Single‐Molecule Magnets DyM2N@C80 and Dy2MN@C80 (M=Sc, Lu): The Impact of Diamagnetic Metals on Dy3+ Magnetic Anisotropy, Dy⋅⋅⋅Dy Coupling, and Mixing of Molecular and Lattice Vibrations

Abstract The substitution of scandium in fullerene single‐molecule magnets (SMMs) DySc2N@C80 and Dy2ScN@C80 by lutetium has been studied to explore the influence of the diamagnetic metal on the SMM performance of dysprosium nitride clusterfullerenes. The use of lutetium led to an improved SMM performance of DyLu2N@C80, which shows a higher blocking temperature of magnetization (T B=9.5 K), longer relaxation times, and broader hysteresis than DySc2N@C80 (T B=6.9 K). At the same time, Dy2LuN@C80 was found to have a similar blocking temperature of magnetization to Dy2ScN@C80 (T B=8 K), but substantially different interactions between the magnetic moments of the dysprosium ions in the Dy2MN clusters. Surprisingly, although the intramolecular dipolar interactions in Dy2LuN@C80 and Dy2ScN@C80 are of similar strength, the exchange interactions in Dy2LuN@C80 are close to zero. Analysis of the low‐frequency molecular and lattice vibrations showed strong mixing of the lattice modes and endohedral cluster librations in k‐space. This mixing simplifies the spin–lattice relaxation by conserving the momentum during the spin flip and helping to distribute the moment and energy further into the lattice.


Introduction
Enclosing lanthanide ions within the fullerenec age is av ersatile route to av ariety of molecular magnets. [1] In particular, when non-metal atoms (C, N, O, S) are capturedb yt he carbon cage together with lanthanides, the strong ionic interactions emerging in such endohedral clusterfullerenes mayl ead to a large magnetic anisotropy. [2] In addition, different magnetic states can emerge from the intramolecular interactions of lanthanidei ons in clusterfullerenes. [2a, e, 3] This combination of properties made lanthanide-clusterfullerenes promising candidates fors ingle-molecule magnets (SMMs). Single-molecule magnetism is ap henomenoni nvolving the slow relaxation of magnetization in molecules with ab istable magnetic ground state and has been ah ot topic in the fieldo fm olecular magnetism during the last decades. [4] Indeed, some lanthanideclusterfullerenese xhibit single-molecule magnetism, [5] which is most robust in dysprosium-clusterfullerenes. [1c, 2d-f, 3a, 6] Nitridec lusterfullerenes (NCFs) with the composition (M 3 + ) 3 N 3À @C 2n 6À ,a nd in particular the speciesw ith C 2n = C 80 -I h (I h denotes the symmetry of the carbon cage) andM= Sc, Y, or heavy lanthanides (Gd-Lu), show the largest syntheticy ield and are therefore the mosts tudied clusterfullerenes to date. [1a, b, 7] The structure of the trimetal nitride cluster,w ith N 3À in its centera nd three M 3 + ions located at the vertices of the trianglew ith rather short MÀNb onds, offers ac onvenient platform to create av arietyo fm olecular magnets with divergent properties. First of all, the nitride ion at ad istance of only 2.0-2.2 from the lanthanide [2d, 6b, 8] generates as trong axial ligand field (LF), whichl eads to an easy-axis magnetic anisotropy for cerium,p raseodymium, neodymium,t erbium, dysprosium, and holmium ions, and an easy-plane anisotropy for erbium and thuliumi ons [2b] (note that we use the term "ligand field" instead of the more commonly used "crystal field" to avoid unnecessary connotations to intermolecular interactions in crystals). The stronga xial LF also ensures that m J is ag ood quantum number,a tl east for several lowest-energy LF states. In particular, the lowest-energy Kramers doubletsf or Dy 3 + in dysprosium-scandium NCFs are essentially pure m J states, with the high-spin J z = AE 15/2 states as the ground Kramers doublet with the magnetic momento riented alongt he DyÀNb ond. [2a, c] The purity of the J z states in terms of m J composition ensures that the magnetic system is weakly susceptible to external perturbationss uch as those introduced by dipolar magnetic fields from neighboring molecules or molecular and lattice vibrations.
Another advantage that lanthanide NCFs offer for tuning magnetic properties is the possibility of combining different metals within one molecule to give the so-called mixed-metal NCFs. [5a, 8c, 9] The importanceo ft his can be besti llustrated with the dysprosium-scandium NCFs Dy x Sc 3Àx N@C 80 -I h (x = 1-3) as an example. [3a] As Sc 3 + is diamagnetic, this series essentially allows analysisofhow two or three dysprosium ions interact magnetically and the effect of these interactions on SMM behavior. DySc 2 N@C 80 -I h was the first endohedralm etallofullerene (EMF) provent ob eaS MM. [6b, d] At temperatures below 7K,t he molecule exhibits magnetic hysteresis with the abrupt drop of the magnetization in zero magnetic field ascribed to the quantum tunneling of magnetization (QTM),w hich is typical for singleion magnets. Dy 2 ScN@C 80 -I h shows magnetich ysteresis below 8K without fast QTM relaxation in zero field, which is explained by ferromagnetic exchange and dipolar coupling between the non-collinear magnetic moments of the two dysprosium ions in the Dy 2 ScN cluster ( Figure 1), thus creatinga na dditional barriera nd preventing zero-field QTM. [3a] The temperature dependence of the magnetization relaxation times in Dy 2 ScN@C 80 -I h revealed ah igh barrier of 1735 K, because of the Orbach relaxation via the fifth Kramers doublet,i ng ood agreement with CASSCF calculations. [2d] Similarf erromagnetic interactions between dysprosium ions are also present in Dy 3 N@C 80 -I h ,b ut the triangular arrangement of the dysprosium ions forbids simultaneousrealizationofferromagnetic coupling for all three Dy···Dy contacts resulting in af rustrated ground state ( Figure 1) with faster relaxation of magnetizationt han in the mono-and dinucleara nalogues. [2a, 3a] Thus, due to different cluster compositions and intramolecular Dy···Dyi nteractions, DySc 2 N@C 80 -I h ,D y 2 ScN@C 80 -I h ,a nd Dy 3 N@C 80 -I h exhibit substantially different SMM behavior at low temperatures.
It is natural to consider that the magnetic properties of the NCFs in the Dy x Sc 3Àx N@C 80 -I h (x = 1-3) series discussed above are determined by the Dy 3 + ions, with the diamagneticS c 3 + ions acting just as placeholders, whicha re needed to keep the trimetallic clusterc omposition. However,i nt his work, we shed more light on the role that the diamagneticm etal can play in the SMM properties of NCFs. For this, we chose lutetiuma sa diamagnetic lanthanide, the ionic radius (R 3 + = 0.86 )o f which is noticeably larger than that of scandium( R 3 + = 0.75 ). Yet, as we reported recently,D yLu 2 N@C 80 -I h and Dy 2 LuN@C 80 -I h can still be synthesized, albeit in lower yields than the dysprosium-scandium analogues. [8c] Thel arger size of Lu 3 + leads to changes in the internal structure of the trimetallic nitride cluster,a nd in particular resultsi ns horter DyÀNb onds than in the dysprosium-scandiumN CFs. The shorteningo ft he DyÀN bonds maya ffect the magnetic anisotropy and the strength of the intramolecular Dy···Dy coupling. Furthermore, lutetium is much heavier than scandium,w hich changes the low-frequency part of the vibrational spectrum.B ecause the relaxation of magnetization involves spin-phonon interactions, alterations of molecular vibrations may also result in changes in the SMM properties. Thus, the goal of this work was to study ap ossible influence of these factors on the SMM properties of the DyM 2 N@C 80 -I h and Dy 2 MN@C 80 -I h NCFs.

Results and Discussion
Magnetization behavior of DyLu 2 N@C 80 and Dy 2 LuN@C 80 The mixed-metal dysprosium-lutetium NCFs with aC 80 -I h fullerene cage (for clarity,t he symmetry designation will be omitted hereafter) were obtained by arc-discharges ynthesis and separated by recycling HPLC as reported earlier. [8c] Similarr etention behavior of Dy 3 N@C 80 ,D y 2 LuN@C 80 ,D yLu 2 N@C 80 ,a nd Lu 3 N@C 80 substantially complicated the separation of the individualc ompounds. However,acompositionalp urity exceeding 90-95 % could be achieved for Dy 2 LuN@C 80 ,a sv erified by MS analysis. Due to the very similar retention behavior,t he separation of DyLu 2 N@C 80 from Lu 3 N@C 80 was not possible, and the relative content of the two NCFs in the studied sample was around 1:1.35. Because Lu 3 N@C 80 is diamagnetic, its presence in the sample does not lead to strong changes in the magnetic properties of DyLu 2 N@C 80 .Ap ossible influence of dilution on the quantum tunneling of magnetization will be specifically considered below.
The magnetization curves of DyLu 2 N@C 80 and Dy 2 LuN@C 80 , measured by SQUID (superconducting quantum interference device) magnetometry,a re shown in Figures2 and 3, respectively.D yLu 2 N@C 80 exhibits hysteresis up to 9Kat as weep rate of 3mTs À1 (Figure 2a). The "butterfly" shape of the hysteresis curves points to efficient zero-field relaxation by QTM similar to that observed in DySc 2 N@C 80 (Figure 2b). [6d] However,i nt he latter,t he openingo ft he hysteresis is narrower and the QTM induces complete loss of magnetization at zero field, whereas in DyLu 2 N@C 80 ,u pon crossing zero-field, the magnetization drops to around 30 %o ft he saturation magnetization value, resultingi nacoercivity of 0.9 Ta t2K. Recently,w es howed that the QTM in DySc 2 N@C 80 is strongly affectedb yd ilution in the diamagnetic matrix, including dilution with Lu 3 N@C 80 . [6b] To Figure 1. Molecular structures of Dy x M 3Àx N@C 80 -I h nitride clusterfullerenes (M = Sc or Lu; x = 1-3) and schematic illustration of the coupling of magnetic moments. Dy is green,Mis magenta, Nisb lue, Ci sl ight gray.The magnetic moments of Dy ions are visualized as green or red arrows. In DyM 2 N@C 80 (left), the magnetic momenti sa lignedalong the DyÀNb ond.In the magnetic ground stateofD y 2 MN@C 80 (middle), the magnetic moments of the two Dy ions are coupled ferromagnetically.InD y 3 N@C 80 (right),t he ground magnetic state is frustrated, and the magnetic moment of the Dy ion illustrated with ared arrow can switch between two isoenergetic orientations without changing the orientation of the two other moments. ensure that the differencei nt he QTM is intrinsicf or the two NCFs and is not caused by the dilutiono fD yLu 2 N@C 80 with Lu 3 N@C 80 ,i nF igure 2b we also show the magnetization curve of DySc 2 N@C 80 diluted with Lu 3 N@C 80 in ar atio of 1:1. This magnetic dilution indeed reducest he QTM step in DySc 2 N@C 80 slightly,but the changes do not reach the magnitude observed for DyLu 2 N@C 80 .T hus, it can be concluded that the zero-field QTM relaxation of magnetization in DyLu 2 N@C 80 is slowert han in DySc 2 N@C 80 .F urthermore, the blockingt emperature of magnetization, T B ,d efined as the temperature of peak magnetic susceptibilitym easured at 0.2 Tf or the sample cooled in zero field, is higher for DyLu 2 N@C 80 (T B = 9.5 K) than for DySc 2 N@C 80 (T B = 6.9 K). Note that the magnetic dilution does not affect the T B value, [6b] and thus the different SMM properties of DyLu 2 N@C 80 and DySc 2 N@C 80 cannotb ec aused by the presence of Lu 3 N@C 80 in the former.T oc onclude, DyLu 2 N@C 80 was found to be as tronger SMM than DySc 2 N@C 80 .T he substitution of scandiumb yl utetium in DyM 2 N@C 80 leads to slower QTM relaxation, slower in-field relaxation, and ab roader magnetic hysteresis with remanence.
For Dy 2 MN@C 80 ,t he influence of the diamagnetic metal on the SMM behavior appears to be weaker than for DyM 2 N@C 80 .
Similarly to Dy 2 ScN@C 80 ,D y 2 LuN@C 80 exhibitso penh ysteresis without ap ronounced QTM step ( Figure 3a). Its blockingt emperatureo fm agnetization, T B = 8K,i st he same as that of Dy 2 ScN@C 80 . [2d] However,a t2 K, the magnetic hysteresis of Dy 2 LuN@C 80 is narrower (Figure 3b)a nd the coercive field of 0.4 Ti ss maller than that of Dy 2 ScN@C 80 (0.7 T). Thus, the substitutiono fs candium by lutetium narrows the magnetic hysteresis but does not change the temperature scale of the slow relaxation. Nevertheless, as we show below,t he temperature dependence of relaxation times reveals that the mechanisms of the relaxation of magnetizationi nD y 2 LuN@C 80 and Dy 2 ScN@C 80 are different.
Relaxation times of magnetization in DyM 2 N@C 80 and Dy 2 MN@C 80 The opening of magnetic hysteresis in the magnetization curveso fm olecular magnets indicates that the magnetization attainsi ts equilibrium value slower than the rate of the magnetic fields weep. The slow relaxation of magnetization is ak ey characteristic of SMMs and needs to be understood in detail. The relaxation of magnetization requires energye xchange between the spin system and at hermalb ath, which is mediated   (b) showst he determinationo ft he blocking temperature, T B ,f rom the temperature dependence of the magnetic susceptibility, c (temperature sweep rate of 5Kmin À1 ). by the phonon system.T he spin-phonon interaction is crucial for such an energy exchange, and the role of vibrational degrees of freedom becomes paramount. Several mechanismso f phonon-mediated spin relaxation have been recognized in studies of paramagnetics alts and adapted for SMMs.
Ad irect mechanism implies as ingle-phonon process in which phonon frequency matches the energy difference between two opposite spins. Because am agnetic field increases the energy gap between the opposite spins, and the phonon density at near-zero frequencyi sl ow and increases with frequency, the relaxation due to ad irect mechanism accelerates in am agnetic field according to Equation (1) in which t M is the magnetization relaxation time, H is the magnetic field, with two terms describing the relaxationo faKramers ion in the absence (ca. H 4 )a nd in the presence (ca. H 2 )o f hyperfine interactions, and A 1 and A 2 are fitting parameters. Thus, the relaxation rate scales linearly with temperature, t À1 M;dir $ T.H owever,w hen at low temperature the phonon density may be very low,t he energy exchange between the phonon systema nd the bath can become the limiting step. This effect, known as ap honon bottleneck, can change the temperature dependence to t À1 M;dir $ T 2 . The Raman mechanism implies as pin flip throught he absorptiona nd emission of two phonons, with the frequency differenceb eing equal to the energy gap between the opposite spins. As the frequencies can be much highert han the Zeemane nergy,a nd the phonond ensity increases with frequency, the Ramanm echanism is more efficient than the direct mechanism once the temperature is sufficiently high to ensure sufficient phonon population. The originalc onsideration with only acoustic phonons in the Debye model gave the power-law temperature dependence expressed by Equation (2) in which n = 9f or Kramers ions and n = 7f or non-Kramers ions, and C is the fitting parameter. [10] However,i fo ptical phonons are also included in the model, powers of 6, 5, and even lower can be expected. [11] As pecial case of the Raman mechanism,i nw hich the absorbed phonon energy corresponds to the real excited spin state, is known as the Orbach mechanism. The temperature dependence of the relaxation rate under the Orbach mechanism has an Arrhenius form, expressed by Equation (3) in which U eff is the effective barrier( corresponding to the energy of the excited magnetic state) and t 0 is the attempt time. [10] Finally,t he QTMi saubiquitous and characteristic relaxation mechanism of SMMs.I nQ TM, the spin flips to the opposite direction without energy transfer.A ss uch, it shouldn ot show temperature dependence. The key condition for QTM is the energy matching of the opposite spin levels, and therefore the application of am agnetic field can quencht he QTM when Zeemansplitting becomes large enough.
It has become common practicet oa nalyze the relaxation of magnetization in SMMsa sacombinationo ft hese processes. [4f, 12] Characteristic temperature and field dependencies of relaxation times allow identification of the prevailing relaxation mechanism. Usually,t he direct mechanism is the most important at the lowest temperatures of af ew K, at somewhat highert emperatures the Raman mechanism becomes dominant, and with furtheri ncreases in temperature, the Orbach mechanism involving LF excited states takes over.N eithert he direct nor Raman mechanism would show an Arrhenius temperatured ependence, so the latter usually serves as an indication of the Orbach mechanism.H owever,b ack in the 1960s, Klemens [13] and others [14] argued that al ocalized vibrational mode can cause an Arrhenius temperature dependencew ith U eff corresponding to the frequencyo ft he mode.F urthermore, it was shown that the direct mechanism can also demonstrate Arrhenius behavior in some conditions. [11a, 15] The relaxation times of magnetization in DyLu 2 N@C 80 and Dy 2 LuN@C 80 in this work were determined at different temperatures by magnetizingt he sample to saturation, quickly sweeping the magnetic field to zero or any other required field value,a nd then following the decay of magnetization with subsequent fitting of the measured decay curvesw ith a stretched exponential( see the Supporting Information for the decay curves and Tables containing all the fitted parameters). Below,t he values obtained here are compared with those of DySc 2 N@C 80 from ref.
[6b],w hereas the relaxation times in Dy 2 ScN@C 80 were re-measured in this work for better consistency.
The zero-field measurements for DyLu 2 N@C 80 are complicated by the relativelyl ong stabilization of the field and fast zerofield QTM. In addition, the decay curves showed two types of behavior:Af ast dropo fm agnetization for around 90 %o ft he sample, followed by am uch slower relaxation of the remaining magnetization. The fit of the decay curvesw ith two stretched exponentsg ave values rangingf rom 54 AE 1s at 1.8 Kt o2 1AE 6s at 5K fort he fast process (Figure4a), which we assigned to QTM. The abrupt change in the relaxation rate may be caused by the redistribution of dipolar fields in the sample when as ignificantn umber of spins flip, the dilution effect of Lu 3 N@C 80 ,and may also reflect different relaxation of the molecules with different dysprosium isotopes. In addition, the slow process may also be caused by slow relaxation of the remnant magnetization in the magnet, or the deviation of the real magnetic field from zero. For comparison, t QTM in nondiluted DySc 2 N@C 80 determined by AC magnetometry in the same temperature range is 1-3 s. [6b] Overall, we can concludet hat the rate of relaxation by the QTM mechanism in DyLu 2 N@C 80 is slower than in DySc 2 N@C 80 .N ote that both compounds show a temperature dependence of the relaxation rate, even in the QTM regime, which mayr eflect the temperature dependence of the phonon collision rate, as suggested by Chilton and coworkers, [16] or result from dipolar intermolecular interactions. Short relaxation times cannot be measured very reliably by DC magnetometry,a nd furtherr elaxation measurements were performed for DyLu 2 N@C 80 in af ield of 0.2 T, applied to quencht he QTM. Over ar ange of only 5K,t he in-fieldr elaxation times in DyLu 2 N@C 80 vary by almost four orders of magnitude, from 4.7 10 5 sa t2Kt o5 8s at 7K ( Figure 4a). When plottedi nA rrhenius coordinates, the temperature dependence of the relaxation times has al inear form below 5K.F or com-parison, the t M values of DySc 2 N@C 80 are systematically shorter than those of DyLu 2 N@C 80 by af actor of six, butt hey also show al inear temperature dependence with av ery similar inclination( Figure 4a). Fitting the relaxation times of DyLu 2 N@C 80 with Equation (3) gave a U eff value of 24.2 AE 0.7K and t 0 of 2.8 AE 0.5 s. The analogous fit for DySc 2 N@C 80 gave U eff = 23.6 AE 1Kand t 0 = 0.6 AE 0.2 s. [6b] Thus, both NCFs have essentially identical U eff values and differ only in their attempt times. The reasonf or the low-temperature U eff barrierso f2 4K in both DyM 2 N@C 80 molecules is not clear.D ue to the very strongL Fs plitting of dysprosium in the NCFs, the energies of the lowest-energy excited states exceed hundreds of K( see the discussion of the ab initio calculations below). [2a-d, f, g] In addition, the t 0 values are many orders of magnitude longert han are usually found for the Orbach mechanism.W et entatively propose that the relaxation of magnetization in SMM EMFs in this temperature range may follow the Raman mechanism with involvement of local vibrations that would also follow Equation (3), but with U eff corresponding to the vibrational frequency. [14a, 17] The vibrational density of states in dysprosium NCFs is discussed further below.
The temperature dependence of the relaxation times of Dy 2 LuN@C 80 measured in zero magnetic field has ac urved shape in the log (t M )v ersus T À1 plot ( Figure 4b)a nd can be described well by using ac ombination of Ramana nd Arrhenius processes,expressed by Equation (4).
The fit of the experimentalz ero-field relaxation times of Dy 2 LuN@C 80 with Equation (4) gives C = (1.14 AE 0.28) 10 À6 s À1 K À5.45 , n 1 = 5.45 AE 0.15, t 0 = 435 AE 50 s, and U eff = 4.3 AE 0.2 K( Ta ble 1). The Raman mechanism dominates above 4K, whereas Arrhenius behaviori sp redominant below 2.5 K, and both mechanismsm ake comparable contributionsi nb etween. As for DyLu 2 N@C 80 ,n either t 0 nor U eff of the Arrhenius part is typical for the standard Orbach relaxation process via ligandfield excited states. Spin relaxationi nD y 2 ScN@C 80 below 8K can be also described by Equation (4) with C = (0.51 AE 0.26) 10 À6 s À1 K À5.99 , n 1 = 5.99 AE 0.33, t 0 = 56 AE 4s,a nd U eff = 8.0 AE 0.1 K (somewhat differentv alues, t 0 = 11.9 AE 1.5 sa nd U eff = 10.7 AE 0.3 K, reported by us in ref. [2d],w ere obtainedw ith as maller   Measurements of magnetizationr elaxation times in Dy 2 LuN@C 80 at 2.5 Ki nd ifferent magnetic fields (Figure 4c)r evealed considerable acceleration of the relaxation with increasing field. The temperature dependence of the relaxation times measured in af ield of 0.2 Talso showedn oticeable deviations from zero-field values below 5K (Figure4b). Such ad ependence of t M on the magnetic field is ac haracteristic of the direct relaxation mechanism [Eq. (1)].T od escribe the temperature dependence of the relaxation times measured in af ield of 0.2 T, we used Equation (4) with the addition of at erm describing the direct process, given by Equation (5) t in which C, n 1 , t 0 ,a nd U eff were fixed to the values determined for zero-field relaxation.
The fit of the experimental data measured in af ield of 0.2 T with Equation (5) gives A(H) = (7.59 AE 0.57) 10 À5 s À1 K À1.67 > and n 2 = 1.67 AE 0.09 (Table 1). The exponent of approximately 1.7 lies between the values expected for normal( n = 1) and bottleneck (n = 2) directp rocesses, and indicates that both are likely to take place. If insteado fu sing the fitting procedure, the n 2 value is fixed to 1a nd A(H)i sd etermined from the field dependence at 2.5 Kbyusing Equation (1) (Figure 4c), then Equation (5) describes well the temperature dependence above 2.5 K, but shows increasing deviations at lower temperature. This indicatest hat the bottleneck process has higher impact at low temperatures, when the number of excited phonons is not sufficient for efficient energyt ransfer.N ote that the direct and Arrhenius processes in Dy 2 LuN@C 80 have similar rates at 0.2 T, and hence in-fieldr elaxation rates are around twice as fast up to 3K,w hen the Ramanp rocess starts to dominate, and the field dependence eventually vanishesby5K.
In striking contrastt oD y 2 LuN@C 80 ,t he relaxation times of Dy 2 ScN@C 80 at 2.5K do not depend on the externalm agnetic field until it exceeds 0.4 T ( Figure 4c). The relaxation times of Dy 2 ScN@C 80 measured at different temperatures in af ield of 0.2 Ta lmostc oincide with the zero-field values and start to show small deviationso nly below 2K.T his shows that the direct mechanism contributes to the spin relaxation in Dy 2 ScN@C 80 at considerably higherf ields and lower temperatures than in Dy 2 LuN@C 80 ,w hich explains whyt he coercive field in the magnetic hysteresis of Dy 2 ScN@C 80 is larger than in Dy 2 LuN@C 80 (Figure 3b).
The temperature dependencies of the relaxation times of DyLu 2 N@C 80 and Dy 2 LuN@C 80 measured in zero field and in a field of 0.2 Ta re compared in Figure 4d.O nce the QTM in DyLu 2 N@C 80 is quenched by the application of af inite field, its relaxation rate is much slower than in Dy 2 LuN@C 80 ,a nd at 2K the difference between mono-and di-dysprosium NCFs exceeds two orders of magnitude.I ntramolecular interactions between dysprosiums pins in Dy 2 MN@C 80 block zero-field QTM and create am anifold of new low-energy coupled spin states. Apparently,s pin relaxation in Dy 2 LuN@C 80 at low temperature proceeds via such coupled states and is therefore much faster than in DyLu 2 N@C 80 ,w hich has only single-ion excited spin states. As imilar difference in the low-temperature relaxation mechanisms was also observedf or DySc 2 N@C 80 and Dy 2 ScN@C 80 . [3a, 6b] If indeed this is the case, the U eff of the Arrhenius process in Dy 2 MN@C 80 may be relatedt ot he energy differenceb etween the ground and the first excited state of the coupled spin system.A th igher temperature, the relaxation rates of DyM 2 N@C 80 and Dy 2 MN@C 80 tend to be more similar, which indicates that relaxation via single-ion states becomes equallyefficient for both types of NCFs.
Single-ion anisotropyo fdysprosium ions in DyM 2 N@C 80 and Dy 2 MN@C 80 (M = Sc, Lu) The central nitride ion is the main source of the magnetic anisotropy in lanthanide NCFs, and the LF is expectedt ob ecome stronger with decreasing distance between N 3À and Dy 3 + .D FT calculations at the PBE level with the 4f-in-core effective potential showedt hat the increasei nt he ionic radius from Sc 3 + to Lu 3 + shortens the DyÀNb ond from 2.156 in DySc 2 N@C 80 to 2.090 in DyLu 2 N@C 80 . [8c] Likewise, the DFT-optimized DyÀN bonds in Dy 2 LuN@C 80 (2.073 and 2.074 )a re shorter than those in Dy 2 ScN@C 80 (2.105 and 2.108 ). These geometrical changes may substantially affect the LF acting on the dysprosium ions.
To explore the influenceo ft his effect, we performed ab initio CASSCF/RASSI calculations on the LF splitting in DyLu 2 N@C 80 and Dy 2 LuN@C 80 molecules and compared the results with those for the dysprosium-scandium analogues. For Dy 2 MN@C 80 molecules, only one dysprosium ion was treated ab initio at at ime, and the other dysprosium was replaced by yttrium. The energies of the Kramers doublets( KDs) in DyM 2 N@C 80 molecules, the pseudo-spin g-tensor of the ground state KD, and the transition probability in the first KD are presented in Figure 5, Table 2, and Ta ble S7 in the SupportingI nformation.T he calculations show that the Dy 3 + ions in DyLu 2 N@C 80 and Dy 2 LuN@C 80 exhibit high magnetic anisotropy with overall LF splitting of 1340-1360 cm À1 .T he quantization axis is aligned along the DyÀNb ond, and the ground Kramers doublet is described as an essentially pure state with j m J j = 15/2. The energy of the second Kramers doublet is predicted to be closet o4 00 cm À1 ,w hiche nsures that the low-temperature magnetic properties of both DyLu 2 N@C 80 and Dy 2 LuN@C 80 are determined solelyb yt he ground state of Dy 3 + ,a si nt he previously studied dysprosium-scandium NCFs. [2d, 3a, 6b,d] Ac omparisono fD ySc 2 N@C 80 and DyLu 2 N@C 80 shows that the increase in metal size from scandiumt ol utetium leads to an increase in LF splitting from 1284 cm À1 (DySc 2 N) to 1348 cm À1 (DyLu 2 N). The energy of the second KD (relative to the first KD) also shows an increasef rom 356 cm À1 in DySc 2 N@C 80 to 391 cm À1 in DyLu 2 N@C 80  are somewhat shorter than that in DyLu 2 N@C 80 ,t he LF splitting for the dysprosium ion in the former is also slightly higher. Likewise, the LF splitting in Dy 2 LuN@C 80 is somewhat higher than in Dy 2 ScN@C 80 ,w hich also correlates with the shorter DyÀ Nb ond lengths (see Ta bles S8 and S9).
For the relaxation of magnetization, not only the energies of the KD states, but also the transition probabilitiesb etween them as well as the composition of the wave functions in the J; m J j i basis are very important (Figure5,s ee also Ta ble S7 in the Supporting Information). The first KD with g z closet o1 9.8 and infinitesimally small g x and g y values has more than 99 % contribution from the jm J j = 15/2 function fora ll the discussed NCFs. The probabilityo faQTM transition within the first KD is only 1.1 10 À9 m B 2 in DySc 2 N@C 80 and 1.7 10 À9 m B 2 in DyLu 2 N@C 80 .T he QTM transition probabilitiesw ithin one KD remain low up to the fourth KD ( Figure 5), and similar values are also found for Dy 2 MN@C 80 molecules. Likewise, transitions between the states of different m J and opposites pin are also not efficient until KD4. The reasons for this situation are rooted in the compositionoft he KD wave functions, which can be de-scribed ase ssentially pure m J states up to KD4-KD5 ( Figure 5). Thus, ab initio calculations predict that the relaxation of magnetization in all dysprosium NCFs should proceed via the KD5, as indeed was observede xperimentally in Dy 2 ScN@C 80 . [2d] There is no considerable difference between dysprosium-scandium and dysprosium-lutetium nitride clusterfullerenes in this regard.
To summarize, the replacement of scandiumb yl utetium in mixed-metal nitride clusterfullerenes shortens the DyÀNb onds and increases the LF splitting by 5-10 %. Otherwise, there is no significant difference in terms of KD composition and the expectedr elaxation pathways via excited KDs.

Intramolecular interactions of dysprosium magnetic momentsi nDy 2 LuN@C 80
The system of two weakly interacting dysprosium centers with magnetic moments b J 1;2 can be described by the effective spin Hamiltonian given by Equation (6) in which b H LF i is the single-ion LF Hamiltonian for the ith dysprosium site, dysprosium moments b J i are treated in the J; m J j i basis, and j 1;2 is the coupling constantb etween the localized dysprosium moments. Here, j 1;2 is treated isotropically in the spirit of the Lines model [18] and includes both exchange and dipolar interactions.
To determine the j 1;2 constant for Dy 2 LuN@C 80 ,w es imulated magnetization curves with different values of j 1;2 and compared them with the experimental data. In these simulations the angle, a,b etween the single-iona nisotropy axes of the dysprosium ions is set to 61.78,a sd eterminedb ya bi nitio calculations.T he best agreement is achieved for j 1;2 = 0.02-0.03 cm À1 ( Figure 6). As follows from Equation (6), for the two dysprosium spins oriented at an angle of a = 61.78,t he energy difference between the states with ferromagnetic (FM) and antiferromagnetic (AF) coupling of the dysprosium ions (J = 15/2) can be calculated from Equation (7) DE FMÀAF ¼ 4j 1;2 J 2 cos a ðÞ ¼ 225j 1;2 cos a ðÞ ð7Þ which gives an estimation of 2.1-3.2 cm À1 (3.1-4.6 K) for Dy 2 LuN@C 80 .T his energy difference is close to the U eff value of 4.3 Kd etermined for the low-temperature relaxation process with Arrhenius behavior (see above). The assumption that the relaxation proceeds by excitationt ot he antiferromagnetically coupled state (i.e., DE FMÀAF = U eff ) [2e, 3a] allows am ore precise estimationo fj 1;2 = 0.028 AE 0.001 cm À1 .T his value also gives ar easonable agreement between the experimental and simulated cT curves(see Figure S8 in the Supporting Information). Magnetic Dy···Dy interactions have two components, dipolar and exchange. The energy difference between the dipolar interactions in the FM and AF states, DE dip FMÀAF ,c an be calculated by using the well-known formula for the energy of dipolari nteractions betweent wo magnetic moments [Eq. 8], Full Paper in which n r ! is the normalo ft he radius vector connecting the two magnetic moments m 1 ! and m 2 ! , R 12 is the distance between them, and m 0 is the vacuum permeability.F or Dy 2 LuN@C 80 with DFT-optimized coordinates and m 1;2 ! = 10 m B ,E quation (8) gives DE dip FMÀAF = 4.8 K. Surprisingly,i ta ppears that the Dy···Dy interactions in Dy 2 LuN@C 80 are solelyo fd ipolar nature,w ith the exchange term vanishinga lmost completely.F or Dy 2 ScN@C 80 ,E quation (8) (Table 3). Thus, substitution of scandium by lutetiumi n the Dy 2 MN cluster results in considerable variation of the coupling constant, mainly because of the negligible exchangei nteractions in Dy 2 LuN@C 80 ,w hich also leads to as maller energy difference between the ferromagnetically and antiferromagnetically coupled states,a nd through this difference has as trong influence on the relaxation of magnetization at low temperature.

Low-frequency molecular and lattice vibrations in dysprosium-metal NCFs
As follows from the ab initio calculations discussed above,d ysprosium-scandium and dysprosium-lutetium NCFs have very similars ingle-ion magnetic anisotropy and ground-state properties.I na ddition, the LF splitting in these NCFs is very large and is not relevant for the low-temperature relaxation of magnetization. Therefore, the difference in their relaxation behavior [a] b is the anglebetween the DyÀNbonds and theq uantization axis of the Dy ions. [b] KD1$KD1' denotes the transition probabilitybetween the two degenerate states in the first KD, that is, the probability of QTM.
[8c] for a comparisono ft he experimental and computedD y ÀNd istances. Figure 6. Experimental magnetization curve of Dy 2 LuN@C 80 measured at 9K (dotted line)c ompared with the curves simulatedu sing Equation (5) with different values of j 1,2 from 0to0.1 cm À1 .The inset compares the experimental curve with the simulated one for j 1,2 = 0.028 cm À1 . The drawbacks of the commonly appliedp henomenological approaches to spin-phonon interactions developed in the 1960s are that they are based on the Debyem odel for vibrations in the crystal and do not provide ac lear connection with the microscopic parameters of the molecules (except for the LF splitting, whent he Orbach process is involved). For instance,a lthough the relaxation times measured in this work can be well fitted by Equations (1)-(3)a nd their combinations, the fitted parameters do not provide sufficient insight into the relationb etween molecular structurea nd the relaxationo f magnetization.
Recently,L unghie tal. [17] analyzed spin-lattice relaxation in molecular magnets by using quantum spin dynamics, considering explicit vibrations of the molecule and deriving the spinphonon coupling parameters from ab initio calculations. They found that spin-phonon relaxationv ia anharmonic phonons may also result in Arrheniusb ehavior,b ut with the U eff corresponding to half of the vibrational frequency.F urthermore, other research teams analyzed the locality of the spin-phonon interaction and concluded that molecular vibrations spatially localized close to the metal center usually have the strongest contribution to the spin relaxation. [17,19] These findings show that the Debye model is oversimplified for the analysiso f spin-phonon relaxation in molecular magnets [20] and an analysis of the real vibrational spectra may give better insighti nto the relaxation mechanism. [21] With this in mind, we decided to analyze the low-frequency part of the vibrational spectra of the NCFs. The experimental Raman spectra of DyM 2 N@C 80 and Dy 2 MN@C 80 (M = Sc, Lu;l imited to frequenciesa bove 50 cm À1 due to instrument limitations) are compared in Figure 7w ith the vibrational density of states (VDOS), computed by DFT for isolated molecules. In addition to thet otal VDOS, Figure7also shows contributions to the VDOS of the whole metal nitride cluster and of only the dysprosium atoms. The computed and experimental frequencies of the cluster-basedm odes are presented in Table 4.
EMFs have rather peculiar vibrational spectra as their molecules consist of two semi-independent units, the vibrations of which show almost no overlap in thee nergy scale, as can be  [a] Designation of clusterm odes:R ,r otation( libration); T, translation; d MNM' ,M ÀNÀM' bending; g N ,n itrogen out-of-planed isplacement; n M-N ,M ÀNs tretching vibration. [b] Experimental Raman spectrum of DySc 2 N@C 80 has ap eak at 145 cm À1 that cannot be assigned on the basis of calculation results. well seen in Figure 7. Thus,t he vibrations of the relativelyr igid carbon cage occur at frequencies exceeding 240 cm À1 ,whereas the frequencieso fm etal-involving modes rarely exceed 230 cm À1 ,b ecause the metal atoms are much heaviert han carbon atoms. Only in the border range of 220-260cm À1 do the squashingc age modes partially mix with the "breathing" mode of the nitride cluster, in which all three metal atoms move radially in one phase along the MÀNb onds. Other metal-based vibrations occur at frequencies below 200 cm À1 . When the nitridec luster is encapsulatedi nside the fullerene, its externald egrees of freedom (i.e.,t ranslations and rotations) are transformed into internal ones (i.e.,m olecular vibrations). The frustratedr otations (i.e.,l ibrations)a re the lowest-frequency intramolecular modes predicted to be close to 30-40 cm À1 in dysprosium-lutetium NCFs and at 40-60 cm À1 in dysprosium-scandium NCFs. The frustrated translationsa re mixed with deformationso ft he cluster( such as in-plane oscillations of the MÀNÀMa ngles). In DyLu 2 N@C 80 and Dy 2 LuN@C 80 ,t hese modes are clustered into two groups close to 80 and1 60 cm À1 (because dysprosium and lutetium have similar atomic masses, both dysprosium-lutetium NCFs have very similarV DOS). In dysprosium-scandiumN CFs, such modes are more uniformly spread in the 80-200 cm À1 range. Finally,t he nitrogen out-ofplane mode also falls in the range of 90-210 cm À1 . The resultso ft he computations agree reasonably well with the experimental Raman spectra. Above 220 cm À1 ,t he spectra of all the NCFs are quasi-continuous because of densely spaced cage vibrations. The calculations seem to overestimate the cage frequencies by around 10 %. In the cluster frequency range, DyLu 2 N@C 80 and Dy 2 LuN@C 80 exhibit only two Raman peaks, at around 80 and 162 cm À1 ,c lose to the predicted frequencies of the mixed translation/deformation modes. The dysprosium-scandiumN CFs exhibit richer spectralp atterns, and most of the observed peaks can be reliably assigned to the computed modes, as listed in Table 4( see refs. [22] for a more detailedd iscussion of the vibrational spectra of MSc 2 N@C 80 NCFs). Due to technical limitations, we cannot record the spectrab elow 50 cm À1 ,w hich precludes experimental observation of the cluster librations in dysprosium-lutetium NCFs. However,g ood agreement between experiment and theory for the cluster modes above 50 cm À1 ensures that the calculated frequencies are not far from reality.I na ddition, in some of the earlierR aman studies of the NCFs,p eaks at around3 0-40cm À1 were reported for Dy 3 N@C 80 ,L u 3 N@C 80 , and some other M 3 N@C 80 molecules. [8a, 23] Librations of the cluster and lattice phononsinspin relaxation The low-frequency vibrations localized on the metal nitride cluster of the M 3 N@C 80 molecules are expected to mediate energy transfer between the spin and athermalbath. The rotational motions of the cluster are especially of interestf rom the point of view of low-temperature spin relaxation because they not only happenatl ow frequencies (i.e.,int he relevant energy range), but also because they may helpt oc onserve the total angular momentum when the spin flips. The relevance of the Einstein-de Haas effect on the single-molecule level was demonstrated by Wernsdorfer and co-workers for TbPc 2 (Pc = phthalocyanine) grafted on ac arbon nanotube (CNT). [24] To conserve the total momentum, the spin reversal of terbium had to result in rotationo ft he TbPc 2 molecule around the terbium quantization axis. However, because the molecule was rigidly bonded to the CNT,t of ulfill the rotational invariance, the rotational momentum had to be transferred to the momentumo ft he phononp ropagating along the nanotube. As a result, the efficient spin reversal proceeded by the direct mechanism when the externalm agnetic field created aZ eeman splitting matching the frequency of the longitudinal nanotube phonon.I nadysprosium nitride cluster, the reversal of the spin aligneda long the DyÀNb ond should induce rotation of the cluster around this bond. But as discussed above,i nteraction with the fullerene cage restricts the rotational motion of the cluster and turns it into avibration, albeit retaining its rotational character.I nD yLu 2 N@C 80 and DySc 2 N@C 80 ,t he frequencies of the corresponding cluster librations are predicted at 27 and 62 cm À1 ,r espectively (Table 4). However,l ocalized molecule vibrations at the G point do not transfer moment either, and hence their dispersions and interactions with lattice phonons should be studied further.
Consideration of the lattice phonons in EMF solids is also necessary because their frequency range is likely to overlap with low-frequency intramolecular vibrations. Unfortunately, experimental information on lattice phononsi nE MFsi sv ery limited. To the best of our knowledge, there has been only one study of monometallofullerenes by inelastic neutron scattering, which showed almost featureless VDOS in the low-frequencyr ange. [25] Far-IR [25,26] as well as the aforementioned Raman studies provedt he presence of somel ow-frequency modes in EMFs, but metal-based intramolecular vibrations could not be distinguished from the lattice modes, and, in addition, the selection rules limit the optical activity only to the G point. Lattice vibrations of empty fullerenes,a nd especially C 60 , are much better studied. According to inelastic neutron scattering and ab initio computations, the lattice phonons of C 60 exhibit rather strong dispersion andc over the range up to 60-70 cm À1 , [27] whereas optical spectroscopic studies revealed the bands of librational modes at 7a nd 18 cm À1 ,a nd those of translational modes at 28, 41, and 59 cm À1 . [28] Thus, we can tentatively suggest that the frequency ranges of lattice modes and the intramolecular clusterv ibrations of EMFs do overlap, which mayl ead to significant mode mixing.
In the absence of experimental information, we performed computational modeling of the lattice phononso fM 3 N@C 80 to analyze their possible interaction with intramolecular vibrations. Complete calculations of the phonons of fullerenec rystals at the DFT level are hardly feasible at this moment, and therefore simulations were performed by using the less demandingd ensity-functional based tight-binding( DFTB) approach. [29] The pair-atomic interaction potentials availablef or Sc 3 N@C 80 showed ar easonable prediction of the molecular geometry and vibrations. [30] As potentials for other lanthanide atoms of interesta re not known, we used the Sc 3 N@C 80 model to determine the Hessian and then computed the dispersion spectra and vibrational eigenvectors for different lanthanidecontaining NCFs by using ap roper mass correction in ad ynamic matrix.
The modell attice of M 3 N@C 80 molecules was simulated with face-centered cubic (FCC)p acking with the optimized unit cell parameter a of 15.5 andt he distance between the centerso f fullerenem olecules of 11 (see Figure S11i nt he Supporting Information). The vibrational spectra computed for isolated DySc 2 N@C 80 and DyLu 2 N@C 80 molecules are compared in Figure 8w ith those of the crystal phase, and dispersion of the phononsa long the high-symmetry line G-X can be seen. In both cases, the DFTB-computed spectra of the isolated molecules start above 50 cm À1 ,w hich is just on the borderline for the acoustic bands of the crystals. The molecular modes away from the frequencyr ange of the acoustic modes shownoticeable Davydovs plitting but negligible k-dispersion. But the local modes close to the acoustic bands intertwine with the dispersed lattice bands giving rise to ad ense phonon structure startingf rom zero frequency on. Three clear acoustic modes are perturbed by as et of what appear to be local modes with fluid character across k-space. To follow the possible mode mixing in k-space, we chose G-point vibrational eigenvectors as ab asis space, in which eigenvectors computed at different k values were projected.
The resultso ft his projection analysis for one pure acoustic mode and foro ne with clusterl ibration charactera re presented in Figure 8a,b using ac olor code (blue for the acoustic and red for the libration) and as catterp lot,f or which the size of the dots is proportional to the magnitude of the projection (Figure 8). In this representation we can see and quantify how strongly the acoustic mode couplest om ore localized modes as af unction of k. This redistribution is also reflected in the projected DOS in Figure8,w ith the acoustic band showinga steadyi ncrease of clusterc ontributions as the energy increases. Thism odel computationc learly shows that substantial mode mixingi ndeed takesp lace in k-space. However,t he 3D modeli sc hallenging to grasp due to extensive mixing and a high density of states. For illustrativep urposes, we simplified the model to one dimension and considered al inear chain of M 3 N@C 80 molecules with a = 10.75 (see Figure S11i nt he Supporting Information). Figure 8c-e shows dispersion relationships computed for 1D chainso ft hree NCF molecules,n amely Sc 3 N@C 80 ,D ySc 2 N@C 80 , and DyLu 2 N@C 80 .T he spectra contain only one prominent acousticb and with al arge dispersion of 50 cm À1 and two bands with as maller dispersion of 10 cm À1 produced by onsite rotational degrees of freedom. In 1D Sc 3 N@C 80 ,t he lattice and clusterm odes are high in frequency,t he clusterl ibration band is flat, and aw eak mixing with the acoustic mode can be detected only around the Xs ymmetry point. With the increase in mass in DySc 2 N@C 80 ,t he cluster-based frequencies decrease faster than those of the lattice phonons, and the degree of mixing increases as manifestedi nt he considerable dispersion of the intramolecular mode propelled by the acoustic band. Furthermore, the local anda coustic mode frequencies are even closer in 1D DyLu 2 N@C 80 ,a nd therefore the mode mixing is much more pronounced.
The mixingo fc luster libration modes with the lattice phonons outside of the G point has profoundc onsequences for momentum transfer.T hese resultss how how in the first instancet he librations, still being local modes, can assist the total conservation of momentum during spin flip and would be able to redistribute the excess angular momentum onto the lattice at some k values. Mixing of the cluster rotations with the lattice phononst hus facilitates spin reversal by interaction of the local and lattice modes in k-space. Based on this conclusion, we tentatively suggestt hat the Arrhenius behavior with a U eff of 24 K( 17 cm À1 )o bservedi nt he temperature dependence of magnetization relaxation times of DySc 2 N@C 80 and DyLu 2 N@C 80 (Figure 4a)may be caused by spin reversal assisted by the rotationalm odes of the M 3 Nc luster. U eff in this case should correspond not to the frequency in the G point, but rather to the frequencies in the high density of states of the phononsw ith strongm ixed character.I na ddition, the differenceinthe masses of the dysprosium-scandium and dysprosium-lutetium nitride clusters will result in ad ifferent degree of mixing between the clusterl ibrations and lattice phonons, thus leading to ad ifferent efficiency of the spin-lattice relaxation.

Conclusions
In this work we have studied the magnetic properties of the nitridec lusterfullerenes DyLu 2 N@C 80 and Dy 2 LuN@C 80 and ana- Figure 8. Phonon spectra for five model systems:3Dcrystal with FCC packing of (a) DySc 2 N@C 80 and (b) DyLu 2 N@C 80 ;1 Dchains of (c) Sc 3 N@C 80 , (d) DySc 2 N@C 80 ,a nd (e) DyLu 2 N@C 80 .F or each system,t he right panelr epresents the frequencies computed for isolated molecules,t he central panel showsp honon dispersionsa long the G-X high-symmetrypath with information on band compositions (see text for details), and the left panel demonstratest he projected VDOS (total,g ray;M 3 Ncluster,p ink;Dyc ontribution, green). lyzed how substitution of scandium by lutetium in the mixedmetal clusterfullerenes DyM 2 N@C 80 and Dy 2 MN@C 80 affects the single-molecule magnetism thereof.D yLu 2 N@C 80 and Dy 2 LuN@C 80 have been found to be SMMs with ab locking temperature of 9.5 and 8K,r espectively.D yLu 2 N@C 80 exhibits a higher blockingt emperature, longer relaxation times, and broader hysteresist han the dysprosium-scandiuma nalogue DySc 2 N@C 80 .B oth DySc 2 N@C 80 and DyLu 2 N@C 80 feature zerofield QTM, and when the QTM is quenched in af inite field of 0.2 T, the magnetization relaxation times of both compounds show Arrhenius behavior with an effective barrier of 24 K. Dy 2 LuN@C 80 and Dy 2 ScN@C 80 have identical blocking temperatures, but show different temperature and field dependence of the relaxation times. In particular, ad irect relaxation mechanism with enhanced field dependence is observed for Dy 2 LuN@C 80 below 5K,w hereas the relaxation times of Dy 2 ScN@C 80 remaini ndependento ffield until the fielde xceeds 0.4 T. The magnetization relaxation times of Dy 2 LuN@C 80 show Arrhenius behaviorw ith an effective barrier of 4.3 K, which has been assigned to the energy of the excited state with antiferromagnetic coupling of the dysprosium moments, DE FMÀAF .I n Dy 2 ScN@C 80 ,t he energy of this state is aroundt wo-fold higher, at 8.0 K. Because Dy 2 LuN@C 80 and Dy 2 ScN@C 80 have almost identical energies of intramolecular dipolar interactions of 4.7 K, the considerable differencei nt heir DE FMÀAF values is attributed to the strong variation in the exchange coupling when scandium is substitutedb yl utetium. Essentially,t he magnetic moments in Dy 2 LuN@C 80 show only dipolar interactions and their exchange coupling vanishes.
To aid the understanding of possible spin-phonon energy exchange, the low-frequency vibrational spectra of the dysprosium-lutetium and dysprosium-scandium NCFs were analyzed experimentally and with the help of DFTcalculations. Enclosing the M 3 Nc luster inside the fullerenec age transforms its rotational degrees of freedom into molecular vibrations, which retain rotational character and are dubbed as librations of the cluster.T he low frequencies of these modes lead to overlap with the frequency range of the lattice phonons. Furthermore, projection analysisa lso revealed the strong mixingo ft he local cluster librations with acoustic phonons of the fullerene lattice in the k-space away from the G point. As ar esult,t hese modes are predicted to facilitate the relaxation of magnetization by helping to conserve momentum during the spin reversal. Thus, the results of our study emphasize that the mixing of local and lattice modes in k-spacem ay be an important mechanism of the spin-lattice relaxation and should be considered for other molecular magnets.

Experimental Section
Powder samples of fullerenes for magnetometry studies were prepared by drop-casting from toluene or CS 2 solutions. The magnetic properties were studied with aQ uantum Design MPMS3 Vibrating Sample Magnetometer (VSM). Modeling of the magnetization curves and the spin Hamiltonian solution was accomplished with the PHI program. [31] Raman spectra were recorded at 77 Kw ith aT64000 triple spectrometer (Jobin Yvon) using an excitation wavelength of l ex = 514 nm (Ar + laser) or l ex = 647 nm (Kr + laser). The samples for Raman measurements were drop-cast onto single-crystal KBr disks.
DFT calculations on isolated M 3 N@C 80 molecules were performed at the PBE-D level with ap lane-wave basis set and the corresponding projector augmented-wave potentials, treating 4f electrons as a part of the core as implemented in the VASP 5.0 package. [32] These calculations employed ac ubic unit cell with al attice parameter of 25 and the atomic cut-off energy was set to 400 eV.T he precision was set to be "accurate" with real-space projector operators optimized down to 10 À4 eV per atom. The G-point Hessian matrix and then the vibration frequencies (or G-point phonons) were determined by using density functional perturbation theory as implemented in VASP. The ab initio energies and wave functions of LF (ligand field) multiplets for the dysprosium-lutetium NCF molecules were calculated at the CASSCF/SO-RASSI level of theory with atomic natural orbital extended relativistic basis set (ANO-RCC) [33] of the valence doublezeta (VDZ) quality using the quantum chemistry package MOLCAS 8.0. [34] The active space of the CASSCF calculations included the 4f shell, that is, 11 active electrons and 7a ctive orbitals. All 21 sextet states and 108 quartets and only 100 doublets were included in the state-averaged CASSCF procedure and further used in the RASSI procedure with as pin-orbit Hamiltonian. The singleion magnetic properties and LF parameters were calculated on the basis of the ab initio data with the use of aS INGLE ANISO module. [35] Phonon spectra modeling and analysis were performed with inhouse Python scripts based on ASE libraries and with the improved version of some functions. [36] The DFTB + software [29b] was used as af orce derivation code in the Calculator class of ASE. The finite differences method was used in phonon calculations with atomic displacement of 0.03 along each Cartesian axis. Prior to phonon calculations, 1D and 3D systems (see Figure S11i nt he Supporting Information) were optimized with aq uasi-Newton algorithm down to 0.0002 eV À1 with 2 k-point sampling in each periodic direction. In all calculations, the non-charge-self-consistent model was employed with the Slater-Koster parameters developed for similar types of system. [37]