Single molecule magnet with an unpaired electron trapped between two lanthanide ions inside a fullerene

Increasing the temperature at which molecules behave as single-molecule magnets is a serious challenge in molecular magnetism. One of the ways to address this problem is to create the molecules with strongly coupled lanthanide ions. In this work, endohedral metallofullerenes Y2@C80 and Dy2@C80 are obtained in the form of air-stable benzyl monoadducts. Both feature an unpaired electron trapped between metal ions, thus forming a single-electron metal-metal bond. Giant exchange interactions between lanthanide ions and the unpaired electron result in single-molecule magnetism of Dy2@C80(CH2Ph) with a record-high 100 s blocking temperature of 18 K. All magnetic moments in Dy2@C80(CH2Ph) are parallel and couple ferromagnetically to form a single spin unit of 21 μB with a dysprosium-electron exchange constant of 32 cm−1. The barrier of the magnetization reversal of 613 K is assigned to the state in which the spin of one Dy centre is flipped.

Supplementary Figure 22. Determination of blocking temperature TB. The sample is first cooled in zerofield (ZFC) to 1.8 K, then χ is measured in the field of 0.2 T with increasing temperature (solid curve), then the measurement is performed at cooling down to 1.8 K (dashed curve). The vertical bars denotes TB values determined with different temperature sweep rate. Figure 23. Determination of magnetization relaxation time from the magnetization decay curve. For the measurement at 0.4 T, magnetization does not decay to zero. To determine equilibrium magnetization (y0), the second measurement, growth of magnetization for a zero-field sample, was used and fitted together with the decay curve. The figure shows the curves measured for T = 7 K. Dots are experimental data, lines are fits with stretched exponential function.    (1) and (2) and contribution of individual mechanisms (Raman, two Orbach processes, and QTM-like temperature-independent regime, which is available only for zero-field measurements). Note that ac-measurements gave virtually identical values for zero-field and the field of 0.4 T.  (1) and (2) and contribution of individual mechanisms (Raman, Orbach processes, and QTM-like temperature-independent regime, which is available only for zero-field measurements).

Supplementary Note 1. Electron paramagnetic resonance (EPR) spectroscopy of Y-DMF extract and benzyl adducts
Supplementary Figure 2 shows the EPR spectrum of the Y-DMF extract in DMF at room temperature. The spectrum reveals the presence of several types of paramagnetic species. The strongest signal with g-factor of 2.0005 presumably corresponds to the anon radicals of empty fullerenes and/or oxidized forms of DMF. Besides, three triplet signals with hyperfine coupling constants of 65-75 G can be well seen. Their g-factors (determined from the position of the central peak in each triplet) are 1.9814, 1.9770, and 1.9744, and corresponding hfc constants are 64.5, 72.1, and 76.2 G. These combinations of g-factors and hfc constants result in coinciding position of the first peak of each of the three triplets. However, the second and the third components of the triplet signals are found at different fields. Due to the strong overlap, it is difficult to estimate the ratio of the peaks for each of the triplet. The ratio of the net integrals of the three triplets considered together is close to 1:2:1 as might be expected for Y2@C2nanion radical with two equivalent 89 Y atoms (nuclear spin 1/2).
Reaction of Y2@C2n − anion radicals with benzyl bromide leads to a mixture of Y2@C2n(CH2Ph) derivatives. Note that each of the three major Y2@C2n − anions present in the DMF extract before the reaction may give several isomers of the benzyl adducts. Besides, Y atoms in such adducts may be nonequivalent. Combination of these two factors leads to a complex EPR spectrum of the mixture of Y2@C2n(CH2Ph) derivatives (Supplementary Figure 3). Despite the complexity, the spectrum clearly shows that Y2@C2n(CH2Ph) derivatives do have large 89 Y hfc constants of 65-80 G, which proves that the singleoccupied Y-Y bonding MO is preserved in Y2@C2n(CH2Ph) derivatives. Note also that the sharp peak near 3490 G indicates that non-Y organic radical are also formed in the reaction. However, it is not practical and not essential for this work to give a detailed interpretation of the whole complex spectrum; subsequent HPLC separation of this mixture gave the pure Y2-I compound, which has simple triplet spectrum and does not contain any other radical impurities (see Figure 3 in the manuscript).

Dy2-I
Assuming the extinctions coefficients of all the fullerenes at 320 nm are the same, then the amount of the fullerenes is linearly related to the HPLC peak area. The relative yield of Y2-I to all the extracted fullerenes is 0.3001 × 0.4008 × 0.4398 ≈ 5%

Supplementary Note 3. X-ray crystallographic analysis of Dy2-I
For the single crystal growth, about 0.1 mg Dy2-I was dissolved in toluene, hexane was layered over the toluene solution, several single crystals was prepared successfully, the dimension of the picked single crystal for the XRD is 0.03 x 0.03 x 0.01 mm 3 . Crystal data and data collection parameters are summarized in Supplementary Table 1.
The Dy2-I density in the single crystal could be calculated as following: There is some disorder with the Dy atoms, two positions of one Dy atom were refined with site occupancies of 0.6983(17) and 0.3016(17) for Dy1A and Dy1B, respectively; three positions for another Dy atom were refined with site occupancies of 0.657 (2), 0.2910(18) and 0.052 (2) for Dy2A, Dy2B and Dy2C, respectively (see Supplementary Figure 11). Two pairs of Dy positions could be used to evaluate the interaction between the two encapsulated Dy atoms based on the site occupancies. The distances for Dy1A-Dy2A and Dy1B-Dy2B are 3.8965(14) and 3.898(3) Å, respectively. The encapsulated Dy atoms are coordinated to hexagons (highlighted green in Supplementary Figure 11) of the fullerene cage in a quasiη 6 fashion with Dy-C distances in the coordinated hexagon being 2.308(8)-2.586(9) Å.
Supplementary Figures 12 and 13 show the packing of the Dy2-I molecules in the single crystal, indicating that the Dy ions in the crystal are well aligned. Fullerene molecules form quasi-hexagonal layers with an AAA staking sequence. Toluene molecules and benzyl groups occupy the voids between the layers. The closest distances between centroids of fullerene fragments within the hexagonal layer are 10.89/10.97/11.11 Å. The distance between the layers is 10.98 Å, and the shortest distance between centroids of fullerene fragments from different layers is 11.04 Å. Some of the neighboring fullerenes within the layer face each other by the hexagons, coordinated by Dy1A/Dy2A atoms with larger occupancies (~0.7). The distance between centroids of these hexagons is 3.55 Å. The distances between Dy atoms of such neighboring molecules are 7.31/7.33 Å (compare to 3.90 Å for the intramolecular Dy-Dy distance).

Supplementary Note 4. Spectroscopic characterization of Dy2-I and Y2-I
UV-Vis-NIR absorption spectra of EMFs are dominated by π-π* excitation in the fullerene cage. Therefore, the spectra are very sensitive to the fullerene structure, but are almost insensitive to the encapsulated metals. Close similarity of the spectra of EMFs with different metal atoms may serve as a proof of the structural similarity. Absorption spectra of Y2-I and Dy2-I are very similar (Supplementary Figure 16), proving that the two compounds have the same fullerene cage structure. Absorption features are extended to about 1100 nm, which indicates that the M2-I molecules have a relatively large optical gap exceeding 1.1 eV despite their open-shell electronic character. The spectra are also similar to that of La2- Similar to the absorption spectra, vibrational spectra of EMFs are also dominated by the vibrations of the fullerene cage. Therefore, close similarity of the vibrational spectra of two EMFs with different metals shows that the two EMFs have the same fullerene cage structure. Due to large masses of endohedral metal atoms, metal-based vibrations occur at low frequencies (below 200 cm -1 ) and can be detected in Raman spectra. Here the difference in the masses of metal atoms is seen as the shift of the characteristic vibrational bands.
Supplementary Figures 17 and 18 show that the IR and Raman spectra of Y2-I and Dy2-I are virtually identical, which additionally confirms that isolated Y2-I and Dy2-I have the same fullerene cage structure. The only noticeable differences are the Raman bands marked with red arrows, which have noticeably different frequencies in Y2-I (181 cm −1 ) and Dy2-I (148 cm −1 ). They are assigned to the meal-cage stretching vibration, and significant metal-dependent shift is due to the larger mass of Dy (162.5 amu) than that of Y (88.9 amu).

Supplementary Note 5. Magnetic moment of Dy2-I
The sample for the measurements of magnetic properties was drop-casted from CS2 solution into propylene capsule and dried under vacuum overnight. The mass of the sample was determined by the change of the mass of the capsule before and after drop-casting/drying, each mass measurement was performed three times: The weight of the empty capsule: 0.19014 g, 0.19014 g, 0.19013 g; The weight of the capsule with Dy2-I: 0.19101 g, 0.19101 g, 0.19102 g.
The mass of the sample used for magnetic measurements is thus determined to be 0.88 mg with weighing uncertainty of 0.01 mg.
Saturated magnetization of the sample at 2 K in the field 7 T is 0.0372 emu, which gives the magnetic moment of 10.5 μB per molecule. Taking into account that the measurement are performed for a disordered powder sample, the value should be doubled, giving the moment of 20.9 μB per Dy2@C80(CH2Ph) molecule. Uncertainty of the mass of 0.01 mg gives uncertainty of the moment of 0.2-0.3 μB. 20.9 μB is very close to 21.0 μB, theoretical value for the [Dy 3+ -e-Dy 3+ ] system with collinear ferromagnetically coupled moments of two Dy ions and one unpaired electron. In case of antiferromagnetic coupling between magnetic moment of Dy and unpaired spin, the total moment of the molecule would be 19.0 μB.

Supplementary Note 6. Determination of relaxation times from decay curves
Long magnetization relaxation times (> 10 sec) were determined from the measurement of magnetization decay using dc-SQUID. The sample was first magnetized to the saturation at 5 Tesla, then the field was swept as fast as possible to zero or 0.4T, and then the decay of magnetization was followed over several hours. Decay curves could not be described by single exponent and were then fitted using stretched exponential function: , where t0 is the relaxation time and y0 is an equilibrium magnetization at the given field and temperature. For the in-field measurements with very long relaxation times, reliable determination of y0 parameter was crucial and could not be accomplished from the decay curve alone. In such cases, the second curve was measured by cooling the sample first in zero-field, then applying the desired finite field (0.4 T), and then following increase of magnetization with time (Supplementary Figure 23). Both decay and growth curves should end-up in the same magnetization value equal y0, and hence the curves were fitted together. Relaxation times determined in dc measurements are listed in Supplementary Tables 2 and 3.

Supplementary Note 7. Contribution of different relaxation mechanisms to the relaxation of magnetization
Experimentally determined relaxation times were fitted using two equations, one for zero-field data (Equation (1)), another one for the measurements in the field of 0.4 T (Equation (2)). The equations are different only in the absence of the QTM term in Equation (2), which describes the relaxation in a finite field. Parameters of both equations were kept identical during the fit. Resulting fitting curves as well as contributions of individual relaxation processes are shown in Supplementary Figure 27.
We also attempted to describe the whole set of experimental points using only one Orbach process: Zero-field: In-field: The fit of experimental points obtained with Supplementary Equations (1) and (2) is substantially worse than with the use of Equation (1) and (2) in the main text, especially between 10 and 25 K (Supplementary Figure 28). Raman relaxation (C = 3.84•10 −10 s −1 K −n , n = 6.7) now dominates all low-temperature data for the in-field relaxation and is switched to the Orbach regime ( eff U =656 K, τ02=8.9•10 −13 s) above 23 K. Thus, we conclude that the terms included in Supplementary Equations (1) and (2) are not sufficient for the description of the whole set of data.

Supplementary Note 8. Ab initio and point-charge calculations of crystal field parameters
Full ab initio treatment of Dy2-I is not possible at this moment, so we performed calculations for single Dy centers and replaced another Dy ions by Y. Besides, the unpaired valence electron was "quenched" by adding one extra electron to the system. Ab initio energies and wave functions of crystal-filed (CF) multiplets for the [DyY-I]molecule (Supplementary Table 6) have been calculated using the quantum chemistry package MOLCAS 8.0. Single point complete active space self-consistent field with spin-orbit interactions calculations (CASSCF/SO-RASSI level of theory) were done to derive ab initio parameters. In all systems, the Dy(III) has 6 H15/2 ground state multiplet, which results in eight low-lying Kramers doublets. The active space of the CASSCF calculations includes nine active electrons and the seven active orbitals (e.g. CAS (9,7)). All 21 sextet states and only 108 quartets and 100 doublets were included in the stateaveraged CASSCF procedure and were further mixed by spin-orbit coupling in the RASSI procedure. Mixed atomic natural extended relativistic basis set (ANO-RCC) was employed with the minimal basis option for C and H atoms, and VDZ-quality for Y and Dy metals in the cluster. The single ion magnetic properties and CF-parameters (Supplementary Table 7 Tables 8, 9). Crystal field parameters from either ab initio or point-charge model calculations (Supplementary Table  10) were then transferred to the PHI code for the further analysis.
In a modeling with the PHI code, we have been using pseudospin model with total spin S=15/2, which produces eight Kramers doublets for a single center in presence of a CF field. The effective g-factors for the coupled spin multiplet were set 1.33 (assuming the 4f 9 -configuration). Transition probabilities were also computed by using the PHI code.

Supplementary Note 9. Crystal field Hamiltonian and Crystal-field potential
The complete crystal field part of Hamiltonian for each atomic center is defined as: where V W are CF-parameters in Steven's notation and X V , Y Z V W the operator equivalent factors and operator equivalents respectively. In case of a simple linear system [Dy 3+ -qm-(+3)], the CF-Hamiltonian is dominated by the single term with k = 2 and q = 0, H ST = ) , where = K < * > and ) = (3:7, _ − 1).
As the radial integral < * > is positive and for linear alignment (_ = a) ) = 1, the sign of the For two point charges with Q and qm, the total CF-potential proportional to: Visualization of the CF potential in Figure 5d and Supplementary Figure 30 shows that even rather small negative charge placed at the midpoint between Dy 3+ ion and a 3+ point charge outweighs the effect of the positive charge. Supplementary Figure 30 shows the variation of the parameter and the change of its sign with the variation of the negative charge. As changes the sign with the increase of the negative charge, the anisotropy type of the Dy ion changes from easy-plane to easy-axis.
where r n is the normal of the radius vector connecting two magnetic moments 1 µ and 2 µ . According to ab initio and simple point charge calculations, the ground state for each Dy 3+ center in Dy2-I is an easyaxis state with μi = 10 μB (Jz = ±15/2, g = 1.33). The dipolar energy difference between parallel and antiparallel alignment of these moments at the distance of 3.96 Å is 4.0 K (2.77 cm −1 ), which gives 12 dip j = 0.012 cm −1 .
Computations of exchange interactions in Gd2-I and estimation of the dipolar term in Dy2-I show that that the strength of the direct coupling between two metal centers, 12 j , is very small compared to the metal-electron interactions ( , i e j ) and in the first approximation can be neglected.

Supplementary Note 11. Magnetic susceptibility of Dy2-I and spin Hamiltonian parameters
In this section we will simulate χmT curves with different parameters of the crystal field and exchange interactions and compare results of simulations to the experimentally measured (M/B)mT functions. The temperature and field dependence of the (M/B)mT function is shown in Supplementary Figure 32. At low magnetic fields, the function is similar to χmT, but in high fields the difference between the derivative M B ∂ ∂ and the ratio M B is significant. This can be clearly seen in Supplementary Figure 33 In all magnetic fields studied, (M/B)mT shows a sharp increase to ca 55 cm 3 mol −1 K with the increase of the temperature. At higher temperatures, the function decreases slowly reaching ca 43 cm 3 mol −1 K at 300 K. But for the low external fields, the low-temperature part of the curve is disturbed by slow relaxation of magnetization (the kink in the 0.2 T curve corresponds to the blocking temperature). For the fields exceeding 1 T the whole curve is measurable. For this reason, and because in the field of 1 T the (M/B)mT function is still close to the χmT function, in analysis of the Hamiltonian parameters below we will use the (M/B)mT measured in the field of 1 T and compare it to the computed χmT curves.
In the simulations described below, the Dy2-I system was modelled using the following effective spin Hamiltonian (Equation (4) of the main text): ( 1) To determine the plausible values of the jDy,e constant, the χmT curves were simulated with the ab initio calculated CF parameters (see Supplementary Note 8) and with different jDy,e constants from 5 to 40 cm −1 . Supplementary Figure 34 shows that small jDy,e values lead to the sharp peak at low temperatures in the predicted χmT curves. With the increase of the jDy,e constant, the peak is becoming broader and is shifting towards higher temperatures. Reasonable agreement with experimental (M/B)mT measured in the field of 1 T is obtained for the jDy,e constant of 30-35 cm −1 .
To evaluate the effect of the single-ion anisotropy, we fixed jDy,e constant to 30 cm −1 and created different sets of CF parameters using the point-charge model described in section Supplementary Note 8 and in Figure 5d of the main text. In particular, Dy ions were placed in the crystal field created by a positive charge of +3 at the distance of 3.96 Å and with the negative charge at the midpoint between Dy ion and the positive charge. Variation of this negative point charge from −0.4 e to −2.0 e gave a set of easy-axis CF parameters with the increasing CF splitting. Supplementary Figure 35 compares the experimental curve to the results of simulations. The small CF splitting results in the peak of the χmT function at low temperature, which is shifting to higher temperature with the increase of the CF splitting. Good agreement between experiment and theory is obtained when the splitting between the ground state and the first excited CF state exceeds 200 cm −1 . The lower limit of 200 cm −1 agrees well with the results of CASSCF calculations (which give very similar curve to the point charge model with the negative charge of −0.8 e).
To summarize, these simulations show that the shape of the experimental curve requires the exchange constant higher than 30 cm −1 and the crystal field with the splitting of the first two CF states exceeding 200 cm −1 . If any of these parameters are smaller, the χmT function has a well-defined peak at low temperature, which contradicts the experimental data due the presence of the low-energy excited state with lower magnetic moment, whose thermal population decreases χmT.

Supplementary Note 12. Magnetization curves at different temperatures
Magnetic moment of 21 μB per molecule determined from the saturation of magnetization at low temperatures indicates that magnetic moment of Dy ions and of the unpaired electron are coupled ferromagnetically. In case of antiferromagnetic (AFM) coupling between Dy and unpaired electron spin, the total moment per molecule would be 19 μB. An alternative proof of the FM coupling in the [Dy 3+ -e-Dy 3+ ] system can be obtained from the shape of the magnetization curves. To exclude the possible errors in the mass determination, comparison was done for the curves normalized to the magnetization value measured at 1.8 K in the field of 7 T.
The curves were computed using spin Hamiltonian in Equation (4) with the jDy,e value of +32 cm −1 (FM coupling) and −32 cm −1 (AFM coupling); jDy,Dy was assumed to be 0 in both cases. Note that in the spectrum of the spin Hamiltonian, the ground state is separated from the first excited state by more than 240 cm -1 , which ensures that magnetization curves measured at temperatures up to 100 K are dominated by the ground state properties. The difference between 19 and 21 μB is not well seen at low temperatures (simulated curves are almost identical at 1.8 K), but the curves diverge more pronouncedly at 35-100 K.
Comparison to the experimental data shows that the FM coupling describes the system considerably better than the AFM coupling, which serves as an independent proof of the magnetic moment of 21 μB.