for the case of small mixing, the diagonal LF terms describe the ab-initio levels very well.

Supplementary Note 3. Magnetic anisotropy and ligand-field states of different lanthanide ions in similar ligand-field environments
In order to estimate the relative importance of the individual axial terms (l = 2, 4, 6), we repeated the calculation with fourth-order term set to zero (curves denoted as axial 2, 6) and finally with both fourthand sixth-order terms set to zero (curves denoted as axial 2). For the Tb ions we find that the quantum chemical data are very well described by considering only the second-order axial LF term; the fourth order provides a minor correction and the sixth order has no influence that would be visible at the scale of the plot. For the Dy ions as well, the second order alone describes the levels reasonably well. However, a minor correction is provided by the fourth-order term and a somewhat larger correction by the sixth order.
The relatively larger influence of the sixth-order term in Dy as compared to Tb can be understood from a combination of two factors. First, the sixth-order term gains a larger importance in Dy due to the larger Jvalue of its ground state multiplet, = 15/2 , compared to = 6 for Tb. This value enters the expectation values of the Stevens operators in the power of l, providing an enhancement by a factor of 2.4 for the importance of order six vs. order two. Second, a factor of modulus 1.5 in favor of the Dy sixth-order contributions is gained by the ratio of the Stevens factors / , see Supplementary Table 12. In a very similar manner one can understand that the fourth-order term has more or less the same influence on the Tb and the Dy ions. Here, a factor of 1.6 due to the different J-values is partly compensated by a factor of 0.8 due to the ratio of the Stevens factors / , see Supplementary Table 12. Summarizing the discussion for the Tb and the Dy ions, the whole LF spectrum and the LF states of Tb 3+ in {Tb2} can be described by a single parameter B(2,0) and the same holds, with a grain of salt, for Dy 3+ in {Dy2}. The sign of this parameter determines, whether the magnetic anisotropy is of uniaxial type, i.e., the ground state is |Jz| = J, or if it is of easy-plane type with a ground state |Jz| = 0 or ½. For the two discussed ions, B(2,0) is negative, implying a uniaxial ground state. The negative sign of B(2,0) for both cases comes about by the same, positive sign of A20 for both systems combined with the same, negative sign of the related Stevens factors.
Turning to the case of the Ho 3+ ions, it can be noted that the second-order Stevens factor αJ is negative as in Tb 3+ and Dy 3+ . Thus, given the same sign of A20 as in the other cases, also B(2,0) is negative and Ho has a high-spin ground state like the previous cases. However, the effect of the sixth-order axial crystal field term is much stronger than in Tb or Dy, since the ratio / provides a factor of 5.3 (as compared with Tb) and the larger J = 8 provides a factor of 3.2. Calculation of the LF levels from the axial terms alone (not shown) shows that the high-spin ground state is (just) not spoilt by the higher-order terms. However, by virtue of the larger pre-factors, the non-axial sixth-order LF parameters now produce a strong mixing of different Jz states, see Supplementary Table 9. This mixing is present even in the ground state. It is responsible for an important contribution to transitions between (quasi) degenerate states. The importance of fourth-order terms is slightly enhanced, if compared to Tb and Dy, but their impact is much smaller than that of other terms.
Finally, the second-order Stevens factor of Er 3+ is positive, opposite to all the other cases. Thus, an easyplane ground state is realized in Er which is not compatible with SMM behavior. The influence of sixthorder contributions to the LF Hamiltonian of Er 3+ in {Er2} is similar to the case of Ho, since the ratio of the Stevens factors / is somewhat larger in Er 3+ than in Ho 3+ , while the value of J = 15/2 is smaller than in Ho. Fourth-order terms have the same, minor importance as in the other systems.
Summarizing all the considered cases, the LF ground states for {Ln2} are solely determined by the sign of the related second-order Stevens factors, encoding the different shapes of the Ln-4f shells. Moreover, it is possible, in a decent approximation, to describe all LF spectra by one single parameter A20, which is determined by the quadrupolar potential at the Ln sites, and appropriately scaled by the radial expectation value of the specific Ln-4f wave function. A refined description, which is needed to account for mixing of the pure Jz states in the cases of Ho and Er, however, has to include all sixth-order terms.
The dominance of the axial second-order LF term can be attributed to the large, almost axially distributed charge on the Ln dimers, including the single-electron σ-bond. The less important, though still significant for the Ho and Er systems, sixth-order contributions (axial and non-axial) can be attributed to the charge distribution on the carbon cage. This means that a transfer of the Ln dimers to other kinds of carbon cages will influence mainly the less important sixth-order terms and keep the present results qualitatively unaffected.

Supplementary Note 4. Spectra of Model Hamiltonians
Although the simple Ising-type Hamiltonian (Supplementary Equation 1) gives insights mostly in the ground state, the formal analysis of the complete Hamiltonian (Supplementary Equation 1) spectra can be made, and transition probability between different states i and f can be estimated as (see code PHI 10 for details):  Equation 1) of 55 cm −1 one can estimate the barrier for the effective Orbach process, through the semi-degenerate exchange-excited doublets to be of 555.8 and 558.5 cm −1 (~800 K). For the non-collinear system, the Hamiltonian S1 is not expected to be fully adequate, but still, the formal spectrum and transition probabilities can be computed and the effective Orbach process barrier can be estimated. For the eff value of 40 cm −1 , there is a state with the energy of 259.6 cm −1 (374 K), which has an almost perpendicular orientation of the main anisotropy axis with respect to the ground state. Experimentally, an Orbach barrier of ~330 K is found for {Ho2}.

Magnetic properties
Magnetic susceptibility in the further discussion is defined as χ = M/H both in experiment and in theory. Note that in high field the M/H ratio is significantly different from differential susceptibility defined as a derivative of magnetization with respect to the external field.
Simulations of magnetic properties are based on the spin Hamiltonian (all calculated curves are powderaveraged): ̂s pin ({Ln 2 }) =̂1 +̂2 − 2 eff̂(̂L n 1 +̂L n 2 ) where ligand field parameters in ̂ are obtained from ab initio calculations, and exchange constant K eff is the only unknown parameter. The value of K eff is varied to find the best fit to the experimental data.
For {TbY}, the spin Hamiltonian is reduced to the following form: where ligand-field parameters in ̂T b are obtained from ab initio calculations. In the fitting of the magnetic data we used the followed strategy: χT curve measured for a given {Ln2} compound was fitted for only one value of the magnetic field, 1 T, to determine K eff . This K eff value was then used to calculate χT curves for other values of the magnetic field (3, 5, and 7 T) as well as to calculate magnetization curves at different temperatures. The agreement of the model and experiment is considered to be good, when a single K eff value can give a good agreement for the whole set of experimental data.
Due to the small mass of the sample, reliable estimations of the absolute magnetization values are possible only for {Dy2} (ref. 1 ), {Er2} and {Ho2}, which show the highest yield in the synthesis. For other sample uncertainties in the mass determination introduced during the sample preparation are too large to allow accurate determination of the absolute magnetization values. We therefore use arbitrary units for these {Ln2} molecules

DFT calculations
We used the VASP code, version 5.0, to optimize the molecular structures at the PBE-D level using PAW pseudopotentials with standard cutoffs as recommended. 3,4,5,6 The 4f shells of the lanthanide elements do not contribute to chemical bonding. Thus, we included the 4f shell in the core potential, i.e., we used the so-called open-core approximation (here, implemented as unpolarized potential). The SCF calculations accounted for spin polarization of the valence states. This procedure is expected to provide realistic results for structures involving Ln ions. 15 The pseudopotential configuration 5p 6 6s 2 5d 1 was used for all Ln atoms. All molecular structures were optimized such that the residual forces for all atoms were below 10 -4 eV/Å.